A Palais-Smale type condition for optimal control problems

We consider an optimal control problem with a general control constraint and a fixed initial state, for which convexity assumptions are absent. To establish the existence of optimal pairs, we introduce a control version of the Palais-Smale type condition adapted from critical point theory and prove that optimal pairs exist under this condition (see Theorem 2.1). Subsequently, we verify that a specific class of optimal control problems satisfies our Palais-Smale type condition (see Proposition 2.1). Furthermore, as an application, we present an example (Example 3.1) and show that, to the best of our knowledge, no existing result can be applied to establish the existence of an optimal pair in this case (see Section 3.1.1). Example 3.2 shows that finitely many Pontryagin maximum principle candidates do not guarantee the existence of an optimal control. Finally, Example 3.3 demonstrates that the proposed Palais-Smale type condition is sufficient, but not necessary, for the existence of optimal controls.