Mariano Mateos

Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems

We study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. The analysis of the approximate control problems is carried out. The uniform convergence of discretized controls to optimal controls is proven under natural assumptions by taking piecewise constant controls. Finally, error estimates are established and some numerical experiments, which confirm the theoretical results, are performed.

Second Order Optimality Conditions for Semilinear Elliptic Control Problems with Finitely Many State Constraints

This paper deals with necessary and sufficient optimality conditions for control problems governed by semilinear elliptic partial differential equations with finitely many equality and inequality state constraints. Some recent results on this topic for optimal control problems based upon results for abstract optimization problems are compared with some new results using methods adapted to the control problems. Meanwhile, the Lagrangian formulation is followed to provide the optimality conditions in the first case; the Lagrangian and Hamiltonian functions are used in the second statement. Finally, we prove the equivalence of both formulations.

Error estimates for the numerical approximation of Neumann control problems

Abstract We continue the discussion of error estimates for the numerical analysis of Neumann boundary control problems we started in Casas et al. (Comput. Optim. Appl. 31:193–219, 2005). In that paper piecewise constant functions were used to approximate the control and a convergence of order O(h) was obtained. Here, we use continuous piecewise linear functions to discretize the control and obtain the rates of convergence in L2(Γ). Error estimates in the uniform norm are also obtained. We also discuss the approach suggested by Hinze (Comput. Optim. Appl. 30:45–61, 2005) as well as the improvement of the error estimates by making an extra assumption over the set of points corresponding to the active control constraints. Finally, numerical evidence of our estimates is provided.

Error Estimates for the Numerical Approximation of a Distributed Control Problem for the Steady-State Navier-Stokes Equations

We obtain error estimates for the numerical approximation of a distributed control problem governed by the stationary Navier-Stokes equations, with pointwise control constraints. We show that the

On saturation effects in the Neumann boundary control of elliptic optimal control problems

L^2

On the Regularity of the Solutions of Dirichlet Optimal Control Problems in Polygonal Domains

-norm of the error for the control is of order

A Paradox in the Approximation of Dirichlet Control Problems in Curved Domains.

h^2

Numerical approximation of elliptic control problems with finitely many pointwise constraints

if the control set is not discretized, while it is of order

Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes

h

Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems. Continuous Piecewise Linear Approximations

if it is discretized by piecewise constant functions. These error estimates are obtained for local solutions of the control problem, which are nonsingular in the sense that the linearized Navier-Stokes equations around these solutions define some isomorphisms, and which satisfy a second order sufficient optimality condition. We establish a second order necessary optimality condition. The gap between the necessary and sufficient second order optimality conditions is the usual gap known for finite dimensional optimization problems.