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

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. 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.

Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs

“…The connection of dissipativity and the turnpike property is also discussed in [22,48]. Recently, turnpike properties for non-observable systems [15,39], for problems arising in deep learning [13] and for fractional parabolic problems [53] were presented. Further, the connection of turnpike properties and long-time behavior of the Hamilton-Jacobi equation was analyzed in [14].…”

“…In this work, we provide a framework to lift the approach of analyzing the extremal equations' solution operator that was considered in [26,27,53] to the nonlinear case. Thus, our approach and the turnpike results results in this paper are similar to [50,51] in the sense that we also consider stabilizability conditions to derive local results via analysis of the extremal equations.…”

Turnpike in optimal control of PDEs, ResNets, and beyond

“…. Many further examples can be fit in this framework, including several classes of degenerate linear parabolic equations ([23,83,68]), evolution equations for the fractional Laplacian with Dirichlet boundary conditions ([182,129]), and so on.…”

Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs

“…The connection of dissipativity and the turnpike property is also discussed in [21,48]. Recently, turnpike properties for non-observable systems [15,39], for problems arising in deep learning [13] and for fractional parabolic problems [53] were presented. Further, the connection of turnpike properties and long-time behavior of the Hamilton-Jacobi equation was analyzed in [14].…”

“…In this work, we provide a framework to lift the approach of analyzing the extremal equations' solution operator that was considered in [25,27,53] to the nonlinear case. Thus, our approach and the turnpike results results in this paper are similar to [50,51] in the sense that we also consider stabilizability conditions to derive local results via analysis of the extremal equations.…”