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
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.
Authors with CRIS profile
Martin Gugat
Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)
Involved external institutions
Croatia (HR)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://dx.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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