Piecewise-constant stabilization of nonlinear systems

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 feedback 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 ∈ R_xⁿ. The piecewise-constant feedback synthesis outlined here has several attractive features: 1) it can be effectively applied to design feedback stabilizers subjected to control constraints; and 2) the designed feedback laws do not cause sliding mode and/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 finite number of jump discontinuities