Set-Valued and Convex Analysis

in Dynamics and Control

Rafal Goebel ∗†

November 12, 2013

Abstract

Many developments in convex, nonsmooth, and set-valued analysis were motivated byoptimization and optimal control theory. Real-life problems frequently yield convex objectivefunctions that need to be minimized; sets of solutions to optimization problems depending ona parameter naturally lead to set-valued mappings; and optimal value functions in optimalcontrol problems, when parameterized by the initial condition, are often nonsmooth. Manyof the theoretical developments then found applications to optimization and optimal control,pushing these fields to places not accessible using classical analysis. Several excellent booksoutline these developments and their applications to optimization and optimal control.

The goal of the lectures is to present several basic elements of convex and set-valued anal-ysis, and motivate them by and apply them to questions not in optimal control but rather indynamical systems and general control theory. Topics from control theory include uncertaindynamics, which involve switching between several linear dynamical systems; uniformityand robustness to perturbations of asymptotic stability in a dynamical system, especiallywhen the dynamics are discontinuous or uncertain; invariance concepts for hybrid dynamicalsystems, that is, dynamical systems which combine continuous-time and discrete-time char-acteristics; and necessary and sufficient conditions for convergence to a consensus, motivatedby problems where several agents or vehicles are to be steered to a common position. Thesetopics call for the introduction of elements of convex analysis, including the construction ofthe conjugate function and the convex subdifferential; and of set-valued analysis, for exam-ple convergence of sets, continuity properties of set-valued mappings, etc. The lectures arenot meant to be a comprehensive treatment of either control theory or set-valued analysisbut, rather, are meant to motivate the audience for further study.

∗Department of Mathematics and Statistics, Loyola University Chicago, 1032 W. Sheridan Road, Chicago, IL60660 rgoebel1@luc.edu. This work was partially supported by the National Science Foundation grant 1008602.

†Rafal Goebel received his M.Sc. degree in mathematics in 1994 from the University of Maria Curie Sklodowskain Lublin, Poland, and his Ph.D. degree in mathematics in 2000 from University of Washington, Seattle. He helda postdoctoral position at the Departments of Mathematics at the University of British Columbia and SimonFraser University in Vancouver, Canada, 2000 – 2002; a postdoctoral and research positions at the Electrical andComputer Engineering Department at the University of California, Santa Barbara, 2002 – 2004; and a part-timeteaching position at the Department of Mathematics at the University of Washington. In 2008, he joined theDepartment of Mathematics and Statistics at Loyola University Chicago. His background includes optimization,optimal control and control theory in general, and convex and nonsmooth analysis. His current research interestsinclude applications of set-valued analysis and of convex duality theory to control problems; generalized conceptsof convexity; and hybrid dynamical systems. Rafal Goebel is the recipient of the 2009 SIAM Activity Group onControl and Systems Theory Prize and a co-recipient of the 2010 IEEE Control Systems Society award for theControl Systems Magazine Outstanding Paper.

1 Introduction

Variational analysis is now a mature and broad discipline of mathematics, which, as the namesuggests, grew out of calculus of variations and subsumes convex, set-valued, and non-smoothanalysis. Convex analysis, the foundations of which can be found in the books Convex analysis byRockafellar [29], has seen motivation from and important applications in optimization, as seen inbooks Convex optimization by Boyd and Vandenberghe [8] or Conjugate duality and optimizationby Rockafellar [30], where convex duality is applied to broadly understood optimization prob-lems beyond finite-dimensional spaces, in finance, control, etc. Nonsmooth analysis, studyingnondifferentiable functions beyond the convex case, has seen motivation from and applicationsto optimal control, as evidenced by the books Optimization and nonsmooth analysis by Clarke[9], Optimal control by Vinter [36], and control theory in general, as seen in Nonsmooth analysisand control theory by Clarke, Ledyaev, Stern, and Wolenski [10]. Set-valued analysis, on its own,has seen treatment in the book Set-valued analysis by Aubin and Frankowsa [4], and applica-tions to dynamical systems described by differential inclusions, as seen in the books DifferentialInclusions by Aubin and Cellina [3] and Viability theory by Aubin [2].

Variational analysis as a whole has now seen extensive treatment in volumes such as Varia-tional analysis by Rockafellar and Wets [31], where finite-dimensional objects are studied, andVariational analysis and generalized differentiation, I and II by Mordukhovich [27], [28], whichdeals with the infinite-dimensional case. Graduate-level textbooks which can serve as a greatintroduction to the area include Convex analysis and nonlinear optimization by Borwein andLewis [7], Introduction to the theory of differential inclusions by Smirnov [33] and Optimal con-trol via nonsmooth analysis by Loewen [23]. New developments in variational analysis abound,motivated by problems where constraints, nonsmoothness, discontinuities, impulsive behavior,and uncertainty are present. Variational analysis can also provide novel insights into classicaltopics, as seen in the book Implicit functions and solution mappings - A view from variationalanalysis by Dontchev and Rockafellar [12].

This set of notes presents several basic topics in variational analysis, mostly from convex andset-valued analysis, and highlights their applications to dynamical and control systems. Whilethe presentation of the topics is necessarily brief, some topics are connected to recent research,by the author and by others.

2 Linear switching systems and convex Lyapunov func-tions

For a linear differential equation x = Ax, asymptotic stability can be equivalently characterizedby the existence of a quadratic Lyapunov function, i.e., a positive definite function which de-creases along every solution to the differential equation. Furthermore, this asymptotic stability isequivalent to that same property for the adjoint system y = AT y. Now consider two — or more— linear differential equations x = A1x, x = A2x, each of which displays asymptotic stability,and look at behaviors resulting from switching between these two differential equations, as oftenas one wants. What happens to asymptotic stability? If asymptotic stability persists, is there aLyapunov function verifying it? Is there a quadratic one? A convex one? What about the adjointswitching system, resulting from switching between y = AT

1 y and y=AT2 y? How are Lyapunov

functions for the switching system and its adjoint related? These questions are addressed in thissection.

2

2.1 Background: linear differential equations and quadratic Lyapunovfunctions

Let f : Rn → Rn be a continuous function with f(0) = 0. The equilibrium x = 0 (or theequilibrium solution x(t) ≡ 0, t ∈ [0,∞)) of the differential equation

x = f(x) (1)

is called stable, if for every ε > 0 there exists δ > 0 such that for every solution ϕ to (1),ϕ(0) ∈ δIB implies rgeϕ ∈ εIB; locally attractive, if there exists δ > 0 such that for every solutionϕ to (1) on [0,∞), ϕ(0) ∈ δIB implies ϕ(t) → 0 as t → ∞; globally attractive, if every solutionϕ to (1) on [0,∞) satisfies ϕ(t) → 0 as t → ∞; and locally, respectively, globally asymptoticallystable if it is both stable and locally, respectively, globally attractive.

Particularly good description of asymptotic stability is possible when the differential equation(1) is linear:

x = Ax. (2)

Fact 2.1 Let A be a n× n matrix. The following are equivalent:

(a) The origin is locally attractive for (2).

(b) The origin is globally asymptotically stable for (2).

(c) Eigenvalues of the matrix A have negative real parts.

(d) There exists a symmetric and positive definite n × n matrix P such that the quadraticfunction

V (x) =1

2x · Px (3)

has the following property: there exists γ > 0 such that, for every solution ϕ to (2),

d

dtV (ϕ(t)) ≤ −γV (ϕ(t)). (4)

(e) There exists a symmetric and positive definite n× n matrix P and γ > 0 such that

PA+ATP ≤ −γP, (5)

in the sense that PA+ATP + γP be negative definite.

(f) The origin is globally asymptotically stable for y = AT y.

Functions satisfying (4) were first introduced in Lyapunov’s dissertation [24] at the end of19th century and are called Lyapunov functions. Proving that the Lyapunov condition (4) impliesasymptotic stability is straightforward. In fact, with a, b > 0 such that a∥x∥2 ≤ V (x) ≤ b∥x∥2for every x ∈ Rn, (4) implies that for every solution ϕ to (2),

a∥ϕ(t)∥2 ≤ V (ϕ(t)) ≤ e−γtV (ϕ(0)) ≤ be−γt∥ϕ(0)∥2,

and so ∥ϕ(t)∥ ≤ Ke−γt∥ϕ(0)∥ for some K > 0, which is referred to as exponential stability. Theconverse implication, from (b) to (d), is far more involved, but can be illustrated in the 2 × 2case.

3

Example 2.2 Let A be a 2 × 2 matrix. Then A = MJM−1 for a nonsingular 2 × 2 matrix Mand a 2× matrix J (the real Jordan form of A) of the following form:(

λ1 00 λ2

),

(λ 00 λ

),

(λ 10 λ

),

(α −ββ α

), (6)

and these forms occur, respectively, when A has two distinct real eigenvalues λ1, λ2; when Ahas a single (repeated) real eigenvalue λ and two distinct eigenvectors; when A has a single(repeated) real eigenvalue λ and one eigenvector; and when A has complex eigenvalues α + iβ,α− iβ. Consider the change of coordinates

z =M−1x, x =Mz.

which turns x = Ax toz = Jz. (7)

Suppose that A is such that 0 is asymptotically stable for (2). Then J is such that 0 is asymp-totically stable for (7). This can be immediately seen by looking at the eigenvalues of A and J ,which are the same, but can also be argued directly from the definition of asymptotic stabilityand the relation z = M−1x. For the first two cases of J in (6), with λ1 ≤ λ2 < 0, and for thelast case, with α < 0, the function W (z) = 1

2∥z∥2 is a Lyapunov function: the reader should

check that for every solution ψ to (7) one obtains ddt∥ψ(t)∥

2 ≤ 2λ2∥ψ(t)∥2 for the first two cases,

while for the last case, ddt∥ψ(t)∥

2 ≤ α∥ψ(t)∥2. For the case of a repeated real eigenvalue λ < 0and one eigenvector, the norm squared fails to be a Lyapunov function, but the reader can verifythat W (z) = 1

2z21 + c

2z22 is a Lyapunov function for z = Jz as long as c > 1/4λ2. Overall, when

J is such that 0 is asymptotically stable for (7), there exists a symmetric and positive definite,in fact diagonal, 2× 2 matrix Q such that

W (z) =1

2z ·Qz

is a Lyapunov function for (7): one has QJ + JTQ ≤ −γQ for some γ > 0. Then, for thesymmetric and positive definite matrix P =M−TQM−1,

PA+ATP = M−TQM−1MJM−1 +M−TJTMTM−TQM−1 =M−T (QJ + JTQ)M−1

≤ M−T (−γQ)M−1 = −γP

and so the function V (x) = 12x · Px is a Lyapunov function for (2).

2.2 Switching systems

Let A1, A2, . . . , Am be n× n matrices. A solution to the switching system

x = Aσx (8)

consists of a piecewise continuous function σ : [0,∞) → 1, 2, . . . ,m and a piecewise differen-tiable function ϕ : [0,∞) → Rn such that

ϕ(t) = Aσ(t)x(t) for almost all t ∈ [0,∞).

Piecewise continuity/differentiability is understood here as finitely many discontinuities/pointsof nondifferentiability on each compact subinterval of [0,∞). A general reference for switchingsystems, which includes control engineering motivation, is the book [22].

4

Example 2.3 With a > 0, consider two 2× 2 matrices given by

A1 =

(0 −1a2 0

), A2 =

(0 a2

−1 0

).

Solutions to x = A1x rotate counterclockwise along the ellipses x21 +x22

a2 = const; solutions to

x = A2x rotate clockwise along the ellipsesx21

a2 + x22 = const. For both differential equations,0 is stable but not asymptotically stable. Note also that for both A1 and A2, eigenvalues areλ = ±ai. Appropriately picking the switching signal σ produces solutions to (8) that converge to0 and solutions that diverge. Furthermore, one can pick σ so that the solution to x ∈ A1x,A2xconverges to 0 but an arbitrarily slow rate. Considering small ε > 0 and matrices

Aε1 =

(0 −1a2 −ε

), Aε

2 =

(0 a2

−1 −ε

)with eigenvalues λ = −ε/2 ±

√a2 − ε2/4i which have negative real parts yields two linear dif-

ferential equations for which 0 is asymptotically stable. Still, picking the switching signal σappropriately produces solutions to (8) given by Aε

1, A2ε that that diverge, and σ can be suchthat the solution converges to 0 arbitrarily slow. Indeed, consider the matrices Aε

1, Aε2 with a = 1

and the function on R2 given by V (x) = 12∥x∥

2. Then, for every solution to (8),

d

dtV (ϕ(t)) = −εϕ2(t)2,

for almost all t, and hence the distance, to 0, of every solution is strictly decreasing. However,the decrease can be slow enough to prevent solutions from converging to 0.

Asymptotic stability of 0 for a linear differential inclusion is understood similarly to what isdone for differential equations, with the requirements placed by stability and attractivity holdingfor all solutions to (8). One conclusion of the example above is that a linear differential inclusionneeds not have 0 asymptotically stable even if for each i = 1, 2, . . . ,m, 0 is asymptotically stablefor x = Aix.

If there exists a Lyapunov function for (8), that is, a differentiable function V : Rn → [0,∞)such that a∥x∥2 ≤ V (x) ≤ b∥x∥2 for some a, b > 0 and all x ∈ Rn and such that, for someγ > 0 and for every solution ϕ to (8), ddtV (ϕ(t)) ≤ −γV (ϕ(t)) for almost all t, then the origin isglobally asymptotically stable for (8); in fact, it is exponentially stable, in the sense that thereexist M,γ > 0 such that every solution ϕ to (8) satisfies

∥ϕ(t)∥ ≤ Ke−γt∥ϕ(0)∥ (9)

The questions that will be answered next, with the help of convex analysis, are:

• If every solution ϕ to (8) satisfies (9), does there exist a Lyapunov function verifying this?

• If every solution ϕ to (8) satisfies (9), what can be said about asymptotic stability for thedual switching system y = AT

σ y?

Remark 2.4 A Lyapunov function for (8), equivalently, a differentiable function V with boundsas above and such that

∇V (x) ·Aix ≤ −γV (x)

for every i = 1, 2, . . . ,m, every x ∈ Rn is callaed a common Lyapunov function. The existenceof a common quadratic Lyapunov function for a family of matrices A1, A2, . . . , Am, when (8)is asymptotically stable, is an active area of research; see [32] for an overview. In general,asymptotic stability of (8) is not sufficient for the existence of a common quadratic Lyapunovfunction; see . This topic is not pursued further here.

5

2.3 Convex conjugate functions

A function f : Rn → R is convex if for every x, y ∈ Rn, every λ ∈ [0, 1],

f ((1− λ)x+ λy) ≤ (1− λ)f(x) + λf(y).

The standard reference for convex analysis is [29]; Fact 2.6 can be found in Chapter 10. In whatfollows, the class of convex, positive definite , and positively homogeneous of degree 2 functionsf : Rn → [0,∞) will play an important role; the class is denoted by C.

Example 2.5 Examples of functions f ∈ C include

(a) f(x) = 12x · Px for a symmetric and positive definite n× n matrix P .

(b) f(x) = g(Ax) for g ∈ C and any n × n matrix A. It is worth noting that, in general, acomposition of a convex function g with an affine mapping x 7→ Ax+b is a convex function;that is x 7→ g(Ax+ b) is a convex function.

(c) f(x) = supγ∈Γ gγ(x), where gγ | γ ∈ Γ is a family of functions gγ ∈ C, under the conditionthat f(x) < ∞ for every x ∈ Rn. This condition is automatically true if the family Γ isfinite. It is worth noting that, in general, a pointwise supremum of convex functions is aconvex function; even when infinite values are allowed.

(d) f(x) = con infγ∈Γ gγ(x), where gγ | γ ∈ Γ is a family of functions gγ ∈ C, con infγ∈Γ gγ(x)denotes the convex hull of infγ∈Γ gγ(x) which is the greatest convex function boundedabove by infγ∈Γ gγ(x), under the condition that f is positive definite. This condition isautomatically true if the family Γ is finite.

Fact 2.6 Every convex function f : Rn → R is continuous, in fact locally Lipschitz continuous.

An important operation on convex functions is that of convex conjugacy. For a functionf : Rn → R, its convex conjugate function is defined, at each y ∈ Rn, by

f∗(y) = supx∈Rn

y · x− f(x) . (10)

In general, f∗ can attain infinite values. In an appropriate sense, f∗ is always convex andlower semicontinuous, because it is a supremum of affine functions. One can also consider (f∗)

∗,

denoted, for simplicity, f∗∗.

Example 2.7 If f(x) = 12x · Px for a symmetric and positive definite n × n matrix P then

f∗(y) = 12y · P

−1y. Indeed,

f∗(y) = supx∈Rn

y · x− 1

2x · Px

and because x 7→ y · x − 1

2x · Px is a strictly concave function which diverges to −∞ when∥x∥ → ∞, its unique supremum is attained when the gradient is 0, so when y = Px. Pluggingin x = P−1y yields the formula for f∗.

For general lower semicontinuous convex functions, the convex conjugacy gives a one to onecorrespondence between a function and its conjugate. Here, only a special case is considered:

Theorem 2.8 Let f ∈ C. Then f∗ ∈ C and f∗∗ = f .

6

Proof. Suppose that f ∈ C. Let µ = 2minf(x) | ∥x∥ = 1. Because f is continuous and positivedefinite, µ > 0. Because f is positively homogeneous of degree 2, for x = 0,

f(x) = ∥x∥2f(

x

∥x∥

)≥ µ

2∥x∥2.

Then f∗(y) is bounded above by the conjugate of the quadratic function x 7→ µ2 ∥x∥

2, which,thanks to Example (2.7), is y 7→ 1

2µ∥y∥2. Hence f∗ is finite-valued and f∗(0) ≤ 0. A symmetric

argument shows that f∗ is bounded below by a multiple of the norm squared. Thus, f∗ is positivedefinite. Now consider λ > 0 and note that, for every y ∈ Rn,

f∗(λy) = supx∈Rn

(λy) · x− f(x) = λ2 supx∈Rn

y · x

λ− 1

λ2f(x)

= λ2 sup

x∈Rn

y · x

λ− f

(xλ

)= λ2 sup

z∈Rn

y · z − f(z) = λ2f∗(y).

Consequently, f∗ ∈ C. To prove that f∗∗ = f , note that by the definition of f∗, for everyx, y ∈ Rn, f∗(y) ≥ y · x − f(x), hence f(x) ≥ x · y − f∗(y). Then for every x ∈ Rn, f(x) ≥supy∈Rn x · y − f∗(y), and so f(x) ≥ f∗∗(x). Suppose then that for some x ∈ Rn, f(x) >

(f∗)∗(x). Then there exists y ∈ Rn and β ∈ R such that

f(x) > x · y + β ∀x ∈ Rn and x · y + β > f∗∗(x).

The first inequality implies that

f∗(y) = supx∈Rn

y · x− f(x) ≤ supx∈Rn

y · x− (y · x+ β) = −β

andf∗∗(x) = sup

y∈Rn

x · y − f∗(y) ≥ x · y − f∗(y) ≥ x · y + β,

which contradicts the second claimed inequality.

2.4 Switching systems, Lyapunov functions, and duality

Consider the discrete-time version of a switching linear system first. For a recent work withextensive bibliography on the topic see [26]. Given n× n matrices A1, A2, . . . , Am, a solution tothe switching system

x+ = Aσx (11)

consists of a function σ : N0 → 1, 2, . . . ,m and a function ϕ : N0 → Rn such that

ϕ(j + 1) = Aσ(j)x(j) for every j ∈ N0.

If there exist M,γ > 0 such that every solution ϕ to (11) satisfies ∥ϕ(j)∥ ≤ Ke−γj∥ϕ(0)∥, then 0is exponentially stable for (11). The result below goes back to [25]. The continuous-time versionincludes smoothness of V , appeared in [11], and will be further mentioned in Section 2.5.

Proposition 2.9 The origin is exponentially stable for (11) if and only if there exists V ∈ Csuch that, for some δ ∈ (0, 1),

V (Aix) ≤ δV (x) for every i = 1, 2, . . . ,m, x ∈ Rn.

7

Proof. Suppose that 0 is exponentially stable for (11). Consider V : Rn → [0,∞) defined by

V (x) = sup∥ϕ(j)∥2e2γj |ϕ is a solution to (11), ϕ(0) = x, j ∈ N0

. (12)

For each fixed j ∈ N0 and each fixed switching signal σ on 0, 1, . . . , j−1, the function x 7→ ϕ(j)is linear, and so x 7→ ∥ϕ(j)∥2 is convex. Then x 7→ ∥ϕ(j)∥2e2γj is convex, and V is the supremumof such functions over all j ∈ N0 and all switching signals. Hence V is convex. It is positivedefinite because V (x) ≥ ∥x∥2, which is seen by considering j = 0, and because the unique solutionfrom 0 is ϕ ≡ 0. Now, pick i ∈ 1, 2, . . .m and note that

V (Aix) = sup∥ϕ(j)∥2e2γj |ϕ is a solution to (11), ϕ(0) = Aix, j ∈ N0

= sup

∥ψ(j)∥2e2γ(j−1) |ψ is a solution to (11), ψ(0) = x, ψ(1) = Aix, j ∈ N

≤ sup

∥ψ(j)∥2e2γ(j−1) |ψ is a solution to (11), ψ(0) = x, j ∈ N0

= e−2γV (x)

The result follows, with δ = e−2γ . The reverse implication follows from the fact that if V iscontinuous (as it is when it is convex and finite-valued), positive definite, and positively homo-geneous of degree 2, then for some a, b > 0 and all x ∈ Rn, a∥x∥2 ≤ V (x) ≤ b∥x∥2.

The next observation appeared in [13] and will lead to equivalent characterization of expo-nential stability for (11) through the dual system

x+ = ATσx. (13)

in Theorem 2.11, obtained first in [5], [6] without the use of convex Lyapunov functions.

Lemma 2.10 Let V ∈ C, A be a n× n matrix, and δ > 0. The following are equivalent:

(a) V (Ax) ≤ δV (x) for every x ∈ Rn;

(b) V ∗(AT y) ≤ δV ∗(y) for every y ∈ Rn;

Proof. Suppose that (a) holds. Then

V ∗(AT y) = supx∈Rn

(AT y

)· x− V (x)

≤ sup

x∈Rn

y ·Ax− 1

δV (Ax)

= sup

z∈rgeA

y · z − 1

δV (z)

≤ sup

x∈Rn

y · x− V (x) =1

δsupx∈Rn

(δy) · x− 1

δV (x)

=

1

δV ∗(δy)

= δV ∗(y).

Hence (a) implies (b). The implication from (b) to (a) follows from considering AT , V ∗, and(V ∗)

∗= V in place of A, V , and V ∗ in the implication from (a) to (b).

Theorem 2.11 Consider n×n matrices A1, A2, . . . , Am. Then 0 is exponentially stable for (11)if and only if 0 is exponentially stable for (13).

Proof. Combine Proposition 2.9 and Lemma 2.10.

8

2.5 More convex analysis, continuous-time switching systems, and du-ality

The concept of a Lyapunov function for a discrete-time switching system, as described in Propo-sition 2.9, does not require the Lyapunov function to be differentiable. For continuous-timeLyapunov functions, their decrease is described using gradients. Hence, to provide continuous-time results similar to those in Lemma 2.10 or Theorem 2.11, one needs some differentiabilityproperties of convex functions and their conjugates. The fact below follows from [29, Theorem26.3].

Fact 2.12 A convex function f ∈ C is differentiable if and only if its conjugate function f∗ ∈ Cis strictly convex.

Let CS be the class of function f ∈ C that are differentiable and strictly convex. The factabove and Theorem 2.8 imply:

Corollary 2.13 f ∈ CS if and only if f∗ ∈ CS.

The lemma below is a special case of [29, Theorem 23.5] which is applicable to nondifferentiableconvex functions.

Lemma 2.14 Assume that f ∈ CS, equivalently, that f∗ ∈ CS. The following are equivalent:

(a) y = ∇f(x);

(b) x = ∇f∗(y);

(c) f(x) + f∗(y) = x · y.

Proof. By definition, for every x, y ∈ Rn, f∗(y) ≥ x · y − f(x). If y = ∇f(x) then x maximizesx · y− f(x) and (c) holds. Thus (a) implies (c). On the other hand, (c) means that x maximizesx · y − f(x), so (a) must hold. The equivalence of (b) and (c) follows by symmetry.

Lemma 2.14 implies the equivalence between V ∈ CS decreasing along solutions to (8) andV ∗ ∈ CS decreasing along solutions to

x = ATσx. (14)

Indeed, for any matrix A, suppose that ∇V (x) · Ax < 0 for every x = 0. Pick any y = 0, letx = ∇V ∗(x) which is equivalent to y = ∇V (x). Then ∇V ∗(y) · AT y = x · AT y = y · Ax =∇V (x) · Ax < 0. Hence ∇V (x) · Ax < 0 for every x = 0 implies ∇V ∗(y) · AT y for every y = 0,and a reverse implication follows by duality. A more useful equivalence, taken from [17], is infact true:

Lemma 2.15 Let V ∈ CS, A be a n× n matrix, and δ > 0. The following are equivalent:

(a) ∇V (x) ·Ax ≤ −δV (x) for every x ∈ Rn;

(b) ∇V ∗(y) ·AT y ≤ −δV ∗(y) for every y ∈ Rn.

Proof. Assume (a) and note that ∇V (x) · Ax ≤ −δV (x) for every x ∈ Rn is equivalent to∇V (x) · Ax ≤ −δ for every x with V (x) = 1. This comes from homogeneity of V , ∇V , and the

fact that for x = 0, V(x/√V (x)

)= 1. For every y ∈ Rn there is x ∈ Rn such that y = ∇V (x);

indeed, one takes x = ∇V ∗(y). If, furthermore, y is such that V ∗(y) = 1 then V (x) = 1. Indeed,

V (x) = x · y − V ∗(y) = maxz∈Rn

x · z − V ∗(z) = maxλ∈R

x · λy − V ∗(λy)

9

= maxλ∈R

λ (V (x) + V ∗(y))− λ2V ∗(y)

= max

λ∈R

λ (V (x) + 1)− λ2

and setting the derivative of the last expression above, in λ, equal to 0 when λ = 1, givesV (x) = 1. Hence, ∇V (x) · Ax ≤ −δ for every x with V (x) = 1 implies ∇V ∗(y) · AT y ≤ −δ forevery y with V ∗(y) = 1, and this implies (b). The opposite implication holds by symmetry.

With this in hand, and with the result on existence of smooth Lyapunov functions [11], onecan now show the equivalence of exponential stability for (8) and (14).

3 Difference and differential inclusions

Besides the pursuit of generalization, there are other natural reasons to consider differential anddifference inclusions rather than equations. Section 3.1 below illustrates this. Basic properties ofset-valued mappings are needed to study differential and difference equations, they are collectedin Section 3.2. Semicontinuity and boundedness properties of the set-valued mapping defining adifference inclusion have implications on uniformity of asymptotic stability, its robustness, etc.,this is presented in Section 3.3. In particular, some descriptions of asymptotic stability for aswitching system and its dual are improved there. A simple example illustrates in Section 3.4how control engineering problems. naturally lead to differential inclusions.

3.1 Some motivation

In Proposition 2.9, it was shown that exponential stability of 0 for a discrete time switchingsystem (11) is equivalent to the existence of a Lyapunov function V ∈ C, which satisfies

V (Aix) ≤ δV (x) for every i = 1, 2, . . . ,m, x ∈ Rn.

This condition in fact guarantees that V is also a Lyapunov function for the linear differenceinclusion

x+ ∈ A(x), (15)

where A(x) is the convex hull of A1x,A2x, . . . , Amx, that is, the smallest convex set containingA1x,A2x, . . . , Amx A solution to (15) is a function ϕ : N0 → Rn such that, for every j ∈ N0,ϕ(j+1) ∈ A(ϕ(j)). Take any w ∈ A(x), which can be equivalently characterized as the set of allconvex combinations

A(x) =

m∑i=1

λiAix |m∑i=1

λi = 1, λi ≥ 0 for i = 1, 2, . . . ,m

.

Then, thanks to convexity of V ,

V (w) = V

(m∑i=1

λiAix

)≤

m∑i=1

λiV (Aix) ≤m∑i=1

λiδV (x) = δV (x).

Hence, exponential stability of 0 for the switching system (11) is equivalent to the existence of aLyapunov function for the linear difference inclusion (15), which in turn implies asymptotic of 0for (15).

The same idea applied to continuous-time systems. If a differentiable V ∈ C is a Lyapunovfunction for (8), that is, if

∇V (x) ·Aix ≤ −γV (x) for every i = 1, 2, . . . ,m, x ∈ Rn,

10

then V is a Lyapunov function for the linear differential inclusion

x+ ∈ A(x), (16)

that is, ∇V · v ≤ −γV (x) for every v ∈ A(x), x ∈ Rn. It was illustrated before that justasymptotic stability is not sufficient for the existence of a Lyapunov function for a the switchingsystem (8). Through the analysis of the set-valued mapping A, it will be shown below that forthe linear differential inclusion (16), asymptotic stability is equivalent to exponential stability,which in turn implies the existence of a convex Lyapunov function.

Another motivation to study difference inclusions and the set-valued mappings defining themcomes from the effect of perturbations on difference equations given by discontinuous functions.For x ∈ R let ⌊x⌋ be the greatest integer less than x; so ⌊3⌋ = 2, ⌊π⌋ = 3, etc. Define f : R → Rto be an odd function such that, f(0) = 0, and for x > 0, f(x) = ⌊x⌋. Then the differenceequation

x+ = f(x)

has 0 globally asymptotically stable. In fact, convergence to 0 occurs in finitely many steps fromany initial condition. For example, the solution ϕ from initial condition π is ϕ(0) = π, ϕ(1) = 3,ϕ(2) = 2, ϕ(3) = 1, ϕ(j) = 0 for all j ≥ 4. Consider arbitrarily small ε > 0 and the differenceequation

x+ = f(x+ ε).

Local asymptotic stability of 0 is preserved, as f(x+ε) = 0 for every x ∈ [−1−ε, 1−ε]. However,convergence to 0 from large initial conditions fails: for example, f(1+ε) = 1, and so the solutionfrom 1 is ϕ ≡ 1. Such dramatic failure of convergence to 0 under small perturbations is dueto discontinuity of f . This behavior can be predicted, without considering perturbations, bylooking at a set-valued “closure” of f , which is the set-valued mapping F given by F (0) = 0,F (x) = f(x) for x ∈ Z, F (x) = x − 1, x for x ∈ N. Then, x+ ∈ F (x) has constant solutionsfrom every integer initial point.

More precisely, one has the following result. Its continuous-time version was predicted by[19], proved by [18], and appears here as Fact 3.13. The discrete-time version is not hard.

Proposition 3.1 Let f : Rn → Rn be a function, I = 0, 1, 2, . . . , J, and ϕi : I → Rn andϕ : I → Rn be functions. The following are equivalent

(a) There exist sequences of functions ϕi, ei, pi : I → Rn such that, for every j ∈ I, ϕi(j) →ϕ(j), ei(j) → 0, pi(j) → 0 as i→ ∞ and, for every j ∈ 0, 1, . . . , J − 1,

ϕi(j + 1) = f(ϕi(j) + ei(j)) + pi(j). (17)

• For every j ∈ 0, 1, . . . , J − 1,

ϕ(j + 1) ∈ F (ϕ(j)),

where, for every x ∈ Rn,

F (x) =∩δ>0

f(x+ δIB). (18)

Proof. Assume (a) and pick j ∈ 0, 1, . . . , J − 1. Then, for every δ > 0 and all large enough i,

ϕi(j + 1) ∈ f(ϕi(j) + δIB) + δIB ⊂ f(ϕ(j) + 2δIB).

Passing to the limit yields ϕ(j + 1) ∈ F (ϕ(j)). Now assume (b) and pick j ∈ 0, 1, . . . , J − 1.Considering δ = 1/i in (18) shows that ϕ(j + 1) ∈ F (ϕ(j)) implies that for every i ∈ N there areei(j), pi(j) ∈ i−1IB such that (17) holds.

Through the analysis of set-valued mappings F and difference inclusions x+ ∈ F (x), robust-ness of asymptotic stability to perturbations can be studied.

11

3.2 Set-valued mappings

The definitions below follow [31]. A set-valued mapping from Rm to Rn assigns, to every x ∈ Rm,a subset of Rn. To distinguish such a mapping F from a function, the notation F : R ⇒ Rn willbe used. The effective domain of F : Rm ⇒ Rn, denoted domF , is the set x ∈ Rm |F (x) = ∅.The mapping F : Rm ⇒ Rn is called

• locally bounded if every point x ∈ Rm has a neighborhood U such that F (U) is a boundedset, where F (U) =

∪u∈U F (u).

• outer semicontinuous if for every convergent sequence xi of points in Rm and everyconvergent sequence yi of points yi ∈ F (xi), one has limn→∞ yi ∈ F (limn→∞ xi).

• inner semicontinuous if for every convergent sequence xi of points in Rm and every y ∈F (limn→∞ xi) there exists a sequence yi of points yi ∈ F (xi) such that limn→∞ yi = y.

• continuous if it is both outer and inner semicontinuous.

The graph of a set-valued mapping F : Rn ⇒ Rn is the set

gphF =(x, y) ∈ R2n | y ∈ F (x)

.

The graph provides a convenient characterization of outer semicontinuity, given below. The proofis straightforward.

Proposition 3.2 A set-valued mapping F : Rn ⇒ Rn is outer semicontinuous if and only ifgphF is closed.

Example 3.3 Examples of set-valued mappings and their continuity properties include:

(a) Reachable sets: Let f : Rn → Rn be continuous and K ⊂ Rn be compact. Define R : R ⇒Rn for τ ≥ 0 by

R(τ) = ϕ(τ) |ϕ : [0, τ ] → Rn solves x = f(x), ϕ(0) ∈ K ,

and extend the definition to τ < 0 by considering solutions on [τ, 0] and their initial pointsϕ(τ). Then R is inner semicontinuous.

(b) Optimal solution mappings: Let f : Rm+n be a function, and define M : Rm ⇒ Rn by

M(x) =

y ∈ Rn | f(x, y) = inf

y∈Rnf(x, y)

.

(c) Inflated functions: Let f : Rm → Rn be a function and pick δ > 0. Define F : Rn ⇒ Rn by

F (x) = f (x+ δIB) + δIB.

If f is locally bounded then so is F . Similarly, if f is continuous then so is F . Note alsothat y ∈ F (x) iff there exist x′, y′ such that y ∈ f(x), x′ ∈ x+ δIB, y ∈ y′ + δIB. Hence,

gphF = gph f + (δIB × δIB) .

This, Proposition 3.2, and the fact that a continuous function has a closed graph can beused to verify that a continuous f leads to an upper semicontinuous F .

12

(d) For any function f : Rn → Rn the set-valued mapping F defined in (18) is outer semi-continuous. Indeed, pick xi → x, yi ∈ F (xi) such that yi → y. Because yi ∈ F (xi),considering δ = 1/i in (18) implies that there exists ui ∈ x + i−1IB and wi ∈ f(ui) suchthat ∥yi − wi∥ < 1/i. Then ui → x and wi → y. Because ui → x, for every δ > 0 andevery large enough i, ui ∈ x + δIB and hence wi ∈ f(x + δIB). Because wi → y, it mustbe that y ∈ f(x+ δIB). Since this holds for every δ > 0, y ∈ F (x), which verifies outersemicontinuity. Furthermore, if f is locally bounded then so is F .

(e) Tangent cones: Let C ⊂ Rn. The tangent cone TC(x) to the set C at x is defined as

TC(x) =

v ∈ Rn | ∃xi ∈ C, xi → x, and λi 0 so that

xi − x

λi→ v

. (19)

Tangent cones play a significant role in Section 4.4.

Proposition 3.4 Consider set-valued mappings F : Rm ⇒ Rn and G : Rk ⇒ Rm and defineM : Rk → Rn by M(x) = F (G(x)) =

∪y∈G(x) F (y).

(a) If F and G are locally bounded then M is locally bounded.

(b) If F and G are outer semicontinuous and G is locally bounded then M is outer semicon-tinuous.

Proof. Local boundedness of F can equivalently be characterized as: for every boundedK ⊂ Rm,F (K) is bounded. Indeed, the closure of a bounded K is compact and relying on the finite opencovering property makes this characterization equivalent to the definition. Using this and asimilar characterization for G proves (a). For (b), pick xi → x and zi → z with zi ∈ F (G(xi)).There exist yi ∈ G(xi) such that zi ∈ F (yi). BecauseG is locally bounded, some subsequence of yiconverges to a point y, and because of outer semicontinuity of G, y ∈ G(x). Outer semicontinuityof F and zik ∈ F (yik) then gives z ∈ F (y), and hence z ∈ F (G(x)).

To see how local boundedness is necessary in (b), consider, for example, a function f : R → Rgiven by f(x) = 1/x for x = 0 and f(0) = 7. This function happens to be an outer semicontinuousmapping which is not locally bounded. Then, consider f composed with itself.

Proposition 3.5 A locally bounded set-valued mapping F : Rn ⇒ Rn is outer semicontinuous ifand only if it has closed values and for every x ∈ R and ε > 0 there exists δ > 0 such that, forevery x′ ∈ x+ δIB,

F (x′) ⊂ F (x) + εIB.

3.3 Difference inclusions and asymptotic stability

A solution to a difference inclusionx+ ∈ F (x), (20)

where F : Rn ⇒ Rn is a set-valued mapping with domF = Rn, is a function ϕ : N0 → Rn suchthat ϕ(j + 1) ∈ F (ϕ(j)) for every j ∈ N0. A sequence of solutions ϕi is locally bounded if forevery J ∈ N there exists M > 0 such that, for every i ∈ N, j ≤ J , one has ϕi(j) ∈MIB.

Theorem 3.6 Suppose that the set-valued mapping F : Rn ⇒ Rn is outer semicontinuous. Then,from every locally bounded sequence ϕi of solutions to (20) one can extract a subsequence thatconverges pointwise to a solution to (20).

13

Proof. Local boundedness ensures that ϕi(0) is a bounded sequence and a convergent subse-quence can be extracted. From that subsequence, one can further extract a subsequence such thatϕi(1) converge, etc. Cantor’s diagonalization argument produces a subsequence of ϕi, whichwe do not relabel, such that ϕi(j) converges for every j ∈ N0 to some point ϕ(j). Now, for eachj ∈ N0, (ϕi(j), ϕi(j + 1)) ∈ gphF , and because the graph of F is closed, (ϕ(j), ϕ(j + 1)) ∈ gphFand hence the mapping ϕ : N0 → Rn is a solution to (20).

Corollary 3.7 Suppose that the set-valued mapping F : Rn ⇒ Rn is outer semicontinuous andlocally bounded. Then, from every sequence ϕi of solutions to (20) such that the sequence ofinitial points ϕi(0) is bounded one can extract a subsequence that converges pointwise to asolution to (20).

Proof. Let K be a compact set such that ϕi(0) ∈ K for every i ∈ N. Then, for every i ∈ N andj ∈ N, ϕi(j) ∈ F j(K). By 3.4, F j is locally bounded, and so F j(K) is a bounded set. Notingthat i ∈ N, j ≤ J , one has ϕi(j) ∈ K ∪ F (K) ∪ · · · ∪ F J(K) concludes the proof.

Lemma 3.8 Suppose that the set-valued mapping F : Rn ⇒ Rn is outer semicontinuous andlocally bounded and let K ⊂ Rn be compact.

(a) Let O ⊂ Rn be open. If, for every solution ϕ to (20) with ϕ(0) ∈ k there exists j ∈ N0 suchthat ϕ(j) ∈ O then there exists J such that, for every solution ϕ to (20) with ϕ(0) ∈ k thereexists j ≤ J such that ϕ(j) ∈ O.

(b) For every J ∈ N0, the reachable set

R≤J (K) = ϕ(j) |ϕ is a solution to (20), ϕ(0) ∈ K, j ∈ 0, 1, 2, . . . , J

is compact.

Proof. Suppose that (a) fails. Then there exists a sequence of solutions ϕi with ϕi(0) ∈ Ksuch that ∥ϕi(j)∥ ∈ O for every i ∈ N, every j ≤ J Because K is compact and the complementof O is closed, Corollary 3.7 yields a solution ϕ which satisfies ∥ϕ(j)∥ ∈ O for every j ∈ N0. Thisis a contradiction. For (b), it is enough to note that

R≤J (K) = K ∪ F (K) ∪ F 2(K) ∪ · · · ∪ F J (K),

that for every j ∈ N0 Fj is outer semicontinuous and locally bounded thanks to Proposition 3.4,

and hence F j(K) is compact for every j ∈ N0.

Proposition 3.9 Suppose that the set-valued mapping F : Rn ⇒ Rn is outer semicontinuousand locally bounded. If 0 is globally asymptotically stable for (20) then

(a) 0 is uniformly (Lyapunov) stable, that is, there exists a continuous and nondecreasingfunction α : [0,∞) → [0,∞) such that α(0) = 0 and, for every ϕ ∈ S and every j ∈ N0,∥ϕ(j)∥ ≤ α (∥ϕ(0)∥).

(b) 0 is uniformly attractive for (20), that is, for every ∆, δ > 0 there exists J > 0 such that,for every ϕ ∈ S(∆IB) and every j > J one has ϕ(j) ∈ δIB.

Proof. For (a), consider a function defined at each r ≥ 0 by

γ(r) = sup ∥ϕ(j)∥ |ϕ ∈ S(rIB), j ∈ N0 .

Clearly, γ(0) = 0; by Lyapunov stability of 0 for (20), γ is continuous at 0; and, by construction,γ is nondecreasing on [0,∞). If γ has finite values for every r, then there exists a continuous α

14

with α(r) ≥ γ(r) which has all of the needed properties. Suppose then, to the contrary, that forsome r > 0 there exists a sequence of solutions ϕi ∈ S(rIB) to (20) and a sequence of ji ∈ N0 suchthat ϕi(ji) → ∞. Local boundedness of F ensures that ϕi is locally bounded. By Theorem 3.6,there exists a subsequence ϕik of ϕi which converges pointwise to a solution ϕ to (20). Thissolution converges to 0 because of the assumed attractivity. Invoke Lyapunov stability of 0 toobtain δ > 0 that corresponds to ε = 1. Then, for some J ∈ N0, ϕ(J) ∈ 0.5δIB, and so, for everylarge enough k, ϕik(J) ∈ δIB. Then, by the choice of δ, ϕik(j) ∈ IB for every j > J . Hence, itmust be that jik ≤ J . But this is impossible because of local boundedness of the sequence ϕi.This proves that γ(r) <∞ for every r ≥ 0. The proof of (b) is similar and left to the reader.

Building on the arguments in the proof of Propostion 3.9 one can show that there exists afunction β : [0,∞)2 → [0,∞) which is 0 at 0 and strictly increasing in the first argument anddecreasing to 0 as the second argument goes to infinity, and such that, for every solution ϕ, onehas ∥ϕ(j)∥ ≤ β (∥ϕ(0)∥, j). In fact, the following can be said, thanks to [34].

Fact 3.10 If 0 is uniformly Lyapunov stable and globally uniformly attractive for (20), then thereexist two functions α1, α2 : [0,∞) → [0,∞) which are continuous, strictly increasing, αi(0) = 0,and limr→∞ αi(r) = ∞, and such that, for every ϕ ∈ S,

α1 (∥ϕ(j)∥) ≤ α2 (∥ϕ(0)∥) e−j . (21)

With the bound (21) one can construct a Lyapunov function for (20), following the idea ofthe proof of Proposition 2.9. For each x ∈ Rn, set

V (x) = supα1 (∥ϕ(j)∥) ej |ϕ ∈ S(x), j ∈ N0

.

Then α1(x) ≤ V (x) ≤ α2(x), so the function V is positive definite, finite-valued, and continuousat 0. Arguments as in the proof of Proposition 2.9 show that V (y) ≤ e−1V (x) for every x, everyy ∈ F (x). Hence this V is a Lyapunov function for (20), but its regularity is questionable. Itcan be argued that V is upper semicontinuous, but an interesting question remains: can onebuild from this V a continuous Lyapunov function for (20), or obtain such a function using othermethods?

Proposition 3.11 Suppose that 0 is locally attractive for the linear difference inclusion (15).Then 0 is globally asymptotically stable, in fact exponentially stable, for (15).

Proof. First, note that the set-valued mapping A defining (15) is outer semicontinuous andlocally bounded, and hence results like Theorem 3.6 and Corollary 3.7 apply. Second, note thatlocal attractivity implies global attractivity for (15), thanks to homogeneity of A. Indeed, letδ > 0 come from the definition of local attractivity of 0 for (15). Because for every solution ϕ to(15) and every λ ∈ R the function λϕ is a solution to (15), and because for every ϕ(0) there existsλ = 0 such that λϕ(0) ∈ δIB, every solution ϕ converges to 0 and hence 0 is globally attractive.

Now, for k ∈ N0 let C0 = x ∈ Rn | 2−k−1 ≤ ∥x∥ ≤ 2−k. By Lemma 3.8 (a), there existsJ such that for every solution ϕ to (15) with ϕ(0) ∈ C0 there exists j ≤ J with ∥ϕ(j)∥ < 1/2.By Lemma 3.8 (b), there exists ρ > 0 such that R≤J(C0) ⊂ ρIB. Homogeneity implies that forevery k ∈ N, for every solution ϕ to (15) with ϕ(0) ∈ Ck there exists j ≤ J with ∥ϕ(j)∥ < 2−k−1

and R≤J (Ck) ⊂ 2−k−1ρIB.Let ϕ be any solution to (15) with ϕ(0) ∈ IB. Pick k0 so that ϕ(0) ∈ Ck0 . There exist

0 < j1 ≤ J , k1 > k0 such that ϕ(j1) ∈ Ck1 , and for j = 0, 1, . . . , j1, ϕ(j) ∈ 2−k0−1ρIB. Similarly,there exist j1 < j2 < j1 + J , k2 > k1 such that ϕ(j2) ∈ Ck2 , and for j = j1, j1 + 2, . . . , j2,ϕ(j) ∈ 2−k1−1ρIB. Repeating this argument shows that ϕ(j) ∈ 2−k0−1ρIB for every j ∈ N0, andhence, any solution to (15) with ϕ(0) ∈ IB satisfies ϕ(j) ∈ ρIB for every j ∈ N0. Homogeneitythen suggests that given ε > 0 one can take δ = ε/ρ to satisfy the conditions for Lyapunovstability. This verifies global asymptotic stability of 0 for (15)

15

The arguments above in fact imply that every solution ϕ to (15) with ϕ(0) ∈ IB satisfiesϕ(j) ∈ ρIB for every j ∈ 0, 1, . . . , J; ϕ(j) ∈ 2−1ρIB for every j ∈ J, J + 1, . . . , 2J; etc. Thenthe exponential bound (9) holds for every such solution, with K = 2ρ and γ = (ln 2)/J , and byhomogeneity, this bound holds for every solution to (15). Hence, 0 is exponentially stable for(15)

Recall that the right-hand side of (15) is given in terms of A, where A(x) is the convex hull ofA1x,A2x, . . . , Amx. DefineAT by settingAT (x) to be the convex hull of AT

1 x,AT2 x, . . . , A

Tmx

and consider the dual linear difference inclusion

x+ ∈ AT (x). (22)

The developments of Proposition 2.9, Lemma 2.10, Proposition 3.11 are summarized in thefollowing theorem.

Theorem 3.12 Consider n× n matrices A1, A2, . . . , Am. The following are equivalent:

• 0 is locally attractive for (15).

• 0 is exponentially stable for (15).

• There exists a convex Lyapunov function V ∈ C for (15), in the sense that, for someδ ∈ (0, 1),

V (x+) ≤ δV (x) for every x+ ∈ A(x), x ∈ Rn.

• There exists a convex Lyapunov function V ∈ C for (22).

• 0 is exponentially stable for (22).

• 0 is locally attractive for (22).

3.4 From discontinuous differential equations to differential inclusions

The following lovely example comes from [1]. In R2, consider the control system

x1 = (x21 − x22)u, x2 = 2x1x2u, (23)

The variable x = (x1, x2) is the state of the system and u is the control variable or input. Thiscontrol system has the following properties:

• For any choice of a function u : [0,∞) → R regular enough to not cause issues with existenceof solutions to the non-autonomous differential equation

x = (x21 − x22)u(t), y = 2x1x2u(t),

the solution x(t) is such that, for some r ∈ R, x21(t) + (x2(t)− r)2= r2 for every t ≥ 0,

except the case when x2(0) = 0.

• For every initial condition x(0) there exists u : [0,∞) → R such that the resulting solutionconverges to 0 as t → ∞. In fact, one can accomplish this with using constant controlvalues u ≡ 1 or u ≡ −1. Setting u ≡ 1 yields a differential equation

x1 = x21 − x22, x2 = 2x1x2.

Solutions with x2(0) > 0 move counterclockwise along circles x21 + (x2 − r)2= r2 and con-

verge to 0; solutions with x2(0) < 0 move clockwise along circles x21 + (x2 − r)2= r2 and

converge to 0; and solutions with x2(0) = 0, x1(0) < 0 clearly converge to 0 as well. Forx2(0) = 0, x1(0) > 0, one picks u ≡ −1 and obtains convergence to 0. This behavior doesnot lead to asymptotic stability of 0; Lyapunov stability is violated.

16

• There exists no continuous function u : R2 → R such that the autonomous differentialequation

x1 = (x21 − x22)u(x), x2 = 2x1x2u(x), (24)

has 0 asymptotically stable. To see this, note first that if u(x) = 0 for some x = 0, that x isan equilibrium and hence attractivity of 0 fails. Furthermore, Lyapunov stability dictatesthat, given r > 0 and x such that x21 + (x2 − r)

2= r2, one must have u(x) < 0 for every

small enough x1 > 0 and u(x) > 0 for every small enough x1 < 0. Hence u(x) = 0 for some

x on x21 + (x2 − r)2= r2.

• There exists a discontinuous function u : R2 → R such that (24) has 0 asymptoticallystable. For example, one can consider

u(x) =

−1 if x1 ≥ 01 if x1 < 0

. (25)

Similarly to what was shown for discrete-time dynamics in Proposition 3.1, the effect ofperturbations on a discontinuous differential equation is captured by passing to a differentialinclusion. The set-valued mapping (27) is the Krasovskii regularization of f . The result belowwas envisioned by [19] and proved by [18]. Alternative approaches are in [9, Theorem 3.1.6] and[15, Section 4.5].

Fact 3.13 Let f : Rn → Rn be a locally bounded function, T > 0, and ϕ : [0, T ] → Rn beabsolutely continuous. The following are equivalent

(a) There exist sequences of function ϕi, ei, pi : [0, T ] → Rn with ϕi absolutely continuous andei, pi measurable such that ϕi → ϕ, ei(j) → 0, pi(j) → 0 uniformly on [0, T ] as i→ ∞ and,for almost every t ∈ [0, T ],

ϕi(t) = f(ϕi(t) + ei(t)) + pi(t). (26)

• For almost every t ∈ [0, T ],ϕ(t) ∈ F (ϕ(t)),

where, for every x ∈ Rn,

F (x) =∩δ>0

con f(x+ δIB). (27)

Consequently, the behavior of the control system (24) with the discontinuous feedback (25)and under small perturbations is reflected in the behavior resulting from

U(x) =

−1 if x1 > 0[−1, 1] if x1 = 0

1 if x1 < 0. (28)

More precisely, the behavior of solutions to x = f(x, u(x)) with f : R3 → R2 defined in (23) andwith u as in (25) under perturbations is reflected in the behavior of solutions to x ∈ f(x,U(x)),where U comes from (28). In particular, each point with x1 = 0 is an equilibrium, in the sensethat there exists a constant solution from that point.

This furthermore implies that, for the differential inclusion resulting from (24), (25), thereexists no smooth Lyapunov function. Indeed, it can be argued, using the continuity of V and ∇V ,that if ∇V (x) · f(x, u(x)) ≤ −γV (x) for every x ∈ Rn then also ∇V (x) · v ≤ −γ/2V (x) for everyx ∈ Rn, and every v ∈ f(x,U(x)). That is, if V is a Lyapunov function for x = f(x, u(x)) then

17

it is also a Lyapunov function for x ∈ f(x,U(x)). The latter is impossible because of equilibriaother than 0.

To conclude this section, some general comments about differential inclusions are given. First,it is noted that if f : Rn → Rn is locally bounded then the set-valued mapping F : R ⇒ Rn

defined in (27) is locally bounded and outer semicontinuous. For differential inclusions givenin terms of such mappings, asymptotic stability theory can be developed similarly to what wasoutlined in Section 3.3 for difference inclusions.

Below, a solution on [0, T ] to the differential inclusion

x ∈ F (x) (29)

is an absolutely continuous function ϕ : [0, T ] → Rn such that, for almost every t ∈ [0, T ],ϕ(t) ∈ F (ϕ(t)). The following result is the key to further results about differential inclusions, forexample about asymptotic stability or reachable sets, just like Theorem 3.6 and Corollary 3.7were the foundations for what was done for difference inclusions.

Theorem 3.14 Suppose that the set-valued mapping F : Rn ⇒ Rn is outer semicontinuous andlocally bounded. Then, from every locally bounded sequence ϕi of solutions to (20) on [0, T ]one can extract a subsequence that converges uniformly to a solution to (20) on [0, T ].

4 Invariance, set convergence, and hybrid dynamics

Invariance of a set under a differential equation or inclusion is the property that solutions fromthe set remain in the set. When solutions are not unique for some initial points, this conceptsplits into weak invariance, also called viability, which requires that some solution from eachinitial point remain in the set, and strong invariance, requiring that every solution remain inthe set. Necessary and sufficient conditions for these properties, utilizing the concept of tangentcones, are stated in Section 4.4.

Invariance plays a role in predicting the asymptotic behavior of solutions to differential equa-tions or inclusions. The classical invariance principle dates back to the work of Krasovskii [20]and LaSalle [21]. It lets one predict the asymptotic behavior of solutions to a differential equationif one has a nondecreasing function, rather than a strictly decreasing Lyapunov function. Theclassical result is recalled below as Theorem 4.6 in Section 4.2. Some ingredients of the classicalresult can be conveniently viewed through the concept of set convergence, summarized in Section4.1 below, and this then lets one extend the result far beyond the classical setting. One suchextension is presented, on an example, in Section 4.3.

4.1 Set convergence

Let Si be a sequence of sets in Rn. The following define an “upper” limit, “lower” limit, andthe limit of the sequence. Terminology follows that of [31].

• the outer limit of the sequence Si, denoted limsupi→∞ Si, is the set of all x ∈ Rn suchthat, for some subsequence Sik and some xk ∈ Sik , x = limk→∞ xi;

• the inner limit of the sequence Si, denoted liminfi→∞ Si, is the set of all x ∈ Rn suchthat, for some xi ∈ Si, x = limi→∞ xi;

• the limit of the sequence Si, denoted limi→∞ Si, is said to exist if the inner and the outerlimits are the same, and then limi→∞ Si = limsupi→∞ Si = liminfi→∞ Si

Example 4.1 The following illustrate set convergence:

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• If Si = xi for every i ∈ N, then limsupSi is the set of cluster points of the sequence xi,liminf Si exists only if limxi exists in the usual sense, and then limsupSi = limSi = limxi.

• If Si = S for every i ∈ N, for some set S ⊂ Rn, then limSi = S. In general, the outer andinner limits are always closed.

• If Si = x ∈ Rn |x2 = aix1 and lim ai = a then limSi = x ∈ Rn |x2 = ax1. Note that,for every ε > 0, it is never the case that Si ⊂ S + εIB.

Concepts of set convergence can be used to describe continuity of set-valued mappings.That is, a set-valued mapping F : Rn ⇒ Rn is outer semicontinuous if for every xi → x,limsupi→∞ F (xi) ⊂ F (x); it is inner semicontinuous if for every xi → x, liminfi→∞ F (xi) ⊃ F (x);and continuous if for every xi → x, limi→∞ F (xi) exists and equals F (x).

For compact sets, one can characterize their convergence through the Hausdorff metric. Withsome care, this can be applied to sets that are not necessarily bounded. The following is a partof [31, Theorem 4.10].

Fact 4.2 Let Si, S ⊂ Rn be closed.

• limsupi→∞ Si ⊂ S if and only if for every ρ > 0, ε > 0 there exists I such that for everyi > I,

Si ∩ ρIB ⊂ S + εIB;

• liminfi→∞ Si ⊃ C if and only if for every ρ > 0, ε > 0 there exists I such that for everyi > I,

S ∩ ρIB ⊂ Si + εIB;

The classical Bolzano-Weierstrass theorem says that a bounded sequence of points in Rn hasa convergent subsequence. This generalizes to sequences of sets, as stated below, following [31,Theorem 4.18]. In particular, a bounded sequence of sets has a convergent subsequence.

Fact 4.3 For every sequence Si of sets Si ⊂ Rn, either the sequence diverges, in the sensethat for every M > 0 there exists I such that for every i > I, Si ∩MIB = ∅; or the sequence hasa subsequence which converges to a nonempty limit.

4.2 The classical invariance principle

Knowing the invariant sets for a differential equation helps one predict the asymptotic behaviorof bounded solutions. The reason for this is as follows. Let f : Rn → Rn be locally Lipschitzcontinuous. Let ϕ : [0,∞) → Rn be a solution. The Ω-limit of ϕ, denoted ω(ϕ), is the set definedby

ω(ϕ) =x ∈ Rn | ∃ ti ≥ 0, lim

i→∞ti = ∞ such that lim

i→∞ϕ(ti) = x

.

Then

• ω(ϕ) is closed and, unless limt→∞ ∥ϕ(t)∥ = ∞, nonempty.

It is easy to show this directly. Alternatively, one can consider sets Si = ϕ(t) | t ≥ i, note thatω(ϕ) = limsupSi, and recall that the outer limit of a sequence of sets is always closed and, unlessthe sets “escape to infinity”, nonempty. Suppose now that ϕ is bounded. Then:

• ω(ϕ) is compact and ϕ converges to ω(ϕ) uniformly, in the sense that for every ε > 0 thereexists T such that for every t > T , ϕ(t) ∈ ω(ϕ) + εIB.

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The proof of convergence by contradiction is simple. Finally,

• ω(ϕ) is invariant: for every x ∈ Ω(ϕ) there exists a solution ψ : R → Rn such thatψ(t) ∈ Ω(ϕ) for every t ∈ R.

To prove this, pick x ∈ Ω(ϕ), find ti → ∞ such that ϕ(ti) → x, define ψi : [0,∞) → Rn byψi(t) = ϕ(t − ti), note that ψi(0) → x, and because solutions to x = f(x) depend continuouslyon initial conditions, ψi converge locally uniformly to a solution ψ with ψ(0) = x. Because ϕconverges uniformly to ω(ϕ), ψ(t) ∈ ω(ϕ) for every t ≥ 0. Hence, ψ verifies forward invariance(the domain of ψ is [0,∞)), and this simpler proof was given for illustration purposes. To proveinvariance, given x ∈ Ω(ϕ), pick ti > 2i such that ϕ(ti) → x and consider ψi : [−i,∞) → Rn

given byψi(t) = ϕ(t− ti). The rest of the argument is the same.

Remark 4.4 Ω-limits are invariant for differential and difference inclusions as well, if the dy-namics are given by outer semicontinuous and locally bounded set-valued mappings. Indeed,the continuous dependence of solutions on initial conditions, as used in the discussion above, isnot essential. Given a bounded sequence of solutions ψi with ψi(0) → x it is enough to knowthat a subsequence converges uniformly and in the limit yields a solution with initial point x.For differential inclusions, this follows from Theorem 3.14; for difference inclusions one invokesTheorem 3.6 and Corollary 3.7. Roughly, outer semicontinuous dependence of solutions on initialconditions is sufficient for the argument to work.

Example 4.5 Consider the differential equation x = f(x) where f : R2 → R2 given by

f(x) =

(−x2 +

(1− x21 − x22

)x1

x1 +(1− x21 − x22

)x2

).

One can see that 0 is an equilibrium, and one can verify that the set x ∈ R2 |x21 + x22 = 1is invariant: on this set, the differential equation reduces to x1 = −x2, x2 = x1. To predictthe behavior of solutions from other initial points, consider how the solution’s distance from 0changes. For every solution ϕ, one has

d

dt(∥ϕ(t)∥)2 = 2

(1− ∥ϕ(t)∥2

)∥ϕ(t)∥2.

This suggests that when ∥ϕ(t)∥ > 1, ∥ϕ(t)∥ is decreasing and when 0 < ∥ϕ(t)∥ < 1, ∥ϕ(t)∥ isincreasing.

Theorem 4.6 Let f : Rn → Rn be locally Lipschitz continuous. Suppose that there existsV : Rn → R such that, for every solution ψ to x = f(x) one has

d

dtV (ψ(t)) ≤ 0.

Let ϕ : [0,∞) → Rn be a bounded solution to x = f(x). Then, for some r ∈ R, ϕ converges tothe union of all invariant subsets of V −1(r).

Proof. Let ϕ : [0,∞) → Rn be a bounded solution to x = f(x). The discussion at the beginningof this section showed that ω(ϕ) is nonempty, compact, invariant, and that ϕ converges to thisset. Because V (ϕ(t)) is nonincreasing on [0,∞) and bounded below, V (ϕ(t)) converges. Letr = limt→∞ V (ϕ(t)). By definition of ω(ϕ), for every x ∈ ω(ϕ) there exist ti → ∞ such thatϕ(ti) → x. Then V (x) = limi→∞ V (ϕ(ti)) = r. This finishes the proof.

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Example 4.7 Consider the differential equation from Example 4.5. The analysis in that examplesuggests that solutions from nonzero initial conditions converge to the unit circle. To fully justifythis, with the help of Theorem 4.6, consider the function V (x) = (x2 + y2 − 1)2. Then, for everysolution ψ, one has

d

dtV (ψ(t)) = −8(x21 + x22 − 1)2(x21 + x22) = −8V (x)∥x∥2.

Theorem 4.6 implies that every solution ϕ converges to the largest invariant subset of V −1(r),for some real r. The set V −1(r) does not have any invariant subsets unless r = 0 or r = 1. Forr = 1, V −1(1) is the unit circle, which is invariant For r = 1, V −1(1) is the union of 0 andthe circle of radius

√2. The only invariant subset of this union is 0. Hence, every solution

converges either to 0 or to the unit circle.

4.3 Invariance in a hybrid system

This section discusses a dynamical system in R2 which can be informally introduced as follows.Let C be the union of the square with vertices (0, 0), (0, 1), (1, 0), (1, 1) and the triangle withvertices (2, 0), (2, 1), and (3, 1). Let D1 be the segment between (1, 0) and (1, 1); let D2 be thesegment between (2, 0) and (3, 1). If x ∈ C and when possible, the solution from x should satisfythe differential equation x = f(x), with f(x) ≡ (1, 0). When x ∈ D1, the solution from x shouldsatisfy the difference equation x+ = (2, x22). When x ∈ D2, the solution from x should satisfyx+ = (0, x2).

For background on such, and similar, dynamical systems, which combine continuous-time anddiscrete-time dynamics, the reader can consult [35], [15], [14] and the references therein.

One can try to parameterize the behaviors of this roughly introduced dynamics by t ∈ R.Then, for example, the “solution” ϕ from (0, 1) is ϕ(t) = (t, 1) for t ∈ [0, 1), ϕ(t) = (t+ 1, 1) fort ∈ [1, 2), and ϕ(t) = ϕ(t− 2) for t ≥ 2. This is a discontinuous function of t, taken to be right-continuous just by choice, as one could use a left-continuous function instead. Another exampleof a “solution”, from (0, .5) is ϕ(t) = (t, .5) for t ∈ [0, 1), ϕ(t) = (t + 1, .25) for t ∈ [1, 1.25),ϕ(t) = (t− 1.25, .25) for t ∈ [1.25, 2.25), ϕ(t) = (t− .25, .0625) for t ∈ [2.25, 2.3125), etc. Noticethat the discontinuities of the two presented “solutions” occur at different times. This can haveunexpected consequences: given the “solution” ϕ from (0, 1) as above, and the “solution” ϕεfrom (0, 1− ε), one has supt∈[0,T ] ∥ϕ(t)− ϕε(t)∥ ≥ 1 for any large enough T .

Questions that may arise after a closer look at the dynamics described above are:

• What is the “solution” from (0, 0)? How to define a solution in general.

• If [0, 1] × 0 ∪ (2, 0) is the Ω-limit of a solution from, say, (0, .5), in what sense is itinvariant? How to state and prove an appropriate invariance principle?

• The behaviors from (0, 1) and (0, 1− ε) appear similar — how should the distance betweenthe corresponding solutions be measured to reflect this?

The remainder of this section outlines one particular approach to the issues above, following [15].Some definitions are simplified for the example at hand.

A hybrid time domain is a subset of R2 given by

∞∪i=0

[ti, ti+1]× i,

where t0 = 0 and ti is a nondecreasing sequence of nonnegative numbers. Recall the set C,let D = D1 ∪D2, consider f : C → R2 given by f(x) = (1, 0) for every x ∈ C, and g : D → R2

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given, for x ∈ D, by

g(x) =

(2, x22) if x ∈ D1

(0, x2) if x ∈ D2.

A solution to the dynamics described above is a function ϕ defined on a hybrid time domain,denoted from now on by domϕ, satisfying the following:

(a) f (t, j) ∈ domϕ is such that, for some ε > 0, (t− ε, t+ ε)× j ⊂ domϕ, then t 7→ ϕ(t, j)is differentiable on (t− ε, t+ ε) and

d

dtϕ(t) = f(ϕ(t));

(b) if (t, j) ∈ domϕ and (t, j + 1) ∈ domϕ then

ϕ(t, j + 1) = g (ϕ(t, j)) .

Hence, for example, the solution ϕ with initial condition (0, 1) has the hybrid time domaindomϕ =

∪∞i=0[i, i + 1] × i and is given by ϕ(t, 0) = (t, 1) if t ∈ [0, 1], ϕ(t, 1) = (t + 1, 1), and

ϕ(t, j) = ϕ(t− 2, j − 2) for t ≥ 2, j ≥ 2 with (t, j) ∈ domϕ.With this definition, there is no ambiguity about the solution from (0, 0). This solution ϕ has

domϕ =∪∞

i=0 [ti, ti+1] × i with t0 = 0, t1 = 1, t2 = 1, t3 = 1, t4 = 2, t5 = 2, t6 = 2, t7 = 3etc., and is given by ϕ(t, 0) = (t, 0) for t ∈ [0, 1], ϕ(1, 1) = (2, 0), ϕ(t, 2) = (t− 1, 0) for t ∈ [1, 2],ϕ(2, 3) = (2, 0), etc.

A natural way to define Ω-limits of solutions is

ω(ϕ) =x ∈ Rn | ∃ (ti, ji) ∈ domϕ, lim

i→∞ti + ji = ∞ such that lim

i→∞ϕ(ti, ji) = x

.

For the remainder of this section, let ϕ be the solution from (0, 0.5). Then

ω(ϕ) = [0, 1]× 0 ∪ (2, 0),

and the forward invariance of this set can be verified by inspection. For example, the previousparagraph described the solution from (0, 0) ∈ ω(ϕ). Backward invariance is not discussed, toavoid the trouble with defining “backward” hybrid time domains. The question now is: whetherforward invariance of ω(ϕ) can be established without inspection, by using an approach similarto the standard one outlined in Section 4.2.

Take any x ∈ ω(ϕ), let (ti, ji) ∈ domϕ be such that ti + ji → ∞ and ϕ(ti, ji) → x. Defineϕi(t, j) = ϕ(t+ti, j+ji) with an appropriate change of domains and note that ϕi form a sequenceof solutions with ϕi(0, 0) → x. Let

gphϕi =(t, j, y) ∈ Rn+2 | (t, j) ∈ domϕi, y = ϕi(t, j)

be the graph of ϕi. Because ϕi(0, 0) → x, Fact 4.3 implies that the sequence gphϕi has aconvergent subsequence gphϕik. That is, gphϕik converge, as sets, to some set L ⊂ Rn+2. IfL is a graph of a solution, say ψ, with ψ(0, 0) = x and if ψ(t, j) ⊂ ω(ϕ) for all (t, j) ∈ domϕ,then ψ verifies forward invariance of ω(ϕ) from x.

The key steps to showing that L is a graph of a solution ψ, in other words that the graphicallimit of the sequence of solutions ϕik is a solution, are as follows:

• Show that the projection of L onto the (t, j)-space, is a hybrid time domain. It is importanthere that the sequence ϕik is bounded.

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• Given that L is the graph of some (possibly set-valued) mapping ψ with domψ being ahybrid time domain, show that ψ is in fact a solution. This can be done directly from thedefinition of a solution, and may require passing from convergence of graphs of solutionsto a differential equation to uniform convergence.

• Argue that ϕ converges to ω(ϕ) and use this to claim that the range of ψ is contained inω(ϕ).

This is left as an exercise to the reader. Details can be found in [16] or [15], where a more generalcase, of systems combining differential inclusions and difference inclusions, was treated. Notethat Theorem 3.14 and Theorem 3.6, Corollary 3.7 are what are needed in such a general setting.

4.4 Tangent cones, invariance, and viability

The tangent cone TC(x) to the set C at the point x was defined in (19). For every pointx ∈ S which is not isolated, TC(x) is nonempty, closed, and contains a nonzero element. Then,automatically, TC(x) is unbounded because it is a cone. For each x ∈ C, TC(x) = ∅. When Cis a smooth manifold, the tangent cone TC amounts to the concept of a tangent space. Moreinteresting examples are given below.

Example 4.8 Let C = [a, b] ⊂ R. Then

TC(x) =

(−∞, 0] if x = a,(−∞,∞) if x ∈ (a, b),[0,∞) if x = b.

For boxes in Rn, i.e., sets given by products of intervals

C = [a1, b1]× [a2, b2]× · · · × [an, bn],

the tangent cone TC(x) can be found coordinate-wise:

TC(x) = T[a1,b1](x1)× T[a2,b2](x2)× · · · × T[an,bn](xn).

Some further examples are

C =x ∈ R2 |x1 ≥ 0, x2 ≤

√x1, TC(0) =

w ∈ R2 |w1 ≥ 0

.

C =x ∈ R2 |x1 ≥ 0, 0 ≤ x2 ≤ sinx1

, TC(0) =

w ∈ R2 |w1 ≥ 0, w2 ≤ w1

.

Tangent cones play a role in the analysis of constrained differential equations and inclusions.Given a set C ⊂ Rn and a set-valued mapping F : Rn ⇒ Rn, consider

x ∈ F (x), x ∈ C, (30)

with ϕ : [0, T ] → Rn being a solution to (30) if ϕ is absolutely continuous, satisfies ϕ(t) ∈ F (ϕ(t))for almost all t ∈ [0, T ], and ϕ(t) ∈ C for every t ∈ [0, T ]. Viability (also called weak invariance)is the property that for every ξ ∈ C there exists a solution to (30). Fact 4.9 gives necessary andsufficient conditions for this property. For details, see [2, Propositions 3.4.1, 3.4.2].

Fact 4.9 Suppose that C ⊂ Rn is closed and F : Rn ⇒ Rn is outer semicontinuous and locallybounded.

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(a) If ϕ : [0, T ] → Rn for some T > 0 is a solution to (30) with ϕ(0) = ξ, then

F (ξ) ∩ TC(ξ) = ∅.

(b) Given ξ ∈ C, if there exists a neighborhood U of ξ such that for every x ∈ U ∩ C,

F (x) ∩ TC(x) = ∅,

then there exists T > 0 and a solution ϕ : I := [0, T ] → Rn to (30) with ϕ(0) = ξ.

The set C is invariant (also called strongly invariant) under x ∈ F (x) if for every ξ ∈ C, everysolution to x ∈ F (x) remains in C. In the situations where solutions to x ∈ F (x) are unique,invariance and viability are the same. For the fact below, see [2, Theorem 4.3.6, Corollary 5.2.3].Some assumptions can be weakened.

Fact 4.10 Suppose that C ⊂ Rn is closed and F : Rn ⇒ Rn is outer semicontinuous and locallybounded.

(a) C is invariant under F if domF = C and, for every x ∈ C,

F (x) ⊂ TC(x).

(b) If the interior of C is nonempty and, for every x on the boundary of C,

F (x) ⊂ Rn \ TRn\C(x),

then for every solution ϕ to x ∈ F (x) with ϕ(0) on the boundary of C there exists T > 0such that ϕ(t) is in the interior of C for every t ∈ (0, T ].

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References

[1] Z. Artstein. Stabilization with relaxed controls. Nonlinear Anal., 7(11):1163–1173, 1983.

[2] J.-P. Aubin. Viability theory. Birkhauser, 1991.

[3] J.-P. Aubin and A. Cellina. Differential Inclusions. Springer-Verlag, 1984.

[4] J.-P. Aubin and H. Frankowska. Set-Valued Analysis. Birkhauser, 1990.

[5] N.E. Barabanov. On the Lyapunov exponent of discrete inclusions. II. Avtomat. i Telemekh.,(3):24–29, 1988. translation in Automat. Remote Control 49 (1988), no. 3, part 1, 283-287.

[6] N.E. Barabanov. Stability of inclusions of linear type. In Proc. 14th American ControlConference, volume 5, pages 3366 – 3370, 1995.

[7] J.M. Borwein and A.S. Lewis. Convex analysis and nonlinear optimization. Springer-Verlag,2000.

[8] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, 2004.available online.

[9] F.H. Clarke. Optimization and Nonsmooth Analysis. Wiley, 1983.

[10] F.H. Clarke, Yu.S. Ledyaev, R.J. Stern, and P.R. Wolenski. Nonsmooth Analysis and ControlTheory. Springer-Verlag, 1998.

[11] W.P. Dayawansa and C.F. Martin. A converse Lyapunov theorem for a class of dynamicalsystems which undergo switching. IEEE Trans. Automat. Control, 44(4):751–760, 1999.

[12] A.L. Dontchev and R.T. Rockafellar. Implicit functions and solution mappings. Springer,2009. A view from variational analysis.

[13] R. Goebel, T. Hu, and A.R. Teel. Dual matrix inequalities in stability and performanceanalysis of linear differential/difference inclusions. In Current trends in nonlinear systemsand control. Birkhauser, 2006.

[14] R. Goebel, R.G. Sanfelice, and A.R. Teel. Hybrid dynamical systems. robust stability andcontrol for systems that combine continuous-time and discrete-time dynamics. IEEE ControlSystems Magazine, 29:28–93, 2009.

[15] R. Goebel, R.G. Sanfelice, and A.R. Teel. Hybrid Dynamical Systems: Modeling, Stability,and Robustness. Princeton University Press, 2012.

[16] R. Goebel and A.R. Teel. Solutions to hybrid inclusions via set and graphical convergencewith stability theory applications. Automatica, 42:573–587, 2006.

[17] R. Goebel, A.R. Teel, T. Hu, and Z. Lin. Conjugate convex Lyapunov functions for duallinear differential inclusions. IEEE Trans. Automat. Contr., 51:661–666, 2006.

[18] O. Hajek. Discontinuous differential equations, I. J. Diff. Eq., 32:149–170, 1979.

[19] H. Hermes. Discontinuous vector fields and feedback control. In Differential Equations andDynamical Systems, pages 155–165. Academic Press, 1967.

[20] N. N. Krasovskiı. Nekotorye zadachi teorii ustoichivosti dvizheniya. Gosudarstv. Izdat.Fiz.-Mat. Lit., Moscow, 1959.

25

[21] J. P. LaSalle. Some extensions of Liapunov’s second method. IRE Trans. Circuit Theory,7(4):520–527, 1960.

[22] D. Liberzon. Switching in Systems and Control. Systems and Control: Foundations andApplications. Birkhauser, 2003.

[23] P.D. Loewen. Optimal Control via Nonsmooth Analysis. American Mathematical Society,1993.

[24] A. M. Lyapunov. The general problem of the stability of motion. Internat. J. Control,55(3):521–790, 1992. Translated by A. T. Fuller from Edouard Davaux’s French translation(1907) of the 1892 Russian original.

[25] A.P. Molchanov and Ye.S. Pyatnitskiy. Criteria of asymptotic stability of differential anddifference inclusions endountered in control theory. System Control Lett., 13:59–64, 1989.

[26] T. Monovich and M. Margaliot. Analysis of discrete-time linear switched systems: a varia-tional approach. SIAM J. Control Optim., 49(2):808–829, 2011.

[27] B.S. Mordukhovich. Variational analysis and generalized differentiation. I. Springer-Verlag,2006. Basic theory.

[28] B.S. Mordukhovich. Variational analysis and generalized differentiation. II. Springer-Verlag,2006. Applications.

[29] R.T. Rockafellar. Convex Analysis. Princeton University Press, 1970.

[30] R.T. Rockafellar. Conjugate Duality and Optimization. Number 16 in CBMS-NSF RegionalConference Series in Applied Mathematics. SIAM, 1974.

[31] R.T. Rockafellar and R. J-B Wets. Variational Analysis. Springer, 1998.

[32] R. Shorten, F. Wirth, O. Mason, K. Wulff, and C. King. Stability criteria for switched andhybrid systems. SIAM Rev., 49(4):545–592, 2007.

[33] G.V. Smirnov. Introduction to the theory of differential inclusions, volume 41 of GraduateStudies in Mathematics. American Mathematical Society, Providence, RI, 2002.

[34] E.D. Sontag. Comments on integral variants of ISS. Systems Control Lett., 34(1-2):93–100,1998.

[35] A. van der Schaft and H. Schumacher. An Intruduction to Hybrid Dynamical Systems,volume 251 of Lect. Notes in Contr. and Inform. Sci. Springer, 2000.

[36] R. Vinter. Optimal control. Systems & Control: Foundations & Applications. Birkhauser,2000.

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