On the convergence of proximal point methods for the optimal control problem with a linear control system

This paper is devoted to study the convergence theory of proximal point algorithm (PPA) for the Mayer-type problem with a finite-dimensional linear control system. We showed that if the cost functional is convex, the iterative sequence of controls of PPA converges weakly to the optimal one. Moreover, under some common assumptions we proved the strong convergence with a linear convergence rate based on the controllability index. In addition, an example is provided with numerical tests to confirm the theoretical results.