Bilinear minimax control problems for a class of parabolic systems with applications to control of nuclear reactors
type: Item
instance of: scholarly article
title: Bilinear minimax control problems for a class of parabolic systems with applications to control of nuclear reactors
zbMATH Open document ID: 1116.49014
author: A. Belmiloudi
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-01-09T00:00:00Z
review text: Consider the following parabolic equation \[ u_t-F(x)u = H(x)u-(f+g)B(x)u \;in \;Q=\Omega \times (0,T), \] subjected to the boundary condition \[ Lu=0 \;in \;\Sigma = \partial \Omega \times (0,T), \] and the initial condition \[ u(0)=u_0 \;in \;\Omega, \] under the pointwise constraints \[ a\leq f(x,t)\leq b, \;c\leq g(x,t)\leq d \;a.e. \;in \;Q. \] Here \(u\) is an \(n\)-dimensional state vector, \(\Omega\) is an open subset of \(\mathbb{R}^N\), \(N\leq 3\), with a \(C^{\infty}\) boundary \(\partial \Omega\), \(0\in [a,b]\cap [c,d]\), and \(B(x)\), \(H(x)\) are matrices with \(L^{\infty}\) entries. In applications \(F\) is an elliptic operator, and the boundary operator \(L\) is of Robin type. Let us define the set of admissible controls, \(U_{ad} = \{f\in L^2(Q); \;a\leq f \leq b \;a.e. \;in \;Q\}\), as well the set of admissible disturbances or noises, \(W_{ad}=\{g\in L^2(Q); \;c\leq g \leq d \;a.e. \;in \;Q\}\). We introduce the cost functional \[ J(f,g) = (1/2){\int}_QK(u-u_{obs})\cdot (u-u_{obs})\, dx\,dt +{\alpha /2}{\| f\| }^2_{L^2(Q)} - {\beta /2}{\| g\| }^2_{L^2(Q)}, \] where \(K\) is symmetric, positive-definite matrix with smooth entries. The paper deals with the following minimax control problem: Find \(f^*\in U_{ad}\), \(g^*\in W_{ad}\) such that \((f^*,g^*)\) is a saddle point of \(J(f,g)\) subject to the above system and constraints. The existence of an optimal solution is proved, and the appropriate necessary conditions for saddle point optimality are derived. Two examples associated with nuclear fission are presented.
Mathematics Subject Classification ID: 49K30
Mathematics Subject Classification ID: 49K20
zbMATH DE Number: 5083301
zbMATH Keywords: bilinear minimax control
zbMATH Keywords: reaction-diffusion
zbMATH Keywords: saddle point
zbMATH Keywords: adjoint equation
zbMATH Keywords: pointwise controllers
zbMATH Keywords: neutron fission
reviewed by: Gheorghe Moroşanu
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.04.037
OpenAlex ID: W2016026883
cites work: Equivalent Norms for Sobolev Spaces
cites work: Nonlinear Uncertain Systems and Necessary Conditions of Optimality
cites work: Minimax Control of Parabolic Systems with State Constraints
cites work: Q4002915
cites work: Controllability for Distributed Bilinear Systems
cites work: Q3725052
cites work: The Pontryagin maximum principle for minimax problems of optimal control
cites work: Robust control problems of vortex dynamics in superconducting films with Ginzburg-Landau complex systems
cites work: Robust and optimal control problems to a phase-field model for the solidification of a binary alloy with a constant temperature
cites work: Robust control problems associated with time-varying delay nonlinear parabolic equations
cites work: Nonlinear robust control problems of parabolic type equations with time-varying delays given in the integral form
cites work: Bilinear spatial control of the velocity term in a Kirchhoff plate equation
cites work: Bilinear optimal control of the velocity term in a Kirchhoff plate equation
cites work: Optimal control of distributed nuclear reactors
cites work: Q4085497
cites work: Robust control of infinite dimensional systems. Frequency domain methods
cites work: Differential stability and robust control of nonlinear systems
cites work: Robustness analysis of nonlinear feedback systems: an input-output approach
cites work: Q4255489
cites work: Nonlinear control systems: An introduction
cites work: Sharp regularity theory for elastic and thermoelastic Kirchhoff equations with free boundary conditions
cites work: Bilinear optimal control for a wave equation with viscous damping
cites work: Optimal control of multigroup neutron fission systems
cites work: Q3278561
cites work: Q5549989
cites work: Min-max game theory and algebraic Riccati equations for boundary control problems with continuous input-solution map. II: The general case
cites work: Q4283320
cites work: Min-max game theory and algebraic Riccati equations for boundary control problems with analytic semigroups—II. The general case
cites work: Minimax control of parabolic systems with Dirichlet boundary conditions and state constraints
cites work: Q4314421
cites work: Minimax control of nonlinear evolution equations.
cites work: A robust control framework for linear, time-invariant, spatially distributed systems
cites work: On Positive Orthant Controllability of Bilinear Systems in Small Codimensions
cites work: Optimal control of distributed nuclear reactors with pointwise controllers
cites work: Q3128879
DOI: 10.1016/J.JMAA.2006.04.037
description: scientific article; zbMATH DE number 5083301