Naslov
Distributed optimal control of parabolic equations by spectral decomposition

Autori
Lazar, Martin ; Molinari, Cesare

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
Croatian German Meeting on Analysis and Mathematical Physics, 2021 / - , 2021, 11-12

Skup
Croatian German Meeting on Analysis and Mathematical Physics

Mjesto i datum
Online, 21.03.2021. - 25.03.2021

Vrsta sudjelovanja
Pozvano predavanje

Vrsta recenzije
Nije recenziran

Ključne riječi
convex optimization ; dual problem ; optimal control ; parabolic equations ; , spectral decomposition
(convex optimization ; dual problem ; , optimal control ; parabolic equations ; spectral decomposition)

Sažetak
We consider the constrained minimisation problem ( P ) min u ∈ L T , U 2 {; ; J ( u ) : x ( T ) ∈ B ε ( x T ) ― }; ; , where x T is some given target state, J is a given cost functional and x is the solution of ( E ) {; ; d d t x(t)+ A x(t)= B t u(t) for t∈ ( 0 , T ) x(0)=0. Here A is an unbounded linear operator allowing for spectral decomposition, while B t is a (time dependent) control operator. If the cost functional J is given by J ( u ) = ‖ u ‖ L 2 the problem ( P ) is reduced to a classical minimal norm control problem which can be solved by Hilbert uniqueness method (HUM). In [1] we allow for a more general cost functional and analyse examples in which, apart from the target state and the control norm, one considers a desired trajectory and penalise a distance of the state from it. Such problem requires a more general approach, and it has been addressed by different methods throughout last decades. In this paper we suggest another method based on the spectral decomposition in terms of eigenfunctions of the operator A . Surprisingly, the problem reduces to a non-linear equation for a scalar unknown, representing a Lagrangian multiplier. The same approach has been recently introduced in [2] for an optimal control problem of the heat equation in which the control was given through the initial datum. This paper generalises the method to the distributed control problems. As can be expected, in this case one has to consider the associated dual problem which makes the calculation more complicated, although the algorithm steps follow a similar structure as in [2]. In the talk basic steps of the method will be explained, followed by numerical examples demonstrating its efficiency.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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