The hamilton-jacobi type equations for nonlinear target control and their approximation

Abstract

This paper gives a comparison principle for first-order PDEs of the Hamilton-Jacobi-Bellman type that arise in problems of nonlinear target control synthesis and reachability analysis under hard bounds on the controls. The emphasis is is on treating backward reachability sets for a system with moving target sets which may also turn to be forward reachability tubes of another system. The target sets are to be reached within preassigned intervals. The exact solutions to this problem, given in set-theoretic terms, are expressed as level sets to the solutions of some specific types of the HJB equation. But these solutions require fairly complicated calculation. The present paper presents an alternative approach that avoids exact solutions in favor of their upper and lower bounds, which in many cases may suffice for solving the required problems. For systems with linear structure ellipsoidal estimates are given, which ensure accurate approximations of nonconvex reachability sets through unions of ellipsoids. © 2008 Springer-Verlag.