Application of penalty function method to computation of reachable sets for control systems with state constraints

Abstract

We study the penalty function type methods for computing the reachable sets of nonlinear control systems with state constraints. The state constraints are given by a finite system of smooth inequalities. The proposed methods are based on removing the state constraints by replacing the original system with an auxiliary system without constraints. This auxiliary system is obtained by modifying the set of velocities of the original system around the boundary of constraints. The right-hand side of the system depends on a penalty parameter. We prove that the reachable sets of the auxiliary system approximate in the Hausdorff metric the reachable set of the original system with state constraints as the penalty parameter tends to zero (infinity) and give the estimates of the rate of convergence. The numerical algorithms for computing the reachable sets, based on Pontryagin’s maximum principle, are also considered.