Budget-constrained infinite horizon optimal control problems with linear dynamics

In this paper a class of infinite horizon optimal control problems with an isoperimetrical constraint, also interpreted as a budget constraint, is considered. Herein a linear both in the state and in the control dynamic is allowed. The problem setting includes a weighted Sobolev space as the state space. We investigate the question of existence of an optimal solution and formulate a corresponding theorem. Which influence the isoperimetrical constraint may have on the feasible set and on the existence of an optimal solution is illustrated in details at hand of a linear-quadratic regulator model.