10 research outputs found
A pointwise tracking optimal control problem for the stationary Navier--Stokes equations
We study a pointwise tracking optimal control problem for the stationary
Navier--Stokes equations; control constraints are also considered. The problem
entails the minimization of a cost functional involving point evaluations of
the state velocity field, thus leading to an adjoint problem with a linear
combination of Dirac measures as a forcing term in the momentum equation, and
whose solution has reduced regularity properties. We analyze the existence of
optimal solutions and derive first and, necessary and sufficient, second order
optimality conditions in the framework of regular solutions for the
Navier--Stokes equations. We develop two discretization strategies: a
semidiscrete strategy in which the control variable is not discretized, and a
fully discrete scheme in which the control variable is discretized with
piecewise constant functions. For each solution technique, we analyze
convergence properties of discretizations and derive a priori error estimates
A DPG method for linear quadratic optimal control problems
The DPG method with optimal test functions for solving linear quadratic
optimal control problems with control constraints is studied. We prove
existence of a unique optimal solution of the nonlinear discrete problem and
characterize it through first order optimality conditions. Furthermore, we
systematically develop a priori as well as a posteriori error estimates. Our
proposed method can be applied to a wide range of constrained optimal control
problems subject to, e.g., scalar second-order PDEs and the Stokes equations.
Numerical experiments that illustrate our theoretical findings are presented
error estimates for semilinear optimal control problems
In two and three dimensional Lipschitz, but not necessarily convex, polytopal domains, we devise and analyze a reliable and efficient a posteriori error estimator for a semilinear optimal control problem; control constraints are also considered. We consider a fully discrete scheme that discretizes the state and adjoint equations with piecewise linear functions and the control variable with piecewise constant functions. The devised error estimator can be decomposed as the sum of three contributions which are associated to the discretization of the state and adjoint equations and the control variable. We extend our results to a scheme that approximates the control variable with piecewise linear functions and also to a scheme that approximates the solution to a nondifferentiable optimal control problem. We illustrate the theory with two and three-dimensional numerical examples