On the penalty function method in the problem of constructing reachable sets for control systems with state constraints

The paper is devoted to the problem of constructing reachable sets for a nonlinear control system with state constraints given in the form of inequalities. An analog of the penalty function method is proposed; it consists in replacing the original system with state constraints by an auxiliary system without state constraints. The right-hand side of the auxiliary system is a modification of the right-hand side of the original system and depends on a scalar parameter (a penalty coefficient). Under certain conditions, we show that a reachable set of the original system can be approximated from within in the Hausdorff metric by reachable sets of auxiliary systems as the penalty coefficient tends to infinity.