: Part 2 continues the study of optimization techniques for elliptic control problems subject to control and state constraints and is devoted to distributed control. Boundary conditions are of mixed Dirichlet and Neumann type. Necessary conditions of optimality are formally stated in form of a local Pontryagin minimum principle. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. The problems are formulated as AMPL [13] scripts and several optimization codes are applied. In particular, it is shown that a recently developed interior point method is able to solve theses problems even for high discretizations. Several numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for dierent types of controls including bang{bang controls. The necessary conditions of optimality are checked numerically in the presence of active control and state constraints....

The optimal boundary control of Navier--Stokes flow is formulated as a constrained optimization problem and a sequential quadratic programming (SQP) approach is studied for its solution. Since SQP methods treat states and controls as independent variables and do not insist on satisfying the constraints during the iterations, care must be taken to avoid a possible incompatibility of Dirichlet boundary conditions and incompressibility constraint. In this paper, compatibility is enforced by choosing appropriate function spaces. The resulting optimization problem is analyzed. Differentiability of the constraints and surjectivity of linearized constraints are verified and adjoints are computed. An SQP method is applied to the optimization problem and compared with other approaches.

ABSTRACT. We develop and analyze a class of overlapping domain decomposition (DD) preconditioners for linear-quadratic elliptic optimal control problems. Our preconditioners utilize the structure of the optimal control problems. Their execution requires the parallel solution of subdomain linear-quadratic elliptic optimal control problems, which are essentially smaller subdomain copies of the original problem. This work extends to optimal control problems the application and analysis of overlapping DD preconditioners, which have been used successfully for the solution of single PDEs. We prove that for a class of problems the performance of the two-level versions of our preconditioners is independent of the mesh size and of the subdomain size. 1.

this paper, these two approaches produce the same control input #. 5. Results As was mentioned in 4, the pressures on the cylinder surface are measured at # 6 # 6 # and the blowing and suction are applied at # 6 # 6 #.In 5.1, sensings and actuations are carried out all over the cylinder surface, i.e. (#, #)=(#,#)=(0, 2#). Local sensings and actuations are performed in 5.2. Finally, open-loop controls are investigated in 5.3. For all cases investigated in this study, we have used the computational time step #t = 0.015, and the control time interval #t c = 0.06. That is, the sensing and actuation are updated at every four computational time steps. We have also 136 C. Min and H. Choi 1.5 1.4 1.3 1.2 1.1 1.0 0 30 60 90 120 t J 3 (c) 1.4 0.8 1.2 1.0 0.4 0 30 60 90 120 J 2 (b) 1.1 1.0 0.9 0.8 0.7 0 30 60 90 120 J 1 (a) 0.6 Figure 4. Time histories of the cost functional with (#, #)=(#,#)=(0, 2#): , # max =0.1; --------, 0.2; ---, 0.3; --------, 0.4. (a) J 1 ;(b) J 2 ;(c) J 3 . investigated a few di#erent combinations of #t and #t c , but the results showed only a slight change compared to those obtained from #t = 0.015 and #t c = 0.06. The Reynolds numbers investigated in this study (two-dimensional computations) are 100 and 160; according to the recent result by Henderson (1997), the two-dimensional wake becomes absolutely unstable to long-wavelength spanwise perturbations and bifurcates to a three-dimensional flow at Re # 190 (mode A; see also Williamson 1988). All controls begin at t = 30 and the maximum blowing/suction value relative to the free-stream velocity, # max = max 06#<2# |#(#)|, is kept constant during the control. 5.1. Sensing and actuation all over the cylinder surface We have applied the actuation values of decreasing J 1 and J 2...

We propose and analyze a semismooth Newton-type method for the solution of a pointwise constrained optimal control problem governed by the time-dependent incompressible Navier-Stokes equations. The method is based on a reformulation of the optimality system as an equivalent nonsmooth operator equation. We analyze the flow control problem and establish q-superlinear convergence of the method. In the numerical implementation, adjoint techniques are combined with a truncated conjugate gradient method. Numerical results are presented that support our theoretical results and confirm the viability of the approach.

Abstract. An optimal control problem for the equations governing the stationary problem of magnetohydrodynamics (MHD) is considered. Control mechanisms by external and injected currents and magnetic fields are treated. An optimal control problem is formulated. First order necessary and second order sufficient conditions are developed. An operator splitting scheme for the numerical solution of the MHD state equations is analyzed. 1.

In this paper, optimal control problems with pointwise state constraints for linear and semilinear elliptic partial differential equations are studied. The problems are subject to perturbations in the problem data. Lipschitz stability with respect to perturbations of the optimal control and the state and adjoint variables is established initially for linear–quadratic problems. Both the distributed and Neumann boundary control cases are treated. Based on these results, and using an implicit function theorem for generalized equations, Lipschitz stability is also shown for an optimal control problem involving a semilinear elliptic equation.

This thesis is concerned with the development of domain decomposition (DD) based preconditioners for linear-quadratic elliptic optimal control problems (LQ-EOCPs), their analysis, and numerical studies of their performance on model problems. The solution of LQ-EOCPs arises in many applications, either directly or as subproblems in Newton or Sequential Quadratic Programming methods for the solution of nonlinear elliptic optimal control problems. After a finite element discretization, convex LQEOCPs lead to large scale symmetric indefinite linear systems. The solution of these large systems is a very time consuming step and must be done iteratively, typically with a preconditioned Krylov subspace method. Developing good preconditioners for these linear systems is an important part of improving the overall performance of the solution method. The DD

A shape design problem for stationary, viscous, incompressible, two-dimensional channel flows is considered. The shape of part of the boundary is determined so that the viscous drag is minimized. The adjoint equation method is used to derive an optimality system and the shape gradient of the design functional.

We consider an optimal control problem of tracking type for Stokes flows in two and three space dimensions. The solution is approximated by a finite element method which is based on the necessary first order optimality conditions. We prove optimal error estimates for the resulting semi-discrete scheme and present numerical examples that confirm our results.