Multitime optimal control with second-order PDEs constraints

In this paper we study a simplified version of multitime optimal control problem for linear second-order partial differential equations (PDE). The multitime multiple integral functional is of Lagrange type. Necessary optimality conditions of multitime maximum principle type are derived. The multitime optimal control with second-order PDE constraints can be analyzed in three ways: as problems governed by (i) explicit m-flows, (ii) implicit m-flows, and (iii) second-order PDEs. All directions lead to variants of multitime maximum principle. The theoretical results are confirmed by solving two significant problems