Numerical Solution of Optimal Feedback Control for Finite Horizon Optimal Control Problem with Growth Condition

In this paper, we study approximation of optimal feedback control for the finite horizon optimal control problem with growth conditions for control systems and cost functionals. The optimal control problems with growth conditions are usually applied in practice. In reducing the boundedness for the dynamics function to the growth conditions, we have some difficulty in proving the convergence of the approximation scheme by the Euler scheme. First, we prove comparison principle of unbounded viscosity solution under the growth condition for the dynamics function and prove that the value function is the unique viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation. We then construct a new approximation scheme using Euler implicit scheme for the finite horizon optimal control problem and prove that the approximation value function converges to the viscosity solution of the HJB equation as the discretization step goes to zero. Based on the approximations of the viscosity solution, we construct the optimal feedback controls for the corresponding discrete control systems. We consider an example to show the efficiency of the time-space approximation algorithm by the Euler implicit scheme through computational results.