Volume 27, 2021
Special issue in honor of Enrique Zuazua's 60th birthday
|Number of page(s)||35|
|Published online||20 January 2021|
Exponential Turnpike property for fractional parabolic equations with non-zero exterior data*
Department of Mathematical Sciences and the Center for Mathematics and Artificial Intelligence (CMAI), George Mason University,
2 Universidad de Santiago de Chile, Facultad de Ciencia, Departamento de Matemática y Ciencia de la Computación, Casilla 307-Correo 2, Santiago, Chile.
** Corresponding author: firstname.lastname@example.org
Accepted: 6 November 2020
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 (- Δ) 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.
Mathematics Subject Classification: 35R11 / 35S15 / 49J20 / 49K20
Key words: Fractional heat equation / Dirichlet and Robin external optimal control problems / admissible control operator / turnpike property / exponential turnpike property.
The work of the first author is partially supported by Air Force Office of Scientific Research (AFOSR) under Award NO [FA9550-18-1-0242] and Army Research Office (ARO) under Award NO: W911NF-20-1-0115. The second author is supported by the Conicyt PAI Convocatoria Nacional Subvención a la Instalación en la Academia Convocatoria 2019 PAI77190106.
© EDP Sciences, SMAI 2021
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