Piecewise-Constant Stabilization

With the help of topological necessary conditions for continuous stabilization it is shown that, in general, in order to stabilize continuous- and discrete-time systems one has to use time-dependent or discontinuous feedback controls. On the other hand, the criterion of stabilization in the class of piecewise-constant feedbacks is established. In the context of this paper a piecewise-constant feedback is associated with a piecewise-constant function of the form u = u(x), where $x in {rm R}^n_x .$ The piecewise-constant feedback synthesis outlined here has several attractive features. First, it can be effectively applied to design feedback stabilizers subjected to control constraints. Second, the designed feedback laws do not cause sliding mode or chattering behavior in the closed loop system; i.e., on a finite interval of time the control in the closed loop system may have only a finite number of jump discontinuities.