Optimality conditions for differential system of Petrowsky type with infinite number of variables and boundary control

In this paper, we study the optimal control problem for an n×nn×n coupled Petrowsky type system involving a 2ℓ2ℓ-th order self-adjoint elliptic operator with an infinite number of variables and constrained boundary control acting through Neumann conditions. Also, we derived the necessary and sufficient conditions of optimality for two types of performance index (quadratic one, general integral form).By using standard Lions’s arguments [J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, vol. 170, Springer-Verlag, 1971] we proved the existence of a solution to the n×nn×n coupled Petrowsky system and we derived optimality conditions for the optimal control problem with a quadratic performance index. In the case of the general integral form of the performance index we applied Dubovitskii–Milyutin’s formalism earlier used in Kotarski [W. Kotarski, Some problems of optimal and pareto optimal control for distributed parameter systems, Reports of Silesian University Katowice, Poland, 1997, no. 1668]. Finally, we provided some special cases.