Exponential Turnpike property for fractional parabolic equations with non-zero exterior data

TL;DR

This paper proves the exponential turnpike property for fractional parabolic equations with non-zero exterior data, showing that optimal controls and states rapidly approach a steady-state as the time horizon increases.

Contribution

It establishes the turnpike property for fractional heat equations with non-zero exterior conditions, including Dirichlet and Robin types, extending previous results to nonlocal operators.

Findings

Proved the turnpike property for fractional heat equations with non-zero exterior data.

Established exponential convergence of optimal controls and states to stationary solutions.

Extended turnpike results to nonlocal Robin boundary conditions.

Abstract

We consider averages convergence as the time-horizon goes to infinity of optimal solutions of time-dependent optimal control problems to optimal solutions of the corresponding stationary optimal control problems. Control problems play a key role in engineering, economics and sciences. To be more precise, in climate sciences, often times, relevant problems are formulated in long time scales, so that, the problem of possible asymptotic behaviors when the time-horizon goes to infinity becomes natural. Assuming that the controlled dynamics under consideration are stabilizable towards a stationary solution, the following natural question arises: Do time averages of optimal controls and trajectories converge to the stationary optimal controls and states as the time-horizon goes to infinity? This question is very closely related to the so-called turnpike property that shows that, often times, the optimal trajectory joining two points that are far apart, consists in, departing from the point of origin, rapidly getting close to the steady-state (the turnpike) to stay there most of the time, to quit it only very close to the final destination and time. In the present paper we deal with heat equations with non-zero exterior conditions (Dirichlet and nonlocal Robin) associated with the fractional Laplace operator $(-\Delta)^s$ ($0<s<1$). We prove the turnpike property for the nonlocal Robin optimal control problem and the exponential turnpike property for both Dirichlet and nonlocal Robin optimal control problems.