Theory of control systems described by differential inclusions
The first chapter introduces the fundamental concepts and conclusions of functional analysis so that readers can have a foundation for going on reading this book successfully and can also understand notations used in the book. The arrangement of this chapter is as follows: The first section deals with normed linear spaces and inner product spaces which both provide a platform for further investigation; the second section introduces convex sets, and the third section considers convex functions; the last section of this chapter introduces semi-continuous functions. These are all necessary for research of set-valued mappings and differential inclusions which are two key concepts in this book. The most materials given in this chapter are referred to [\textit{J. B. Conway}, A course in functional analysis. New York, NY: Springer-Verlag (1985; Zbl 0558.46001)] which has been widely used in Chinese universities. Chapter 2 deals with two fundamental concepts for the control systems described by differential inclusions. They are set-valued mappings and differential inclusions. The first two sections introduce the set-valued mappings and the succeeding four sections involve the differential inclusions. In Chapter 3, the authors deal with several kinds of differential inclusions and their control. They recall that mathematical investigation in many fields starts from the linear case; for example, at the beginning of control theory, one deals with the linear system, the functional analysis starts with the linear normed spaces and linear mappings. Consequently, the authors consider convex processes which can be treated as an extension of the linear single-valued mapping to set-valued mapping. They consider linear polytopic differential inclusions and the Lur'e systems. This can also be thought as extensions of linear control systems to differential inclusions and nonlinear differential inclusions. The organization of this chapter is as follows. At the first section, convex processes in Banach spaces are defined and verified that the convex processes hold the same properties as linear mappings in Banach spaces. The second section deals with the convex processes in finite dimensional spaces and shows that they have similar construction as matrices. The third section considers the differential inclusion yielded by convex processes. The controllability is discussed. And the last section investigates the stability of convex process differential inclusions. A class of differential inclusion systems ``the linear polytope systems'' is discussed in the next chapter. This kind of system can be viewed as another extension of the linear control systems to the set-valued mappings. This chapter contains four sections. In the first section, the definition of the linear polytope system and the motivation of investigation are presented. Section 4.2 deals with the convex hull Lyapunov function which is the main tool in this chapter. Then, Section 4.3 considers the control of the linear polytope system. Conclusions of Section 4.3 are applied to deal with saturated control at the last section of this chapter. In the theory of differential inclusion control systems, the linear polytope system is relatively simple. Moreover, only the linear polytope control system and the Lur'e differential inclusion system discussed in the next chapter are investigated comparatively deeply by so far. In the former chapter, control problems of linear polytope systems are considered by using the convex hull Lyapunov function. The linear polytope system is a convex combination of a finite set of finite linear systems. Involved set-valued mapping in the differential inclusion is the convex combination; hence, convex theory can be applied to deal with these control problems. In this last chapter, the Luré differential inclusion system and its relative control problems are considered. This kind of differential inclusions is different from linear convex hulls; the set-valued mapping satisfies a so-called sector condition, i.e., the image of the set-valued mapping is in a cone. Hence, it is a naturally nonlinear mapping.
scientific article; zbMATH DE number 6603154