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Paper details
Number 3 - September 2023
Volume 33 - 2023
Optimal control problems without terminal constraints: The turnpike property with interior decay
Martin Gugat, Martin Lazar
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.
Keywords
optimal control, turnpike property, system with hyperbolic PDEs, interior decay