Computing the reachable set boundary for an abstract control problem

Abstract

The problems of reachability for control systems with integral constraints on the control variables have been studied in the literature on the optimal control theory. For a nonlinear control-affine system on a finite time interval it was shown that every admissible control that steers the control system to the boundary of its reachable set is a local solution of some auxiliary optimal control problem with an integral cost. In this paper we prove an analog of this result for an abstract control system defined by a differentiable mapping in Banach spaces with a constraint, given as a level set of some continuous functional. The proof is based on the covering mappings theorem for differentiable mappings of Banach spaces. We consider a nonlinear affine-control system with integral constraints on the state and the control variables given by the joint integral inequality with an integrant quadratic in the control variables. Assuming the controllability of the linearized system, we prove that any admissible control, that steers the control system to the boundary of its reachable set, is a local solution to an optimal control problem with integral cost functional. Necessary conditions are obtained for the optimality of controls taking the system to the boundary of the reachable set in the form of Pontryagin’s maximum principle. An algorithm for computing the reachable sets based on the maximum principle is proposed.