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

Gugat M, Lazar M (2023)


Publication Language: English

Publication Type: Journal article, Original article

Publication year: 2023

Journal

Book Volume: 33

Pages Range: 429–438

Journal Issue: 3

URI: https://www.amcs.uz.zgora.pl/?action=paper&paper=1715

DOI: 10.34768/amcs-2023-0031

Open Access Link: https://sciendo.com/fr/article/10.34768/amcs-2023-0031

Abstract

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 [r T,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.

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How to cite

APA:

Gugat, M., & Lazar, M. (2023). Optimal Control Problems without Terminal Constraints: The Turnpike Property with Interior Decay. International Journal of Applied Mathematics and Computer Science, 33(3), 429–438. https://doi.org/10.34768/amcs-2023-0031

MLA:

Gugat, Martin, and Martin Lazar. "Optimal Control Problems without Terminal Constraints: The Turnpike Property with Interior Decay." International Journal of Applied Mathematics and Computer Science 33.3 (2023): 429–438.

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