On the convergence of difference approximations in distributed parameter optimal control problems

Since optimal control problems for distributed parameter systems are intimately connected to optimal control problems described by partial differential equations and require a treatment in function spaces, there is a great difficulty to solve these problems directly. Consequently it is considered to be necessary to obtain approximating solutions by nummerical computation. In computing a numerical solution of a partial differential equation, an approximating method in which differential operator is replaced by a difference operator is widely used. A similar approximating procedure is commonly used in calculating an optimal control in systems with distributed parameters. Convergency conditions of a sequence of approximating solutions for the optimal control problems of distributed parameter systems are given in this paper.