11 research outputs found
Experimental Study of the HUM Control Operator for Linear Waves
We consider the problem of the numerical approximation of the linear
controllability of waves. All our experiments are done in a bounded domain
\Omega of the plane, with Dirichlet boundary conditions and internal control.
We use a Galerkin approximation of the optimal control operator of the
continuous model, based on the spectral theory of the Laplace operator in
\Omega. This allows us to obtain surprisingly good illustrations of the main
theoretical results available on the controllability of waves, and to formulate
some questions for the future analysis of optimal control theory of waves.Comment: 38 figure
Numerical controllability of the wave equation through primal methods and Carleman estimates
This paper deals with the numerical computation of boundary null controls for
the 1D wave equation with a potential. The goal is to compute an approximation
of controls that drive the solution from a prescribed initial state to zero at
a large enough controllability time. We do not use in this work duality
arguments but explore instead a direct approach in the framework of global
Carleman estimates. More precisely, we consider the control that minimizes over
the class of admissible null controls a functional involving weighted integrals
of the state and of the control. The optimality conditions show that both the
optimal control and the associated state are expressed in terms of a new
variable, the solution of a fourth-order elliptic problem defined in the
space-time domain. We first prove that, for some specific weights determined by
the global Carleman inequalities for the wave equation, this problem is
well-posed. Then, in the framework of the finite element method, we introduce a
family of finite-dimensional approximate control problems and we prove a strong
convergence result. Numerical experiments confirm the analysis. We complete our
study with several comments
Numerical null controllability of the 1D heat equation: Carleman weights an duality
This paper deals with the numerical computation of distributed null controls for the 1D heat equation. The goal is to compute a control that drives (a numerical approximation of) the solution from a prescribed initial state at t = 0 exactly to zero at t = T. We extend the earlier contribution of Carthel, Glowinski and Lions [5], which is devoted to the computation of minimal L2-norm controls. We start from some constrained extremal problems introduced by Fursikov and Imanuvilov [15]) and we apply appropriate duality techniques. Then, we introduce numerical approximations of the associated dual problems and we apply conjugate gradient algorithms. Finally, we present several experiments, we highlight the in uence of the weights and we analyze this approach in terms of robustness and e fficiency
Spacetime finite element methods for control problems subject to the wave equation
We consider the null controllability problem for the wave equation, and analyse a stabilized finite element method formulated on a global, unstructured spacetime mesh. We prove error estimates for the approximate control given by the computational method. The proofs are based on the regularity properties of the control given by the Hilbert Uniqueness Method, together with the stability properties of the numerical scheme. Numerical experiments illustrate the results
Experimental Study of the HUM Control Operator for Linear Waves
National audienc
Experimental Study of the HUM control operator for Linear Waves
International audienc
Experimental Study of the HUM Control Operator for Linear Waves
National audienc