Subsystems on nonlinear control dynamical systems

Abstract

A control dynamical system is a very complex object because it essentially is a union of a set of ordinary differential equations (dynamical systems). Therefore, many solutions (phase trajectories) pass through the initial point of the system’s phase space. The task of control is to choose in this (generally, infinite) set a solution that satisfies certain conditions (for example, boundary conditions as in the terminal control problem). This is clearly not a trivial task. The concept of subsystem can facilitate the solution of control problems by reducing them to simpler ones. The reduction is based on the fact that subsystems are simpler than the original system because they only deal with a part of solutions of the original system. At the same time, solutions of the problem for a subsystem can be useful in solving the problem for the entire system. The concept of subsystem in the theory of control systems is similar to the concept of subspace in the theory of linear spaces or to a subgroup in group theory. More precisely, if the original system is defined on the set M (phase space), then a subsystem is defined on a subset N ⊂ M , and the solutions of the subsystem belonging to N are also solutions of the original system. In this paper, we construct subsystems for a class of nonlinear control systems (affine systems). In particular, we discuss some types of subsystems and their application in control problems.