The connection between discrete and continuous state constrained optimal control systems

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