Francisco Fuica

Francisco Fuica, Enrique Otarola, Abner J. Salgado

We propose an a posteriori error estimator for a sparse optimal control problem: the control variable lies in the space of regular Borel measures. We consider a solution technique that relies on the discretization of the control variable as a linear combination of Dirac measures. The proposed a posteriori error estimator can be decomposed into the sum of two contributions: an error estimator in the maximum norm for the discretization of the adjoint equation and an estimator in the $L^2$-norm that accounts for the approximation of the state equation. We prove that the designed error estimator is locally efficient and we explore its reliability properties. The analysis is valid for two and three-dimensional domains. We illustrate the theory with numerical examples.

Alejandro Allendes, Francisco Fuica, Enrique Otárola

We propose and analyze reliable and efficient a posteriori error estimators for an optimal control problem that involves a nondifferentiable cost functional, the Poisson problem as state equation and control constraints. To approximate the solution to the state and adjoint equations we consider a piecewise linear finite element method whereas three different strategies are used to approximate the control variable: piecewise constant discretization, piecewise linear discretization and the so-called variational discretization approach. For the first two aforementioned solution techniques we devise an error estimator that can be decomposed as the sum of four contributions: two contributions that account for the discretization of the control variable and the associated subgradient, and two contributions related to the discretization of the state and adjoint equations. The error estimator for the variational discretization approach is decomposed only in two contributions that are related to the discretization of the state and adjoint equations. On the basis of the devised a posteriori error estimators, we design simple adaptive strategies that yield optimal rates of convergence for the numerical examples that we perform.

Francisco Fuica, Nicolai Jork

We consider an unregularized optimal control problem subject to the steady-state Navier-Stokes equations. We derive the existence of optimal solutions and prove first- and -- necessary and sufficient -- second-order optimality conditions. To approximate solutions to the optimal control problem, we consider the variational discretization scheme. We analyze convergence properties of the discretization and prove a priori error estimates for locally optimal controls that are nonsingular and which satisfy a growth condition which implies a bang-bang structure.

Alejandro Allendes, Francisco Fuica, Enrique Otárola et al.

We propose and analyze a reliable and efficient a posteriori error estimator for the pointwise tracking optimal control problem of the Stokes equations. This linear-quadratic optimal control problem entails the minimization of a cost functional that involves point evaluations of the velocity field that solves the state equations. This leads to an adjoint problem with a linear combination of Dirac measures as a forcing term and whose solution exhibits reduced regularity properties. We also consider constraints on the control variable. The proposed a posteriori error estimator can be decomposed as the sum of four contributions: three contributions related to the discretization of the state and adjoint equations, and another contribution that accounts for the discretization of the control variable. On the basis of the devised a posteriori error estimator, we design a simple adaptive strategy that illustrates our theory and exhibits a competitive performance.