Stabilization by Constrained Controls∗

A stabilization problem for a general nonlinear control system is considered. In particular the control corresponding to the equilibrium position may belong to the boundary of the control set. A linear control system is considered as a first approximation for the original problem. The right-hand side of the linear system generates a set-valued map of a special type known as a convex process. This set-valued map has a number of properties similar to those of a linear operator. They allow one to establish necessary and sufficient conditions for solvability of the regulator design problem for the first approximation and to construct a Lyapunov function. Based on these results the nonlinear stabilization problem is investigated. Different statements of the regulator design problem are studied. Stabilization problems for some mechanical systems are considered to illustrate the regulator design techniques. The properties of transient characteristics (the “peak” effect) are discussed for a linear stabilization problem under controllability conditions.