Feedback Stabilization and Lyapunov Functions

F. H. Clarke

Institut Girard Desargues (Bât 101) Université Claude Bernard Lyon I (La Doua)

69622 Villeurbanne, France (clarke@desargues.univ-lyon1.fr)

Yu. S. Ledyaev^{∗} Steklov Institute of Mathematics

Moscow 117966, Russia (ledyaev@math-stat.wmich.edu)

L. Rifford

Institut Girard Desargues (Bât 101) Université Claude Bernard Lyon I (La Doua)

69622 Villeurbanne, France (rifford@desargues.univ-lyon1.fr)

R. J. Stern^{†}

Department of Mathematics and Statistics

Concordia University

Montreal, Quebec H4B 1R6, Canada (stern@vax2.concordia.ca)

January 13, 1999

Abstract

Given a locally defined, nondifferentiable but Lipschitz Lyapunov function, we employ it in order to construct a (discontinuous) feedback law which stabilizes the underlying system to any given tolerance. A converse result shows that suitable Lyapunov functions of this type exist under mild assumptions. We also establish that the feedback in question possesses a robustness property relative to measurement error, despite the fact that it may not be continuous.

0Key words: Asymptotic stabilizability, discontinuous feedback law, system sampling, locally Lipschitz Lyapunov function, nonsmooth analysis, robustness.

Mathematical Subject Classification: 93D05, 93D20, 34D20, 26B05

∗Supported in part by Russian Fund for Fundamental Research Grant 96-01-00219. Current address: Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA.

†Research supported by the Natural Sciences Engineering Research Council of Canada and Fonds pour la Formation de Chercheurs et l’Aide à la Recherche du Québec.

0 Introduction

Consider a standard control system of the form

ẋ(t) = f(x(t), u(t)) a.e., u(t) ∈ U,  (1)

and let V be a smooth Lyapunov function for the system; thus we have

V (x) ≥ 0, V (x) = 0 iff x = 0, V (x) → ∞ as ∥x∥ → ∞,

and (for some function W) the Infinitesimal Decrease Condition

min_{u∈U} ⟨∇V (x), f(x, u)⟩ ≤ −W (x) < 0, x ≠ 0.  (2)

It is well-known (but true) that the existence of such a “control-Lyapunov function” (V, W) (a framework introduced by Eduardo Sontag) implies (open-loop) asymptotic controllability to the origin: for every α ∈ IR^{n}, there is a control u(t) such that the solution x(·) of (1) with initial condition x(0) = α satisfies x(t) → 0 as t → ∞. (In addition, convergence to zero takes place in a certain uniform and stable manner that we will not dwell upon here.) A related and important goal in many situations is to produce a state feedback k(·) : IR^{n} → U which stabilizes the system; i.e., such that the system ẋ = f(x, k(x)) is globally asymptotically stable. This article explores the question of how to define such a feedback law through the use of a given Lyapunov function V.

The ideal case, a well-known heuristic useful for motivational purposes, is the one in which we can find a continuous function k(x) that selects a value of u ∈ U attaining (or almost) the minimum in (2):

⟨∇V (x), f(x, k(x))⟩ ≤ −W (x) ∀x ≠ 0.

Then any solution of ẋ = f(x, k(x)) is such that

\frac{d}{dt}V (x(t)) = ⟨∇V (x(t)), ẋ(t)⟩ ≤ −W (x(t)) < 0,

a monotonicity conclusion that, together with the growth property of V, assures that x(t) → 0 as t → ∞.

There are two fundamental difficulties with this ideal picture, and both concern regularity issues. The first is that a differentiable Lyapunov function may not exist, and the second is that even when a smooth V exists, the continuous selection k(·) does not generally exist. If we have recourse to a discontinuous feedback k(·), then the issue arises of how to interpret the discontinuous differential equation ẋ = f(x, k(x)).

The primary goal of this article is to give a general answer to the problem of defining a (discontinuous) stabilizing feedback based upon a given (nondifferentiable) Lyapunov function, one for which a nonsmooth version of infinitesimal decrease is known to hold only on a restricted set. The construction is described in Section 1, while Section 2 establishes the converse result that under mild conditions, a Lyapunov function of the type required in the previous section always exists. In the final section, we address the issue of robustness of the feedback with respect to measurement error and small perturbations of the dynamics, a particularly important issue when discontinuity is present. Some works and general references related to this article include [2, 3, 8, 11, 12, 13, 15, 17, 18, 20, 23, 24, 26, 28, 29, 31, 32]. We proceed now to situate our results with respect to the literature.

The possible nonexistence of continuous stabilizing feedback was brought to light in the seminal work of Sontag and Sussmann [30] and of Brockett [4]. The latter developed a necessary condition for continuous stabilizability and adduced the following example, the “non-holonomic integrator”:

\{ \begin{matrix}ẋ_{1} & = u_{1} \\ ẋ_{2} & = u_{2} \\ ẋ_{3} & = x_{1}u_{2} − x_{2}u_{1} \end{matrix} (u_{1}, u_{2}) ∈ B̄ =: U.

This system is globally asymptotically controllable yet fails to admit a continuous stabilizing feedback (by Brockett’s condition). In considering the use of discontinuous feedback laws k(·), one could have recourse to the Filippov solution concept [14]: x is said to be a solution of ẋ = f(x, k(x)) =: g(x) provided that we have

ẋ ∈ ⋂_{meas(Ω)=0^{δ>0}} cl co(g([x + δB] \ Ω)).

However, as shown by Ryan [25] and by Coron and Rosier [12], Brockett’s condition continues to hold for this solution concept, so that the nonholonomic integrator (for example) cannot be stabilized by a discontinuous feedback in the Filippov sense.

In [6] it was shown that any globally asymptotically controllable system is stabilizable by a (possibly discontinuous) feedback if the trajectory x(·) associated to the feedback is defined in a natural way that involves discretizing the control law (closed-loop system sampling) in a manner similar to that used in differential games by Krasovskii and Subbotin [19]. We proceed now to describe this concept, which is the one used in this article.

Let π = {t_{i}}_{i≥0} be a partition of [0, ∞), by which we mean a countable, strictly increasing sequence t_{i} with t_{0} = 0 such that t_{i} → ∞ as i → ∞. The diameter of π, denoted diam(π), is defined as sup_{i≥0}(t_{i+1} − t_{i}). Given an initial condition x_{0}, the π-trajectory x(·) corresponding to π and an arbitrary feedback law k : IR^{n} → U is defined in a step-by-step fashion as follows. Between t_{0} and t_{1}, x is a classical solution of the differential equation

ẋ(t) = f(x(t), k(x_{0})), x(0) = x_{0}, t_{0} ≤ t ≤ t_{1}.

(Of course in general we do not have uniqueness of the solution, nor is there necessarily even one solution, although nonexistence will be ruled out by the feedback constructed in Section 1, which will preclude blow-up of the solution in finite time.) We then set x_{1} := x(t_{1}) and restart the system with control value k(x_{1}):

ẋ(t) = f(x(t), k(x_{1})), x(t_{1}) = x_{1}, t_{1} ≤ t ≤ t_{2},

and so on in this fashion. The resulting trajectory x is a physically meaningful one that corresponds to a natural sampling procedure and piecewise constant controls; the smaller diam(π), the greater the sampling rate. Since our results are couched in term of π-trajectories, the issue of defining a solution concept for discontinuous differential equations is effectively sidestepped. Our approach will lead to precise estimates of how small the step size diam(π) must be for a prescribed stabilization tolerance to ensue, and of the resulting stabilization time, in terms of the given data.

The next major point to address concerns the nonsmoothness of the Lyapunov function V. An early and important result of Artstein [1] implies in particular that the nonholonomic integrator fails to admit a smooth V (see [7] for related results). It has been shown by Sontag [26], however, that globally asymptotically controllable systems always admit a continuous Lyapunov function V satisfying the following nonsmooth version of the Infinitesimal Decrease Condition:

inf_{u∈U} DV (x; f(x, u)) ≤ −W (x) < 0, x ≠ 0,  (3)

where the lower Dini derivate DV is defined by

DV (x; v) := lim inf_{v′→v^{t↓0}} \frac{V (x + tv′) − V (x)}{t}.  (4)

Among the several important ways in which the theory of nonsmooth analysis intervenes in this article is that of asserting the equivalence to (3) of another, and for our purposes more useful, form of the Infinitesimal Decrease Condition:

inf_{u∈U}⟨f(x, u), ζ⟩ ≤ −W (x) < 0 ∀x ≠ 0, ∀ζ ∈ ∂_{P} V (x).  (5)

Here ∂_{P} V (x) refers to the proximal subdifferential of V at x (which may very well be empty); ζ belongs to ∂_{P} V (x) iff there exists σ and η > 0 such that

V (y) − V (x) + σ∥y − x∥^{2} ≥ ⟨ζ, y − x⟩ ∀y ∈ x + ηB.

(B denotes the open unit ball, and the open ball of radius r centered at x is written as either x + rB or B(x, r).) The equivalence of (3) and (5) is a consequence of Subbotin’s Theorem, which links Dini derivates to proximal calculus (see for example [6] or [9], our principal source for the theory of nonsmooth analysis).

The essential reason for which proximal calculus is well-suited to our approach is because of its relation to metric projection onto sets, upon which is based the “proximal aiming” method that we employ. The crux is this: when x(t_{i}) = x lies outside a level set S = S(c) := {V ≤ c} and admits closest point (or projection) s in S, then x − s is a “proximal normal” to S at s, and for some λ > 0 we have λ(x − s) ∈ ∂_{P} V (s). Then (5) can be invoked at s to find a suitable value of the control u which moves the state toward S, in the sense that the Euclidian distance d_{S} decreases at a certain positive rate Δ:

d_{S}(x(t)) − d_{S}(x(t_{i})) ≤ −Δ(t − t_{i}), t_{i} ≤ t ≤ t_{i+1},

provided x(t_{i}) is close enough to S to start with, and provided diam(π) is small enough. A sequence of such feedbacks is amalgamated in the first section to produce the stabilizing feedback k(·) that is sought.

Our approach requires the Lyapunov function V to be Lipschitz (in the zone under consideration). In the case of a system which is globally asymptotically controllable to the origin, which has received considerable attention, it is not known whether a suitable V exists that is Lipschitz near 0, but that is somewhat beside the point in the setting in which we work. Theorem 1 derives finite-time stabilizability, to a close approximation of some level set S(a), as a consequence of the supposed existence of a Lipschitz Lyapunov function. When applied to the special case of global asymptotic controllability, this requires only a Lipschitz V defined on the complement of a small ball around the origin, and the stabilization takes place not asymptotically to the origin, but rather in finite time to a small neighborhood of it. (This has been called “practical” stabilization.) In contrast, [6] obtains asymptotic stabilizability to the origin (the case S(a) = {0}); the proof uses Moreau-Yosida inf convolution to make a continuous Lyapunov function Lipschitz as an intermediate step. This methodology was also employed earlier [10], in a differential game setting. The direct use of a Lipschitz Lyapunov function, when it is possible, leads to a far more transparent feedback construction with direct stabilization estimates, and has the important consequence of yielding robustness, as we discuss presently. The fact that under mild assumptions, suitable Lipschitz Lyapunov functions do exist leading to practical stabilization to any required tolerance, is proven in Section 2.

Ledyaev and Sontag [21] have recently proved that there is a close relationship between the issues of “how regular a Lyapunov function does the system admit” and “how robust a stabilizing feedback does the system admit”. Consider for example a perturbed equation ẋ = f(x, k(x + p)), where p represents a measurement error. Full robustness of the feedback k is taken to mean that for any ϵ, there is a δ > 0 such that whenever the perturbation p(t) satisfies ∥p(t)∥ ≤ δ for all t, then stabilization to the ϵ-ball takes place. Then [21] asserts that the system admits a fully robust stabilizing feedback iff it admits a smooth (C^{1} or C^{∞}) Lyapunov function. Thus the nonholonomic integrator, which can be stabilized by a discontinuous feedback (in view of [6]), does not admit a fully robust stabilizing feedback. It is possible to recover robustness, however, through the use of a dynamic feedback; see [22]. The issue of the robustness of discontinuous feedbacks with respect to measurement error seems to have been raised first by Hermes [16].

The above concept of full robustness, unrelated as it is to the system sampling method that we employ, is not the one discussed in this article. Instead, we introduce a type of relative robustness in which we require the size of the measurement error to be limited as a function of the maximum step size δ of the underlying partition. This step size δ must still be small enough (for stabilization), but at the same time the individual steps must be big enough to preclude a possible chattering phenomenon, even in the presence of small errors. This consideration, which leads us to specify “reasonably uniform” sampling in Section 3, appears to be new in this context. The term “reasonably uniform” is taken here to mean that the following holds:

\frac{δ}{2} ≤ t_{i+1} − t_{i} ≤ δ ∀i ≥ 0,

although it is possible to replace the factor 1/2 by any constant in (0, 1).

To conclude with the nonholonomic integrator, then, it turns out that the system does admit a relatively robust stabilizing feedback to within any prescribed tolerance r, in the sense that for all initial conditions in a bounded set, we will have ∥x(t)∥ ≤ r for all t ≥ T, whenever x is a π-trajectory, if π is a reasonably uniform partition whose diameter is sufficiently small, and whenever measurement and external error do not exceed a critical level related to the sampling rate. It is possible to exhibit a Lipschitz Lyapunov function for the system, and to make explicit the resulting feedback, as will be shown in a forthcoming article.

1 A feedback construction

For a given function V : IR^{n} ⟶ (−∞, ∞], we shall deal frequently with the sublevel sets S(r) defined as follows:

S(r) := {x ∈ IR^{n} : V (r) ≤ r}.

In addition, the following sets are considered:

S(a, b) := {x ∈ IR^{n} : a ≤ V (x) ≤ b}.

Let a and b be two given numbers with a < b. The following hypotheses are made concerning the function V and the system function f:

(H1) V is lower semicontinuous, S(b) ≠ ∅, and for some η > 0, V is Lipschitz of rank L_{V} on S(a, b) + ηB:

|V (x) − V (y)| ≤ L_{V} ∥x − y∥ ∀(x, y) ∈ S(a, b) + ηB.

(H2) ∃δ_{1} ∈ (0, b − a) and δ_{2} > 0 such that

S(a + δ_{1}) + δ_{2}B ⊂ S(b).

(H3) f(x, u) is continuous on S(b) + ηB as a function of x for each u ∈ U, and ∃m > 0 such that

∥f(x, u)∥ ≤ m ∀x ∈ S(b) + ηB, ∀u ∈ U.

(H4) f is Lipschitz in x of rank L_{f} on S(a, b) + ηB:

∥f(x, u) − f(y, u)∥ ≤ L_{f}∥x − y∥ ∀(x, y) ∈ S(a, b) + ηB, ∀u ∈ U.

(H5) There exists ω > 0 such that, for every x ∈ S(a, b) + ηB, we have

inf_{v∈cof(x,U)} DV (x; v) ≤ −ω.

Remark 1.1 We do not require that f and V be defined except on S(b) + ηB; the Lipschitz conditions on these functions, as well as the Infinitesimal Decrease Condition (H5), are posited only on a neighborhood of S(a, b). No hypotheses are made concerning the abstract set U, nor on the nature of the dependence of f on the control variable. It is shown in §2 that (H2) automatically holds when S(b) is compact and V is continuous on S(b) + ηB; see Lemma 2.9. The set S(a) is not assumed to be nonempty a priori, but that fact is a consequence of the following theorem.

Theorem 1 For any γ > 0 sufficiently small, there exist positive numbers δ, T and a feedback k : S(b) + ηB ⟶ U such that whenever a partition π satisfies diam(π) < δ, then any π-trajectory x(·) having x(0) ∈ S(b) + γB satisfies

x(t) ∈ S(b) + γB ∀t ≥ 0,
x(t) ∈ S(a) + γB ∀t ≥ T.

Remark 1.2 Thus we almost recover the conclusion of the “ideal case” discussed in the Introduction (with {0} replaced by the more general S(a)), but in approximate terms (to tolerance γ), with a discontinuous feedback, and for a nonsmooth Lyapunov function satisfying only local hypotheses. The proof is constructive and provides explicit estimates of γ, δ and T in terms of the given data.

Remark 1.3 Taking W (x) = ω for purposes of comparison, note that (H5) is an apparently weaker hypothesis than (3); in actual fact, each is equivalent to (5). Because V is Lipschitz and f continuous in x near the points in question, this in turn is equivalent to the following:

inf_{u∈U}⟨ζ, f(x, u)⟩ ≤ −ω ∀x ∈ S(a, b) + ηB, ∀ζ ∈ ∂_{L}V (x),

where ∂_{L}V is the limiting subdifferential of V (see [9]).

Proof of Theorem 1

The proof of Theorem 1 is based upon defining a feedback control via projections. The first two lemmas below guarantee that the projections lie in the set where the hypotheses are active.

Lemma 1.1 Let ϵ lie in [0, δ_{1}] and suppose that x is a point in the set

[S(a + ϵ) + min{δ_{2}, η}B] \ S(a + ϵ).

Then x ∈ S(a, b), and if s ∈ proj(x, S(a + ϵ)), then s ∈ S(a, b) + ηB.

Proof Since we have S(a+δ_{1})+δ_{2}B contained in S(b) by hypothesis, it follows that x lies in S(b). Since x does not belong to S(a + ϵ), we deduce x ∈ S(a, b). Finally, we have

∥s − x∥ < min{δ_{2}, η} ≤ η,

whence s ∈ S(a, b) + ηB. □

Lemma 1.2 Let 0 < γ < η/2, and suppose that for some r′ and r with a ≤ r′ < r ≤ b we have x ∈ [S(r) + γB] \ [S(r′) + γB]. Then x ∈ S(a, b) + γB, and if s ∈ proj(x, S(r)), then s ∈ S(a, b) + ηB.

Proof There exists y ∈ S(r) having ∥y − x∥ < γ. Since x does not belong to S(r′)+γB, we have V (y) > r′ necessarily. Thus y ∈ S(a, b) and x ∈ S(a, b)+γB. Finally, we note ∥x − s∥ < γ, whence ∥y − s∥ < 2γ and s lies in S(a, b) + ηB. □

The next “solvability” result is central to our approach. The notation u_{+} stands for max{u, 0}.

Lemma 1.3 For any r ∈ [a, b], for any x ∈ S(a, b), we have

d(x, S(r)) ≤ \frac{m}{ω}(V (x) − r)_{+}.

Proof We shall invoke results (and terminology) from [9] to give a short proof of this result, whose proof from first principles would be lengthy.

We define a lower semicontinuous function g : IR^{n} ⟶ [0, ∞] as follows:

g(x) := (V (x) − r)_{+} + I_{S(b)}(x),

where I_{S(b)}(·) is the indicator function of the set S(b). At any point x in the open set C := {y : g(y) > 0} at which g is finite, we have x ∈ S(a, b), and the Infinitesimal Decrease Condition implies that

inf{Dg(x; v) : v ∈ cof(x, U)} ≤ −ω.

It follows from this that for any ϵ > 0, for any x ∈ C, for any ζ ∈ ∂_{P} g(x), there exists u ∈ U such that

⟨ζ, f(x, u)⟩ ≤ −ω + ϵ.

Since ∥f(x, u)∥ ≤ m and since ϵ > 0 is arbitrary, we derive ∥ζ∥ ≥ ω/m. This verifies the hypothesis of the Solvability Theorem [9, Theorem 3.3.1] (with the sets labeled there as V and Ω both taken to be IR^{n}), whose conclusion is precisely the desired one since S(r) = {x : g(x) = 0}. We remark that an alternate proof can be based upon weak monotonicity: the Infinitesimal Decrease Condition implies the existence of a trajectory x with x(0) = x and along which V (x(t)) + tω is decreasing (see [9, Theorem 4.5.7] ), which implies the result. □

We now proceed to fix γ > 0 such that

γ < min \{δ_{1}, \frac{η}{2} , \frac{ω}{12L_{f}L_{V}} \} ,  (6)

and we define

β := min \{δ_{1}, \frac{(b − a)}{2}, \frac{γω}{4m}\} .  (7)

Let N be the first integer such that

b − Nβ > a ≥ b − (N + 1)β.

Note that N ≥ 1 since β < b − a. We proceed to define certain sets Ω_{i}(i = 0, 1, ⋯, N + 1) that lie at the heart of our construction.

For 0 ≤ i ≤ N − 1, we set

Ω_{i} := [S(b − iβ) + γB] \ [S(b − (i + 1)β) + γB];

for i = N we set

Ω_{N} := [S(b − Nβ) + γB] \ [S(b − Nβ) + \frac{γ}{4} B];

and finally, we define

Ω_{N+1} := S(b − Nβ) + \frac{γ}{4} B.

We now gather some facts about these sets.

Lemma 1.4

(a) The Ω_{i} are disjoint, and Ω_{i} is contained in S(a, b) + γB for i ≤ N.

(b) ⋃_{i=0}^{N+1} Ω_{i} = S(b) + γB.

(c) If x ∈ Ω_{i} for some i ∈ {0, 1, ⋯, N} and s ∈ proj(x, S(b − iβ)), then s ∈ S(a, b) + ηB.

(d) S(b − iβ) + \frac{γ}{4} B ⊂ S(b − (i + 1)β)) + γB (i = 0, 1, ⋯, N − 1).

(e) For every i ∈ \{0, 1, ⋯, N\}, ∀x ∈ Ω_{i}, we have \frac{γ}{4} ≤ d(x, S(b − iβ)) < γ.

(f) S(b − Nβ) + \frac{γ}{2} B ⊂ S(a) + γB, so that Ω_{N+1} ⊂ S(a) + γB.

Proof

(a): That the Ω_{i} are disjoint is evident; that they lie in S(a, b) + γB for i ≤ N follows from Lemma 1.2 for i < N and from Lemma 1.1 for i = N (recall that b − Nβ − a ≤ β ≤ δ_{1} and γ < δ_{2}).

(b): Evident.

(c): Direct from Lemma 1.2 (i < N) or Lemma 1.1 (i = N).

(d): Let x lie in S(b − iβ) + γ/4B, and let s ∈ S(b − iβ) satisfy ∥x − s∥ < γ/4. Then V (s) ≤ b−iβ, and if V (s) ≤ b−(i+1)β the conclusion is immediate. Otherwise we have

V (s) > b − (i + 1)β > a,

so that s ∈ S(a, b). By Lemma 1.3 there exists y ∈ S(b − (i + 1)β) such that

∥s − y∥ ≤ \frac{m}{ω}[V (s) − b + (i + 1)β] ≤ \frac{mβ}{ω} ≤ \frac{γ}{2} ,

in view of (7). Then

∥x − y∥ ≤ ∥x − s∥ + ∥s − y∥ < \frac{γ}{4} + \frac{γ}{2} < γ,

which establishes the desired conclusion.

(e): For i = N, this is immediate from the definition of Ω_{N}; for i < N, it is a consequence of (d).

(f): Let x belong to S(b−Nβ)+γ/2B, and let s ∈ S(b−Nβ) satisfy ∥x−s∥ < γ/2. If V (s) ≤ a, then x ∈ S(a) + γB. Otherwise, s belongs to S(a, b), and Lemma 1.3 implies the existence of y ∈ S(a) such that

∥y − s∥ ≤ \frac{m}{ω}[V (s) − a] ≤ \frac{m}{ω}[b − Nβ − a] ≤ \frac{mβ}{ω} ≤ \frac{γ}{2} .

But then ∥x − y∥ < γ, so again x ∈ S(a) + γB.

□

Lemma 1.5 Let x ∈ Ω_{i}(i = 0, 1, ⋯, N), and let s ∈ proj(x, S(b − iβ)). Then there exists u ∈ U such that

⟨x − s, f(s, u)⟩ ≤ \frac{−ω}{2L_{V}} ∥x − s∥.

Proof By definition, x − s lies in the proximal normal cone N^{P} (s, S(b − iβ)). Note that s lies in S(a, b)+ηB (by Lemma 1.2 for i < N, Lemma 1.1 for i = N), so that V is Lipschitz of rank L_{V} in a neighborhood of s. A basic calculus result [9, 1.11.26] yields the existence of λ > 0 such that λ(x − s) ∈ ∂_{L}V (s), and necessarily λ∥x − s∥ ≤ L_{V}. In accord with Remark 1.1, there exists u ∈ U such that

⟨λ(x − s), f(s, u)⟩ ≤ \frac{−ω}{2} .

The result follows. □

Defining the feedback

We now define a feedback k(·) on S(b) + γB as follows. If x ∈ Ω_{i} for some i ∈ {0, 1, ⋯, N}, then we set k(x) = u, where u is one of the points corresponding to x (and a projection s) as in Lemma 1.5. There remain the points x in Ω_{N+1} to consider (see Lemma 1.4(b)). For such x, we define k(x) to be any point in U. (We remark that we have phrased the definition of k(x) in such a way that the choice of u as indicated above is made once and for all, but in fact a different choice could be made if the same state x recurred subsequently, without at all affecting what follows; this fact is relevant for real-time control.)

The remainder of the proof consists in establishing that for suitably small mesh size, any π-trajectory generated by k(·) with initial condition in S(b)+γB remains in S(b) + γB, enters S(a) + γB within a certain (uniform) time, and then remains in that set subsequently.

We consider countable partitions {t_{j}} such that t_{0} = 0, lim_{j→∞} t_{j} = ∞, and such that 0 < t_{j+1} − t_{j} ≤ δ ∀j ≥ 0, where δ is any positive number satisfying

δ < min \{\frac{γ}{4m}, \frac{ω}{6mL_{f}L_{V}} , \frac{γω}{48m^{2}L_{V}} , 1\} .  (8)

For such a partition, let x_{0} be any point in S(b) + γB, and let x(·) be a πtrajectory with x(0) = x_{0}. We denote x(t_{j}) by x_{j}, and we set

Δ := \frac{ω}{60L_{V}}

.

Lemma 1.6 For some t_{j} ∈ π, suppose that x_{j} ∈ Ω_{i}, i ∈ {0, 1, ⋯, N}. Then

x(t) ∈ S(b) + γB ∀t ∈ [t_{j}, t_{j+1}], and
d(x(t), S(b − iβ)) ≤ d(x_{j}, S(b − iβ)) − Δ(t − t_{j}) ∀t ∈ [t_{j}, t_{j+1}].

Proof We have x_{j} ∈ S(a, b) + γB by Lemma 1.4(a), and ∥ẋ(t)∥ ≤ m while x(t) lies in S(b) + ηB. Since δm < γ/4 by (8) and γ < η/2, it follows that x(t) lies in S(a, b) + ηB for t ∈ [t_{j}, t_{j+1}], as does the point s that figures in the definition of k(x_{j}); this was pointed out in the proof of Lemma 1.5, where we also deduced the inequality

⟨x_{j} − s, f(s, k(x_{j}))⟩ ≤ \frac{−ω}{2L_{V}} ∥x_{j} − s∥.  (9)

We fix t ∈ (t_{j}, t_{j+1}) and set

Ψ := \frac{x(t) − s}{∥x(t) − s∥}.

Note that x(t) ≠ s, since ∥x_{j} − s∥ ≥ γ/4 by Lemma 1.4(e), and since

∥x(t) − x_{j}∥ < δm < γ/4.

We now observe two inequalities:

d(x(t), S(b − iβ)) ≤ ∥x(t) − s∥ = ⟨Ψ, x(t) − s⟩,
d(x_{j}, S(b − iβ)) = ∥x_{j} − s∥ ≥ ⟨Ψ, x_{j} − s⟩.

These together imply

d(x(t), S(b − iβ)) − d(x_{j}, S(b − iβ)) ≤ ⟨Ψ, x(t) − x_{j}⟩
= τ⟨Ψ, f_{j}⟩,  (10)

where we introduce the notation τ := t − t_{j},

x(t) = x_{j} + τf_{j}, f_{j} := \frac{1}{τ} ∫_{t_{j}}^{t} f(x(r), k(x_{j}))dr.

We also set

f̂_{j} := f(s, k(x_{j})) = \frac{1}{τ} ∫_{t_{j}}^{t} f(s, k(x_{j}))dr.

Note that

∥f_{j} − f̂_{j}∥ ≤ \frac{1}{τ} ∫_{t_{j}}^{t} L_{f}∥x(r) − s∥dr

(the Lipschitz condition holds since we are in S(a, b) + ηB)

≤ \frac{1}{τ} ∫_{t_{j}}^{t} L_{f}(∥x(r) − x_{j}∥ + ∥x_{j} − s∥)dr
≤ L_{f}(τm + d(x_{j}, S(b − iβ))).

It follows from this and (9) that we have

⟨x_{j} − s, f_{j}⟩ = ⟨x_{j} − s, f̂_{j} + f_{j} − f̂_{j}⟩
≤ \frac{−ω}{2L_{V}} ∥x_{j} − s∥ + L_{f}d(x_{j}, S(b − iβ))\{τm + d(x_{j}, S(b − iβ))\}
≤ d(x_{j}, S(b − iβ))[\frac{−ω}{2L_{V}} + L_{f}δm + γL_{f}]

(since τ < δ and d(x_{j}, S(b − iβ)) ≤ γ)

≤ d(x_{j}, S(b − iβ))[\frac{−ω}{2L_{V}} + \frac{ω}{6L_{V}} + \frac{ω}{6L_{V}} ]

(we have invoked (8) and (6))

≤ −\frac{γω}{24L_{V}}

(since d(x_{j}, S(b − iβ)) ≥ γ/4 by Lemma 1.4(e)).

We shall use this bound on ⟨x_{j} − s, f_{j}⟩ to derive one on ⟨x(t) − s, f_{j}⟩ as follows:

⟨x(t) − s, f_{j}⟩ = ⟨x_{j} + τf_{j} − s, f_{j}⟩ ≤ ⟨x_{j} − s, f_{j}⟩ + τ∥f_{j}∥^{2}
≤ \frac{−γω}{24L_{V}} + δm^{2} ≤ \frac{−γω}{48L_{V}}  (11)

(in light of (8)). We also have

∥x(t) − s∥ = ∥x_{j} + τf_{j} − s∥ ≤ ∥x_{j} − s∥ + τ∥f_{j}∥ ≤ γ + δm
< \frac{5γ}{4} (by (8)).

Combining this with (11) we arrive at

⟨Ψ, f_{j}⟩ = ⟨\frac{x(t) − s}{∥x(t) − s∥}, f_{j}⟩ ≤ \frac{−γω}{48L_{V}} /\frac{5γ}{4} = −Δ.

Together with (10), this gives the inequality asserted by the Lemma. Since this inequality evidently implies

d(x(t), S(b − iβ)) < γ,

it also follows that x(t) ∈ S(b) + γB. □

Lemma 1.7 If x_{j} ∈ Ω_{i} where 0 ≤ i ≤ N, then x_{j+1} lies in Ω_{i′} for some i′ ≥ i.

Proof Since x_{j} ∈ Ω_{i}, we have d(x_{j}, S(b − iβ)) < γ, and (by Lemma 1.6) d(x_{j+1}, S(b − iβ)) < γ. Now let 1 ≤ k < i. Since S(b − iβ) ⊂ S(b − (k + 1)β), we deduce d(x_{j+1}, S(b − (k + 1)β)) < γ. But then x_{j+1} ∉ Ω_{k} by definition of Ω_{k}. Since x_{j+1} ∈ S(b) + γB by Lemma 1.6, we must have x_{j+1} ∈ Ω_{i′} for some i′ ≥ i, in view of Lemma 1.4(b). □

Lemma 1.8 If x(τ) ∈ Ω_{N+1} for some τ ∈ π, then x(t) ∈ S(a) + γB ∀t ≥ τ.

Proof We know that x(τ) lies in the interior of S(a) + γB by Lemma 1.4(f). For t > τ, as long as d(x(t), S(b − Nβ)) does not attain or exceed γ/2, then x(t) remains in S(a) + γB. Thus ∥ẋ(t)∥ remains bounded by m and no blow-up occurs (i.e., x(t) is well-defined).

It suffices therefore to prove that the continuous function

g(t) := d(x(t), S(b − Nβ))

does not become greater than or equal to γ/2 for some t_{0} > τ. We have g(τ) < γ/4.

Since we have chosen δ to satisfy δm < γ/4, at the next node τ_{1} following τ we have g(τ_{1}) < γ/2, and two cases arise. The first is when γ/4 ≤ g(τ_{1}) < γ/2; in that case, x(τ_{1}) belongs to Ω_{N}, and Lemma 1.6 shows that at the next node τ_{2}, g(τ_{2}) will have decreased relative to g(τ_{1}). The other case is when g(τ_{1}) < γ/4; but then we are in the same situation as we were with τ. The conclusion is that g(t) never exceeds γ/2, as required. □

One last lemma and the proof is complete. We set

T := (1 + \frac{b − a}{β})(1 + \frac{45γL_{V}}{ω} ).

Lemma 1.9

x(t) ∈ S(a) + γB ∀t ≥ T.

Proof In view of Lemma 1.8 and Lemma 1.4(f), it suffices to prove that there is a node τ ∈ π with τ ≤ T for which x(τ) ∈ Ω_{N+1}. Note that x(0) belongs to some Ω_{i}(0 ≤ i ≤ N + 1) by Lemma 1.4(b); if i = N + 1 we are done, so assume i ≤ N. Since δ < 1 by (8), there is a node τ_{1} lying in the open interval (σ, σ + 1), where σ := (3γ)/(4Δ). By Lemma 1.7, x(τ_{1}) belongs either to Ω_{i} or to Ω_{i′} for some i′ > i. In the former case, it follows that x(t) lies in Ω_{i} for every node t ∈ π lying between 0 and τ_{1}, and the inequality of Lemma 1.6 applies to give

d(x(τ_{1}), S(b − iβ)) ≤ d(x(0), S(b − iβ)) − Δτ_{1}
< γ − \frac{3γ}{4} = \frac{γ}{4} .

However, the left side is no less than γ/4 by Lemma 1.4(e). This contradiction shows that, in fact, x(τ_{1}) must belong to some Ω_{i′} for an index i′ > i. If i′ = N + 1, we are done; otherwise, the same argument, beginning now at (τ_{1}, x(τ_{1})), yields the existence of a node τ_{2} ∈ π with τ_{2} ≤ 2σ + 2 such that x(τ_{2}) belongs to Ω_{i′′}, where i′′ > i′. Continuing in this manner, we find that (since there are at most N +1 steps as above prior to landing in Ω_{N+1}), there is a node τ ∈ π with τ ≤ (N +1)(σ+1) such that x(τ) ∈ Ω_{N+1}. But N < (b−a)/β implies that T as defined above is greater than (N + 1)(σ + 1). □

Remark 1.4 It is a consequence of the construction that for suitably small δ > 0, the set A := S(a + δ) is attained in a time that approaches 0 as x(0) approaches A. This is the key property that is needed in the converse Lyapunov theorem that we now proceed to develop.

2 Construction of a Lyapunov function

We show in this section that under reasonable assumptions, there always exist Lyapunov functions having the properties required for the feedback construction of the preceding section, and giving rise to practical feedback stabilization of arbitrarily prescribed range. While the result below appears to be new and the approach to proving it has some novel features, there is a familiar heuristic at work: the Lyapunov function is constructed as the value function associated with a parametrized family of optimal control problems.

The function f(x, u) describing the dynamics is supposed in this section to satisfy much the same regularity conditions as before. Specifically, we require that for any bounded subset S of IR^{n}, there exist constants m = m(S) and L = L(S) such that

∥f(x, u)∥ ≤ m ∀x ∈ S, ∀u ∈ U
∥f(x, u) − f(y, u)∥ ≤ L∥x − y∥ ∀x, y ∈ S, ∀u ∈ U.

(As before, U is just an abstract set, and no hypotheses are made concerning the nature of the dependence of f on u.)

In addition, we require “nice” controllability to a given compact set A via relaxed trajectories. Let us now proceed to make this precise. We define a multifunction Γ on IR^{n} by

Γ(x) := cl co{f(x, u) : u ∈ U}.

By “trajectory” (or Γ-trajectory) we mean an absolutely continuous function x(·) on an interval [0, T] such that

ẋ(t) ∈ Γ(x(t)) a.e. t ∈ [0, T].

Given α ∈ IR^{n}, we define T_{A}(α) as the least time required for a trajectory to go from α to the set A:

T_{A}(α) := inf{T ≥ 0 : x(·) is a trajectory on [0, T], x(0) = α, x(T) ∈ A}.

The controllability hypothesis that we make is that every α admits a trajectory steering it to A in finite time, a time which goes to zero as α approaches A. Equivalently:

(CH) T_{A}(α) < ∞ ∀α ∈ IR^{n}, and lim_{d(α,A)↓0} T_{A}(α) = 0.

We remark that proximal criteria exist ensuring that A satisfies (CH); see [9, 4.6.7]. Fix now a point a_{0} ∈ A. Below, diam(A) refers to the usual diameter of A as a subset of IR^{n}.

Theorem 2 For any r > 0 and R > diam(A) + r, there exist numbers a, b, γ, η and a function V : IR^{n} ⟶ IR such that

S(a) + γB ⊂ A + rB ⊂ B̄(a_{0}, R) ⊂ S(b) + γB,

and such that all the hypotheses of Theorem 1 are satisfied and, in addition, S(b) is compact. The feedback defined in Theorem 1 stabilizes S(b) + γB to A + rB.

The following addresses the issue of practical semi-global stabilization of globally asymptotically controllable systems.

Corollary 1 Let the system be globally asymptotically controllable to the origin. Then, for any 0 < r < R, there exists a feedback of the type constructed in Theorem 1 which is defined on a neighborhood of B̄(0, R), and which stabilizes every initial point in B(0, R) to the ball B(0, r).

Proof It is known that a continuous global Lyapunov function exists for the problem of stabilization to the origin [26], and every level set S(a) for a > 0 of that function satisfies (CH) [9, 4.6.7]. It suffices now to take such a level set A contained in the ball B(r/2, 0), and to apply Theorem 2 with r := r/2 and a_{0} := 0. □

Proof of Theorem 2

We begin the proof by defining another multifunction Γ̃ (more useful than Γ for being uniformly bounded):

Γ̃(x) := cl co\{\frac{v}{1 + ∥v∥} : v ∈ Γ(x)\}.

We set

T̃_{A}(α) := inf{T ≥ 0 : x(·) is a Γ̃-trajectory on [0, T], x(0) = α, x(T) ∈ A}.

Evidently (or by convention) we have T̃_{A} = 0 on A.

Lemma 2.1

(a) Γ̃ is locally Lipschitz and has nonempty convex compact values in B̄(0, 1).

(b) T̃_{A}(α) is finite ∀α ∈ IR^{n}.

(c) lim_{d(α,A)↓0} T̃_{A}(α) = 0.

(d) There exists a positive number ϵ such that whenever α ∈ A + ϵB, and whenever the Γ̃-trajectory x(·) has x(0) = α and x(T) ∈ A for some T ≤ T̃_{A}(α) + ϵ, then we have ∥x − a_{0}∥_{∞} ≤ diam(A) + 1. We can suppose ϵ < 1, ϵ < r, and

sup\{T̃_{A}(α) : α ∈ A + ϵB\} < \frac{r^{2}}{4(1 + diam(A))}.  (12)

Proof We omit the routine proof of (a). For (b), let α ∈ IR^{n} be given. By assumption, there is a Γ-trajectory x on an interval [0, T] such that x(0) = α, x(T) ∈ A. We set

T̃ := ∫_{0}^{T} (1 + ∥ẋ(t)∥)dt

and we define a function x̃ on [0, T̃] by

x̃(τ) := x(t),

where t = t(τ) is determined in [0, T] by

τ = ∫_{0}^{t}(1 + ∥ẋ(r)∥)dr

(this change of variables or time scale is known as the Erdmann Transform.) Then

\frac{dx̃}{dτ} = \frac{ẋ(t)}{1 + ∥ẋ(t)∥} ∈ Γ̃(x̃(τ)) a.e.,

so that x̃ is a Γ̃-trajectory. Hence T̃_{A}(α) ≤ T̃ < ∞.

We turn now to (c). Let α_{i} be a sequence for which d(α_{i}, A) decreases to 0. Then T_{A}(α_{i}) → 0 by assumption. Let m be such that ∥f(x, u)∥ ≤ m for (x, u) ∈ (A + B̄) × U. Then, as soon as T_{A}(α_{i}) is strictly less than 1/m, there is a Γ-trajectory x_{i} on an interval [0, T_{i}] such that

x_{i}(0) = α_{i}, x_{i}(T_{i}) ∈ A, T_{i} < \frac{1}{m}, T_{i} < T_{A}(α_{i}) + \frac{1}{i}.

It follows that x_{i}(t) ∈ A + B̄ for t ∈ [0, T_{i}]. Now let x̃_{i} be the Erdmann Transform of x_{i} as given above. Then

T̃_{A}(α_{i}) ≤ T̃_{i} = ∫_{0}^{T_{i}}(1 + ∥ẋ_{i}(t)∥)dt ≤ (1 + m)T_{i} < (1 + m)(T_{A}(α_{i}) + \frac{1}{i}).

It follows that T̃_{A}(α_{i}) → 0, as required.

We now examine (d). If the assertion is false, there exist a sequence α_{i} with d(α_{i}, A) ↓ 0 and corresponding Γ̃-trajectories x_{i} with x_{i}(0) = α_{i}, x_{i}(T_{i}) ∈ A, such that

T_{i} ≤ T̃_{A}(α_{i}) + \frac{1}{i}, ∥x_{i} − a_{0}∥_{∞} > diam(A) + 1.

Since T̃_{A}(α_{i}) → 0 by (c), we have T_{i} → 0. On the other hand, there is a subinterval of [0, T_{i}] in which ∥x_{i} − a_{0}∥ goes from being diam(A) + 1 to at most diam(A), and since ∥ẋ_{i}(t)∥ ≤ 1 the length of that subinterval (and hence, T_{i}) is at least 1. This contradiction establishes the first part of (d); the rest follows immediately by shrinking ϵ as required, in light of (c). □

Defining a value function

We proceed now to define a new multifunction F (x) whose effect is to enlarge the set Γ̃(x) for d(x, A) < ϵ. We set

F (x) := \{ \begin{matrix}Γ̃(x) & for d(x, A) ≥ ϵ \\ Γ̃(x) + 2[\frac{ϵ−d(x,A)}{ϵ}]B̄ & for d(x, A) ≤ ϵ \end{matrix}

Having done this, we define a value function V (·) on IR^{n} in terms of the trajectories of F as follows :

V (α) := inf{∫_{0}^{T} ∥x(t) − a_{0}∥dt : T ≥ 0, x(0) = α, ẋ ∈ F (x) a.e., x(T) ∈ A}.

We stress that T is a choice variable here, in this free time problem.

Lemma 2.2

(a) F is compact and convex-valued, uniformly bounded and locally Lipschitz.

(b) V (·) is nonnegative, finite-valued and lower semicontinuous, and the infimum defining V (α) is attained for every α.

(c) V (α) = 0 iff α ∈ A, and lim_{d(α,A)↓0} V (α) = 0.

(d) The sublevel sets S(b) := {α : V (α) ≤ b} of V are compact.

Proof The assertions of (a) are immediate. Since F (x) is uniformly bounded, the attainment and the lower semicontinuity asserted in (b) follow from standard “compactness of trajectories” arguments; see [9, Chapter 4] for details. The first assertion of (c) is clear, and the other one stems from Lemma 2.1 as follows. Let α ∈ A + ϵB, and let the Γ̃-trajectory x satisfy x(0) = α, x(T) ∈ A, and T ≤ T̃_{A}(α) + δ, for some δ ∈ (0, ϵ). Then ∥x − a_{0}∥_{∞} ≤ diam(A) + 1 (by choice of ϵ), and we deduce

V (α) ≤ ∫_{0}^{T} ∥x(t) − a_{0}∥dt ≤ (T̃_{A}(α) + δ)(diam(A) + 1).

Since T̃_{A}(α) ↓ 0 as d(α, A) ↓ 0, (c) follows. Finally we turn to (d). If ∥α − a_{0}∥ > diam(A) + ϵ, then the time required for a trajectory x to go from x = α to the boundary of A + ϵB is at least ∥α − a_{0}∥ − diam(A) − ϵ. But then V (α) ≥ ϵ(∥α − a_{0}∥ − diam(A) − ϵ). This implies assertion (d). □

The next step invokes Hamiltonian conditions for optimal control, and uses the lower Hamiltonian h associated with F:

h(x, p) := min{⟨p, v⟩ : v ∈ F (x)}.

Lemma 2.3 Let ζ ∈ ∂_{P} V (α), where α does not lie in A. Let x be a trajectory solving the problem that defines V (α), with associated time T. Then there exists an absolutely continuous function p on [0, T] such that

(−ṗ − \frac{x − a_{0}}{∥x − a_{0}∥}, ẋ) ∈ ∂_{C}h(x, p) a.e. t ∈ [0, T]  (13)
p(0) = ζ  (14)
h(x(t), p(t)) + ∥x(t) − a_{0}∥ = 0 ∀t ∈ [0, T].  (15)

Proof By definition of ∂_{P} V (α), we have for some σ ≥ 0 and for all α′ near α,

V (α′) + σ∥α′ − α∥^{2} − ⟨ζ, α′⟩ ≥ −⟨ζ, α⟩.

Let x′ be a trajectory near x (in the L^{∞} norm), put α′ = x′(0) and α = x(0) and rearrange to derive that x′(·) = x(·) solves locally the problem of minimizing

∫_{0}^{T′} ∥x′(t) − a_{0}∥dt − ⟨ζ, x′(0)⟩ + σ∥x′(0) − x(0)∥^{2}

over the trajectories x′ for F satisfying x′(T′) ∈ A. (Here T′ and x′(0) are free.) We apply the corollary of Theorem 3.6.1 of [5] (with time reversed) to deduce the existence of an absolutely continuous function q on [0, T] satisfying

(−q̇, ẋ) ∈ ∂_{C}[H(x, q) − ∥x − a_{0}∥](x, q) a.e. t ∈ [0, T], (16)
q(0) = −ζ,  (17)
H(x(t), q(t)) = ∥x(t) − a_{0}∥, t ∈ [0, T],  (18)

where H(x, p) is the function −h(x, −p) and ∂_{C} denotes the generalized gradient. The Hamiltonian inclusion above implies

(q̇ − \frac{x − a_{0}}{∥x − a_{0}∥}, ẋ) ∈ ∂_{C}h(x, −q) a.e., t ∈ [0, T].

Now putting p := −q gives to these conclusions the form asserted in the statement of the lemma. □

Lemma 2.4 For any constant c > 0, there is a constant M_{c} with the following property. If α ∈ S(c) and if the trajectory x on [0, T] attains the infimum defining V (α), then ∥x − a_{0}∥_{∞} ≤ M_{c}, T ≤ M_{c}.

Proof If ∥x − a_{0}∥_{∞} > c + diam(A) + 1, then the time required for ∥x − a_{0}∥ to attain the value diam(A) + 1 exceeds c (since ∥ẋ∥ ≤ 1). But then V (α) ≥ (1 + diam(A))c > c. This shows that ∥x − a_{0}∥_{∞} is bounded by c + diam(A) + 1. By Lemma 2.1(c), lim_{d(α,A)↓0} T̃_{A}(α) = 0. So there exists ρ > 0 such that

T̃_{A}(x) ≤ 1 ∀x ∈ A + ρB̄.

Now take α outside A + ρB̄, and let τ_{ρ} denote the first time t that x(t) attains A + ρB̄. Then V (α) ≥ ρτ_{ρ}, whence τ_{ρ} ≤ c/ρ for α ∈ S(c).

We deduce that

T̃_{A}(α) ≤ τ_{ρ} + 1 (by choice of ρ)
≤ \frac{c}{ρ} + 1.

If α ∈ A + ρB̄, the same bound evidently holds. It suffices now to set

M_{c} := max\{\frac{c}{ρ} + 1, c + diam(A) + 1\}.

□

Lemma 2.5 V is locally Lipschitz on IR^{n}.

Proof We prove first that V is locally Lipschitz on the open set {V > 0} = comp(A). Let α_{0} belong to this set, take any δ > 0 such that δ < d(α_{0}, A), and any element ζ ∈ ∂_{P} V (α), where

∥α − α_{0}∥ < δ, V (α) ≤ V (α_{0}) + δ =: c.  (19)

The conclusions of Lemma 2.4 are available for any trajectory solving the V (α) problem. If K is a Lipschitz constant for F on the ball B(0, M_{c} + 1 + ∥a_{0}∥) (where M_{c} comes from Lemma 2.4), then the Hamiltonian inclusion (13) implies

∥ṗ∥ ≤ K∥p∥ + 1.  (20)

The condition (15) at t = T gives ∥p(T)∥ ≤ diam(A) since x(T) ∈ A, and since F (x(T)) = Γ̃(x(T))+2B̄ ⊃ B̄. This, together with (20) and Gronwall’s Lemma, leads to

∥ζ∥ = ∥p(0)∥ ≤ e^{KT} ∥p(T)∥ + ∫_{0}^{T} e^{K(T −s)}ds
≤ diam(A)e^{KT} + \frac{e^{KT} − 1}{K}
≤ diam(A)e^{KM_{c}} + \frac{e^{KM_{c}} − 1}{K},

since T ≤ M_{c} by Lemma 2.4. This establishes a uniform bound on elements of ∂_{P} V (α) whenever α satisfies (19), which proves that V is Lipschitz on a neighborhood of α_{0} [9, 1.11.11]. Thus V is locally Lipschitz on the set where it is strictly positive.

There is a neighborhood N of A on which V is bounded above, in view of Lemma 2.2(c). The argument above therefore yields a bound L on elements of ∂_{P} V (α) for all α ∈ N \ A, so that V is uniformly Lipschitz of rank L on α ∈ N \ A by [9, Theorem 1.7.3]. Of course, V = 0 on A, and is continuous at each point of A in view of Lemma 2.2(c). That V is Lipschitz on N, and hence locally Lipschitz on IR^{n}, now follows. □

Lemma 2.6

sup{V (α) : α ∈ A + ϵB} < inf{V (α) : d(α, A) ≥ r}

Proof Let d(α, A) < ϵ, fix δ ∈ (0, ϵ) and let the trajectory x on [0, T] satisfy x(0) = α, x(T) ∈ A, T < T̃_{A}(α) + δ. Then by Lemma 2.1 we have ∥x − a_{0}∥_{∞} ≤ diam(A) + 1 and so

V (α) ≤ ∫_{0}^{T} ∥x(t) − a_{0}∥dt ≤ (T̃_{A}(α) + δ)(diam(A) + 1).

We derive V (α) ≤ (diam(A) + 1)T̃_{A}(α), and (from (12))

V (α) ≤ sup\{T̃_{A}(α) : d(α, A) ≤ ϵ\}(diam(A) + 1) < \frac{r^{2}}{4} .

Now let d(α, A) ≥ r, and let x solve the problem defining V (α). There is an interval of length at least r/2 during which ∥x(t) − a_{0}∥ ≥ r/2 (since ∥ẋ∥ ≤ 1), whence

V (α) > \frac{r^{2}}{4} .

The result follows. □

Lemma 2.7 There exist positive numbers a, b, η with a < b such that

S(a) + ηB ⊂ A + rB ⊂ B̄(a_{0}, R) ⊂ S(b)

and

S(a, b) + ηB ⊂ {α : d(α, A) > ϵ}.

Proof Pick a number a > 0 lying between the two quantities in the statement of Lemma 2.6. Then evidently the compact set S(a) satisfies

S(a) ⊂ comp{d(α, A) ≥ r} = A + rB,

whence S(a) + ηB ⊂ A + rB for η > 0 suitably small. It also follows that (for any b > a) the compact set S(a, b) is contained in the open set {α : d(α, A) > ϵ}. Any b suitably large will satisfy B̄(a_{0}, R) ⊂ S(b), since V is bounded on bounded sets. Finally, by shrinking η further if necessary, we will have the final conclusion of the Lemma as well. □

Lemma 2.8 The Infinitesimal Decrease Condition (H5) of §1 holds on S(a, b)+ ηB, with ω := ϵ.

Proof As pointed out in Remark 1.3, it suffices to show that for any α ∈ S(a, b) + ηB, for any ζ ∈ ∂_{P} V (α), one has:

inf{⟨ζ, f(α, u)⟩ : u ∈ U} ≤ −ϵ.  (21)

Let x be a trajectory solving the problem defining V (α). Then, by Lemma 2.3, we have (at t = 0):

h(α, ζ) + ∥α − a_{0}∥ = 0.

Since d(α, A) > ϵ by Lemma 2.7, we have F (α) = Γ̃(α), so that the preceding equality yields, for any δ > 0, the existence of some element v ∈ Γ(α) such that

⟨ζ, \frac{v}{1 + ∥v∥}⟩ ≤ −∥α − a_{0}∥ + δ < −ϵ + δ.

For δ small enough, the right side is negative, whence

⟨ζ, v⟩ < −ϵ + δ.

Given that Γ(α) := cl cof(α, U), this yields the existence of u ∈ U for which

⟨ζ, f(α, u)⟩ < −ϵ + 2δ.

Since δ is arbitrarily small, (21) ensues. □

Since S(b) is compact, f is Lipschitz in x and bounded on S(b) + ηB, in accord with hypotheses (H3) (H4) of §1. When the level sets are compact and V is continuous, (H2) always holds. The verification of this fact is the last property to confirm.

Lemma 2.9 Hypothesis (H2) holds.

Proof If (H2) fails, then there exist sequences α_{i} ∈ IR^{n}, ϵ_{i} ↓ 0, and u_{i} ∈ B(0, 1) such that

V (α_{i}) ≤ a + ϵ_{i} and V (α_{i} + ϵ_{i}u_{i}) > b.

Since S(b) is compact, we can suppose by passing to a subsequence that α_{i} → α_{0}. Then, since V is continuous, we have V (α_{0}) ≥ b > a ≥ V (α_{0}), a contradiction. □

The setting of Theorem 1 is established, and Theorem 2 is proved.

3 Robustness

We prove in this section that the feedback constructed in §1 is robust with respect to small measurement error and persistent external disturbance, in a precise sense that requires two stipulations. The first is that the measurement error must not exceed in order of magnitude the step size of the underlying discretization, a condition which appears to be rather natural. The second requirement is perhaps more surprising, and surfaces from the nature of the feedback construction. It dictates that each step be “big enough” (while continuing to be “small enough”) so as to counteract the measurement error by means of the attractive effect inherent in the construction. Thus the partitions used to discretize the effect of the control are taken to be “reasonably uniform”. Our perturbed system is modeled by

ẋ = f(x, k(x + p)) + q,

where the external disturbance q : [0, ∞) ⟶ IR^{n} is a bounded measurable function:

∥q(t)∥ ≤ E_{q} t ≥ 0 a.e.

Given a partition π = {t_{i}}_{i≥0} of [0, ∞) and the initial condition x_{0}, the resulting π-trajectory of our perturbed system is defined by successively solving the differential equation

ẋ(t) = f(x(t), k(x(t_{i}) + p_{i})) + q(t), t ∈ [t_{i}, t_{i+1}],

with x(0) = x_{0}. The continuous function x(t) is the real state of the system, while the sequence {x(t_{i}) + p_{i}} corresponds to the inexact measurements used to select control values.

Theorem 3 The feedback k : S(b) + γB → U constructed in Theorem 1 is robust in the sense that there exist positive numbers δ_{0}, T and E_{q} such that, for every δ ∈ (0, δ_{0}) there exists E_{p}(δ) > 0 having the following property: for any partition π = {t_{i}}_{i≥0} having

\frac{δ}{2} ≤ t_{i+1} − t_{i} ≤ δ i ≥ 0,

where 0 < δ < δ_{0}, for any set of measurement errors {p_{i}}_{i≥0} having

∥p_{i}∥ ≤ E_{p}(δ), i ≥ 0,

for any initial condition x_{0} such that x_{0} + p_{0} ∈ S(b) + γB, for any disturbance q having ∥q∥_{∞} ≤ E_{q}, the resulting π-trajectory x satisfies

x(t_{i}) + p_{i} ∈ S(b) + γB ∀i ≥ 0,
x(t) ∈ S(b) + 2γB ∀t ≥ 0,
x(t) ∈ S(a) + γB ∀t ≥ T.

Remarks 3.1 (a) Note that unlike T and E_{q}, the maximum admissible measurement error E_{p} depends on δ. Note also that (in contrast to Theorem 1) x(t) may not lie in S(b) + γB for all t, although for large t it must do so. We prove, however, that the “observed values” of the state, namely the values x(t_{i}) + p_{i}(i ≥ 0), all fall in S(b) + γB, the domain of definition of k.

(b) Certain other kinds of error, for example a disturbance d(·) entering into the dynamics in the form ẋ = f(x, k(x) + d), can be reduced to that of external disturbance by positing suitable continuity of f in the control variable.

(c) The maximum admissible disturbance measure E_{q} will be seen to be proportional to ω/L_{V}. This has a natural physical meaning, as can easily be seen in the case of smooth V and a continuous feedback k(x) such that

⟨∇V (x), f(x, k(x))⟩ ≤ −W (x).

Then we see that the perturbed system

ẋ = f(x, k(x)) + q

is stabilized by k if ∥q∥_{∞} < W (x)/∥∇V (x)∥ for every x, a bound akin to that involving ω/L_{V}.

Proof of Theorem 3

We adapt the proof of Theorem 1, whose first five lemmas hold with no change whatever, as does the definition of k(·). Recall that γ, β and N were introduced earlier; see (6) (7). We now define our upper bound for δ:

δ_{0} := min \{\frac{6γL_{V}}{ω} , 1, \frac{γω}{24L_{V} (m + \frac{ω}{6L_{V}} + 1)^{2}} , \frac{γ}{20m}\}  (22)
E_{q} := \frac{ω}{6L_{V}}  (23)
T := (1 + \frac{b − a}{β}) (1 + \frac{81γL_{V}}{ω} )  (24)

(this T differs slightly from the one in Theorem 1) and we let π = {t_{i}}_{i≥0} be a partition as described in the statement of Theorem 3, with corresponding measurement errors {p_{i}}_{i≥0} having ∥p_{i}∥ ≤ E_{p} for some E_{p} > 0 satisfying

E_{p} < min\{\frac{3γ}{80} , δ, \frac{δω}{432L_{V}} \}.  (25)

We also admit any disturbance q(·) for which ∥q∥_{∞} ≤ E_{q}, and we take x_{0} such that x_{0} + p_{0} ∈ S(b) + γB. We shall show that the corresponding π-trajectory has the required properties. We introduce the notation

x_{i} := x(t_{i}), y_{i} := x_{i} + p_{i}

for the actual and the measured state values at time t_{i}, and proceed to develop modified versions of the four last lemmas figuring in the proof of Theorem 1. We set

Δ̃ := \frac{ω}{108L_{V}} .

Lemma 3.1 For some t_{j} ∈ π, suppose that y_{j} ∈ Ω_{i}, i ∈ {0, 1, ⋯, N}. Then

x(t) ∈ S(b) + 2γB ⊂ S(b) + ηB, t_{j} ≤ t ≤ t_{j+1}
y_{j} ∈ S(b) + γB, y_{j+1} ∈ S(b) + γB,
d(y_{j+1}, S(b − iβ)) ≤ d(y_{j}, S(b − iβ)) − Δ̃(t_{j+1} − t_{j}).

Proof Note that y_{j} ∈ S(b) + γB by Lemma 1.4; it will follow from the last conclusion of the current Lemma that y_{j+1} ∈ S(b) + γB. Also, ∥x_{j} − y_{j}∥ = ∥p_{j}∥ ≤ E_{p}, together with ∥x(t) − x_{j}∥ ≤ δm, yield

x(t) ∈ S(a, b) + γB + (E_{p} + δm)B ⊂ S(b) + 2γB,

since E_{p} + δm < γ in view of (22) and (25). Since 2γ < η by (6), this gives x(t) ∈ S(b) + ηB. By Lemma 1.5 we have

⟨y_{j} − s, f(s, k(y_{j}))⟩ ≤ −\frac{ω}{2L_{V}} ∥y_{j} − s∥,  (26)

where s ∈ proj(y_{j}, S(b − iβ)). Fix t ∈ (t_{j}, t_{j+1}) and set

Ψ := \frac{x(t) − s}{∥x(t) − s∥}.

Note that x(t) ≠ s since ∥y_{j} − s∥ ≥ γ/4 by Lemma 1.4(e), while

∥y_{j} − x(t)∥ ≤ ∥y_{j} − x_{j}∥ + ∥x_{j} − x(t)∥ < E_{p} + δm < \frac{γ}{4} ,

as already noted. We observe the relations

d(x(t), S(b − iβ)) ≤ ∥x(t) − s∥ = ⟨Ψ, x(t) − s⟩,
d(y_{j}, S(b − iβ)) = ∥y_{j} − s∥ ≥ ⟨Ψ, y_{j} − s⟩,

whence

d(x(t), S(b − iβ)) − d(y_{j}, S(b − iβ)) ≤ ⟨Ψ, x(t) − y_{j}⟩
= ⟨Ψ, x_{j} + τ(f_{j} + q_{j}) − y_{j}⟩
≤ τ⟨Ψ, f_{j} + q_{j}⟩ + ∥p_{j}∥,  (27)

where we have introduced

τ := t − t_{j},
f_{j} := \frac{1}{τ} ∫_{t_{j}}^{t} f(x(r), k(y_{j}))dr,

q_{j} := \frac{1}{τ} ∫_{t_{j}}^{t} q(r)dr.

We also set f̂_{j} := f(s, k(y_{j})); note ∥f̂_{j}∥ ≤ m. We have

∥f_{j} − f̂_{j}∥ ≤ \frac{1}{τ} ∫_{t_{j}}^{t} L_{f}∥x(r) − s∥dr ≤ \frac{L_{f}}{τ} ∫_{t_{j}}^{t}(∥x(r) − x_{j}∥ + ∥x_{j} − s∥)dr
≤ L_{f}(δm + ∥x_{j} − y_{j}∥ + ∥y_{j} − s∥)
≤ L_{f}(δm + E_{p} + γ) ≤ \frac{5}{4}L_{f}γ.  (28)

We deduce

⟨x(t) − s, f_{j} + q_{j}⟩ = ⟨x_{j} + τ(f_{j} + q_{j}) − s, f_{j} + q_{j}⟩
= ⟨y_{j} − s − p_{j}, f_{j} + q_{j}⟩ + τ∥f_{j} + q_{j}∥^{2}
≤ ⟨y_{j} − s, f_{j}⟩ + E_{p}(m + E_{q}) + δ(m + E_{q})^{2}
= ⟨y_{j} − s, f̂_{j}⟩ + ⟨y_{j} − s, f_{j} − f̂_{j}⟩ + (m + E_{q})[δm + δE_{q} + E_{p}]
≤ −\frac{ω}{2L_{V}} ∥y_{j} − s∥ + \frac{5}{4}∥y_{j} − s∥L_{f}γ + (m + E_{q})[δm + δE_{q} + E_{p}]

(where we have used (26) and (28) )

≤ d(y_{j}, S(b − iβ)) [−\frac{ω}{2L_{V}} + \frac{ω}{6L_{V}} ] + δ[m + E_{q} + 1]^{2}

(by (6), and since E_{p} < δ by (25))

≤ \frac{γ}{4} [−\frac{ω}{3L_{V}} ] + \frac{γω}{24L_{V}} = −\frac{γω}{24L_{V}}

(since d(y_{j}, S(b − iβ)) ≥ γ/4 by Lemma 1.4, and since δ < δ_{0} defined by (22)). Note also that

∥x(t) − s∥ = ∥y_{j} − p_{j} + τ(f_{j} + q_{j}) − s∥
≤ γ + E_{p} + δ(m + E_{q}) ≤ \frac{5γ}{4} + δE_{q} < \frac{9γ}{4}

(note δE_{q} < γ because of δ < δ_{0}, in view of (22) (23)).

It follows that

⟨Ψ, f_{j} + q_{j}⟩ = ⟨\frac{x(t) − s}{∥x(t) − s∥}, f_{j} + q_{j}⟩ ≤ −\frac{(γω)/(24L_{V})}{(9γ)/4} = −\frac{ω}{54L_{V}} .

Substituting into (27) leads to

d(x(t), S(b − iβ)) − d(y_{j}, S(b − iβ)) ≤ −2Δ̃(t − t_{j}) + E_{p}.

We obtain from this

d(y_{j+1}, S(b − iβ)) − d(y_{j}, S(b − iβ)) ≤ ∥y_{j+1} − x(t_{j+1})∥ − 2Δ̃(t − t_{j}) + E_{p}
≤ −Δ̃(t_{j+1} − t_{j}) + [2E_{p} − Δ̃(t_{j+1} − t_{j})]
≤ −Δ̃(t_{j+1} − t_{j}),

by (25), and since t_{j+1} − t_{j} ≥ δ/2. □

Lemma 3.2 If y_{j} ∈ Ω_{i}, where i ≤ N, then y_{j+1} lies in Ω_{k} for some k ≥ i.

Proof We know that y_{j+1} ∈ Ω_{k} for some k, since y_{j+1} belongs to S(b)+γB by Lemma 3.1. Suppose that k < i. We have d(y_{j}, S(b−iβ)) < γ by definition of Ω_{i}, and Lemma 3.1 implies d(y_{j+1}, S(b − iβ)) < γ. But S(b − iβ) ⊂ S(b − (k + 1)β), so that d(y_{j+1}, S(b − (k + 1)β)) < γ. But then y_{j+1} ∉ Ω_{k} by definition of Ω_{k} (note that k ≤ N). This contradiction proves the Lemma. □

Lemma 3.3 If τ ∈ π is such that y(τ) ∈ Ω_{N+1}, then

x(t) ∈ S(a) + γB ∀t ≥ τ.

Proof We first establish

d(y(τ′), S(b − Nβ)) ≤ \frac{2γ}{5} for all nodes τ′ ≥ τ.  (29)

We consider first τ′ = τ + 1. We have d(y(τ), S(b − Nβ)) ≤ γ/4, whence

d(y(τ + 1), S(b − Nβ)) ≤ d(x(τ + 1), S(b − Nβ)) + E_{p}
≤ d(x(τ), S(b − Nβ)) + E_{p} + mδ
≤ d(y(τ), S(b − Nβ)) + 2E_{p} + mδ
< \frac{γ}{4} + \frac{3γ}{20} = \frac{2γ}{5} ( by (25)).

If d(y(τ + 1), S(b − Nβ)) is in fact ≤ γ/4, then this same argument yields

d(y(τ + 2), S(b − Nβ)) ≤ \frac{2γ}{5} .

If however d(y(τ + 1), S(b − Nβ)) > γ/4, then y(τ + 1) lies in Ω_{N} by definition, and Lemma 3.1 yields again d(y(τ + 2), S(b − Nβ)) < 2γ/5. Continuing in this way, we obtain (29) for all nodes τ′ ≥ τ.

We use (29) to argue as follows: let t ≥ τ, and let τ′ ≥ τ be a node adjacent to t; then

d(x(t), S(b − Nβ)) ≤ d(x(τ′), S(b − Nβ)) + δm
≤ d(y(τ′), S(b − Nβ)) + E_{p} + δm
< \frac{2γ}{5} + \frac{γ}{10} = \frac{γ}{2} (by (25)).

This gives x(t) ∈ S(a) + γB by Lemma 1.4(f). □

Lemma 3.4 Let

T := (1 + \frac{b − a}{β}) (1 + \frac{81γL_{V}}{ω} ) .

Then x(t) ∈ S(a) + γB ∀t ≥ T.

Proof In view of Lemma 3.3, it suffices to prove that some node τ ∈ π with τ ≤ T is such that y(τ) ∈ Ω_{N+1}. The argument is identical to that used to prove Lemma 1.8, with Δ replaced by Δ̃, and applied to the y_{i} rather than the x_{i}. □

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