Introduction to the theory of differential inclusions (Q2767615)

The book presents a systematic treatment of the theory of differential inclusions written at the graduate level. The text consists of two parts. The first part (Chapters 1-3) contains a brief introduction to convex analysis, theory of set-valued maps and nonsmooth analysis. The second part is organized as follows. Basic existence theorems as well as results regarding dependence on initial conditions are stated in Chapter 4. For simplicity only differential inclusions with a convex-valued right-hand side are considered in this text. Chapter 5 is devoted to the problem of selection of a solution to a differential inclusion, which is contained in a given set (viability problem). Chapter 6 focuses on the controllability problem. Sufficient conditions for the local controllability are developed in terms of a convex process that approximate the differential inclusion. The above controllability conditions at first approximation are illustrated by a couple of mechanical examples. Optimal control problems are considered in Chapter 7. The next chapter is devoted to the issues of weak and strong stability in differential inclusions. Stability conditions based on the Lyapunov direct method and concepts of nonsmooth analysis are presented in the text. In addition, constructive algebraic criteria for the stability are obtained by means of the first approximation technique. At that first approximations of two types are considered: a linear system of differential equations which is a selector of the given inclusion, and a differential inclusion with a convex process in the right-hand side. These linearization techniques are applied to exhibit the weak asymptotic stability of a model of a missile moving in the plane. Chapter 9 deals with the problem of stabilization for nonlinear control systems. The crucial point here is the relationship between stabilizability of a control system and weak asymptotic stability of the corresponding differential inclusion. By applying the Lyapunov direct method and selection theorems, the author obtains sufficient conditions for the stabilizability. These conditions are applied to study the stabilizability of some mechanical systems. In conclusion, some bibliographical comments connected with the material of the book are given. The book is well written and contains a number of excellent problems.