Optimal Control Problems without Terminal Constraints: The Turnpike Property with Interior Decay

We show a turnpike result for problems of optimal control with possibly nonlinear systems as well as pointwise-in-time state and control constraints. The objective functional is of integral type and contains a tracking term which penalizes the distance to a desired steady state. In the optimal control problem, only the initial state is prescribed. We assume that a cheap control condition holds that yields a bound for the optimal value of our optimal control problem in terms of the initial data. We show that the solutions to the optimal control problems on the time intervals [0,T ] have a turnpike structure in the following sense: For large T the contribution to the objective functional that comes from the subinterval [T/2,T ], i.e., from the second half of the time interval [0,T ], is at most of the order 1/T . More generally, the result holds for subintervals of the form [rT,T ], where r ∈ (0, 1/2) is a real number. Using this result inductively implies that the decay of the integral on such a subinterval in the objective function is faster than the reciprocal value of a power series in T with positive coefficients. Accordingly, the contribution to the objective value from the final part of the time interval decays rapidly with a growing time horizon. At the end of the paper we present examples for optimal control problems where our results are applicable.