In this paper the optimal control problems with constraints in which the state and control variables are allowed to range in Banach spaces are studied by means of a penalty function approach. The constraints treated herein are described by equalities and inequalities which are defined in Banach spaces.
At first, the existence of the optimal and approximate solutions and the convergence of a series of the approximate solutions to the optimal one are discussed in the problems in general Banach space.
Next, the problems in L^p or Euclidean space are treated and the necessary and sufficient conditions for the existence of the optimal solution are derived. These conditions are analogous to those in the Kuhn-Tucher's theorem generalized in the infinite dimensional space.
The results obtained are thought to be applicable extensively to the optimal control problems for the distributed-parameter, lumped-parameter and discrete time systems in which the state and control variables are constrained.