We consider semi-discrete approximations of optimal control problems for linear distributed parameter dynamical systems, with cost functionals in Bolza or infinite horizon form. We give conditions for the convergence of approximate value functions and prove that the approximate optimal controls are a minimizing sequence for the continuous problem. We also show some concrete applications.
FERRETTI R (1997). Internal Approximation Schemes for Optimal Control Problems in Hilbert Spaces. JOURNAL OF MATHEMATICAL SYSTEMS, ESTIMATION, AND CONTROL, 7(1), 1-25.