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

• Published in: International journal of applied mathematics and computer science 2023-09, Vol.33 (3), p.429-438
• Main Authors: Gugat, Martin, Lazar, Martin
• Format: Article
• Language: English
• Subjects: Decay; interior decay; Nonlinear control; Nonlinear systems; Optimal control; Power series; Real numbers; system with hyperbolic PDEs; Terminal constraints; Toll roads; turnpike property
• ISSN: 1641-876X; 2083-8492
• DOI: 10.34768/amcs-2023-0031

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 ] have a turnpike structure in the following sense: For large the contribution to the objective functional that comes from the subinterval [ ], i.e., from the second half of the time interval [0 ], is at most of the order 1 . More generally, the result holds for subintervals of the form [ ], where (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 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.