Optimal Control of Parabolic Systems with Boundary Conditions Involving Time Delays

Various optimal control problems with quadratic cost functionals for parabolic systems with Neumann boundary conditions involving time delays are considered. Necessary and sufficient conditions which the optimal controls must satisfy are derived. Estimates and a sufficient condition for the boundedness of solutions are obtained for systems with specified forms of feedback controls. Similar problems for systems with more complex boundary conditions involving time delays are discussed briefly.