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Bilinear Optimal Control of an Advection-Reaction-Diffusion System

We consider the bilinear optimal control of an advection-reaction-diffusion system, where the control arises as the velocity field in the advection term. Such a problem is generally challenging from both theoretical analysis and algorithmic design perspectives, mainly because the state variable depends nonlinearly on the control variable and, an additional divergence-free constraint on the control is coupled together with the state equation. Mathematically, the proof of the existence of optimal solutions is delicate, and, up to now, only some results have been known for a few special cases where additional restrictions are imposed on the space dimension and the regularity of the control. We prove the existence of optimal controls and derive the first-order optimality conditions in general settings without any extra assumptions. Computationally, the well-known conjugate gradient (CG) method can be applied conceptually. However, due to the additional divergence-free constraint on the control variable and the nonlinear relation between the state and control variables, it is challenging to compute the gradient and the optimal stepsize at each CG iteration, and thus nontrivial to implement the CG method. To address these issues, we advocate a fast inner preconditioned CG method to ensure the divergence-free constraint and an efficient inexactness strategy to determine an appropriate stepsize. An easily implementable nested CG method is thus proposed for solving such a complicated problem. For the numerical discretization, we combine finite difference methods for the time discretization and finite element methods for the space discretization. Efficiency of the proposed nested CG method is promisingly validated by the results of some preliminary numerical experiments.

Ketersediaan

Informasi Detail

Judul Seri

SIAM REVIEW; Vol.64 No.2 June 2022

No. Panggil

-

Penerbit

: .,

Deskripsi Fisik

p. 392-421

Bahasa

English

ISBN/ISSN

-

Klasifikasi

NONE

Tipe Isi

-

Tipe Media

-

Tipe Pembawa

-

Edisi

-

Subjek

FINITE ELEMENT METHOD
FINITE DIFFERENCE METHOD
CONJUGATE GRADIENT METHOD
BILINEAR OPTIMAL CONTROL
ADVECTION-REACTION-DIFFUSION SYSTEM
NESTED ITERATION

Info Detail Spesifik

https://doi.org/10.1137/21M1389778

Pernyataan Tanggungjawab

Roland Glowinski, Yongcun Song, Xiaoming Yuan, Hangrui Yue

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