The Pontryagin Maximum Principle for Nonlinear Optimal Control Problems with Infinite Horizon

The famous proof of the Pontryagin maximum principle for control problems on a finite horizon bases on the needle variation technique, as well as the separability concept of cones created by disturbances of the trajectories. In this paper, we follow the approach to state the optimal control problem as an extremal problem in function spaces, and then realizing the Lagrange principle for this extremal problem via the separation technique. The result is the Pontryagin maximum principle as necessary condition for a strong local minimizer in infinite horizon optimal control problems. This approach yields the existence of the adjoint and the validity of the transversality conditions at infinity. To get the Lagrange multipliers rule in this particular context, we introduce a multiple needle variation technique on the infinite horizon.