Wyniki wyszukiwania - BazTech

1

Optimality conditions for state-constrained PDE control problems with time-dependent controls

Los Reyes J. C. de, Merino P., Rehberg F., Troltzsch F.

Control and Cybernetics

|

2008

|

Vol. 37, no 1

5-38

EN

The paper deals with optimal control problems for semilinear elliptic and parabolic PDEs subject to pointwise state constraints. The main issue is that the controls are taken from a restricted control space. In the parabolic case, they are Rm -vector-valued functions of time, while they are vectors of Rm in elliptic problems. Under natural assumptions, first- and second-order sufficient optimality conditions are derived. The main result is the extension of second-order sufficient conditions to semilinear parabolic equations in domains of arbitrary dimension. In the elliptic case, the problems can be handled by known results of semi-infinite optimization. Here, different examples are discussed that exhibit different forms of active sets and where second-order sufficient conditions are satisfied at the optimal solution.

2

On convergence of regularization methods for nonlinear parabolic optimal control problems with control and state constraints

Neitzel I., Troltzsch F.

Control and Cybernetics

|

2008

|

Vol. 37, no 4

1013-1043

EN

Moreau-Yosida and Lavrentiev type regularization methods are considered for nonlinear optimal control problems governed by semilinear parabolic equations with bilateral pointwise control and state constraints. The convergence of optimal controls of the regularized problems is studied for regularization parameters tending to infinity or zero, respectively. In particular, the strong convergence of global and local solutions is addressed. Moreover, strong regularity of the Lavrentiev-regularized optimality system is shown under certain assumptions, which, in particular, allows to show that locally optimal solutions of the Lavrentiev regularized problems are locally unique. This analysis is based on a second-order sufficient optimality condition and a separation assumption on almost active sets.

3

Lipschitz stability of optimal controls for the steady-state Navier-Stokes equations

Roubicek T., Troltzsch F.

Control and Cybernetics

|

2003

|

Vol. 32, no 3

683-705

EN

An optimal control problem with quadratic cost functional for the steady-state Navier-Stokes equations with no-slip boundary condition is considered. Lipschitz stability of locally optimal controls with respect to certain perturbations of both the cost functional and the equation is proved provided a second-order sufficient optimality condition holds. For a sufficiently small Reynolds number, even global Lipschitz stability of the unique optimal control is shown.