Finite element approximation of sparse parabolic control problems

Abstract

We study the finite element approximation of an optimal control problem governed by a semilinear partial differential equation and whose objective function includes a term promoting space sparsity of the solutions. We prove existence of solution in the absence of control bound constraints and provide the adequate second order sufficient conditions to obtain error estimates. Full discretization of the problem is carried out, and the sparsity properties of the discrete solutions, as well as error estimates, are obtained.

Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity

“…Proceeding in this way, we recover the same sparsity pattern exhibited by the continuous solution of the control problem. This sparse structure was also proved in [10] for approximations of the controls by piecewise constant functions in time and space. This numerical integration leads to new difficulties in the derivation of error estimates.…”

“…In this work, we continue our study, started in [10], about finite element approximations of problems where the optimal solutions are directionally sparse.…”

“…Both in the paper at hand and in [10] the state equation is a parabolic partial differential equation and the functional involves a tracking-type term for the state, a Tikhonov regularization term for the control and a non differentiable term that promotes spatial sparsity. Also in both works, the state and the adjoint state equations are discretized using a discontinuous in time and continuous in space Galerkin scheme, namely dG0-cG1, and the controls are discretized in time using piecewise constant functions.…”

“…While in [10] the controls are discretized in space using piecewise constant functions, in this work we discretize the controls using functions that are piecewise constant in time and continuous piecewise linear in space. For easier comparison of the results, we will refer to the space of approximation functions used in [10] as Uσ and to the space of approximation functions used in this work as Vσ.…”

“…Numerical experiments in Section 8 confirm this obtained order of convergence. This order seems optimal for our approach, regarding the H 1 (Q)-regularity of the optimal control -see [10,Theorem 3.3]-, which limits the order of convergence we can achieve. This is not a special situation in sparse control; see e.g.…”