Journal of Computational and Applied Mathematics
A posteriori error estimates for mixed finite element solutions of convex optimal control problems
Journal of Computational and Applied Mathematics
Journal of Scientific Computing
Robust error estimates for the finite element approximation of elliptic optimal control problems
Journal of Computational and Applied Mathematics
Mixed Finite Element Method for Dirichlet Boundary Control Problem Governed by Elliptic PDEs
SIAM Journal on Control and Optimization
Error Estimates of Stochastic Optimal Neumann Boundary Control Problems
SIAM Journal on Numerical Analysis
A Legendre-Galerkin Spectral Method for Optimal Control Problems Governed by Stokes Equations
SIAM Journal on Numerical Analysis
Domain decomposition methods for PDE constrained optimization problems
VECPAR'04 Proceedings of the 6th international conference on High Performance Computing for Computational Science
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An abstract framework for the analysis and approximation of a class of nonlinear optimal control and optimization problems is constructed. Nonlinearities occur in both the objective functional and the constraints. The framework includes an abstract nonlinear optimization problem posed on infinite-dimensional spaces, an approximate problem posed on finite-dimensional spaces, together with a number of hypotheses concerning the two problems. The framework is used to show that optimal solutions exist, to show that Lagrange multipliers may be used to enforce the constraints, to derive an optimality system from which optimal states and controls may be deduced, and to derive existence results and error estimates for solutions of the approximate problem. The abstract framework and the results derived from that framework are then applied to three concrete control or optimization problems and their approximation by finite-element methods. The first involves the von K\'arm\'an plate equations of nonlinear elasticity, the second the Ginzburg--Landau equations of superconductivity, and the third the Navier--Stokes equations for incompressible, viscous flows.