A stable fixed focus and its immunity zone in a second-order controlled dynamical system

Abstract

The qualitative analysis of second-order controlled dynamical systems with bounded control necessitates the study of auxiliary smooth autonomous systems corresponding to boundary values of the control as well as of sewed systems of one-sided intersection obtained from the auxiliary systems by smooth sewing along the contact curve. In the presence of singular points on the singular curve, the dynamical systems corresponding to constant values of the control functions can have equilibria whose coordinates are independent of the values of these functions (xed equilibria). Such equilibria always sit at singular points of the contact curve. In the present paper, we consider a xed equilibrium O of the type of a stable focus located at a node singular point of the contact curve. Under specic constraints on the control, the equilibrium of each of the sewed systems of one-sided intersection coinciding with such a singular point has the type of a focus. We analyze the stability of this equilibrium. We show that the control can be subjected to constraints such that the values of control functions belong to the stability domain of the focus O but the controllability set into the neighborhood of the equilibrium does not contain safe zones. 2. STATEMENT OF THE PROBLEM We consider the controlled dynamical system of the form dx=dt = P (x )+ u(t)Q(x); (1) where x =( x1;x2), P (x )=( P1(x);P2(x)), and Q(x )=( Q1(x);Q2(x)) are vector functions of the class C k (k 5) and u(t) fR 1 ! R 1 g (the control) is a piecewise continuous bounded function on each nite interval, m u(t) n. Suppose that foru(t) , =c onst,m