Optimality of the boundary control for n×n parabolic lag system

Optimal boundary control problem for n×n coupled system of second order parabolic lag partial differential equation with infinitely many variables is considered. By Lions scheme [J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Grundlehren Math. Wiss., vol. 170, Springer-Verlag, 1971], necessary and sufficient condition of optimality for the Neumann problem with quadratic performance functional and constrained control is derived. Finally, several mathematical examples for derived optimality conditions are presented.