The purpose of this paper is to establish an approximate method for solving constrained optimal control problems defined in Banach space by means of an interior penalty method.
The constraints treated herein are described by equalities and inequalities. These constraints, the control variables and the state variables are defined in general real Banach space.
At first, the case in which the equality constraints and the inequality constraints are embeded in the penalty function to obtain the approximate solutions is considered, and it is shown that a series of the approximate solutions converges to the optimal solution weakly or strongly and a series of the approximate values of the cost functional, converges to the optimal value.
Next, the case in which the inequality constraints are embeded only in the penalty function is treated. In this case, the results obtained for some functionals are symmetrical to the results obtained by the exterior penalty method. Especially the upper bounds for the optimal value of the cost functional are obtained, while the lower bounds are obtained generally by the exterior method.