Loss control regions in optimal control problems

Abstract

This paper addresses optimal control problems with loss control regions. In that context the state space is partitioned into disjoint subsets, referred to as regions, which are classified into two types: control regions and loss control regions. When the state belongs to a control region, the control is permanent (i.e. the control value is authorized to be modified at any time). On the contrary, when the state belongs to a loss control region, the control must remain constant as long as the state belongs to this region. The objective of this paper is twofold. First, we reformulate the above setting into a hybrid optimal control problem (with spatially heterogeneous dynamics) involving moreover a regionally switching parameter, and we prove a corresponding hybrid maximum principle: hence first-order necessary optimality conditions in a Pontryagin form are obtained. Second, this paper proposes a two-steps numerical scheme to solve optimal control problems with loss control regions. It is based on a direct numerical method (applied to a regularized problem) which initializes an indirect numerical method (applied to the original problem and based on the aforementioned necessary optimality conditions). This numerical approach is applied to several illustrative examples.