Error estimates for linear-quadratic control problems with control constraints

An abstract linear-quadratic optimal control problem is investigated with pointwise control constraints. This paper is concerned in discretization of the control by piecewise linear functions. Under the assumption that the optimal control and the optimal adjoint state is Lipschitz continuous and piecewise of class $C^2$ an approximation of order $h^{3/2}$ is proved for the solution of the control discretized problem with respect to the solution of the continuous one. Numerical tests are presented after the theoretical part

Error estimates for parabolic optimal control problems with control constraints

An optimal control problem for the 1-d heat equation is investigated with pointwise control constraints. This paper is concerned with the discretization of the control by piecewise linear functions. The connection between the solutions of the discretized problems and the continuous one is investigated. Under an additional assumption on the adjoint state an approximation order $\sigma^{3/2}$ is proved for uniform discretizations. In the general case it is shown that a non-uniform control discretization ensure an approximation of order $\sigma^{3/2}$. Numerical tests confirm the theoretical part

SUFFICIENT OPTIMALITY CONDITIONS FOR THE MOREAU-YOSIDA TYPE REGULARIZATION CONCEPT APPLIED TO SEMILINEAR ELLIPTIC OPTIMAL CONTROL PROBLEMS WITH POINTWISE STATE CONSTRAINTS

We develop sufficient optimality conditions for a Moreau-Yosidaregularized optimal control problem governed by a semilinear ellipticPDE with pointwise constraints on the state and the control. We makeuse of the equivalence of a setting of Moreau-Yosida regularization to a special setting of the virtual control concept,for which standard second order sufficient conditions have been shown. Moreover, we present a numerical example,solving a Moreau-Yosida regularized model problem with an SQP method

A priori error analysis for state constrained boundary control problems. Part II: Full discretization

This is the second of two papers concerned with a state-constrained optimal control problems with boundary control, where the state constraints are only imposed in an interior subdomain. We apply the virtual control concept introduced in [26] to regularize the problem. The arising regularized optimal control problem is discretized by finite elements and linear and continuous ansatz functions for the boundary control. In the first part of the work, we investigate the errors induced by the regularization and the discretization of the boundary control. The second part deals with the error arising from discretization of the PDE. Since the state constraints only appear in an inner subdomain, the obtained order of convergence exceeds the known results in the field of a priori analysis for state-constrained problems. The theoretical results are illustrated by numerical computations

A priori error analysis for state constrained boundary control problems : Part II: Full discretization

This is the second of two papers concerned with a state-constrained optimal control problems with boundary control, where the state constraints are only imposed in an interior subdomain. We apply the virtual control concept introduced in [26] to regularize the problem. The arising regularized optimal control problem is discretized by finite elements and linear and continuous ansatz functions for the boundary control. In the first part of the work, we investigate the errors induced by the regularization and the discretization of the boundary control. The second part deals with the error arising from discretization of the PDE. Since the state constraints only appear in an inner subdomain, the obtained order of convergence exceeds the known results in the field of a priori analysis for state-constrained problems. The theoretical results are illustrated by numerical computations

Regularization error estimates for semilinear elliptic optimal control problems with pointwise state and control constraints

In this paper a class of semilinear elliptic optimal control problem with pointwise state and control constraints is studied. A sufficient second order optimality condition and uniqueness of the dual variables are assumed for that problem. Sufficient second order optimality conditions are shown for regularized problems with small regularization parameter. Moreover, error estimates with respect to the regularization parameter are derived

A priori error analysis for state constrained boundary control problems : Part I: Control discretization

This is the first of two papers concerned with a state-constrained optimal control problems with boundary control, where the state constraints are only imposed in an interior subdomain. We apply the virtual control concept introduced in [20] to regularize the problem. The arising regularized optimal control problem is discretized by finite elements and linear and continuous ansatz functions for the boundary control. In the first part of the work, we investigate the errors induced by the regularization and the discretization of the boundary control. The second part deals with the error arising from discretization of the PDE. Since the state constraints only appear in an inner subdomain, the obtained order of convergence exceeds the known results in the field of a priori analysis for state-constrained problem