The connection between discrete and continuous state constraints optimal control systems: the deterministic case

An optimal control problem driven by an ordinary differential equation under continuous state constraints is considered in this study. From an operational point of view, we introduce a discrete state constraints optimal control problem and prove that this discrete state constraints optimal control problem is a near-optimal control problem of the original problem. Furthermore, we show that the optimal solution of the near-optimal control problem converges to the optimal solution of the original one. Finally, we use a linear quadratic optimal problem to verify the main results of this study.