Optimal control of 2-D wave differential inclusions with state constraints

The paper discusses the optimization of 2-D wave differential inclusions (DFIs) with the Laplacian in a bounded cuboid and the first mixed initial-boundary-value problem. Particular attention is paid to problems with state constraints, for which optimality conditions are formulated in terms of the Euler-Lagrange adjoint inclusions. Next, using the well-known Green?s formula, the obtained results are generalized to the multidimensional case. In problems with convex inequalities, dual cones are calculated, which are an integral part of Euler-Lagrange inclusions. The examples show that when constructing an adjoint inclusion, it is very important to consider the phase boundary?.