Optimal distributed control of linear parabolic equations by spectral decomposition. (15th February 2021)

• Record Type: Journal Article
• Authors: Lazar, Martin; Molinari, Cesare
• Is Part Of: Optimal control applications and methods. Volume 42:Number 4(2021)
• Page Start: 891
• Page End: 926
• Publication Date: 2021-02-15
• Subjects: convex optimization -- dual problem -- optimal control -- parabolic equations -- spectral decomposition; Control theory -- Periodicals; Mathematical optimization -- Periodicals; 629.8312
• DOI: 10.1002/oca.2708
• Languages: English
• ISSNs: 0143-2087

Abstract

Abstract: We construct an algorithm for solving a constrained optimal control problem for a first‐order evolutionary system governed by a positive self‐adjoint operator. The problem consists in identifying distributed control that minimizes a given cost functional, which comprises a cost of the control and a trajectory regulation term, while steering the final state close to a given target. The approach explores the dual problem and it generalizes the Hilbert Uniqueness Method (HUM). The practical implementation of the algorithm is based on a spectral decomposition of the operator determining the dynamics of the system. Once this decomposition is available – which can be done offline and saved for future use – the optimal control problem is solved almost instantaneously. It is practically reduced to a scalar nonlinear equation for the optimal Lagrange multiplier. The efficiency of the algorithm is demonstrated through numerical examples corresponding to different types of control operators and penalization terms.