Differential Inclusions

Frontmatter

Introduction

Abstract

A great impetus to study differential inclusions came from the development of Control Theory, i.e. of dynamical systems

$${\text{x'(t) = f}}\left( {{\text{t,x}}\left( t \right){\text{,u}}\left( t \right)} \right){\text{, x(0) = }}{{\text{x}}_0}$$

(*)

“controlled” by parameters u(t) (the “controls”). Indeed, if we introduce the set-valued map

$$F(t,x)\dot = {\left\{ {f\left( {t,x,u} \right)} \right\}_{u \in U}}$$

then solutions to the differential equations (*) are solutions to the “differential inclusion”

$$x'\left( t \right) \in F\left( {t,x\left( t \right)} \right),{\mkern 1mu} x\left( 0 \right) = {x_0}$$

(**)

in which the controls do not appear explicitely.

Jean-Pierre Aubin, Arrigo Cellina

Chapter 0. Background Notes

Abstract

We gather several known (or less known) results which are used in several distinct places of this book. We prove the existence of locally Lipschitzean partitions of unity subordinated to a locally finite open covering of a metric space and we recall the definition and the characterization of absolutely continuous functions.

Jean-Pierre Aubin, Arrigo Cellina

Chapter 1. Set-Valued Maps

Abstract

We gather in this chapter the properties of set-valued maps which are needed for the study of differential inclusions.

Jean-Pierre Aubin, Arrigo Cellina

Chapter 2. Existence of Solutions to Differential Inclusions

Abstract

In what follows we shall deal with the existence and properties of solutions to differential inclusions of the form

$$x'\left( t \right) \in F\left( {x\left( t \right)} \right)$$

(1)

or

$$x'(t) \in F(t,x(t)).$$

(2)

Jean-Pierre Aubin, Arrigo Cellina

Chapter 3. Differential Inclusions with Maximal Monotone Maps

Abstract

We devote this chapter to a very important class of differential inclusions

$$x'\left( t \right) \in - A\left( {x\left( t \right)} \right)$$

(1)

where A(x) ≐ −F(x)is a so-called “maximal monotone” set-valued map.

Jean-Pierre Aubin, Arrigo Cellina

Chapter 4. Viability Theory: The Nonconvex Case

Abstract

We devote this chapter to general Viability Theory and we postpone to the next chapter the further results obtained when we assume that the viability subset is convex.

Jean-Pierre Aubin, Arrigo Cellina

Chapter 5. Viability Theory and Regulation of Controled Systems: The Convex Case

Abstract

When we assume that the viability subset K is convex and compact, we obtain many more properties. The most striking one is that the tangential condition

$$\forall x \in K,\,F\left( x \right) \cap {T_K}\left( x \right) \ne \phi $$

(1)

which is necessary and sufficient when F has convex values for the differential inclusion

$$\begin{array}{*{20}{c}} {i)x\prime \left( t \right) \in F\left( {x\left( t \right)} \right),} \\ {ii)x\left( 0 \right) = {x_0},{x_0}{\text{given in K}},} \end{array}{\text{ }}$$

(2)

to have viable trajectories for all initial states x₀ in K, is also a sufficient condition for F to have an equilibrium state x̄ in K.

Jean-Pierre Aubin, Arrigo Cellina

Chapter 6. Liapunov Functions

Abstract

We shall investigate whether differential inclusions

$$x'\left( t \right) \in F\left( {x\left( t \right)} \right),{\text{ }}x\left( 0 \right) = {x_0}$$

(1)

do have trajectories satisfying the property

$$\forall t > s,{\text{ }}V\left( {x\left( t \right)} \right) - V\left( {x\left( s \right)} \right) + \int\limits_s^1 {W\left( {x\left( \tau \right),x'\left( \tau \right)} \right)} dx \leqslant 0$$

(2)

where

$$\left\{ {\begin{array}{*{20}{c}} {i)\,V\,is\,function\,from\,K\dot = Dom\,F\,to\,{R_ + }} \\ {ii)\,W\,is\,a\,function\,from\,Graph\,(F)\,to\,{R_ + }} \end{array}} \right.$$

(3)

Trajectories x(·) of differential inclusion (1) satisfying (2) will be called “monotone trajectories” (with respect to V and W).

Jean-Pierre Aubin, Arrigo Cellina

Comments

Abstract

The definitions and most results of Section 1 are classical. Some related material appears in the book by Berge [1959]. Theorem 1.2 is stated without proof (and without the assumption of completeness of Y) in Choquet [1948]. For the history of the concepts of continuity of set valued maps we refer to the forthcoming book by Rockafellar and Wets. For Theorem 2.2 we refer to the book by Spanier [1966]. Proposition 2.2 is taken from Aubin [1979c], while Proposition 2.3 comes from the book by Ekeland and Teman [1974]. Theorems 2.4 and 2.5 are well known theorems from Berge [1959]. The important results of Section 3 were obtained independently by Robinson [1976a] and Ursescu [1975].

Jean-Pierre Aubin, Arrigo Cellina

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