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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