The Experts below are selected from a list of 111 Experts worldwide ranked by ideXlab platform

Michael K Giles - One of the best experts on this subject based on the ideXlab platform.

Rensheng Dou - One of the best experts on this subject based on the ideXlab platform.

Abderrahim Hantoute - One of the best experts on this subject based on the ideXlab platform.

Arnaud Munch - One of the best experts on this subject based on the ideXlab platform.

  • numerical approximation of bang bang controls for the heat equation an optimal design approach
    Systems & Control Letters, 2013
    Co-Authors: Arnaud Munch, Francisco Periago
    Abstract:

    This work is concerned with the numerical computation of null controls of minimal $L^{\infty}$-norm for the linear heat equation with a bounded potential. Both, the cases of internal and boundary (Dirichlet and Neumann) controls are considered. Dual arguments allow to reduce the search of controls to the unconstrained minimization of a Conjugate Function with respect to the initial condition of a backward heat equation. However, as a consequence of the regularizing property of the heat operator, this initial (final) condition lives in a huge space, that can not be approximated with robustness. For this reason, very specific to the parabolic situation, the minimization is severally ill-posed. On the other hand, the optimality conditions for this problem show that, in general, the unique control $v$ of minimal $L^{\infty}$-norm has a bang-bang structure as he takes only two values: this allows to reformulate the problem as an optimal design problem where the new unknowns are the amplitude of the bang-bang control and the space-time regions where the control takes its two possible values. This second optimization variable is modeled through a characteristic Function. Since the admissibility set for this new control problem is not convex, we obtain a relaxed formulation of it which leads to a well-posed relaxed problem and lets use a gradient descent method for the numerical resolution of the problem. Numerical experiments, for the inner and boundary controllability cases, are described within this new approach.

  • NUMERICAL NULL CONTROLLABILITY OF THE HEAT EQUATION THROUGH A LEAST SQUARES AND VARIATIONAL APPROACH
    2012
    Co-Authors: Arnaud Munch, Pablo Pedregal
    Abstract:

    This work is concerned with the numerical computation of null controls for the heat equation. The goal is to compute an approximation of controls that drives the solution from a prescribed initial state at t = 0 to zero at t = T. In spite of the diffusion of the heat equation, recent developments indicate that this issue is difficult and still largely open. Most of the existing literature, concerned with controls of minimal L2-norm, make use of dual convex arguments and introduce backward adjoint system. In practice, the null control problem is then reduced to the minimization of a dual Conjugate Function with respect to the final condition of the adjoint state. As a consequence of the highly regularizing property of the heat kernel, this final condition - which may be seen as the Lagrange multiplier for the null controllability condition - does not belongs to L2, but to a much larger space than can hardly be approximated by finite (discrete) dimensional basis. This phenomenon, unavoidable whatever be the numerical approximation used, strongly deteriorates the efficiency of minimization algorithms. In this work, we do not use duality arguments and in particular do not introduce any backward heat equation. For the boundary case, the approach consists, first, in introducing a class of Functions satisfying a priori the boundary conditions in space and time - in particular the null controllability condition at time T-, and then finding among this class one element satisfying the heat equation. This second step is done by minimizing a convex Functional, among the admissible corrector Functions of the heat equation. The inner case is performed in a similar way. We present the (variational) approach, discuss the main features of it, and then describe some numerical experiments highlighting the interest of the method. The method holds in any dimension but, for the sake of simplicity, we provide details in the one-space dimensional case.

Francisco Periago - One of the best experts on this subject based on the ideXlab platform.

  • numerical approximation of bang bang controls for the heat equation an optimal design approach
    Systems & Control Letters, 2013
    Co-Authors: Arnaud Munch, Francisco Periago
    Abstract:

    This work is concerned with the numerical computation of null controls of minimal $L^{\infty}$-norm for the linear heat equation with a bounded potential. Both, the cases of internal and boundary (Dirichlet and Neumann) controls are considered. Dual arguments allow to reduce the search of controls to the unconstrained minimization of a Conjugate Function with respect to the initial condition of a backward heat equation. However, as a consequence of the regularizing property of the heat operator, this initial (final) condition lives in a huge space, that can not be approximated with robustness. For this reason, very specific to the parabolic situation, the minimization is severally ill-posed. On the other hand, the optimality conditions for this problem show that, in general, the unique control $v$ of minimal $L^{\infty}$-norm has a bang-bang structure as he takes only two values: this allows to reformulate the problem as an optimal design problem where the new unknowns are the amplitude of the bang-bang control and the space-time regions where the control takes its two possible values. This second optimization variable is modeled through a characteristic Function. Since the admissibility set for this new control problem is not convex, we obtain a relaxed formulation of it which leads to a well-posed relaxed problem and lets use a gradient descent method for the numerical resolution of the problem. Numerical experiments, for the inner and boundary controllability cases, are described within this new approach.