In this paper we investigate a space-time finite element approximation of parabolic optimal control problems. The first order optimality conditions are transformed into an elliptic equation of fourth order in space and second order in time involving only the state or the adjoint state in the space-time domain. We derive a priori and a posteriori error estimates for the time discretization of the state and the adjoint state. Furthermore, we also propose a space-time mixed finite element discretization scheme to approximate the space-time elliptic equations, and derive a priori error estimates for the state and the adjoint state. Numerical examples are presented to illustrate our theoretical findings and the performance of our approach.Keywords: parabolic optimal control problems, space-time finite elements, mixed finite elements, a priori error estimates, a posteriori error estimates.

In this paper, we study an edge-stabilization Galerkin approximation scheme for the constrained optimal-control problem governed by convection-dominated diffusion equation. The method uses least-square stabilization of the gradient jumps across element edges. A priori and a posteriori error estimates are derived for both the state, co-state and the control. The theoretical results are illustrated by two numerical experiments.

In this paper, the problem of switching stabilization for a class of delayed switched systems is studied by using state-dependent switching control. Some sufficient conditions for asymptotic stability of delayed switched systems are derived, which are based on the Lyapunov functional method. The derived stability conditions can be applied for delayed switched systems with partial or all unstable modes. Moreover, a typical time delay called leakage delay existing in the negative feedback term of a system are also considered. Two numerical examples, that is, a switched system with two unstable modes and a switched system with delayed feedback control, are given to show the effectiveness of the main results. such that the following inequality holds:

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Finite element approximations of Dirichlet boundary control problems governed by parabolic PDEs on convex polygonal domains are studied in this paper. The existence of a unique solution to optimal control problems is guaranteed based on very weak solution of the state equation and L 2 (0, T ; L 2 (Γ)) as control space. For the numerical discretization of the state equation we use standard piecewise linear and continuous finite elements for the space discretization of the state, while a dG(0) scheme is used for time discretization. The Dirichlet boundary control is realized through a space-time L 2 -projection. We consider both piecewise linear, continuous finite element approximation and variational discretization for the controls and derive a priori L 2 -error bounds for controls and states. We finally present numerical examples to support our theoretical findings.