Constructive solution of a bilinear control problem

2006 ◽  
Vol 342 (2) ◽  
pp. 119-124 ◽  
Author(s):  
Lucie Baudouin ◽  
Julien Salomon
Keyword(s):  
Control Problem ◽  
Bilinear Control ◽  
Constructive Solution ◽  
Bilinear Control Problem

scholarly journals Optimal bilinear control problem related to a chemo-repulsion system in 2D domains

2020 ◽  
Vol 26 ◽  
pp. 29 ◽  
Author(s):  
Francisco Guillén-González ◽  
Exequiel Mallea-Zepeda ◽  
María Ángeles Rodríguez-Bellido
Keyword(s):  
Optimal Control ◽  
Control Problem ◽  
Optimal Solution ◽  
Strong Solutions ◽  
Global Optimal Solution ◽  
Production Term ◽  
Bilinear Control ◽  
First Order ◽  
Global Optimal ◽  
Bilinear Control Problem

In this paper, we study a bilinear optimal control problem associated to a chemo-repulsion model with linear production term in a bidimensional domain. The existence, uniqueness and regularity of strong solutions of this model are deduced, proving the existence of a global optimal solution. Afterwards, we derive first-order optimality conditions by using a Lagrange multipliers theorem.

scholarly journals Numerical study of polynomial feedback laws for a bilinear control problem

2018 ◽  
Vol 8 (3) ◽  
pp. 557-582 ◽  
Author(s):  
Tobias Breiten ◽  
◽  
Karl Kunisch ◽  
Laurent Pfeiffer ◽  
Keyword(s):  
Control Problem ◽  
Numerical Study ◽  
Bilinear Control ◽  
Feedback Laws ◽  
Bilinear Control Problem

Numerical Approximation of Bilinear Control of the Schroedinger Equation

2005 ◽  
Author(s):  
Katherine A. Kime
Keyword(s):  
Control Problem ◽  
Numerical Approximation ◽  
Schroedinger Equation ◽  
Quantum Systems ◽  
One Dimensional ◽  
Bilinear Control ◽  
The One ◽  
Localized Initial Data ◽  
Bilinear Control Problem ◽  
Crank Nicolson

We consider the one-dimensional Schroedinger equation in which the control is a time-dependent rectangular potential barrier/well. This is a bilinear control problem, as the potential multiplies the state. Differential geometric methods have been used to treat the bilinear control of systems of finitely many ODEs, and have been applied to the Schroedinger equation (quantum systems). In this paper we will calculate, using MATLAB, explicit controls which steer localized initial data to localized terminal data. These will be obtained using the Crank-Nicolson approximation, in which both space and time are discretized. If one semi-discretizes, in space, one obtains a bilinear control problem for a system of finitely many ODEs. One may pass from the semi-discretized system to Crank-Nicolson using the trapezoid rule. Thus the controls we calculate may be used to construct approximations to controls for the system of ODEs.

scholarly journals Optimal bilinear control of eddy current equations with grad–div regularization

2015 ◽  
Vol 23 (1) ◽  
Author(s):  
Irwin Yousept
Keyword(s):  
Optimal Control ◽  
Control Problem ◽  
Electric Conductivity ◽  
Eddy Current ◽  
Regularity Property ◽  
Control Approach ◽  
Bilinear Control ◽  
Time Harmonic ◽  
Eddy Current Equations ◽  
Higher Regularity

AbstractAn optimal bilinear control problem governed by time-harmonic eddy current equations is considered to estimate the electric conductivity of a 3D bounded isotropic domain. The model problem is mainly complicated by the possible presence of non-conducting materials in the domain. We introduce an optimal control approach based on grad-div regularization and divergence penalization. The estimation for the electric conductivity obtained by solving the optimal control problem is allowed to be discontinuous. Here, no higher regularity property can be derived from the corresponding optimality conditions. We analyze the approach and present various numerical results exhibiting its numerical performance

Constrained bilinear control problem: Application to a cancer chemotherapy model

2017 ◽  
Vol 10 (04) ◽  
pp. 1750054 ◽  
Author(s):  
El Hassan Zerrik ◽  
Nihale El Boukhari
Keyword(s):  
Optimal Control ◽  
Control Problem ◽  
Cancer Chemotherapy ◽  
Sufficient Conditions ◽  
Treatment Protocol ◽  
Bilinear Model ◽  
Bilinear Control ◽  
Finite Dimensional ◽  
Quadratic Cost ◽  
Admissible Controls

The aim of this paper is to investigate the optimal control problem for finite-dimensional bilinear systems and its application to a chemotherapy model. We characterize an optimal control that minimizes a quadratic cost functional in two cases of constrained admissible controls, then we give sufficient conditions for the uniqueness of such a control, and we derive useful algorithms for the computation of the optimal control. The established results are applied to a cancer chemotherapy bilinear model in order to simulate the optimal treatment protocol using two different approaches: one based on a limited instant toxicity, and the other on a limited cumulative toxicity along the therapy session.

Effect of the Spatial Extent of the Control in a Bilinear Control Problem for the Schroedinger Equation

2009 ◽  
Author(s):  
Katherine A. Kime
Keyword(s):  
Differential Geometry ◽  
Finite Difference ◽  
Lie Algebras ◽  
Grid Point ◽  
Schroedinger Equation ◽  
Time Varying ◽  
One Dimensional ◽  
Bilinear Control ◽  
The One ◽  
Bilinear Control Problem

We consider control of the one-dimensional Schroedinger equation through a time-varying potential. Using a finite difference semi-discretization, we consider increasing the extent of the potential from a single central grid-point in space to two or more gridpoints. With the differential geometry package in Maple 8, we compute and compare the corresponding Control Lie Algebras, identifying a trend in the number of elements which span the Control Lie Algebras.

Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem

2020 ◽  
Vol 26 ◽  
pp. 78
Author(s):  
Thirupathi Gudi ◽  
Ramesh Ch. Sau
Keyword(s):  
Finite Element ◽  
Control Problem ◽  
Dirichlet Boundary ◽  
A Priori ◽  
Diffusion Problem ◽  
Discrete Control ◽  
Optimal Order ◽  
Control Constraints ◽  
Element Analysis ◽  
Dirichlet Boundary Control

We study an energy space-based approach for the Dirichlet boundary optimal control problem governed by the Laplace equation with control constraints. The optimality system results in a simplified Signorini type problem for control which is coupled with boundary value problems for state and costate variables. We propose a finite element based numerical method using the linear Lagrange finite element spaces with discrete control constraints at the Lagrange nodes. The analysis is presented in a combination for both the gradient and the L2 cost functional. A priori error estimates of optimal order in the energy norm is derived up to the regularity of the solution for both the cases. Theoretical results are illustrated by some numerical experiments.

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