A Turnpike Result for Boundary Control Problems with the Transport Equation under Uncertainty

Abstract

Turnpike results hold a central position in the area of applied mathematics, offering profound insights into the long-term behavior of optimal control systems. They play an important role in guiding decision-making processes in dynamic systems. Often dynamic systems depend on uncertain data and/or parameters. In this paper, optimal boundary control problems for the transport equation with random initial data and random source term are considered. The paper commences with a presentation of a novel direct proof for a integral turnpike result for the optimal control and the corresponding optimal state in the deterministic setting, that does not require any restrictions on the source term of the transport equation. Further the analysis is extended to the stochastic setting, where the initial data is perturbed by a Wiener process and the source term is multiplied with a random variable. Integral turnpike results for the optimal controls and the corresponding expected optimal states are presented for this setting as well and the turnpike constant is further specified for Gaussian and uniformly distributed random variables.