On Deterministic Control Problems: An Approximation Procedure for the Optimal Cost I. The Stationary Problem

We study deterministic optimal control problems having stopping time, continuous and impulse controls in each strategy. We obtain the optimal cost, considered as the maximum element of a suitable set of subsolutions of the associated Hamilton–Jacobi equation, using an approximation method. A particular derivative discretization scheme is employed. Convergence of approximate solutions is shown taking advantage of a discrete maximum principle which is also proved. For the numerical solutions of approximate problems we use a method of relaxation type. The algorithm is very simple; it can be run on computers with small central memory. In Part I we study the stationary case, in Part II [SIAM J. Control Optim., 23 (1985), pp. 267–285] we study the nonstationary case.