Sensitivity Analysis of Optimal Control for a Class of Parabolic PDEs Motivated by Model Predictive Control

Abstract

We analyze the sensitivity of the extremal equations that arise from the first order optimality conditions for time dependent optimization problems. More specifically, we consider parabolic PDEs with distributed or boundary control and a linear quadratic performance criterion. We prove the solution's boundedness with respect to the right-hand side of the first order optimality condition which includes initial data. If the system fulfills a particular stabilizability and detectability assumption, the bound is independent of the time horizon. As a consequence, the influence of a perturbation of the right-hand side decreases exponentially backward in time. We use this property for the construction of efficient numerical discretizations in a Model Predictive Control scheme. Moreover, a quantitative turnpike theorem in the W ([0, T ])-norm is derived.

Exponential sensitivity and turnpike analysis for linear quadratic optimal control of general evolution equations

“…Our paper generalizes recent results in this direction from parabolic equations [14], which enjoy very favorable smoothing properties, to general evolution equations with much less structure, which in particular comprises hyperbolic equations. This makes the analysis much more delicate.…”

“…This makes the analysis much more delicate. For example, in [14], the boundary control or boundary observation case could be dealt with in a straightforward fashion, as the state and adjoint belong to a Sobolev space for a.a. times on which boundary control or observation operators are bounded. This is no longer the case when moving to general evolution equations.…”

“…The analysis in this work is based on analyzing the extremal equations of the optimal control problem, avoiding the use of Riccati equations. Compared to [14] we use weaker assumptions and thus obtain estimates in weaker norms, concerning spatial regularity, which is to be expected when going from parabolic to e.g. hyperbolic equations.…”

“…[25,24,21,22,6,5]. The proofs establishing the turnpike property in the recent works [5,24,14] are based on a stabilizability and detectability assumption on the system. Turnpike theorems in Hilbert spaces were given in [24] for general semigroups with bounded control and observation operators, and for boundary controlled parabolic equations.…”

On the Turnpike Property and the Receding-Horizon Method for Linear-Quadratic Optimal Control Problems

“…The main novelty of our result is the estimate (11) in weighted spaces, with a constant M which is independent of T . Let us mention that a similar result has been obtained in [12,Theorem 2], for negative weights. The proof is based on a Neumann-series argument.…”