Phase Portraits of Control Dynamical Systems

Abstract

UNDOUBTEDLY ONE OF the most often studied problems in mathematics is to describe and characterize the solutions of a general set of first order nonlinear differential equations. The reason for this interest is perhaps not to investigate whether or not our solar system is stable, but certainly the fact that many concrete physical systems can be modelled as a system of differential equations has vitalized the above question over the years. Today much of the research connected with the study of the equation ~ ( t )= F(x(t), t) goes under the title Dynamical Systems Theory, see Abraham and Marsden (1978) and Arnold (1989) as two excellent references in this field. An extremely important part of Dynamical Systems Theory is formed by the so-called qualitative theory, where the interest lies not so much in obtaining explicit solutions of the differential equation but much more in generating answers to a question like: is a specific solution of Yc(t)=F(x(t), t) asymptotically stable or not. The phase portrait of a dynamical system forms another example of such a qualitative analysis. In the phase portrait of Yc(t) = F(x(t), t) a rough sketch of trajectories (solutions of the differential equation) is drawn in the plane--no matter if the system lives on a higher dimensional space; some nice, illustrative examples may be found in, for instance, the above-mentioned references (Abraham and Marsden, 1978; Arnold, 1989). For low-dimensional systems such a phase portrait provides a useful information about the (stability) behavior of solutions of the system. The book Phase Portraits of Control Dynamical Systems forms an attempt to describe a phase portrait of a nonlinear control system .f(t) =f(x(t) , u(t)), i.e. a family of differential equations parametrized by the admissible input functions u(t) ~ U. As in case of a dynamical system without inputs, Yc(t) = F(x(t), t), the phase portrait of a control system gives a rough picture of the behavior of the system. Phase Portraits of Control Dynamical Systems is an English translation from an originally Russian text of 1984. The book presents in a non-rigorous way an introduction to the geometric representation of nonlinear control systems, or, stated in Soviet style, the phase portraits of differential inclusions ~?(t) ~ q~(x(t)), with • a set valued map defined as • (x) = {f(x, u) [u E U}. The material is presented in 40 sections of 2-5 pages each and is completed with an extensive list of references of mainly Soviet publications. Since it is impossible to summarize each of the 40 sections of Phase Portraits of Control Dynamical Systems only a short description of the contents will follow. The book gives a general mathematical introduction to nonlinear control systems described as J?(t)=f(x(t) , u(t)) where the controls u(.) belong to a predefined given set U. An essential assumption is that at each point x in the state space the set • (x) forms a convex subset of the tangent space at x. This allows for the--a t some places very useful--transition to the corresponding Hamiitonian H(p, x) which is defined as the supremum (maximum) of p.f where .f runs over @(x). The basic question that is studied throughout the book is whether or not the control system is controllable or not. Or, more specifically, given the system ~( t )= f (x ( t ) , u(t)) with some set of admissible controls U and an initial state xo, is it possible to construct for a given endpoint xt an input function u(t) such that the corresponding solution will reach