A smooth converse lyapunov theorem for r

A Smooth Converse Lyapunov Theorem for Robust StabilityYuan Wang†Department of MathematicsFlorida Atlantic UniversityBoca Raton, FL 33431ywang@polya.math.fau.edu Abstract.. In aunified a

A Smooth Converse Lyapunov Theorem for Robust Stability

Yuan Wang†Department of MathematicsFlorida Atlantic UniversityBoca Raton, FL 33431ywang@polya.math.fau.edu

Abstract This paper presents a Converse Lyapunov Function Theorem motivated by robust control

anal-ysis and design Our result is based upon, but generalizes, various aspects of well-known classical theorems In aunified and natural manner, it (1) allows arbitrary bounded time-varying parameters in the system description,(2) deals with global asymptotic stability, (3) results in smooth (infinitely differentiable) Lyapunov functions,and (4) applies to stability with respect to not necessarily compact invariant sets

1 Introduction This work is motivated by problems of robust nonlinear stabilization One of our main

contributions is to provide a statement and proof of a Converse Lyapunov Function Theorem which is in a formparticularly useful for the study of such feedback control analysis and design problems We provide a single (andnatural) unified result that:

1 applies to stability with respect to not necessarily compact invariant sets;

2 deals with global (as opposed to merely local) asymptotic stability;

3 results in smooth (infinitely differentiable) Lyapunov functions;

4 most importantly, applies to stability in the presence of bounded time varying parameters in the system.(This latter property is sometimes called “total stability” and it is equivalent to the stability of an associateddifferential inclusion.)

The interest in stability with respect to possibly non-compact sets is motivated by applications to areas such

as output-control (one needs to stabilize with respect to the zero set of the output variables) and Luenberger-typeobserver design (“detectability” corresponds to stability with respect to the diagonal set{(x, x)}, as a subset of

the composite state/observer system) Such applications and others are explored in [16], Chapter 5

Smooth Lyapunov functions, as opposed to merely continuous or once-differentiable, are required in order

to apply “backstepping” techniques in which a feedback law is built by successively taking directional derivatives

of feedback laws obtained for a simplified system (See for instance [9] for more on backstepping design.)Finally, the effect of parameter uncertainty, and the study of associated Lyapunov functions, are topics ofinterest in robust control theory An application of the result proved in this paper to the study of “input to statestability” is provided in [27]

∗Supported in part by US Air Force Grant AFOSR-91-0346

†Supported in part by NSF Grant DMS-9108250

Keywords: Nonlinear stability, Stability with respect to sets, Lyapunov function techniques, Robust stability.Running head: Converse Lyapunov Theorem for Robust Stability

AMS(MOS) subject classifications: 93D05, 93D09, 93D20, 34D20

1

1.1 Organization of Paper The paper is organized as follows.

The next section provides the basic definitions and the statement of the main result Actually, two versionsare given, one that applies to global asymptotic stability with respect to arbitrary invariant sets, but assumingcompleteness of the system —that is, global existence of solutions for all inputs— and another version whichdoes not assume completeness but only applies to the special case of compact invariant sets (in particular, tothe usual case of global asymptotic stability with respect to equilibria)

Equivalent characterizations of stability by means of decay estimates have proved very useful in controltheory –see e.g [25]– and this is the subject of Section 3 Some technical facts about Lyapunov functions,including a result on the smoothing of such functions around an attracting set, are given in Section 4 After this,Section 5 establishes some basic facts about complete systems needed for the main result

Section 6 contains the proof of the main result for the general case Our proof is based upon, and follows

to a great extent the outline of, the one given by Wilson in [31], who provided in the late 1960s a converseLyapunov function theorem for local asymptotic stability with respect to closed sets There are however somemajor differences with that work: we want a global rather than a local result, and several technical issues appear

in that case; moreover, and most importantly, we have to deal with parameters, which makes the careful analysis

of uniform bounds of paramount importance (In addition, even for the case of no parameters and local stability,several critical steps in the proof are only sketched in [31], especially those concerning Lipschitz properties andsmoothness around the attracting set Later the author of [21] rederived the results, but only for the casewhen the invariant set is compact Thus it seems useful to have an expository detailed and self-contained proof

in the literature for the more general cases.) A needed technical result on smoothing functions, based closelyalso on [31], is placed in an Appendix for convenience Section 7 deals with the compact case, essentially byreparameterization of trajectories

An example, motivated by related work of Tsinias and Kalouptsidis, is given in Section 8 to show that theanalogous theorems are false for unbounded parameters

Obviously in a topic such as this one, there are many connections to previous work While it is likely that

we have missed many relevant references, we discuss in Section 9 some relationships between our work and otherresults in the literature Relations to work using “prolongations” are particularly important, and are the subject

of some more detail in Section 10

2 Definitions and Statements of Main Results Consider the following system:

˙

x(t) = f (x(t), d(t)) ,

(1)

where for each t ∈ IR, x(t) ∈ IR n and d(t) ∈ D, and where D is a compact subset of IR m, for some positive

integers n and m The map f : IR n × D → IR n

is assumed to satisfy the following two properties:

• f is continuous.

• f is locally Lipschitz on x uniformly on d, that is, for each compact subset K of IR n there is some

constant c so that |f(x, d) − f(z, d)| ≤ c |x − z| for all x, z ∈ K and all d ∈ D, where |·| denotes the

usual Euclidian norm

Note that these properties are satisfied, for instance, if f extends to a continuously differentiable function on a

neighborhood of IRn × D.

LetM D be the set of all measurable functions from IR toD We will call functions d ∈ M D time varying parameters For each d ∈ M D , we denote by x(t, x0, d) (and sometimes simply by x(t) if there is no ambiguity from the context) the solution at time t of (1) with x(0) = x0 This is defined on some maximal interval

Sometimes we will need to consider time varying parameters d that are defined only on some interval I ⊆ IR

with 0 ∈ I In those cases, by abuse of notation, x(t, x0, d) will still be used, but only times t ∈ I will be

considered

2

The system is said to be forward complete if T x0,d= +∞ for all x0 all d ∈ M D It is backward complete if

T x −

0,d=−∞ for all x0 all d ∈ M D , and it is complete if it is both forward and backward complete.

We say that a closed setA is an invariant set for (1) if

where F (x) = {f(x, d), d ∈ D} The set A is invariant for (1) if and only if it is invariant with respect to (2)

(see e.g [1]) The notions of stability to be considered later can be rephrased in terms of (2) as well 2

We will use the following notation: for each nonempty subsetA of IR n

the common point-to-set distance, and|ξ| {0}=|ξ| is the usual norm.

LetA ⊆ IR nbe a closed, invariant set for (1) We emphasize that we do not requireA to be compact We

will assume throughout this work that the following mild property holds:

state-important special case in whichA reduces to an equilibrium point.)

Definition 2.2 System (1) is (absolutely) uniformly globally asymptotically stable (UGAS) with respect

to the closed invariant setA if it is forward complete and the following two properties hold:

1 Uniform Stability There exists a K ∞ -function δ( ·) such that for any ε ≥ 0,

|x(t, x0, d) | A ≤ ε for all d ∈ M D , whenever |x0| A ≤ δ(ε) and t ≥ 0

Observe that when A is compact the forward completeness assumption is redundant, since in that case

property (4) already implies that all solutions are bounded

In the particular case in which the setD consists of just one point, the above definition reduces to the

standard notion of set asymptotic stability of differential equations (Note, however, that this definition differsfrom those in [3], and [31], which are not global.) If, in addition,A consists of just an equilibrium point x0, this

is the usual notion of global asymptotic stability for the solution x(t) ≡ x0

Remark 2.3 It is an easy exercise to verify that an equivalent definition results if one replacesM Dby the

Remark 2.4 Note that the uniform stability condition is equivalent to: there is aK ∞ -function ϕ so that

|x(t, x0, d) | A ≤ ϕ(|x0| A ), ∀x0, ∀t ≥ 0, and ∀d ∈ M D

3

The following characterization of the UGAS property will be extremely useful.

Proposition 2.5 The system (1) is UGAS with respect to a closed, invariant set A ⊆ IR n

Observe that when A is compact the forward completeness assumption is again redundant, since in that

case property (6) implies that solutions are bounded

Next we introduce Lyapunov functions with respect to sets For any differentiable function V : IR n −→ IR,

we use the standard Lie derivative notation

is a function V : IR n −→ IR such that V is smooth on IR n \A and satisfies

1 there exist two K ∞ -functions α1 and α2 such that for any ξ ∈ IR n ,

α1(|ξ| A)≤ V (ξ) ≤ α2(|ξ| A) ;(7)

2 there exists a continuous, positive definite function α3 such that for any ξ ∈ IR n \A, and any d ∈ D,

L fdV (ξ) ≤ −α3(|ξ| A )

(8)

A smooth Lyapunov function is one which is smooth on all of IR n 2

Remark 2.7 Continuity of V on IR n \A and property 1 in the definition imply:

• V is continuous on all of IR n

;

• V (x) = 0 ⇐⇒ x ∈ A; and

• V : IR n onto −→ IR ≥0(recall the assumption in equation (3)) 2

Our main results will be two converse Lyapunov theorems The first one is for general closed invariant setsand assumes completeness of the system

Theorem 2.8 Assume that the system (1) is complete Let A ⊆ IR n be a nonempty, closed invariant subset for this system Then, (1) is UGAS with respect to A if and only if there exists a smooth Lyapunov function V with respect to A.

The following result does not assume completeness but instead applies only to compactA:

Theorem 2.9 Let A ⊆ IR n

be a nonempty, compact invariant subset for the system (1) Then, (1) is UGAS with respect to A if and only if there exists a smooth Lyapunov function V with respect to A.

3 Some Preliminaries about UGAS It will be useful to have a restatement of the second condition

in the definition of UGAS stated in terms of uniform attraction times:

Lemma 3.1 The uniform attraction property defined in Definition 2.2 is equivalent to the following: There exists a family of mappings {T r } r>0 with

• for each fixed r > 0, T r : IR>0

onto

−→ IR >0 is continuous and is strictly decreasing;

• for each fixed ε > 0, T r (ε) is (strictly) increasing as r increases and lim r →∞ T r (ε) = ∞;

such that, for each d ∈ M D ,

|x(t, x0, d) | A < ε whenever |x0| A < r and t ≥ T r (ε)

(9)

4

Proof Sufficiency is clear Now we show the necessity part For any r, ε > 0, let

¯

T r (s) ds

+1

Then it follows that

• for any fixed r, T r(·) is continuous, maps IR >0

onto

−→ IR >0, and is strictly decreasing;

• for any fixed ε, T r (ε) is increasing as r increases, and lim r→∞ T r (ε) = ∞.

So the only thing left to be shown is that T r defined by (14) satisfies (9) To do this, pick any x0 and t with

|x0| A < r and t ≥ T r (ε) Then

t ≥ T r (ε) > ˜ T r (ε) ≥ ¯ T r (ε)

Hence, by the definition of ¯T r (ε), |x(t, x0, d) | A < ε , as claimed.

3.1 Proof of Characterization via Decay Estimate We now provide a proof of Proposition 2.5.

[⇐=] Assume that there exists a KL-function β such that (6) holds Let

c1 def

= sup β( ·, 0) ≤ ∞ , and choose δ( ·) to be any K ∞-function with

δ(ε) ≤ ¯ β −1 (ε), any 0 ≤ ε < c1,

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where ¯β −1denotes the inverse function of ¯β( ·) = β( ·, 0) (If c1=∞, we can simply choose δ(ε) = β¯−1 (ε).) Clearly δ(ε) is the desired K ∞-function for the uniform stability property.

The uniform attraction property follows from the fact that for every fixed r, lim

t →∞ β(r, t) = 0.

[=⇒] Assume that (1) is UGAS with respect to the closed set A, and let δ be as in the definition Let ϕ( ·) be the K-function δ −1(·) As mentioned in Remark 2.4, it follows that |x(t, x0, d)) | A ≤ ϕ(|x0| A) for any

x0∈ IR n

, any t ≥ 0, and any d ∈ M D

Let {T r } r ∈(0, ∞) be as in Lemma 3.1, and for each r ∈ (0, ∞) denote ψ r

def

= T r −1 Then, for each

r ∈ (0, ∞), ψ r : IR>0 −→IR >0 is again continuous, onto, and strictly decreasing We also write ψ r(0) = +∞,

which is consistent with that fact that

lim

t →0+ψ r (t) = + ∞ (Note: The property that T(·) (t) increases to ∞ is not needed here.)

Claim: For any |x0| A < r, any t ≥ 0 and any d ∈ M D,|x(t, x0, d) | A ≤ ψ r (t).

Proof: It follows from the definition of the maps T r that, for any r, ε > 0, and for any d ∈ M D,

|x0| A < r, t ≥ T r (ε) =⇒ |x(t, x0, d) | A < ε

As t = T r (ψ r (t)) if t > 0, we have, for any such x0 and d,

|x(t, x0, d)| A < ψ r (t) , ∀t > 0

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The claim follows by combining (15) and the fact that ψ r(0) = +∞.

Now for any s ≥ 0 and t ≥ 0, let

for any fixed s, ψ(s, t) decreases to 0 as t¯ → ∞.

Next we construct a function ˜ψ : IR [0, ∞) × IR ≥0 −→ IR ≥0with the following properties:

• for any fixed t ≥ 0, ˜ ψ(·, t) is continuous and strictly increasing;

• for any fixed s ≥ 0, ˜ ψ(s, t) decreases to 0 as t → ∞;

dε = ¯ ψ(s, t)

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t →∞

¯

ψ(ε, t) dε = 0

Now we see that the function ˆψ(s, t) satisfies all of the requirements for ˜ ψ(s, t) except possibly for the strictly

increasing property We define ˜ψ as follows:

which concludes the proof of the Proposition

4 Some Preliminaries about Lyapunov Functions In this section we provide some technical results

about set Lyapunov functions A lemma on differential inequalities is also given, for later reference

Remark 4.1 One may assume in Definition 2.6 that all of α1, α2, α3 are smooth in (0, + ∞) and of class

K ∞ For α1 and α2, this is proved simply by finding two functions ˜α1, ˜ α2 inK ∞ , smooth in (0, + ∞) so that

˜

α1(s) ≤ α1(s) ≤ α2(s) ≤ ˜α2(s) , for all s.

For α3, a new Lyapunov function W and a function ˜ α3 which satisfies (8) with respect to W , but is smooth

in (0, + ∞) and of class K ∞, can be constructed as follows First, pick ˜α3 to be anyK ∞-function, smooth in

Now define β(s) def=

We claim that this is bounded by−˜α3(|ξ| A ) Indeed, if s def= |ξ| A ≤ α −1

1 (1), then from the first item above andthe definition of ˜α3,

In either case, γ(α1(s))α3(s) ≥ ˜α3(s) , as desired From now on, whenever necessary, we assume that α1, α2, α3

4.1 Smoothing of Lyapunov Functions When dealing with control system design, one often needs to

know that V can be taken to be globally smooth, rather than just smooth outside of A.

Proposition 4.2 If there is a Lyapunov function for (1) with respect to A, then there is also a smooth such Lyapunov function.

The proof relies on constructing a smooth function of the form W = β ◦ V , where

i=1 % i The following regularization result will be needed; it generalizes to arbitraryA the analogous (but

simpler, due to compactness) result for equilibria given in [13, Theorem 6]

Lemma 4.3 Assume that V : IR n −→ IR ≥0 is C0, the restriction V |IRn \A is C ∞ , and also V | A =

0 , V |IRn \A > 0 Then there exists a K ∞ -function β, smooth on (0, ∞) and so that β (i)

(t) → 0 as t → 0+

for each i = 0, 1, and having β 0 (t) > 0, ∀t > 0, such that

W def= β ◦ V

is a C ∞ function on all of IR n

Proof Let K1, K2, , be compact subsets of IR nsuch thatA ⊆S∞

i=1 int (K i ) For any k ≥ 1, let

ThenG k is compact (because of compactness of the sets K i and continuity of V ) Observe that each derivative

γ k (i) has a compact support included in clos I k , so it is bounded For each k = 1, 2, , let c k ∈ IR satisfy

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1 c k ≥ 1;

2 c k ≥ | (D %

V ) (x) | for any multi-index |%| ≤ k and any x ∈ G k; and

3 c k ≥ |γ (i)

k (t) |, for any i ≤ k and any t ∈ IR >0

Choose the sequence d k to satisfy

o, it follows that

γ (i) (t) ≤ d k −1 γ k (i) −1 (t) + d k γ k (i) (t) ,

and noticing that

general, see e.g [4] (p 52), ifA ⊆ IR n

is closed and ϕ : IR n −→ IR satisfies that ϕ| A = 0, ϕ |IRn \A is C ∞, and for

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each boundary point a of A and all multi-indices % = (%1, %2, , % n ) , it holds that lim

x →a

x 6∈A

D ϕ(x) = 0 , then ϕ is

C ∞on IRn.)

Pick one such %0 and any sequence {x n } with x n → ¯x ∈ ∂A If |%0| = 0, one only needs to show that

W (x n)→ 0, which follows easily from the facts that β ∈ K ∞ and V (x n)→ 0 So from now on, we can assume

k + 2 Observe that if t < T , then t 6∈ I1∪ · · · ∪ I k

As V is C0 everywhere, V = 0 at A, V (x n)→ V (¯x) = 0 So there exists N such that V (x n ) < T whenever

n > N Fix an N like this Then for any n > N ,

γ s (j) (V (x n )) = 0, ∀j, ∀s = 1, 2, , k, (since γ s vanishes outside I s ) Pick any j ∈ IN with j ≤ i, any h ∈ IN with h ≤ i, and %1, , % h multi-indicessuch that|% µ | ≤ i, ∀µ = 1, , h Then for any q ∈ IN with q > k, by the way we chose c k,

γ q (j) (V (x n)) ≤ c q , since q > k > i ≥ j Also, if V (x n)∈ I q , then again by the properties of the sequence c k,

|D %µ

V (x n)| ≤ c q , (since q > k > l and x n ∈ K l imply x n ∈ K1∪ · · · ∪ K q, and |% µ | ≤ i < k < q) Therefore, for such q, if

(k + 1)! =

1

2k (k + 1)! <

ε (k + 1)! .

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10

Now observe that

(D %0W ) (x) = (D %0(β ◦ V )) (x)

is a sum of≤ i! terms (recall 0 < i = |%0|), each of which is of the form

β (p) (V (x)) (D %1V ) (x) · · · (D % h

V ) (x) , where 0 < p ≤ i, h ≤ i, and each |% µ | ≤ i Each

Now let us return to the proof of the Proposition 4.2

Proof of Proposition 4.2 Assume A, V and α1, α2, α3 are as defined in Definition 2.6 Let β, W be as in Lemma 4.3 We show that W is a smooth Lyapunov function as required.

which concludes the proof of the Proposition

4.2 A Useful Estimate The following lemma establishes a useful comparison principle.

Lemma 4.4 For each continuous and positive definite function α, there exists a KL-function β α (s, t) with the following property: if y( ·) is any (locally) absolutely continuous function defined for t ≥ 0 and with y(t) ≥ 0 for all t, and y(·) satisfies the differential inequality

Proof Define for any s > 0, η(s) def= −

Z s

1

dr α(r) This is a strictly decreasing differentiable function on(0, ∞) Without loss of generality, we will assume that lim s→0+η(s) = +∞ If this were not the case, we could

consider instead the following function:

¯

α(s) def=

(min{s, α(s)} , if 0 ≤ s < 1 , α(s) , if s ≥ 1

11

This function is again continuous, positive definite, satisfies ¯α(s) ≤ α(s) for any s ≥ 0, and

We claim that for any y( ·) satisfying the conditions in the Lemma,

y(t) ≤ β α (y0, t) , for all t ≥ 0

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As ˙y(t) ≤ −α(y(t)), it follows that y(t) is nonincreasing, and if y(t0) = 0 for some t0≥ 0, then y(t) ≡ 0, ∀t ≥ t0

Without loss of generality, assume that y0> 0 Let

t0 def

= inf{t : y(t) = 0} ≤ +∞

It is enough to show (26) holds for t ∈ [0, t0)

As η is strictly decreasing, we only need to show that η(y(t)) ≥ η(y0) + t , that is,

−

Z y(t)

1

dr α(r) ≥ −

Z y0

1

dr α(r) + t ,

which is equivalent to

Z y0y(t)

dr α(r) ≥ t

Z t

0

dτ = −t

Changing variables in the integral, this gives (27)

It only remains to show that β α is of classKL The function β α is continuous since both η and η −1 arecontinuous in their domains, and lim

r →∞ η

−1 (r) = 0 It is strictly increasing in s for each fixed t since since both

η and η −1 are strictly decreasing Finally, β α (s, t) → 0 as t → ∞ by construction So β α is aKL-function.

5 Some Properties of Complete Systems We need to first establish some technical properties that

hold for complete systems, and in particular a Lipschitz continuity fact

In what follows we use S to denote the closure of S for any subset S of IR

Proposition 5.1 Assume that (1) is forward complete Then for any compact subset K of IR n and any

T > 0, the set R ≤T (K) is compact.

To prove Proposition 5.1, we first need to make a couple of technical observations

Lemma 5.2 Let K be a compact subset of IR n and let T > 0 Then the set R ≤T (K) is compact if and only

if R ≤T (ξ) is compact for each ξ ∈ K.

Proof It is clear that the compactness of R ≤T (K) implies the compactness of R ≤T (ξ) for any ξ ∈ K Now assume, for T > 0 and a compact set K, that R ≤T (ξ) is compact for each ξ ∈ K Pick any ξ ∈ K, and

letU = {η : d(η, R ≤T (ξ)) < 1 } Then U is compact Let C be a Lipschitz constant for f with respect to x on U, and let r = e −CT For each d ∈ M D and each η with |η − ξ| < r, let ˜t= inf{t ≥ 0 : |x(t, η, d) − x(t, ξ, d)| ≥ 1}.

Then, using Gronwall’s Lemma, one can show that ˜t ≥ T , from which it follows that

R ≤T (η) ⊆ U, ∀|η − ξ| < r.

Thus, for each ξ ∈ K, there is a neighborhood V ξ of ξ such that R ≤T(V ξ ) is compact By compactness of K, it

follows thatR ≤T (K) is compact.

Lemma 5.3 For any subset S of IR n and any T > 0,

Proof The first conclusion follows from the continuity of solutions on initial states; see [26], Theorem 1.

The second is immediate from there

We now return to the proof of Proposition 5.1 By Lemma 5.2, it is enough to show thatR ≤T

(ξ) is compact for each ξ ∈ IR n

and each T > 0 Pick any ξ0∈ IR n

, and let

τ = sup{T ≥ 0 : R ≤T (ξ0) is compact}

Note that τ > 0 This is because |x(t, ξ0, d) − ξ0| ≤ 1 for any 0 ≤ t < 1/M and any d ∈ M D, where

M = max {|f(ξ, d)| : |ξ − ξ0| ≤ 1, d ∈ D}

We must show that τ = ∞.

Assume that τ < ∞ Using the same argument used above, one can show that if R ≤t (ξ0) is compact for

some t > 0 then there is some δ > 0 such that R ≤(t+δ)

(ξ0) is compact From here it follows thatR ≤τ

(ξ0) is notcompact By definition,R ≤t (ξ0) is compact for any t < τ

Let τ1 = τ /2 Then there is some η1 ∈ R τ1(ξ0) such that R ≤(τ−τ1) (η1) is not compact; otherwise, byLemma 5.2,R ≤(τ−τ1) R τ1(ξ0)

would be compact This, in turn, would imply thatR ≤τ

Since η1 ∈ R τ1(ξ0), there exists a sequence {z n } → η1 with z n ∈ R τ1(ξ0) Assume, for each n, that

z n = x(τ1, ξ0, d n ) for some d n ∈ M D For each d ∈ M D , and each s ∈ IR, we use d s to denote the function

defined by d s (t) = d(s + t) Then by uniqueness, one has that for each n, x(s, z n , (d n)τ1) ∈ K1 for any

−τ1≤ s ≤ 0, where K1=R ≤τ1 (ξ0) We want to claim next that, by compactness of K1and Gronwall’s Lemma,

|x(−τ1, η1, (d n)τ1)− ξ0| = |x(−τ1, η1, (d n)τ1)− x(−τ1, z n , (d n)τ1)| −→ 0, as n → ∞

13

The only potential problem is that the solution x( −τ1, η1, (d n)τ1) may fail to exist a priori However, it is possible

to modify f (x, d) outside a neighborhood of K1× D so that it now has compact support and is hence globally bounded The modified dynamics is complete Now the above limit holds for the modified system, and a fortriori

it also holds for the original system

Choose n0 such that

|x(−τ1, η1, (d n0)τ1)− ξ0| < 1/2.

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Let v1= d n0, and let η0= x( −τ1, η1, (d n0)τ1) Then, by continuity on initial conditions, there is a neighborhood

U1of η1 contained in B(η1, 1) such that

Let τ2= τ1/2 = (τ − τ1)/2 Applying the above argument with ξ0 replaced by η1, τ replaced by (τ − τ1),

and τ1 replaced by τ2, one shows that there exists some η2 ∈ R τ2(η1) such thatR ≤t (η2) is compact for any

0≤ t < τ − σ2, andR ≤(τ−σ2) (η2) is not compact, where σ2= τ1+ τ2, and there exist some v2 defined on [0, τ2)and some neighborhoodU2 of η2 contained in B(η2, 1), such that

function v does not exist for time τ , contradicting forward completeness.

First notice that for any compact set S, there exists some k such that η k 6∈ S Otherwise, assume that there exists some compact set S such that η k ∈ S for all k Let S1={η : d(η, S) ≤ 1} The compactness of S implies that there exists some δ > 0 such that

R ≤t (η) ⊆ S1

for any η ∈ S, and any t ∈ [0, δ] In particular, it implies that R ≤(τ−σk) (η k)⊆ S1 for k large enough so that

τ − σ k < δ This contradicts the fact that R ≤(τ−σk) (η k ) is not compact for each k.

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Assume that x(τ, ζ0, v) is defined By continuity on initial conditions, this would imply that x(t, ζ k , v) is defined for all t ≤ τ and for all k large enough, and it converges uniformly to x(t, ζ0, v) Thus, x(t, ζ k , v) remains

in a compact for all t ∈ [0, τ] and all k But

x(σ k , ζ k , v) = x(σ k , ζ k , v k ) = η k , contradicting what was just proved So x(τ, ζ0, v) is not defined, which contradicts the forward completeness of

R ≥−T (K)[

R ≤T (K)

Combining the above conclusion and Gronwall’s Lemma, one has the following fact:

Proposition 5.5 Assume that (1) is complete For any fixed T > 0 and any compact K ⊆ IR n

, there

is a constant C > 0 (which only depends on the set K and T ), such that for the trajectories x(t, x0, d) of the system (1),

|x(t, ξ, d) − x(t, η, d)| ≤ C|ξ − η|

6 Proof of the First Converse Lyapunov Theorem.

Proof [ ⇐=] Pick any x0 ∈ IR n and any d ∈ M D , and let x( ·) be the corresponding trajectory Then we

have

dV (x(t))

dt ≤ −α3(|x(t)| A)≤ −α(V (x(t))) , a.e t ≥ 0 , where α is the K ∞-function defined by

α( ·) def

= α3(α −12 (·)) Now let β α be theKL-function as in Lemma 4.4 with respect to α, and define

Therefore the system (1) is UGAS with respect toA, by Proposition 2.5.

[=⇒] We will show the existence of a not necessarily smooth Lyapunov function; then the existence of a

smooth function will follow from Proposition 4.2 Assume that the system is UGAS with respect to the setA Let δ and T r be as in Definition 2.2 and Lemma 3.1

ξ ∈ K ε, r , d ∈ M D , and t < q ε, r =⇒ |x(t, ξ, d)| A ≥ r

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Proof: If the statement were not true, then there would exist ε, r with 0 < ε < r and three sequences {ξ k } ⊆ K ε, r,{t k } ⊆ IR and d k ∈ M Dwith limk→∞ t k=−∞ such that for all k:

|x(t k , ξ k , d k)| A < r

Pick k large enough so that −t k > T r (ε), then by the uniform attraction property,

|ξ k | A=|x(−t k , x(t k , ξ k , d k ), (d k)t k)| A < ε ,

which is a contradiction This proves the fact

Therefore, for any ξ ∈ K ε, r,

g(ξ) = inf {|x(t, ξ, d)| A : t ∈ [q ε, r , 0], d ∈ M D }.

Lemma 6.1 The function g(ξ) is locally Lipschitz on IR n \A, and continuous everywhere.

Proof Fix any ξ0∈ IR n \A, and let s = |ξ0| A

2 Let ¯B (ξ0, s) denote the closed ball centered at ξ0 and with

radius s Then ¯ B (ξ0, s) ⊆ K σ, r for some 0 < σ < r Pick a constant C as in Proposition 5.5 with respect to this closed ball and T = |q σ, r | Pick any ζ, η ∈ ¯ B (ξ0, s) For any ε > 0, there exist some d η,ε and t η,ε ∈ [q σ,r , 0] such that g(η) ≥ |x(t η,ε , η, d η,ε)| A − ε Thus

g(ζ) − g(η) ≤ |x(t η,ε , ζ, d η,ε)| A − |x(t η,ε , η, d η,ε)| A + ε ≤ C|ζ − η| + ε

(35)

Note that (35) holds for all ε > 0, so it follows that

g(ζ) − g(η) ≤ C |ζ − η| Similarly, g(η) − g(ζ) ≤ C|ζ − η| This proves that g is locally Lipschitz on IR n \A.

Note that g is 0 on A, and for ξ ∈ A, η ∈ IR n

:

|g(η) − g(ξ)| = |g(η)| ≤ |η| A ≤ |η − ξ| , thus g is globally continuous (We are not claiming that g is locally Lipschitz on IR n, though.)

where k : R ≥0 −→ IR >0is any strictly increasing, smooth function that satisfies:

• there are two constants 0 < c1 < c2< ∞, such that k(t) ∈ [c1, c2] for all t ≥ 0;

• there is a bounded positive decreasing continuous function τ(·), such that

hold for complete systems, and in particular a Lipschitz continuity fact

In...

Proof It is clear that the compactness of R ≤T (K) implies the compactness of R ≤T (ξ) for any ξ ∈ K Now assume, for T > and a compact set K, that R ≤T... contradicts the fact that R ≤(τ−σk) (η k ) is not compact for each k.

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