engleski

Optimal distributed control of the heat-type equations by spectral decomposition

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 minimises 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 generalises 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 o✏ine and saved for future use−, the optimal control problem is solved almost instantaneously. It is practically reduced to a scalar non-linear equation for the optimal Lagrange multiplier. The efficiency of the algorithm is demonstrated through numerical examples corresponding to different types of control operators and penalisation terms.

optimal control ; parabolic equations ; spectral decomposition ; convex optimisation ; dual problem

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