text: 10 { } and V =∆ V (F ). The sequence V is non-negative (since To apply this bound effectively, we need a lower bound on 0 0 n each V is a sum of varentropies), and nonincreasing by P (δ). To derive such a lower bound, we observe that, for any n { } ≥ n≥ Theorem 2. Thus V converges to a limit c 0. Our goal δ 0, n is to prove that c = 0. The analysis in the proof of Theorem 2 covers the present ≤ − ≤ case as a special instance. In the present notation, the recursive V n [1 Pn(δ)]δ + Pn(δ)M δ + Pn(δ)M (51) relation (38) takes the form (Fn,i, Fn,i+2n−1) = ψ2(Fn−1,i, Fn−1,i), 1 ≤ i ≤ 2n−1, bywLheemremMa 1=∆. L2e.t34n34 bies tshuechbotuhnatd foonr avlalrenntr≥opyn p,roVvide≥d { } 0 ≥ 0 n since here we have Fn−1,i = Fn−1,i+2n−1 due to i.i.d. BDEs c/2. Since V n converges to c 0, n0 exists and is ﬁnite. at the transform input. Using this relation, we obtain readily This, combined with (51), implies the following bound on the an explicit formula for the incremental change in normalized fraction of bad indices. varentropy from generation n t(cid:88)o (n + 1), namely, − − ≥ V δ ≥ c/2 δ ≥ − − 2n ≥ Pn(δ) n , n n0. (52) D =∆ V V = C(F ), n 0, (44) M M n+1 n+1 n n,i (cid:2) i=1 (cid:3) where Using (52) in (50) with δ = c/4 gives − ≤ − · ≥ C(Fn,i) =∆ V (Fn,i) V (Fn+1,i) + V (Fn+1,i+2n) /2. (45) Dn (c/4M) ∆(c/4), n n0. (53) If we denote the conditional entropy random variables in the { } polar transform as hn,i , it can be seen that From (46), we see that having c > 0 is incompatible with (53). This completes the proof that c = 0 (subject to the C(Fn,i) = Cov(hn+1,i, hn+1,i+2n). assumption that (49) holds, which is proved in the Appendix). Thus, we have C(F ) ≥ 0 by Theorem ??(cid:48), implying that VI. CONCLUDING REMARKS ≤ ≥ n,i (cid:88) Dn 0 for all n 1. It is useful to note here that One of the implications of the convergence of average ∞ varentropy to zero is that the entropy random variables “con- − c =∆ l→im∞ V n = V (F0) Dn, (46) centrate” around their means along almost all trajectories of n i=1 the polar transform. This concentration phenomenon provides showing explicitly that c is the limit of a monotone nonin- a theoretical basis for understanding why polar decoders are creasing sequence of sums. robust against quantization of likelihoood ratios [10]. ≥ For δ 0, let Theorem 3 may be seen as an alternative version of the “polarization” results of [4]. In [4], the analysis was centered M { ∈ M ≥ } =∆ F : V (F) δ . (47) around the mutual information function and martingale meth- δ ods were used to establish asymptotic results. The present and { ∈ M } study is centered around the varentropy and uses weak conveg- ∆(δ) =∆ inf C(F) : F δ . (48) ence of probability distributions. The use of weak convergence in such problems is not new; Richardson and Urbanke [6, As we will see in a moment, the main technical problem that pp. 187-188] used similar methods to deal with problems of remains is to show that convergence of functionals deﬁned on the space of binary ⇒ memoryless channels. δ > 0 = ∆(δ) > 0. (49) We should mention that Alsan and Telatar [11] have given an elementary proof of polarization that avoids martingale While this proposition seems plausible in view of the fact theory, and instead, uses Mrs. Gerber’s lemma [12]. It appears (cid:48) that C(F) = 0 iff V (F) = 0 (by Theorem ?? ), there is the possible to adopt the method of [11] to establish Theorem 3 technical question of whether the “inf” in (48) is achieved as without using weak convergence. ∈ M a “min” by some F . We will ﬁrst complete the proof APPENDIX δ of Theorem 3 by assuming that (49) holds. Then, we will give PROOF OF (49) a proof of (49) in the Appendix. M { ≤ ≤ ∈ M } Lemma 10. The space of CDFs on [0, 1] is a compact Let J (δ) =∆ 1 i 2n : F , and P (δ) =∆ metric space. | | n n,i δ n J (δ) /2n. For δ > 0, we may think of J (δ) as the set of “bnad” BDEs in the nth generation and P (δn) as their fraction Proof. This follows from a general result about probability n in the same population. From (44), we obtain the bound measures on compact metric spaces. Theorem 6.4 in [14, p. 45] M states that, for any compact metric space X, the space (X) D ≤ −P (δ)∆(δ), δ ≥ 0. (50) of all probability measures deﬁned on theMσ-algebra of Borel n n sets in X is compact. Our deﬁnition of above coincides M with the (X) with X = [0, 1]. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: V

text: V

text: (F

text: The

text: sequence

text: V

text: is

text: non-negative

text: (since

text: To

text: apply

text: this

text: bound

text: effectively,

text: we

text: need

text: a

text: lower

text: bound

text: on

text: n

text: each

text: V

text: is

text: a

text: sum

text: of

text: varentropies),

text: and

text: nonincreasing

text: by

text: P

text: (δ).

text: To

text: derive

text: such

text: a

text: lower

text: bound,

text: we

text: observe

text: that,

text: for

text: any

text: n

text: n≥

text: Theorem

text: Thus

text: V

text: converges

text: to

text: a

text: limit

text: c

text: Our

text: goal

text: δ

text: n

text: is

text: to

text: prove

text: that

text: c

text: The

text: analysis

text: in

text: the

text: proof

text: of

text: Theorem

text: covers

text: the

text: present

text: case

text: as

text: a

text: special

text: instance.

text: In

text: the

text: present

text: notation,

text: the

text: recursive

text: V

text: n

text: Pn(δ)]δ

text: Pn(δ)M

text: δ

text: Pn(δ)M

text: relation

text: takes

text: the

text: form

text: (Fn,i,

text: i

text: bywLheemremMa

text: bies

text: tshuechbotuhnatd

text: foonr

text: avlalrenntr≥opyn

text: p,roVvide≥d

text: n

text: since

text: here

text: we

text: have

text: due

text: to

text: i.i.d.

text: BDEs

text: Since

text: V

text: n

text: converges

text: to

text: c

text: exists

text: and

text: is

text: ﬁnite.

text: at

text: the

text: transform

text: input.

text: Using

text: this

text: relation,

text: we

text: obtain

text: readily

text: This,

text: combined

text: with

text: implies

text: the

text: following

text: bound

text: on

text: the

text: an

text: explicit

text: formula

text: for

text: the

text: incremental

text: change

text: in

text: normalized

text: fraction

text: of

text: bad

text: indices.

text: varentropy

text: from

text: generation

text: n

text: (n

text: namely,

text: V

text: δ

text: δ

text: Pn(δ)

text: n

text: n

text: D

text: V

text: V

text: C(F

text: n

text: M

text: M

text: n

text: n,i

text: where

text: Using

text: in

text: with

text: δ

text: gives

text: C(Fn,i)

text: V

text: (Fn,i)

text: V

text: V

text: Dn

text: n

text: If

text: we

text: denote

text: the

text: conditional

text: entropy

text: random

text: variables

text: in

text: the

text: polar

text: transform

text: as

text: hn,i

text: it

text: can

text: be

text: seen

text: that

text: From

text: we

text: see

text: that

text: having

text: c

text: is

text: incompatible

text: with

text: This

text: completes

text: the

text: proof

text: that

text: c

text: (subject

text: to

text: the

text: C(Fn,i)

text: assumption

text: that

text: holds,

text: which

text: is

text: proved

text: in

text: the

text: Appendix).

text: Thus,

text: we

text: have

text: C(F

text: by

text: Theorem

text: implying

text: that

text: VI.

text: CONCLUDING

text: REMARKS

text: n,i

text: Dn

text: for

text: all

text: n

text: It

text: is

text: useful

text: to

text: note

text: here

text: that

text: One

text: of

text: the

text: implications

text: of

text: the

text: convergence

text: of

text: average

text: varentropy

text: to

text: zero

text: is

text: that

text: the

text: entropy

text: random

text: variables

text: “con-

text: c

text: l→im∞

text: V

text: n

text: V

text: Dn,

text: centrate”

text: around

text: their

text: means

text: along

text: almost

text: all

text: trajectories

text: of

text: n

text: the

text: polar

text: transform.

text: This

text: concentration

text: phenomenon

text: provides

text: showing

text: explicitly

text: that

text: c

text: is

text: the

text: limit

text: of

text: a

text: monotone

text: nonin-

text: a

text: theoretical

text: basis

text: for

text: understanding

text: why

text: polar

text: decoders

text: are

text: creasing

text: sequence

text: of

text: sums.

text: robust

text: against

text: quantization

text: of

text: likelihoood

text: ratios

text: For

text: δ

text: let

text: Theorem

text: may

text: be

text: seen

text: as

text: an

text: alternative

text: version

text: of

text: the

text: “polarization”

text: results

text: of

text: In

text: the

text: analysis

text: was

text: centered

text: M

text: M

text: F

text: V

text: (F)

text: δ

text: around

text: the

text: mutual

text: information

text: function

text: and

text: martingale

text: meth-

text: δ

text: ods

text: were

text: used

text: to

text: establish

text: asymptotic

text: results.

text: The

text: present

text: and

text: M

text: study

text: is

text: centered

text: around

text: the

text: varentropy

text: and

text: uses

text: weak

text: conveg-

text: ∆(δ)

text: inf

text: C(F)

text: F

text: δ

text: ence

text: of

text: probability

text: distributions.

text: The

text: use

text: of

text: weak

text: convergence

text: in

text: such

text: problems

text: is

text: not

text: new;

text: Richardson

text: and

text: Urbanke

text: As

text: we

text: will

text: see

text: in

text: a

text: moment,

text: the

text: main

text: technical

text: problem

text: that

text: pp.

text: used

text: similar

text: methods

text: to

text: deal

text: with

text: problems

text: of

text: remains

text: is

text: to

text: show

text: that

text: convergence

text: of

text: functionals

text: deﬁned

text: on

text: the

text: space

text: of

text: binary

text: memoryless

text: channels.

text: δ

text: ∆(δ)

text: We

text: should

text: mention

text: that

text: Alsan

text: and

text: Telatar

text: have

text: given

text: an

text: elementary

text: proof

text: of

text: polarization

text: that

text: avoids

text: martingale

text: While

text: this

text: proposition

text: seems

text: plausible

text: in

text: view

text: of

text: the

text: fact

text: theory,

text: and

text: instead,

text: uses

text: Mrs.

text: Gerber’s

text: lemma

text: It

text: appears

text: that

text: C(F)

text: iff

text: V

text: (F)

text: (by

text: Theorem

text: there

text: is

text: the

text: possible

text: to

text: adopt

text: the

text: method

text: of

text: to

text: establish

text: Theorem

text: technical

text: question

text: of

text: whether

text: the

text: “inf”

text: in

text: is

text: achieved

text: as

text: without

text: using

text: weak

text: convergence.

text: M

text: a

text: “min”

text: by

text: some

text: F

text: We

text: will

text: ﬁrst

text: complete

text: the

text: proof

text: APPENDIX

text: δ

text: of

text: Theorem

text: by

text: assuming

text: that

text: holds.

text: Then,

text: we

text: will

text: give

text: PROOF

text: OF

text: a

text: proof

text: of

text: in

text: the

text: Appendix.

text: M

text: M

text: Lemma

text: The

text: space

text: of

text: CDFs

text: on

text: is

text: a

text: compact

text: Let

text: J

text: (δ)

text: i

text: F

text: and

text: P

text: (δ)

text: metric

text: space.

text: n

text: n,i

text: δ

text: n

text: J

text: (δ)

text: For

text: δ

text: we

text: may

text: think

text: of

text: J

text: (δ)

text: as

text: the

text: set

text: of

text: “bnad”

text: BDEs

text: in

text: the

text: nth

text: generation

text: and

text: P

text: (δn)

text: as

text: their

text: fraction

text: Proof.

text: This

text: follows

text: from

text: a

text: general

text: result

text: about

text: probability

text: n

text: in

text: the

text: same

text: population.

text: From

text: we

text: obtain

text: the

text: bound

text: measures

text: on

text: compact

text: metric

text: spaces.

text: Theorem

text: in

text: p.

text: M

text: states

text: that,

text: for

text: any

text: compact

text: metric

text: space

text: X,

text: the

text: space

text: (X)

text: D

text: −P

text: (δ)∆(δ),

text: δ

text: of

text: all

text: probability

text: measures

text: deﬁned

text: on

text: theMσ-algebra

text: of

text: Borel

text: n

text: n

text: sets

text: in

text: X

text: is

text: compact.

text: Our

text: deﬁnition

text: of

text: above

text: coincides

text: M

text: with

text: the

text: (X)

text: with

text: X

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: PTH MOMENT NOISE-TO-STATE STABILITY OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH PERSISTENT NOISE∗ DAVID MATEOS-NU´N˜EZ AND JORGE CORTE´S† Abstract. This paper studies the stability properties of stochastic diﬀerential equations subject to persistent noise (including the case of additive noise), which is noise that is present even at the equilibria of the underlying diﬀerential equation and does not decay with time. The class of systems we consider exhibit disturbance attenuation outside a closed, not necessarily bounded, set. We identify conditions, based on the existence of Lyapunov functions, to establish the noise- to-state stability in probability and in pth moment of the system with respect to a closed set. As part of our analysis, we study the concept of two functions being proper with respect to each other formalized via pair of inequalities with comparison functions. We show that such inequalities deﬁne several equivalence relations for increasingly strong reﬁnements on the comparison functions. We also provide a complete characterization of the properties that a pair of functions must satisfy to belong to the same equivalence class. This characterization allows us to provide checkable conditions to determine whether a function satisﬁes the requirements to be a strong NSS-Lyapunov function in probability or a pth moment NSS-Lyapunov function. Several examples illustrate our results. Stochastic diﬀerential equations (SDEs) go beyond ordinary diﬀerential equations (ODEs) to deal with systems subject to stochastic perturbations of a particular type, known as white noise. Applications are numerous and include option pricing in the stock market, networked systems with noisy communication channels, and, in general, scenarios whose complexity cannot be captured by deterministic models. In this paper, we study SDEs subject to persistent noise (including the case of additive noise), i.e., systems for which the noise is present even at the equilibria of the underlying ODE and does not decay with time. Such scenarios arise, for instance, in control-aﬃne systems when the input is corrupted by persistent noise. For these systems, the presence of persistent noise makes it impossible to establish in general a stochastic notion of asymptotic stability for the (possibly unbounded) set of equilibria of the underlying ODE. Our aim here is to develop notions and tools to study the stability properties of these systems and provide probabilistic guarantees on the size of the state of the system. Literature review: In general, it is not possible to obtain explicit descriptions of the solutions of SDEs. Fortunately, the Lyapunov techniques used to study the qualitative behavior of ODEs [6, 10] can be adapted to study the stability properties of SDEs as well [7, 13, 27]. Depending on the notion of stochastic convergence used, there are several types of stability results in SDEs. These include stochastic stability (or stability in probability), stochastic asymptotic stability, almost sure exponential sta- bility, and pth moment asymptotic stability, see e.g., [13, 14, 26, 27]. However, these notions are not appropriate in the presence of persistent noise because they require the eﬀect of the noise on the set of equilibria to either vanish or decay with time. To deal with persistent noise, as well as other system properties like delays, a concept of ultimate boundedness is required that generalizes the notion of convergence. As an ∗ A preliminary version of this manuscript was presented as [15] at the 2013 American Control Conference, Washington, D.C. This manuscript is a revision of the version submitted to SIAM Journal on Control and Optimization 52 (4) (2014), 2399-2421. † The authors are with the Department of Mechanical and Aer{ospace Engineering,}University of California, San Diego, 9500 Gilman Dr, La Jolla, CA 92093, USA, dmateosn,cortes @ucsd.edu. 1 ##LTLine##

text: PTH

label: paragraph

text: MOMENT

text: NOISE-TO-STATE

text: STABILITY

text: OF

text: STOCHASTIC

text: DIFFERENTIAL

text: EQUATIONS

text: WITH

text: PERSISTENT

text: NOISE∗

text: DAVID

text: MATEOS-NU´N˜EZ

text: AND

text: JORGE

text: CORTE´S†

text: Abstract.

text: This

label: abstract

text: paper

text: studies

text: the

text: stability

text: properties

text: of

text: stochastic

text: diﬀerential

text: equations

text: subject

text: to

text: persistent

text: noise

text: (including

text: the

text: case

text: of

text: additive

text: noise),

text: which

text: is

text: noise

text: that

text: is

text: present

text: even

text: at

text: the

text: equilibria

text: of

text: the

text: underlying

text: diﬀerential

text: equation

text: and

text: does

text: not

text: decay

text: with

text: time.

text: The

text: class

text: of

text: systems

text: we

text: consider

text: exhibit

text: disturbance

text: attenuation

text: outside

text: a

text: closed,

text: not

text: necessarily

text: bounded,

text: set.

text: We

text: identify

text: conditions,

text: based

text: on

text: the

text: existence

text: of

text: Lyapunov

text: functions,

text: to

text: establish

text: the

text: noise-

text: to-state

text: stability

text: in

text: probability

text: and

text: in

text: pth

text: moment

text: of

text: the

text: system

text: with

text: respect

text: to

text: a

text: closed

text: set.

text: As

text: part

text: of

text: our

text: analysis,

text: we

text: study

text: the

text: concept

text: of

text: two

text: functions

text: being

text: proper

text: with

text: respect

text: to

text: each

text: other

text: formalized

text: via

text: pair

text: of

text: inequalities

text: with

text: comparison

text: functions.

text: We

text: show

text: that

text: such

text: inequalities

text: deﬁne

text: several

text: equivalence

text: relations

text: for

text: increasingly

text: strong

text: reﬁnements

text: on

text: the

text: comparison

text: functions.

text: We

text: also

text: provide

text: a

text: complete

text: characterization

text: of

text: the

text: properties

text: that

text: a

text: pair

text: of

text: functions

text: must

text: satisfy

text: to

text: belong

text: to

text: the

text: same

text: equivalence

text: class.

text: This

text: characterization

text: allows

text: us

text: to

text: provide

text: checkable

text: conditions

text: to

text: determine

text: whether

text: a

text: function

text: satisﬁes

text: the

text: requirements

text: to

text: be

text: a

text: strong

text: NSS-Lyapunov

text: function

text: in

text: probability

text: or

text: a

text: pth

text: moment

text: NSS-Lyapunov

text: function.

text: Several

text: examples

text: illustrate

text: our

text: results.

text: Stochastic

text: diﬀerential

text: equations

text: (SDEs)

text: go

text: beyond

text: ordinary

text: diﬀerential

text: equations

text: (ODEs)

text: to

text: deal

text: with

text: systems

text: subject

text: to

text: stochastic

text: perturbations

text: of

text: a

text: particular

text: type,

text: known

text: as

text: white

text: noise.

text: Applications

text: are

text: numerous

text: and

text: include

text: option

text: pricing

text: in

text: the

text: stock

text: market,

text: networked

text: systems

text: with

text: noisy

text: communication

text: channels,

text: and,

text: in

text: general,

text: scenarios

text: whose

text: complexity

text: cannot

text: be

text: captured

text: by

text: deterministic

text: models.

text: In

text: this

text: paper,

text: we

text: study

text: SDEs

text: subject

text: to

text: persistent

text: noise

text: (including

text: the

text: case

text: of

text: additive

text: noise),

text: i.e.,

text: systems

text: for

text: which

text: the

text: noise

text: is

text: present

text: even

text: at

text: the

text: equilibria

text: of

text: the

text: underlying

text: ODE

text: and

text: does

text: not

text: decay

text: with

text: time.

text: Such

text: scenarios

text: arise,

text: for

text: instance,

text: in

text: control-aﬃne

text: systems

text: when

text: the

text: input

text: is

text: corrupted

text: by

text: persistent

text: noise.

text: For

text: these

text: systems,

text: the

text: presence

text: of

text: persistent

text: noise

text: makes

text: it

text: impossible

text: to

text: establish

text: in

text: general

text: a

text: stochastic

text: notion

text: of

text: asymptotic

text: stability

text: for

text: the

text: (possibly

text: unbounded)

text: set

text: of

text: equilibria

text: of

text: the

text: underlying

text: ODE.

text: Our

text: aim

text: here

text: is

text: to

text: develop

text: notions

text: and

text: tools

text: to

text: study

text: the

text: stability

text: properties

text: of

text: these

text: systems

text: and

text: provide

text: probabilistic

text: guarantees

text: on

text: the

text: size

text: of

text: the

text: state

text: of

text: the

text: system.

text: Literature

text: review:

text: In

text: general,

text: it

text: is

text: not

text: possible

text: to

text: obtain

text: explicit

text: descriptions

text: of

text: the

text: solutions

text: of

text: SDEs.

text: Fortunately,

text: the

text: Lyapunov

text: techniques

text: used

text: to

text: study

text: the

text: qualitative

text: behavior

text: of

text: ODEs

text: can

text: be

text: adapted

text: to

text: study

text: the

text: stability

text: properties

text: of

text: SDEs

text: as

text: well

text: Depending

text: on

text: the

text: notion

text: of

text: stochastic

text: convergence

text: used,

text: there

text: are

text: several

text: types

text: of

text: stability

text: results

text: in

text: SDEs.

text: These

text: include

text: stochastic

text: stability

text: (or

text: stability

text: in

text: probability),

text: stochastic

text: asymptotic

text: stability,

text: almost

text: sure

text: exponential

text: sta-

text: bility,

text: and

text: pth

text: moment

text: asymptotic

text: stability,

text: see

text: e.g.,

text: However,

text: these

text: notions

text: are

text: not

text: appropriate

text: in

text: the

text: presence

text: of

text: persistent

text: noise

text: because

text: they

text: require

text: the

text: eﬀect

text: of

text: the

text: noise

text: on

text: the

text: set

text: of

text: equilibria

text: to

text: either

text: vanish

text: or

text: decay

text: with

text: time.

text: To

text: deal

text: with

text: persistent

text: noise,

text: as

text: well

text: as

text: other

text: system

text: properties

text: like

text: delays,

text: a

text: concept

text: of

text: ultimate

text: boundedness

text: is

text: required

text: that

text: generalizes

text: the

text: notion

text: of

text: convergence.

text: As

text: an

text: A

text: preliminary

text: version

text: of

text: this

text: manuscript

text: was

text: presented

text: as

text: at

text: the

text: American

text: Control

text: Conference,

text: Washington,

text: D.C.

text: This

text: manuscript

text: is

text: a

text: revision

text: of

text: the

text: version

text: submitted

text: to

text: SIAM

text: Journal

text: on

text: Control

text: and

text: Optimization

text: The

text: authors

text: are

text: with

text: the

text: Department

text: of

text: Mechanical

text: and

text: Aer{ospace

text: Engineering,}University

text: of

text: California,

text: San

text: Diego,

text: Gilman

text: Dr,

text: La

text: Jolla,

text: CA

text: USA,

text: dmateosn,cortes

text: @ucsd.edu.

text: ##LTLine##


text: α (s) ≥ φ (s) for all s ∈ [0, b ] and with α (b ) = M . (This is possible because the 2 1 1 2 1 2 only function that cannot be upper bounded by an arbitrary continuous function in some arbitrarily small interval [0, b ] is the function that has a jump at 0.) The rest 1 of the construction is explicit. We deﬁne α as a piecewise linear function in (b , ∞) 2 1 in the following way: for each k ≥ 2, we deﬁne (cid:44) M − α (b ) α (s) α (b ) + k+1 2 k−1 (s − b ), ∀s ∈ (b , b ]. 2 2 k−1 b − b k−1 k−1 k k k−1 The resulting α is continuous by construction. Also, α (b ) = M , so that, induc- 2 2 1 2 tively, α (b ) = M for k ≥ 2. Two facts now follow: ﬁrst, M − α (b ) = 2 k−1 k k+1 2 k−1 M − M ≥ 1/(k + 1)2 for k ≥ 2, so α has positive slope in each interval (b , b ] k+1 k 2 k−1 k and thus is strictly increasing in (b , ∞); second, α (s) > α (b ) = M ≥ φ (b ) ≥ 1 2 2 k−1 k 1 k φ (s) for all s ∈ (b , b ], for each k ≥ 2, so α (s) ≥ φ (s) for all s ∈ (b , ∞). 1 k−1 k 2 1 1 R We have left to show the existence of α ∈ K such that α (s) ≤ ψ (s) for all s ∈ . 1 1 1 ≥0 First, since 0 ≤ ψ (s) ≤ φ (s) for all s ≥ 0 by deﬁnition and by P0, using the 1 1 sandwich theorem [11, p. 107], we derive that ψ is right continuous at 0 the same 1 as φ . In addition, since ψ is nondecreasing, it can only have a countable number of 1 1 jump discontinuities (none of them at 0). Therefore, we can pick c > 0 such that a 1 continuous and nondecreasing function ψˆ can be constructed in [0, c ) by removing 1 1 the jumps of ψ , so that ψˆ (s) ≤ ψ (s). Moreover, since ψ is positive deﬁnite and 1 1 1 1 right continuous at 0, then ψˆ is also positive deﬁnite. Thus, there exists α in [0, c ) 1 1 1 continuous, positive deﬁnite, and strictly increasing, such that, for some r < 1, (4.5) α (s) ≤ rψˆ (s) ≤ rψ (s) 1 1 1 R for all s ∈ [0, c ). To extend α to a function in class K in , we follow a sim- 1 1 ≥0 R ilar strategy as for α . Given a strictly increasing sequence {c } ⊆ with 2 R k k≥2 ≥0 lim c = ∞, we deﬁne a sequence {m } ⊆ in the following way: k→∞ k k k≥1 ≥0 (cid:44) (4.6) mk inf ψ1(s) − ψ1(c11)+−kα21(c1) = ψ1(ck) − ψ1(c11)+−kα21(c1). s∈[ck,ck+1) Next we deﬁne α in [c , ∞) as the piecewise linear function 1 1 (cid:44) m − α (c ) α (s) α (c ) + k 1 k (s − c ), ∀s ∈ [c , c ), 1 1 k c − c k k k+1 k+1 k for all k ≥ 1, so α is continuous by construction. It is also strictly increasing because 1 α (c ) = m = (ψ (c ) + α (c ))/2 > α (c ) by (4.5), and also, for each k ≥ 2, 1 2 1 1 1 1 1 1 1 the slopes are positive because m − α (c ) = m − m > 0 (due to the fact k 1 k k k−1 that {m } in (4.6) is strictly increasing because ψ is nondecreasing). Finally, k k≥1 1 α (s) < α (c ) = m < ψ (c ) ≤ ψ (s) for all s ∈ [c , c ), for all k ≥ 1 by (4.6). 1 1 k+1 k 1 k 1 k k+1 Equipped with α , α as deﬁned above, and as a consequence of (4.2), we have that 1 2 19 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: α

text: (s)

text: φ

text: (s)

text: for

text: all

text: s

text: b

text: and

text: with

text: α

text: (b

text: M

text: (This

text: is

text: possible

text: because

text: the

text: only

text: function

text: that

text: cannot

text: be

text: upper

text: bounded

text: by

text: an

text: arbitrary

text: continuous

text: function

text: in

text: some

text: arbitrarily

text: small

text: interval

text: b

text: is

text: the

text: function

text: that

text: has

text: a

text: jump

text: at

text: The

text: rest

text: of

text: the

text: construction

text: is

text: explicit.

text: We

text: deﬁne

text: α

text: as

text: a

text: piecewise

text: linear

text: function

text: in

text: (b

text: in

text: the

text: following

text: way:

text: for

text: each

text: k

text: we

text: deﬁne

text: M

text: α

text: (b

text: α

text: (s)

text: α

text: (b

text: (s

text: b

text: ∀s

text: (b

text: b

text: b

text: b

text: k

text: k

text: The

text: resulting

text: α

text: is

text: continuous

text: by

text: construction.

text: Also,

text: α

text: (b

text: M

text: so

text: that,

text: induc-

text: tively,

text: α

text: (b

text: M

text: for

text: k

text: Two

text: facts

text: now

text: follow:

text: ﬁrst,

text: M

text: α

text: (b

text: k

text: M

text: M

text: for

text: k

text: so

text: α

text: has

text: positive

text: slope

text: in

text: each

text: interval

text: (b

text: b

text: k

text: k

text: and

text: thus

text: is

text: strictly

text: increasing

text: in

text: (b

text: second,

text: α

text: (s)

text: α

text: (b

text: M

text: φ

text: (b

text: k

text: k

text: φ

text: (s)

text: for

text: all

text: s

text: (b

text: b

text: for

text: each

text: k

text: so

text: α

text: (s)

text: φ

text: (s)

text: for

text: all

text: s

text: (b

text: k

text: R

text: We

text: have

text: left

text: to

text: show

text: the

text: existence

text: of

text: α

text: K

text: such

text: that

text: α

text: (s)

text: ψ

text: (s)

text: for

text: all

text: s

text: ≥0

text: First,

text: since

text: ψ

text: (s)

text: φ

text: (s)

text: for

text: all

text: s

text: by

text: deﬁnition

text: and

text: by

text: using

text: the

text: sandwich

text: theorem

text: p.

text: we

text: derive

text: that

text: ψ

text: is

text: right

text: continuous

text: at

text: the

text: same

text: as

text: φ

text: In

text: addition,

text: since

text: ψ

text: is

text: nondecreasing,

text: it

text: can

text: only

text: have

text: a

text: countable

text: number

text: of

text: jump

text: discontinuities

text: (none

text: of

text: them

text: at

text: Therefore,

text: we

text: can

text: pick

text: c

text: such

text: that

text: a

text: continuous

text: and

text: nondecreasing

text: function

text: ψˆ

text: can

text: be

text: constructed

text: in

text: c

text: by

text: removing

text: the

text: jumps

text: of

text: ψ

text: so

text: that

text: ψˆ

text: (s)

text: ψ

text: (s).

text: Moreover,

text: since

text: ψ

text: is

text: positive

text: deﬁnite

text: and

text: right

text: continuous

text: at

text: then

text: ψˆ

text: is

text: also

text: positive

text: deﬁnite.

text: Thus,

text: there

text: exists

text: α

text: in

text: c

text: continuous,

text: positive

text: deﬁnite,

text: and

text: strictly

text: increasing,

text: such

text: that,

text: for

text: some

text: r

text: α

text: (s)

text: rψˆ

text: (s)

text: rψ

text: (s)

text: R

text: for

text: all

text: s

text: c

text: To

text: extend

text: α

text: to

text: a

text: function

text: in

text: class

text: K

text: in

text: we

text: follow

text: a

text: sim-

text: ≥0

text: R

text: ilar

text: strategy

text: as

text: for

text: α

text: Given

text: a

text: strictly

text: increasing

text: sequence

text: {c

text: with

text: R

text: k

text: ≥0

text: lim

text: c

text: we

text: deﬁne

text: a

text: sequence

text: {m

text: in

text: the

text: following

text: way:

text: k→∞

text: k

text: k

text: ≥0

text: mk

text: inf

text: Next

text: we

text: deﬁne

text: α

text: in

text: [c

text: as

text: the

text: piecewise

text: linear

text: function

text: m

text: α

text: (c

text: α

text: (s)

text: α

text: (c

text: k

text: k

text: (s

text: c

text: ∀s

text: [c

text: c

text: k

text: c

text: c

text: k

text: k

text: k

text: for

text: all

text: k

text: so

text: α

text: is

text: continuous

text: by

text: construction.

text: It

text: is

text: also

text: strictly

text: increasing

text: because

text: α

text: (c

text: m

text: (ψ

text: (c

text: α

text: (c

text: α

text: (c

text: by

text: and

text: also,

text: for

text: each

text: k

text: the

text: slopes

text: are

text: positive

text: because

text: m

text: α

text: (c

text: m

text: m

text: (due

text: to

text: the

text: fact

text: k

text: k

text: k

text: that

text: {m

text: in

text: is

text: strictly

text: increasing

text: because

text: ψ

text: is

text: nondecreasing).

text: Finally,

text: k

text: α

text: (s)

text: α

text: (c

text: m

text: ψ

text: (c

text: ψ

text: (s)

text: for

text: all

text: s

text: [c

text: c

text: for

text: all

text: k

text: by

text: k

text: k

text: k

text: Equipped

text: with

text: α

text: α

text: as

text: deﬁned

text: above,

text: and

text: as

text: a

text: consequence

text: of

text: we

text: have

text: that

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: respect to a closed set, implies the corresponding NSS property of the system with respect to the set. In particular, pth moment NSS with respect to a set provides a bound, at each time, for the pth power of the distance from the state to the set, and this bound is the sum of an increasing function of the size of the noise covariance and a decaying eﬀect of the initial conditions. This bound can be achieved regardless of the possibility that inside the set some combination of the states accumulates the variance of the noise. This is a meaningful stability property for the aforementioned class of systems because the presence of persistent noise makes it impossible to estab- lish in general a stochastic notion of asymptotic stability for the set of equilibria of the underlying diﬀerential equation. We have also studied in depth the inequalities between pairs of functions that appear in the various notions of Lyapunov functions mentioned above. We have shown that these inequalities deﬁne equivalence relations and have developed a complete characterization of the properties that two functions must satisfy to be related by them. Finally, building on this characterization, we have provided an alternative statement of our stochastic stability results. Future work will include the study of the eﬀect of delays and impulsive right-hand sides in the class of SDEs considered in this paper. The ﬁrst author would like to thank Dean Richert for useful discussions. In addition, the authors would like to thank Dr. Fengzhong Li for his kind observations that have made possible an important correction of the proof of Theorem 3.6. The research was supported by NSF award CMMI-1300272. References. [1] V. S. Borkar, Probability Theory: An Advanced Course, Springer, New York, 1995. [2] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2009. [3] H. Deng and M. Krstic´, Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance, Systems & Control Letters, 39 (2000), pp. 173–182. [4] H. Deng, M. Krstic´, and R. J. Williams, Stabilization of stochastic non- linear systems driven by noise of unknown covariance, IEEE Transactions on Automatic Control, 46 (2001), pp. 1237–1253. [5] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985. [6] H. K. Khalil, Nonlinear Systems, Prentice Hall, 3 ed., 2002. [7] R. Khasminskii, Stochastic Stability of Diﬀerential Equations, vol. 66 of Stochastic Modelling and Applied Probability, Springer, 2012. [8] F. C. Klebaner, Introduction to Stochastic Calculus With Applications, Impe- rial College Press, 2005. [9] P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, Jour- nal of Diﬀerential Equations, 253 (2012), pp. 1422–1438. [10] V. Lakshmikantham, V. M. Matrosov, and S. Sivasundaram, Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems, vol. 63 of 25

text: respect

text: to

text: a

text: closed

text: set,

text: implies

text: the

text: corresponding

text: NSS

text: property

text: of

text: the

text: system

text: with

text: respect

text: to

text: the

text: set.

text: In

text: particular,

text: pth

text: moment

text: NSS

text: with

text: respect

text: to

text: a

text: set

text: provides

text: a

text: bound,

text: at

text: each

text: time,

text: for

text: the

text: pth

text: power

text: of

text: the

text: distance

text: from

text: the

text: state

text: to

text: the

text: set,

text: and

text: this

text: bound

text: is

text: the

text: sum

text: of

text: an

text: increasing

text: function

text: of

text: the

text: size

text: of

text: the

text: noise

text: covariance

text: and

text: a

text: decaying

text: eﬀect

text: of

text: the

text: initial

text: conditions.

text: This

text: bound

text: can

text: be

text: achieved

text: regardless

text: of

text: the

text: possibility

text: that

text: inside

text: the

text: set

text: some

text: combination

text: of

text: the

text: states

text: accumulates

text: the

text: variance

text: of

text: the

text: noise.

text: This

text: is

text: a

text: meaningful

text: stability

text: property

text: for

text: the

text: aforementioned

text: class

text: of

text: systems

text: because

text: the

text: presence

text: of

text: persistent

text: noise

text: makes

text: it

text: impossible

text: to

text: estab-

text: lish

text: in

text: general

text: a

text: stochastic

text: notion

text: of

text: asymptotic

text: stability

text: for

text: the

text: set

text: of

text: equilibria

text: of

text: the

text: underlying

text: diﬀerential

text: equation.

text: We

text: have

text: also

text: studied

text: in

text: depth

text: the

text: inequalities

text: between

text: pairs

text: of

text: functions

text: that

text: appear

text: in

text: the

text: various

text: notions

text: of

text: Lyapunov

text: functions

text: mentioned

text: above.

text: We

text: have

text: shown

text: that

text: these

text: inequalities

text: deﬁne

text: equivalence

text: relations

text: and

text: have

text: developed

text: a

text: complete

text: characterization

text: of

text: the

text: properties

text: that

text: two

text: functions

text: must

text: satisfy

text: to

text: be

text: related

text: by

text: them.

text: Finally,

text: building

text: on

text: this

text: characterization,

text: we

text: have

text: provided

text: an

text: alternative

text: statement

text: of

text: our

text: stochastic

text: stability

text: results.

text: Future

text: work

text: will

text: include

text: the

text: study

text: of

text: the

text: eﬀect

text: of

text: delays

text: and

text: impulsive

text: right-hand

text: sides

text: in

text: the

text: class

text: of

text: SDEs

text: considered

text: in

text: this

text: paper.

text: The

text: ﬁrst

text: author

text: would

text: like

text: to

text: thank

text: Dean

text: Richert

text: for

text: useful

text: discussions.

text: In

text: addition,

text: the

text: authors

text: would

text: like

text: to

text: thank

text: Dr.

text: Fengzhong

text: Li

text: for

text: his

text: kind

text: observations

text: that

text: have

text: made

text: possible

text: an

text: important

text: correction

text: of

text: the

text: proof

text: of

text: Theorem

text: The

text: research

text: was

text: supported

text: by

text: NSF

text: award

text: References.

text: V.

text: S.

text: Borkar,

text: Probability

text: Theory:

text: An

text: Advanced

text: Course,

text: Springer,

text: New

text: York,

text: S.

text: Boyd

text: and

text: L.

text: Vandenberghe,

text: Convex

text: Optimization,

text: Cambridge

text: University

text: Press,

text: H.

text: Deng

text: and

text: M.

text: Krstic´,

text: Output-feedback

text: stabilization

text: of

text: stochastic

text: nonlinear

text: systems

text: driven

text: by

text: noise

text: of

text: unknown

text: covariance,

text: Systems

text: Control

text: Letters,

text: pp.

text: H.

text: Deng,

text: M.

text: Krstic´,

text: and

text: R.

text: J.

text: Williams,

text: Stabilization

text: of

text: stochastic

text: non-

text: linear

text: systems

text: driven

text: by

text: noise

text: of

text: unknown

text: covariance,

text: IEEE

text: Transactions

text: on

text: Automatic

text: Control,

text: pp.

text: R.

text: A.

text: Horn

text: and

text: C.

text: R.

text: Johnson,

text: Matrix

text: Analysis,

text: Cambridge

text: University

text: Press,

text: H.

text: K.

text: Khalil,

text: Nonlinear

text: Systems,

text: Prentice

text: Hall,

text: ed.,

text: R.

text: Khasminskii,

text: Stochastic

text: Stability

text: of

text: Diﬀerential

text: Equations,

text: vol.

text: of

text: Stochastic

text: Modelling

text: and

text: Applied

text: Probability,

text: Springer,

text: F.

text: C.

text: Klebaner,

text: Introduction

text: to

text: Stochastic

text: Calculus

text: With

text: Applications,

text: Impe-

text: rial

text: College

text: Press,

text: P.

text: E.

text: Kloeden

text: and

text: T.

text: Lorenz,

text: Mean-square

text: random

text: dynamical

text: systems,

text: Jour-

text: nal

text: of

text: Diﬀerential

text: Equations,

text: pp.

text: V.

text: Lakshmikantham,

text: V.

text: M.

text: Matrosov,

text: and

text: S.

text: Sivasundaram,

text: Vector

text: Lyapunov

text: Functions

text: and

text: Stability

text: Analysis

text: of

text: Nonlinear

text: Systems,

text: vol.

text: of


text: Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991. [11] J. Lewin and M. Lewin, An Introduction to Mathematical Analysis, BH math- ematics series, The Random House, 1988. [12] S.-J. Liu, J.-F. Zhang, and Z.-P. Jiang, A notion of stochastic input-to-state stability and its application to stability of cascaded stochastic nonlinear systems, Acta Mathematicae Applicatae Sinica, English Series, 24 (2008), pp. 141–156. [13] X. Mao, Stochastic versions of the LaSalle theorem, Journal of Diﬀerential Equa- tions, (1999), pp. 175–195. [14] , Stochastic Diﬀerential Equations and Applications, Woodhead Publishing, 2nd ed., 2011. [15] D. Mateos-Nu´n˜ez and J. Corte´s, Stability of stochastic diﬀerential equa- tions with additive persistent noise, in American Control Conference, Washing- ton, D.C., June 2013, pp. 5447–5452. [16] J. N. McDonald and N. A. Weiss, A Course in Real Analysis, Elsevier, Oxford, UK, 1999. [17] J. R. Movellan, Tutorial on stochastic diﬀerential equations, tutorial, MPLab, UCSD, 2011. [18] B. O¨ksendal, Stochastic Diﬀerential Equations - An Introduction with Appli- cations, Universitext, Springer-Verlag, 2010. [19] L. Praly and Y. Wang, Stabilization in spite of matched unmodeled dynamics and an equivalent deﬁnition of input-to-state stability, Mathematics of Control, Signals and Systems, 9 (1996), pp. 1–33. [20] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970. [21] H. Schurz, On moment-dissipative stochastic dynamical systems, Dynamic sys- tems and applications, 10 (2001), pp. 11–44. [22] E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, vol. 6 of TAM, Springer, 2 ed., 1998. [23] , Input to state stability: Basic concepts and results, Nonlinear and Optimal Control Theory, 1932 (2008), pp. 163–220. [24] E. D. Sontag and Y. Wang, On characterizations of the input-to-state stability property, Systems & Control Letters, 24 (1995), pp. 351–359. [25] , New characterizations of input-to-state stability, IEEE Transactions on Automatic Control, 41 (1996), pp. 1283 – 1294. [26] T. Taniguchi, The asymptotic behaviour of solutions of stochastic functional diﬀerential equations with ﬁnite delays by the Lyapunov-Razumikhin method, in Advances in Stability Theory at the End of the 20th Century, A. A. Martynyuk, ed., vol. 13 of Stability and Control: Theory, Methods and Applications, Taylor and Francis, 2003, pp. 175–188. [27] U. H. Thygesen, A survey of Lyapunov techniques for stochastic diﬀerential equations, Tech. Rep. 18-1997, Department of Mathematical Modeling, Technical University of Denmark, 1997. [28] F. Wu and P. E. Kloeden, Mean-square random attractors of stochastic delay diﬀerential equations with random delay, Discrete and Continuous Dynamical Systems - Series B (DCDS-B), 18 (2013), pp. 1715–1734. [29] Z.-J. Wu, X.-J. Xie, and S.-Y. Zhang, Adaptive backstepping controller de- sign using stochastic small-gain theorem, Automatica, 43 (2007), pp. 608–620. The next result is used in the proof of Theorem 4.6. 26 ##LTLine## ##LTLine## ##LTLine##

text: Mathematics

label: reference

text: and

text: its

text: Applications,

text: Kluwer

text: Academic

text: Publishers,

text: Dordrecht,

text: The

text: Netherlands,

text: J.

text: Lewin

text: and

text: M.

text: Lewin,

text: An

text: Introduction

text: to

text: Mathematical

text: Analysis,

text: BH

text: math-

text: ematics

text: series,

text: The

text: Random

text: House,

text: S.-J.

text: Liu,

text: J.-F.

text: Zhang,

text: and

text: Z.-P.

text: Jiang,

text: A

text: notion

text: of

text: stochastic

text: input-to-state

text: stability

text: and

text: its

text: application

text: to

text: stability

text: of

text: cascaded

text: stochastic

text: nonlinear

text: systems,

text: Acta

text: Mathematicae

text: Applicatae

text: Sinica,

text: English

text: Series,

text: pp.

text: X.

text: Mao,

text: Stochastic

text: versions

text: of

text: the

text: LaSalle

text: theorem,

text: Journal

text: of

text: Diﬀerential

text: Equa-

text: tions,

text: pp.

text: Stochastic

text: Diﬀerential

text: Equations

text: and

text: Applications,

text: Woodhead

text: Publishing,

text: ed.,

text: D.

text: Mateos-Nu´n˜ez

text: and

text: J.

text: Corte´s,

text: Stability

text: of

text: stochastic

text: diﬀerential

text: equa-

text: tions

text: with

text: additive

text: persistent

text: noise,

text: in

text: American

text: Control

text: Conference,

text: Washing-

text: ton,

text: D.C.,

text: June

text: pp.

text: J.

text: N.

text: McDonald

text: and

text: N.

text: A.

text: Weiss,

text: A

text: Course

text: in

text: Real

text: Analysis,

text: Elsevier,

text: Oxford,

text: UK,

text: J.

text: R.

text: Movellan,

text: Tutorial

text: on

text: stochastic

text: diﬀerential

text: equations,

text: tutorial,

text: MPLab,

text: UCSD,

text: B.

text: O¨ksendal,

text: Stochastic

text: Diﬀerential

text: Equations

text: An

text: Introduction

text: with

text: Appli-

text: cations,

text: Universitext,

text: Springer-Verlag,

text: L.

text: Praly

text: and

text: Y.

text: Wang,

text: Stabilization

text: in

text: spite

text: of

text: matched

text: unmodeled

text: dynamics

text: and

text: an

text: equivalent

text: deﬁnition

text: of

text: input-to-state

text: stability,

text: Mathematics

text: of

text: Control,

text: Signals

text: and

text: Systems,

text: pp.

text: R.

text: T.

text: Rockafellar,

text: Convex

text: Analysis,

text: Princeton

text: University

text: Press,

text: H.

text: Schurz,

text: On

text: moment-dissipative

text: stochastic

text: dynamical

text: systems,

text: Dynamic

text: sys-

text: tems

text: and

text: applications,

text: pp.

text: E.

text: D.

text: Sontag,

text: Mathematical

text: Control

text: Theory:

text: Deterministic

text: Finite

text: Dimensional

text: Systems,

text: vol.

text: of

text: TAM,

text: Springer,

text: ed.,

text: Input

text: to

text: state

text: stability:

text: Basic

text: concepts

text: and

text: results,

text: Nonlinear

text: and

text: Optimal

text: Control

text: Theory,

text: pp.

text: E.

text: D.

text: Sontag

text: and

text: Y.

text: Wang,

text: On

text: characterizations

text: of

text: the

text: input-to-state

text: stability

text: property,

text: Systems

text: Control

text: Letters,

text: pp.

text: New

text: characterizations

text: of

text: input-to-state

text: stability,

text: IEEE

text: Transactions

text: on

text: Automatic

text: Control,

text: pp.

text: T.

text: Taniguchi,

text: The

text: asymptotic

text: behaviour

text: of

text: solutions

text: of

text: stochastic

text: functional

text: diﬀerential

text: equations

text: with

text: ﬁnite

text: delays

text: by

text: the

text: Lyapunov-Razumikhin

text: method,

text: in

text: Advances

text: in

text: Stability

text: Theory

text: at

text: the

text: End

text: of

text: the

text: Century,

text: A.

text: A.

text: Martynyuk,

text: ed.,

text: vol.

text: of

text: Stability

text: and

text: Control:

text: Theory,

text: Methods

text: and

text: Applications,

text: Taylor

text: and

text: Francis,

text: pp.

text: U.

text: H.

text: Thygesen,

text: A

text: survey

text: of

text: Lyapunov

text: techniques

text: for

text: stochastic

text: diﬀerential

text: equations,

text: Tech.

text: Rep.

text: Department

text: of

text: Mathematical

text: Modeling,

text: Technical

text: University

text: of

text: Denmark,

text: F.

text: Wu

text: and

text: P.

text: E.

text: Kloeden,

text: Mean-square

text: random

text: attractors

text: of

text: stochastic

text: delay

text: diﬀerential

text: equations

text: with

text: random

text: delay,

text: Discrete

text: and

text: Continuous

text: Dynamical

text: Systems

text: Series

text: B

text: (DCDS-B),

text: pp.

text: Z.-J.

text: Wu,

text: X.-J.

text: Xie,

text: and

text: S.-Y.

text: Zhang,

text: Adaptive

text: backstepping

text: controller

text: de-

text: sign

text: using

text: stochastic

text: small-gain

text: theorem,

text: Automatica,

text: pp.

text: The

text: next

text: result

text: is

text: used

text: in

text: the

text: proof

text: of

text: Theorem

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Remark 3.4. (Itˆo formula and exponential dissipativity). Interestingly, the condi- tions (3.3) and (3.4) are equivalent to (cid:0) (cid:1) (3.5) L[V](x, t) ≤ −η−1(V(x)) + σ |Σ(t)| , F R for all x ∈ n, where η−1 ∈ K is convex. Note that, since L[V] is not the Lie ∞ derivative of V (as it contains the Hessian of V), one cannot directly deduce from (3.5) the existence of a continuously twice diﬀerentiable function V˜ such that (cid:0) (cid:1) (3.6) L[V˜](x, t) ≤ −cV˜(x) + σ˜ |Σ(t)| , F as instead can be done in the context of ISS, see e.g. [19]. • R R Example 3.5. (A noise-dissipative Lyapunov function). Assume that h : n → is continuously diﬀerentiable and veriﬁes (3.7) γ((cid:107)x − x(cid:48)(cid:107)2) ≤ (x − x(cid:48))T(∇h(x) − ∇h(x(cid:48))) 2 R for some convex function γ ∈ K for all x, x(cid:48) ∈ n. In particular, this implies that h ∞ is strictly convex. (Incidentally, any strongly convex function veriﬁes (3.7) for some choice of γ linear and strictly increasing.) Consider now the dynamics (cid:0) (cid:1) (3.8) dx(ω, t) = − δLx(ω, t) + ∇h(x(ω, t)) dt + Σ(t) dB(ω, t), R for all t ∈ [t , ∞), where x(ω, t ) = x with probability 1 for some x ∈ n, and δ > 0. 0 R 0 0 0 Here, the matrix L ∈ n×n is symmetric and positive semideﬁnite, and the matrix- R valued function Σ : [t , ∞) → n×m is continuous. This dynamics corresponds to the 0 (cid:44) (cid:44) R SDE (2.1) with f(x, t) −δLx − ∇h(x) and G(x, t) I for all (x, t) ∈ n × [t , ∞). n 0 R Let x∗ ∈ n be the unique solution of the Karush-Kuhn-Tucker [2] condition δLx∗ = (cid:44) −∇h(x∗), corresponding to the unconstrained minimization of F(x) δxTLx + h(x). R R 2 (cid:44) Consider then the candidate Lyapunov function V ∈ C2( n; ) given by V(x) R ≥ 0 1(x − x∗)T(x − x∗). Using (2.2), we obtain that, for all x ∈ n, 2 (cid:16) (cid:17) (cid:16) (cid:17) L[V](x, t) = −(x − x∗)T δLx + ∇h(x) + 1 trace Σ(t)TΣ(t) 2 (cid:16) (cid:17) = −δ(x − x∗)TL(x − x∗) − (x − x∗)T ∇h(x) − ∇h(x∗) + 1|Σ(t)|2 2 F ≤ −γ((cid:107)x − x∗(cid:107)2) + 1|Σ(t)|2 . 2 2 F 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Remark

text: (Itˆo

text: formula

text: and

text: exponential

text: dissipativity).

text: Interestingly,

text: the

text: condi-

text: tions

text: and

text: are

text: equivalent

text: to

label: equation

text: L[V](x,

text: t)

text: σ

text: |Σ(t)|

text: F

text: R

text: for

text: all

text: x

text: n,

text: where

text: η−1

text: K

text: is

text: convex.

text: Note

text: that,

text: since

text: L[V]

text: is

text: not

text: the

text: Lie

text: derivative

text: of

text: V

text: (as

text: it

text: contains

text: the

text: Hessian

text: of

text: V),

text: one

text: cannot

text: directly

text: deduce

text: from

text: the

text: existence

text: of

text: a

text: continuously

text: twice

text: diﬀerentiable

text: function

text: V˜

text: such

text: that

text: L[V˜](x,

text: t)

text: −cV˜(x)

text: σ˜

text: |Σ(t)|

text: F

text: as

text: instead

text: can

text: be

text: done

text: in

text: the

text: context

text: of

text: ISS,

text: see

text: e.g.

text: R

text: R

text: Example

text: (A

text: noise-dissipative

text: Lyapunov

text: function).

text: Assume

text: that

text: h

text: n

text: is

text: continuously

text: diﬀerentiable

text: and

text: veriﬁes

text: (x

text: R

text: for

text: some

text: convex

text: function

text: γ

text: K

text: for

text: all

text: x,

text: n.

text: In

text: particular,

text: this

text: implies

text: that

text: h

text: is

text: strictly

text: convex.

text: (Incidentally,

text: any

text: strongly

text: convex

text: function

text: veriﬁes

text: for

text: some

text: choice

text: of

text: γ

text: linear

text: and

text: strictly

text: increasing.)

text: Consider

text: now

text: the

text: dynamics

text: dx(ω,

text: t)

text: δLx(ω,

text: t)

text: ∇h(x(ω,

text: t))

text: dt

text: Σ(t)

text: dB(ω,

text: t),

text: R

text: for

text: all

text: t

text: [t

text: where

text: x(ω,

text: t

text: x

text: with

text: probability

text: for

text: some

text: x

text: n,

text: and

text: δ

text: R

text: Here,

text: the

text: matrix

text: L

text: n×n

text: is

text: symmetric

text: and

text: positive

text: semideﬁnite,

text: and

text: the

text: matrix-

text: R

text: valued

text: function

text: Σ

text: [t

text: n×m

text: is

text: continuous.

text: This

text: dynamics

text: corresponds

text: to

text: the

text: R

text: SDE

text: with

text: f(x,

text: t)

text: −δLx

text: ∇h(x)

text: and

text: G(x,

text: t)

text: I

text: for

text: all

text: (x,

text: t)

text: n

text: [t

text: n

text: R

text: Let

text: x∗

text: n

text: be

text: the

text: unique

text: solution

text: of

text: the

text: Karush-Kuhn-Tucker

text: condition

text: δLx∗

text: −∇h(x∗),

text: corresponding

text: to

text: the

text: unconstrained

text: minimization

text: of

text: F(x)

text: δxTLx

text: h(x).

text: R

text: R

text: Consider

text: then

text: the

text: candidate

text: Lyapunov

text: function

text: V

text: n;

text: given

text: by

text: V(x)

text: R

text: x∗)T(x

text: x∗).

text: Using

text: we

text: obtain

text: that,

text: for

text: all

text: x

text: n,

text: L[V](x,

text: t)

text: −(x

text: x∗)T

text: δLx

text: ∇h(x)

text: trace

text: Σ(t)TΣ(t)

text: −δ(x

text: x∗)TL(x

text: x∗)

text: (x

text: x∗)T

text: ∇h(x)

text: ∇h(x∗)

text: F

text: F

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: R R (cid:44) We note that W ∈ C( n; ) deﬁned by W(x) γ((cid:107)x − x∗(cid:107)2) veriﬁes ≥ 0 2 (cid:0) (cid:1) R V(x) = 1γ−1 W(x) ∀x ∈ n, 2 where γ−1 is concave and belongs to the class K as explained in Section . Therefore, ∞ V is a noise-dissipative Lyapunov function for (3.8), with concave η ∈ K given by (cid:44) ∞ η(r) = 1/2γ−1(r) and σ ∈ K given by σ(r) 1/2 r2. • The next result generalizes [4, Th. 4.1] to positive semideﬁnite Lyapunov functions that satisfy weaker dissipativity properties (cf. (3.5)) than the typical exponential-like inequality (3.6), and characterizes the overshoot gain. Theorem 3.6. (Noise-dissipative Lyapunov functions have an NSS dynamics). Under Assumption 2.1, and further assuming that Σ is continuous, suppose that V is a noise- dissipative Lyapunov function for (2.1). Then, E(cid:2) (cid:3) (cid:0) (cid:1) (cid:16) (cid:0) (cid:1)(cid:17) (3.9) V(x(t)) ≤ µ˜ V(x ), t − t + η 2 σ max |Σ(s)| , 0 0 F t0≤s≤t for all t ≥ t , where the class KL function (r, s) (cid:55)→ µ˜(r, s) is well deﬁned as the 0 solution y(s) to the initial value problem (3.10) y˙(s) = −1η−1(y(s)), y(0) = r. 2 Proof. Recall that Assumption 2.1 guarantees the global existence and uniqueness of solutions of (2.1). Given the process {V(x(t))} , the proof strategy is to obtain a E(cid:2) (cid:3) t≥t0 diﬀerential inequality for V(x(t)) using Itˆo formula (2.3), and then use a compari- son principle to translate the problem into one of standard input-to-state stability for an appropriate choice of the input. To carry out this strategy, we consider Itˆo formula (2.3) with respect to an arbitrary reference time instant t(cid:48) ≥ t , 0 (3.11) (cid:90) (cid:90) t t V(x(t)) = V(x(t(cid:48))) + L[V](x(s), s)ds + ∇V(x(s))TG(x(s), s)Σ(s)dB(s), (cid:48) (cid:48) t t and we ﬁrst ensure that the expectation of the integral against Brownian motion is 0. R Let S = {x ∈ n : (cid:107)x(cid:107) ≤ k} be the ball of radius k centered at the origin. Fix Rk 2 x ∈ n and denote by τ the ﬁrst exit time of x(t) from S for integer values of k 0 k (cid:44) k greater than (cid:107)x(t )(cid:107) , namely, τ inf {s ≥ t : (cid:107)x(s)(cid:107) ≥ k}, for k > (cid:100)(cid:107)x(t )(cid:107) (cid:101). 0 2 k 0 2 0 2 Since the event {ω ∈ Ω : τ ≤ t} belongs to F for each t ≥ t , it follows that τ is an k t 0 k {F }-stopping time for each t ≥ t . Now, for each k ﬁxed, if we consider the random t (cid:44) 0 variable t (t) min{t, τ } and deﬁne I(t(cid:48), t) as the stochastic integral in (3.11) for any k k ﬁxed t(cid:48) ∈ [t , t (t)], then the process I(t(cid:48), t (t)) has zero expectation as we show next. 0 k k 9 ##LTLine## ##LTLine##

text: R

text: R

text: We

text: note

text: that

text: W

text: C(

text: n;

text: deﬁned

text: by

text: W(x)

text: veriﬁes

text: R

text: V(x)

text: 1γ−1

text: W(x)

text: ∀x

text: n,

text: where

text: γ−1

text: is

text: concave

text: and

text: belongs

text: to

text: the

text: class

text: K

text: as

text: explained

text: in

text: Section

text: Therefore,

text: V

text: is

text: a

text: noise-dissipative

text: Lyapunov

text: function

text: for

text: with

text: concave

text: η

text: K

text: given

text: by

text: η(r)

text: and

text: σ

text: K

text: given

text: by

text: σ(r)

text: The

text: next

text: result

text: generalizes

text: Th.

text: to

text: positive

text: semideﬁnite

text: Lyapunov

text: functions

text: that

text: satisfy

text: weaker

text: dissipativity

text: properties

text: (cf.

text: than

text: the

text: typical

text: exponential-like

text: inequality

text: and

text: characterizes

text: the

text: overshoot

text: gain.

text: Theorem

text: (Noise-dissipative

text: Lyapunov

text: functions

text: have

text: an

text: NSS

text: dynamics).

text: Under

text: Assumption

text: and

text: further

text: assuming

text: that

text: Σ

text: is

text: continuous,

text: suppose

text: that

text: V

text: is

text: a

text: noise-

text: dissipative

text: Lyapunov

text: function

text: for

text: Then,

text: V(x(t))

text: µ˜

text: V(x

text: t

text: t

text: η

text: σ

text: max

text: |Σ(s)|

text: F

text: for

text: all

text: t

text: t

text: where

text: the

text: class

text: KL

text: function

text: (r,

text: s)

text: µ˜(r,

text: s)

text: is

text: well

text: deﬁned

text: as

text: the

text: solution

text: y(s)

text: to

text: the

text: initial

text: value

text: problem

text: y˙(s)

text: r.

text: Proof.

text: Recall

text: that

text: Assumption

text: guarantees

text: the

text: global

text: existence

text: and

text: uniqueness

text: of

text: solutions

text: of

text: Given

text: the

text: process

text: {V(x(t))}

text: the

text: proof

text: strategy

text: is

text: to

text: obtain

text: a

text: diﬀerential

text: inequality

text: for

text: V(x(t))

text: using

text: Itˆo

text: formula

text: and

text: then

text: use

text: a

text: compari-

text: son

text: principle

text: to

text: translate

text: the

text: problem

text: into

text: one

text: of

text: standard

text: input-to-state

text: stability

text: for

text: an

text: appropriate

text: choice

text: of

text: the

text: input.

text: To

text: carry

text: out

text: this

text: strategy,

text: we

text: consider

text: Itˆo

text: formula

text: with

text: respect

text: to

text: an

text: arbitrary

text: reference

text: time

text: instant

text: t

text: t

text: t

text: V(x(t))

text: L[V](x(s),

text: s)ds

text: ∇V(x(s))TG(x(s),

text: s)Σ(s)dB(s),

text: t

text: t

text: and

text: we

text: ﬁrst

text: ensure

text: that

text: the

text: expectation

text: of

text: the

text: integral

text: against

text: Brownian

text: motion

text: is

text: R

text: Let

text: S

text: {x

text: n

text: k}

text: be

text: the

text: ball

text: of

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text: dwarfs that extends out to 10 pc, which are ob- mass RV planet searches using new instruments servable from most facilities in the northern and (e.g. ISHELL with use of a gas cell, SPIROU and southern hemispheres. Since this time, two of CARMENES Rayner et al. 2012; Reshetov et al. these stars have been found to have spectroscopic 2012; Quirrenbach et al. 2010) coming online, as companions (GJ 867BD, LHS 1610AB; Davison et these instruments should be sensitive enough to al. 2014; Bonﬁls et al. 2013). For the remaining search for Earth-mass planets around mid to late 58 isolated stars in the sample, we provide new M-dwarfs. V, R and I photometry and new low dispersion ´ optical (6000 9000 ˚A) spectra, from which we The authors would like to thank Guillem measure the Hα equivalent widths, Na I indices Anglada-Escud´e, Jeremy Jones, John Rayner, and spectral types. We determine that 35 stars Robert Parks, Peter Plavchan, Angelle Tanner are emission-line stars, and note the trend of later and our anonymous referre for intellectual insight. type M-stars being more likely to be emission-line Also, the authors would like to note that the as- stars than earlier type M-stars. trometric data were obtained by the RECONS For a smaller sub-sample of 12 stars, we present team through the NOAO Surveys Program and rotational velocities and absolute radial velocities. the SMARTS Consortium. The infrared spec- Ten of these 12 stars have vsini values below our troscopic data were obtained from the Infrared ´ detection limit (3 km s 1), while G 99-49 and Telescope Facility, which is operated by the Uni- ˘ ´ GJ 729 have vsini values of 5.8 0.3 km s 1 and versity of Hawaii under Cooperative Agreement ˘ ´ 3.8 0.6 km s 1, respectively. For these 12 stars no. NNX-08AE38A with the National Aeronau- ´ our observed RV dispersion is 99 m s 1 with a tics and Space Administration, Science Mission ´ standard deviation of 27 m s 1. We also demon- Directorate, Planetary Astronomy Program. The strate the achievable RV precision using our tech- authors are grateful for the telescope time granted „ ´ nique is 90 m s 1 over timescales from 13 days to them. This publication makes use of data prod- to 5 years. Our spectroscopic results indicate that ucts from the Two Micron All Sky Survey, which is on average we would have detected companions a joint project of the University of Massachusetts with masses greater than 1.5 M in 10 day or- and the Infrared Processing and Analysis Cen- JUP bital periods and greater than 7 M in 100 day ter/California Institute of Technology, funded by JUP orbital periods. the National Aeronautics and Space Administra- ´ Over baselines of 9 13 years, we provide as- tion and the National Science Foundation. This trometry with typical residuals (after determining work was supported by the National Science Foun- „ parallaxes and proper motions) of 3 mas. This dation through a Graduate Research Fellowship to allows us to exclude the presence of companions Cassy Davison. This project was funded in part with masses greater than 11.5 M with orbital by NSF/AAG grant no. 0908018 and NSF grants JUP periods of 4 years and 7.5 M with orbital pe- AST05-07711 and AST09-08402. JUP riods of 8 years. Although we do not detect com- Facilities: IRTF (CSHELL), CTIO:2MASS, panions around any of the stars in our sub-sample, CTIO:0.9m. these results do show that we could easily detect REFERENCES brown dwarfs with wide orbits around low mass Allers, K. N., Jaﬀe, D. T., Luhman, K. L., et al. 2007, ApJ, stars. 657, 511 An ensemble result of this work is that all 12 Bailey, J. I., White, R. J., Blake, C. H., et al. 2012, ApJ, 749, stars studied here with both infrared spectroscopy 16 and astrometry and in the literature with various Barnes, R., Meadows, V. S., Domagal-Goldman, S. D., et al. other techniques are found to be single stars with- 2011, 16th Cambridge Workshop on Cool Stars, Stellar Sys- tems, and the Sun, 448, 391 out stellar, brown dwarf, or giant planet compan- ions within the respective detection limits of the Barnes, J. R., Jenkins, J. S., Jones, H. R. A., et al. 2012, MNRAS, 424, 591 studies. These results provide a ﬁrst step in the process of vetting our nearest neighbors for planet Barnes, J. R., Jenkins, J. S., Jones, H. R. A., et al. 2014, MNRAS, 439, 3094 searches. We suggest these slowly rotating, single stars would be prime targets for upcoming Earth Batalha, N. M., Borucki, W. J., Koch, D. G., et al. 2010, ApJ, 713, L109 14

text: dwarfs

text: that

text: extends

text: out

text: to

text: pc,

text: which

text: are

text: ob-

text: mass

text: RV

text: planet

text: searches

text: using

text: new

text: instruments

text: servable

text: from

text: most

text: facilities

text: in

text: the

text: northern

text: and

text: (e.g.

text: ISHELL

text: with

text: use

text: of

text: a

text: gas

text: cell,

text: SPIROU

text: and

text: southern

text: hemispheres.

text: Since

text: this

text: time,

text: two

text: of

text: CARMENES

text: Rayner

text: et

text: al.

text: Reshetov

text: et

text: al.

text: these

text: stars

text: have

text: been

text: found

text: to

text: have

text: spectroscopic

text: Quirrenbach

text: et

text: al.

text: coming

text: online,

text: as

text: companions

text: (GJ

text: LHS

text: Davison

text: et

text: these

text: instruments

text: should

text: be

text: sensitive

text: enough

text: to

text: al.

text: Bonﬁls

text: et

text: al.

text: For

text: the

text: remaining

text: search

text: for

text: Earth-mass

text: planets

text: around

text: mid

text: to

text: late

text: isolated

text: stars

text: in

text: the

text: sample,

text: we

text: provide

text: new

text: M-dwarfs.

text: V,

text: R

text: and

text: I

text: photometry

text: and

text: new

text: low

text: dispersion

text: optical

text: ˚A)

text: spectra,

text: from

text: which

text: we

text: The

text: authors

text: would

text: like

text: to

text: thank

text: Guillem

text: measure

text: the

text: Hα

text: equivalent

text: widths,

text: I

text: indices

text: Anglada-Escud´e,

text: Jeremy

text: Jones,

text: John

text: Rayner,

text: and

text: spectral

text: types.

text: We

text: determine

text: that

text: stars

text: Robert

text: Parks,

text: Peter

text: Plavchan,

text: Angelle

text: Tanner

text: are

text: emission-line

text: stars,

text: and

text: note

text: the

text: trend

text: of

text: later

text: and

text: our

text: anonymous

text: referre

text: for

text: intellectual

text: insight.

text: type

text: M-stars

text: being

text: more

text: likely

text: to

text: be

text: emission-line

text: Also,

text: the

text: authors

text: would

text: like

text: to

text: note

text: that

text: the

text: as-

text: stars

text: than

text: earlier

text: type

text: M-stars.

text: trometric

text: data

text: were

text: obtained

text: by

text: the

text: RECONS

text: For

text: a

text: smaller

text: sub-sample

text: of

text: stars,

text: we

text: present

text: team

text: through

text: the

text: NOAO

text: Surveys

text: Program

text: and

text: rotational

text: velocities

text: and

text: absolute

text: radial

text: velocities.

text: the

text: SMARTS

text: Consortium.

text: The

text: infrared

text: spec-

text: Ten

text: of

text: these

text: stars

text: have

text: vsini

text: values

text: below

text: our

text: troscopic

text: data

text: were

text: obtained

text: from

text: the

text: Infrared

text: detection

text: limit

text: km

text: s

text: while

text: G

text: and

text: Telescope

text: Facility,

text: which

text: is

text: operated

text: by

text: the

text: Uni-

text: GJ

text: have

text: vsini

text: values

text: of

text: km

text: s

text: and

text: versity

text: of

text: Hawaii

text: under

text: Cooperative

text: Agreement

text: km

text: s

text: respectively.

text: For

text: these

text: stars

text: no.

text: with

text: the

text: National

text: Aeronau-

text: our

text: observed

text: RV

text: dispersion

text: is

text: m

text: s

text: with

text: a

text: tics

text: and

text: Space

text: Administration,

text: Science

text: Mission

text: standard

text: deviation

text: of

text: m

text: s

text: We

text: also

text: demon-

text: Directorate,

text: Planetary

text: Astronomy

text: Program.

text: The

text: strate

text: the

text: achievable

text: RV

text: precision

text: using

text: our

text: tech-

text: authors

text: are

text: grateful

text: for

text: the

text: telescope

text: time

text: granted

text: nique

text: is

text: m

text: s

text: over

text: timescales

text: from

text: days

text: to

text: them.

text: This

text: publication

text: makes

text: use

text: of

text: data

text: prod-

text: to

text: years.

text: Our

text: spectroscopic

text: results

text: indicate

text: that

text: ucts

text: from

text: the

text: Two

text: Micron

text: All

text: Sky

text: Survey,

text: which

text: is

text: on

text: average

text: we

text: would

text: have

text: detected

text: companions

text: a

text: joint

text: project

text: of

text: the

text: University

text: of

text: Massachusetts

text: with

text: masses

text: greater

text: than

text: M

text: in

text: day

text: or-

text: and

text: the

text: Infrared

text: Processing

text: and

text: Analysis

text: Cen-

text: JUP

text: bital

text: periods

text: and

text: greater

text: than

text: M

text: in

text: day

text: ter/California

text: Institute

text: of

text: Technology,

text: funded

text: by

text: JUP

text: orbital

text: periods.

text: the

text: National

text: Aeronautics

text: and

text: Space

text: Administra-

text: Over

text: baselines

text: of

text: years,

text: we

text: provide

text: as-

text: tion

text: and

text: the

text: National

text: Science

text: Foundation.

text: This

text: trometry

text: with

text: typical

text: residuals

text: (after

text: determining

text: work

text: was

text: supported

text: by

text: the

text: National

text: Science

text: Foun-

text: parallaxes

text: and

text: proper

text: motions)

text: of

text: mas.

text: This

text: dation

text: through

text: a

text: Graduate

text: Research

text: Fellowship

text: to

text: allows

text: us

text: to

text: exclude

text: the

text: presence

text: of

text: companions

text: Cassy

text: Davison.

text: This

text: project

text: was

text: funded

text: in

text: part

text: with

text: masses

text: greater

text: than

text: M

text: with

text: orbital

text: by

text: NSF/AAG

text: grant

text: no.

text: and

text: NSF

text: grants

text: JUP

text: periods

text: of

text: years

text: and

text: M

text: with

text: orbital

text: pe-

text: and

text: JUP

text: riods

text: of

text: years.

text: Although

text: we

text: do

text: not

text: detect

text: com-

text: Facilities:

text: IRTF

text: (CSHELL),

text: panions

text: around

text: any

text: of

text: the

text: stars

text: in

text: our

text: sub-sample,

text: these

text: results

text: do

text: show

text: that

text: we

text: could

text: easily

text: detect

text: REFERENCES

text: brown

text: dwarfs

text: with

text: wide

text: orbits

text: around

text: low

text: mass

text: Allers,

text: K.

text: N.,

text: Jaﬀe,

text: D.

text: T.,

text: Luhman,

text: K.

text: L.,

text: et

text: al.

text: ApJ,

text: stars.

text: An

text: ensemble

text: result

text: of

text: this

text: work

text: is

text: that

text: all

text: Bailey,

text: J.

text: I.,

text: White,

text: R.

text: J.,

text: Blake,

text: C.

text: H.,

text: et

text: al.

text: ApJ,

text: stars

text: studied

text: here

text: with

text: both

text: infrared

text: spectroscopy

text: and

text: astrometry

text: and

text: in

text: the

text: literature

text: with

text: various

text: Barnes,

text: R.,

text: Meadows,

text: V.

text: S.,

text: Domagal-Goldman,

text: S.

text: D.,

text: et

text: al.

text: other

text: techniques

text: are

text: found

text: to

text: be

text: single

text: stars

text: with-

text: Cambridge

text: Workshop

text: on

text: Cool

text: Stars,

text: Stellar

text: Sys-

text: tems,

text: and

text: the

text: Sun,

text: out

text: stellar,

text: brown

text: dwarf,

text: or

text: giant

text: planet

text: compan-

text: ions

text: within

text: the

text: respective

text: detection

text: limits

text: of

text: the

text: Barnes,

text: J.

text: R.,

text: Jenkins,

text: J.

text: S.,

text: Jones,

text: H.

text: R.

text: A.,

text: et

text: al.

text: MNRAS,

text: studies.

text: These

text: results

text: provide

text: a

text: ﬁrst

text: step

text: in

text: the

text: process

text: of

text: vetting

text: our

text: nearest

text: neighbors

text: for

text: planet

text: Barnes,

text: J.

text: R.,

text: Jenkins,

text: J.

text: S.,

text: Jones,

text: H.

text: R.

text: A.,

text: et

text: al.

text: MNRAS,

text: searches.

text: We

text: suggest

text: these

text: slowly

text: rotating,

text: single

text: stars

text: would

text: be

text: prime

text: targets

text: for

text: upcoming

text: Earth

text: Batalha,

text: N.

text: M.,

text: Borucki,

text: W.

text: J.,

text: Koch,

text: D.

text: G.,

text: et

text: al.

text: ApJ,


text: Table 3—Continued Name SpType Date H-α NaI YYYYMMDD EW ˚A Index GJ 1111 M6.0Ve 20100305 -8.0 1.47 LHS 2090 M6.0Ve 20020402 -15.8 1.16 20061209 -7.9 1.33 LHS 2206 M4.0Ve 20020401 -4.4 1.14 20061209 -3.3 1.28 LHS 292 M6.5Ve 19901123 -0.8 1.37 20031207 -2.4 1.43 20060531 -9.5 1.36 GJ 402 M3.0V 20090203 -0.2 1.26 20100305 0.0 1.28 20110517 0.3 1.21 GJ 406 M5.0Ve 19951203 -16.1 1.37 20030717 -11.6 1.33 20090505 -8.0 1.38 GJ 447 M4.0V 19930315 1.0 1.21 20040608 0.2 1.23 20090505 -0.1 1.26 GJ 1154 M4.5Ve 20040313 -6.6 1.29 GJ 1156 M4.5Ve 19930314 -5.6 1.25 20040608 -6.5 1.27 GJ 486 M4.0V 20040312 0.3 1.16 GJ 493.1 M4.5Ve 20040312 -4.9 1.27 GJ 555 M3.5V 19930314 0.6 1.18 20040608 0.5 1.20 LHS 3003 M7.0Ve 19930317 -28.5 1.29 20040311 -7.5 1.40 20060527 -19.4 1.35 20090505 0.0 1.32 20110727 2.4 1.00 GJ 609 M4.0V 20090506 -0.2 1.16 20110727 0.1 1.16 GJ 628 M3.0V 19930315 0.9 1.13 20040608 0.0 1.17 GJ 643 M3.0V 20040608 0.0 1.19 GJ 644C M7.0Ve 19950812 -11.6 1.44 20060525 -7.1 1.42 20060526 -10.4 1.40 GJ 1207 M3.5Ve 20020402 -3.9 1.11 20060526 -4.9 1.17 20090505 -7.2 1.27 GJ 699 M3.5V 19950814 0.0 1.22 20041001 0.0 1.21 20060527 0.6 1.17 GJ 1224 M4.0Ve 19931101 -4.6 1.25 20040608 -4.3 1.26 GJ 1230B M4.5Ve 19931101 -0.1 1.26 20040930 -1.2 1.29 GJ 729 M3.5Ve 19931101 -2.4 1.20 GJ 729 M3.5Ve 20040929 -2.5 1.25 GJ 752B M8.0Ve 19950812 -2.6 1.31 22 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Table

text: Name

text: SpType

text: Date

text: H-α

text: NaI

text: YYYYMMDD

text: EW

text: ˚A

text: Index

text: GJ

text: LHS

text: LHS

text: LHS

text: GJ

text: GJ

text: GJ

text: GJ

text: GJ

text: GJ

text: GJ

text: GJ

text: LHS

text: GJ

text: GJ

text: GJ

text: GJ

text: GJ

text: GJ

text: GJ

text: GJ

text: GJ

text: GJ

text: GJ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Table 5 Spectroscopic Results Name Model Teff Abs. ´RVa N ∆ time σR´V vsin´ib (K) (m s 1) (# nights) (days) (m s 1) (km s 1) ˘ G 99-49 3200 30190 11 855 98 5.8˘ 0.3 GJ 300 3200 9043 10 1881 110 0.9˘0.71 GJ 406 3000 ´19537 12 638 132 1.6˘0.71 GJ 555 3200 ´ 1413 8 396 47 0.6˘0.71 GJ 628 3000 ´21109 9 1215 98 1.2 ˘0.71 GJ 729 3200 ´10244 6 433 125 3.8˘ 0.6 GJ 1002 3000 ´39871 6 368 84 1.4˘0.31 GJ 1065 3200 ´ 9099 5 13 68 1.0˘0.81 GJ 1224 3000 ´32673 7 374 139 1.4˘0.41 GJ 1286 3000 40803 5 367 96 0.7˘0.91 LHS 1723 3000 ´42430 6 367 115 1.0˘0.81 LHS 3799 3000 1641 6 602 74 1.5 0.51 „ ´ aThe error on the absolute RV measurements is estimated to be 100 m s 1, based on systematic uncertainties. bThe vsini values reported are the values used in the best ﬁt model. Since values below ´ 3 km s 1 cannot be conﬁdently measured at our resolution, we adopt a vsini upper limit ´ value of 3 km s 1 for these stars. 26 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Table

text: Spectroscopic

text: Results

text: Name

text: Model

text: Teff

text: Abs.

text: ´RVa

text: N

text: time

text: σR´V

text: vsin´ib

text: (K)

text: (m

text: s

text: nights)

text: (days)

text: (m

text: s

text: (km

text: s

text: G

text: GJ

text: GJ

text: GJ

text: GJ

text: GJ

text: GJ

text: GJ

text: GJ

text: GJ

text: LHS

text: LHS

text: aThe

text: error

text: on

text: the

text: absolute

text: RV

text: measurements

text: is

text: estimated

text: to

text: be

text: m

text: s

text: based

text: on

text: systematic

text: uncertainties.

text: bThe

text: vsini

text: values

text: reported

text: are

text: the

text: values

text: used

text: in

text: the

text: best

text: ﬁt

text: model.

text: Since

text: values

text: below

text: km

text: s

text: cannot

text: be

text: conﬁdently

text: measured

text: at

text: our

text: resolution,

text: we

text: adopt

text: a

text: vsini

text: upper

text: limit

text: value

text: of

text: km

text: s

text: for

text: these

text: stars.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 2 TABLE I. Constraints used for the global ﬁt, and the main inputs involved (more information can be found in ref. [6]). When two errors are quoted, the ﬁrst one is statistical, the second one systematic. The lattice inputs are our own averages obtained as described in the text. CKM Process Observables Theoretical inputs | | → | | ± ± V 0+ 0+ transitions V = 0.97425 0 0.00022 [12] Nuclear matrix elements | ud| → | | ud nu→cl ± → ± ± V K π(cid:96)ν V fK π(0) = 0.21664 0.00048 [13] fK π(0) = 0.9641 0.0015 0.045 us → Bus SL→+ ± · − + ± ± K eν (K eν ) = (1.581 0.008) 10 5 [13] f = 155.2 0.2 0.6 MeV → e B → e ± K K µν (K µν ) = 0.6355 0.0011 [13] → µ B → µ ± · − τ Kν (τ Kν ) = (0.6955 0.0096) 10 2 [13] τ B → τ | | → → (K µν ) ± ± ± |VVuuds| K µν/π µν BB(π→→ µνµµ) = 1.3365 0.0032 [13] fK/fπ = 1.1942 0.0009 0.0030 → → (τ Kν ) ± · − τ Kν/τ πν B → τ = (6.43 0.09) 10 2 [13] (τ πν ) | | | | τ ± V νN V = 0.230 0.011 [13] cd → B cd→νN ± · − ± ± D µν (D µν) = (3.74 0.17) 10 4 [15] f /f = 1.201 0.004 0.010 → | | → ± D→s D ± ± D π(cid:96)ν V fD π(0) = 0.148 0.004 [14] fD π(0) = 0.666 0.020 0.048 | | → |cd |+ ± + V W cs¯ V → = 0.94+−0.32 0.13 [13] cs → B cs W→ cs¯ ±0.26 · − ± ± D τν (D τν) = (5.55 0.24) 10 2 [15] f = 245.3 0.5 4.5 MeV s → B s→ ± · − Ds D µν (D µν ) = (5.57 0.24) 10 3 [15] s→ | |s → µ ± → ± ± D K(cid:96)ν V fD K(0) = 0.712 0.007 [14, 16] fD K(0) = 0.747 0.011 0.034 | | cs| + | ± ± · − + V semileptonic decays V = (3.70 0.12 0.26) 10 3 [15] form factors, shape functions ub → B u→b SL ± · − ± ± B τν (B τν) = (1.08 0.21) 10 4 [15, 17] f /f = 1.205 0.004 0.007 | | | | ± ± · − Bs B V semileptonic decays V = (41.00 0.33 0.74) 10 3 [15] form factors, OPE matrix elements cb → cb SL α B ππ, ρπ, ρρ branching ratios, CP asymmetries [15] isospin symmetry → ± β B (cc¯)K sin(2β) = 0.682 0.019 [15] → ∗ ∗ [cc¯] γ B D( )K( ) inputs for the 3 methods [15] GGSZ, GLW, ADS methods → − ± φ B J/ψ(KK, ππ) φ = 0.015 0.035 [15] ∗ s s s ± − ± ± V V (cid:48) ∆m ∆m = 0.510 0.003 ps 1 [15] Bˆ /Bˆ = 1.023 0.013 0.014 tq tq d d ± − Bs Bd ± ± ∆m ∆m = 17.757 0.021 ps 1 [15] Bˆ = 1.320 0.017 0.030 →s B →s · − Bs ± ± B µµ (B µµ) = (2.8+−0.7) 10 9 [18] f = 225.6 1.1 5.4 MeV ∗ s s| | ±0.6 · − Bs ± ± V V (cid:15) (cid:15) = (2.228 0.011) 10 3 [13] Bˆ = 0.7615 0.0027 0.0137 t∗d ts K K K ± ± V V κ = 0.940 0.013 0.023 cd cs (cid:15) parameters. We rely mainly on lattice QCD simulations systematics. for the determination of these quantities, as they provide well-established methods to compute these observables The combination of lattice values is a critical point of with a controlled accuracy. Some of the uncertainties most global analyses of the ﬂavour physics data, even have a clear statistical interpretation. Lattice simulations though there is no universal deﬁnition of theoretical un- evaluate Green functions in a Euclidean metric expressed certainties (and hence how to combine them). Several ap- as path integrals using Monte Carlo methods, and their proaches have been proposed to perform such a combina- accuracy depends on the size of the sample of gauge con- tion [9, 19], and we have also proposed our own scheme, ﬁgurations used for the computation. The remaining systematic, reproducible and to some extent conserva- uncertainties are systematic: they are now dominant in tive [20]. We have collected the relevant lattice calcula- most cases and they depend on the computational strate- tions of the meson decay constants, as→well as→the Bd, gies chosen by competing lattice collaborations: discreti- Bs →and K bag parameters, and the K π, D π and sation methods used to describe gauge ﬁelds and fermions D K vector form factors at zero momentum trans- on a lattice, interpolating ﬁelds, parameters of the sim- fer. We base our set of calculations on the latest FLAG ulations, such as the size of the (ﬁnite) volumes and lat- (Flavour Lattice Averaging Group) report [19], with the tice spacings, the masses of the quarks that can be sim- addition of new results published since that report was ulated, and the number of dynamical ﬂavours included written [6]. We perform our averages considering values as sea quarks. These simulations must often be extrapo- from lattice simulations with diﬀerent numbers of dy- lated to obtain physical quantities, relying in particular namical ﬂavours (Nf = 2, 2 + 1, 2 + 1 + 1). Even though on eﬀective theories such as chiral perturbation theory the diﬀerent collaborations attempt at assessing the cor- and heavy-quark eﬀective theory which induce further responding systematics in a careful way, one cannot ex- clude that such combinations are aﬀected by further sys- ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TABLE

text: I.

text: Constraints

label: caption

text: used

text: for

text: the

text: global

text: ﬁt,

text: and

text: the

text: main

text: inputs

text: involved

text: (more

text: information

text: can

text: be

text: found

text: in

text: ref.

text: When

text: two

text: errors

text: are

text: quoted,

text: the

text: ﬁrst

text: one

text: is

text: statistical,

text: the

text: second

text: one

text: systematic.

text: The

text: lattice

text: inputs

text: are

text: our

text: own

text: averages

text: obtained

text: as

text: described

text: in

text: the

text: text.

text: CKM

label: table

text: Process

text: Observables

text: Theoretical

text: inputs

text: V

text: transitions

text: V

text: Nuclear

text: matrix

text: elements

text: ud|

text: ud

text: nu→cl

text: V

text: K

text: V

text: fK

text: fK

text: us

text: Bus

text: SL→+

text: K

text: eν

text: (K

text: eν

text: f

text: MeV

text: e

text: B

text: e

text: K

text: K

text: µν

text: (K

text: µν

text: µ

text: B

text: µ

text: τ

text: Kν

text: (τ

text: Kν

text: τ

text: B

text: τ

text: (K

text: µν

text: |VVuuds|

text: K

text: µν/π

text: µν

text: BB(π→→

text: µνµµ)

text: fK/fπ

text: (τ

text: Kν

text: τ

text: Kν/τ

text: πν

text: B

text: τ

text: (τ

text: πν

text: τ

text: V

text: νN

text: V

text: cd

text: B

text: cd→νN

text: D

text: µν

text: (D

text: µν)

text: f

text: /f

text: D→s

text: D

text: D

text: V

text: fD

text: fD

text: |cd

text: V

text: W

text: cs¯

text: V

text: cs

text: B

text: cs

text: W→

text: cs¯

text: D

text: τν

text: (D

text: τν)

text: f

text: MeV

text: s

text: B

text: s→

text: Ds

text: D

text: µν

text: (D

text: µν

text: s→

text: |s

text: µ

text: D

text: V

text: fD

text: fD

text: cs|

text: V

text: semileptonic

text: decays

text: V

text: form

text: factors,

text: shape

text: functions

text: ub

text: B

text: u→b

text: SL

text: B

text: τν

text: (B

text: τν)

text: f

text: /f

text: Bs

text: B

text: V

text: semileptonic

text: decays

text: V

text: form

text: factors,

text: OPE

text: matrix

text: elements

text: cb

text: cb

text: SL

text: α

text: B

text: ππ,

text: ρπ,

text: ρρ

text: branching

text: ratios,

text: CP

text: asymmetries

text: isospin

text: symmetry

text: β

text: B

text: (cc¯)K

text: [cc¯]

text: γ

text: B

text: D(

text: )K(

text: inputs

text: for

text: the

text: methods

text: GGSZ,

text: GLW,

text: ADS

text: methods

text: φ

text: B

text: J/ψ(KK,

text: ππ)

text: φ

text: s

text: s

text: s

text: V

text: V

text: ∆m

text: ∆m

text: ps

text: Bˆ

text: /Bˆ

text: tq

text: tq

text: d

text: d

text: Bs

text: Bd

text: ∆m

text: ∆m

text: ps

text: Bˆ

text: →s

text: B

text: →s

text: Bs

text: B

text: µµ

text: (B

text: µµ)

text: f

text: MeV

text: s

text: s|

text: Bs

text: V

text: V

text: Bˆ

text: t∗d

text: ts

text: K

text: K

text: K

text: V

text: V

text: κ

text: cd

text: cs

text: parameters.

text: We

text: rely

text: mainly

text: on

text: lattice

text: QCD

text: simulations

text: systematics.

text: for

text: the

text: determination

text: of

text: these

text: quantities,

text: as

text: they

text: provide

text: well-established

text: methods

text: to

text: compute

text: these

text: observables

text: The

text: combination

text: of

text: lattice

text: values

text: is

text: a

text: critical

text: point

text: of

text: with

text: a

text: controlled

text: accuracy.

text: Some

text: of

text: the

text: uncertainties

text: most

text: global

text: analyses

text: of

text: the

text: ﬂavour

text: physics

text: data,

text: even

text: have

text: a

text: clear

text: statistical

text: interpretation.

text: Lattice

text: simulations

text: though

text: there

text: is

text: no

text: universal

text: deﬁnition

text: of

text: theoretical

text: un-

text: evaluate

text: Green

text: functions

text: in

text: a

text: Euclidean

text: metric

text: expressed

text: certainties

text: (and

text: hence

text: how

text: to

text: combine

text: them).

text: Several

text: ap-

text: as

text: path

text: integrals

text: using

text: Monte

text: Carlo

text: methods,

text: and

text: their

text: proaches

text: have

text: been

text: proposed

text: to

text: perform

text: such

text: a

text: combina-

text: accuracy

text: depends

text: on

text: the

text: size

text: of

text: the

text: sample

text: of

text: gauge

text: con-

text: tion

text: and

text: we

text: have

text: also

text: proposed

text: our

text: own

text: scheme,

text: ﬁgurations

text: used

text: for

text: the

text: computation.

text: The

text: remaining

text: systematic,

text: reproducible

text: and

text: to

text: some

text: extent

text: conserva-

text: uncertainties

text: are

text: systematic:

text: they

text: are

text: now

text: dominant

text: in

text: tive

text: We

text: have

text: collected

text: the

text: relevant

text: lattice

text: calcula-

text: most

text: cases

text: and

text: they

text: depend

text: on

text: the

text: computational

text: strate-

text: tions

text: of

text: the

text: meson

text: decay

text: constants,

text: as→well

text: as→the

text: Bd,

text: gies

text: chosen

text: by

text: competing

text: lattice

text: collaborations:

text: discreti-

text: Bs

text: →and

text: K

text: bag

text: parameters,

text: and

text: the

text: K

text: π,

text: D

text: π

text: and

text: sation

text: methods

text: used

text: to

text: describe

text: gauge

text: ﬁelds

text: and

text: fermions

text: D

text: K

text: vector

text: form

text: factors

text: at

text: zero

text: momentum

text: trans-

text: on

text: a

text: lattice,

text: interpolating

text: ﬁelds,

text: parameters

text: of

text: the

text: sim-

text: fer.

text: We

text: base

text: our

text: set

text: of

text: calculations

text: on

text: the

text: latest

text: FLAG

text: ulations,

text: such

text: as

text: the

text: size

text: of

text: the

text: (ﬁnite)

text: volumes

text: and

text: lat-

text: (Flavour

text: Lattice

text: Averaging

text: Group)

text: report

text: with

text: the

text: tice

text: spacings,

text: the

text: masses

text: of

text: the

text: quarks

text: that

text: can

text: be

text: sim-

text: addition

text: of

text: new

text: results

text: published

text: since

text: that

text: report

text: was

text: ulated,

text: and

text: the

text: number

text: of

text: dynamical

text: ﬂavours

text: included

text: written

text: We

text: perform

text: our

text: averages

text: considering

text: values

text: as

text: sea

text: quarks.

text: These

text: simulations

text: must

text: often

text: be

text: extrapo-

text: from

text: lattice

text: simulations

text: with

text: diﬀerent

text: numbers

text: of

text: dy-

text: lated

text: to

text: obtain

text: physical

text: quantities,

text: relying

text: in

text: particular

text: namical

text: ﬂavours

text: (Nf

text: Even

text: though

text: on

text: eﬀective

text: theories

text: such

text: as

text: chiral

text: perturbation

text: theory

text: the

text: diﬀerent

text: collaborations

text: attempt

text: at

text: assessing

text: the

text: cor-

text: and

text: heavy-quark

text: eﬀective

text: theory

text: which

text: induce

text: further

text: responding

text: systematics

text: in

text: a

text: careful

text: way,

text: one

text: cannot

text: ex-

text: clude

text: that

text: such

text: combinations

text: are

text: aﬀected

text: by

text: further

text: sys-

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