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Parametric Sensitivity Analysis for Optimal Boundary Control of a 3D ReactionDiffusion System
 LARGE SCALE NONLINEAR OPTIMIZATION
, 2005
"... A boundary optimal control problem for an instationary nonlinear reactiondiffusion equation system in three spatial dimensions is presented. The control is subject to pointwise control constraints and a penalized integral constraint. Under a coercivity condition on the Hessian of the Lagrange functi ..."
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A boundary optimal control problem for an instationary nonlinear reactiondiffusion equation system in three spatial dimensions is presented. The control is subject to pointwise control constraints and a penalized integral constraint. Under a coercivity condition on the Hessian of the Lagrange function, an optimal solution is shown to be a directionally differentiable function of perturbation parameters such as the reaction and diffusion constants or desired and initial states. The solution’s derivative, termed parametric sensitivity, is characterized as the solution of an auxiliary linearquadratic optimal control problem. A numerical example illustrates the utility of parametric sensitivities which allow a quantitative and qualitative perturbation analysis of optimal solutions.
Differential Stability of Control Constrained Optimal Control Problems for the NavierStokes Equations
, 2005
"... Distributed optimal control problems for the timedependent and the stationary NavierStokes equations subject to pointwise control constraints are considered. Under a coercivity condition on the Hessian of the Lagrange function, optimal solutions are shown to be directionally differentiable functio ..."
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Distributed optimal control problems for the timedependent and the stationary NavierStokes equations subject to pointwise control constraints are considered. Under a coercivity condition on the Hessian of the Lagrange function, optimal solutions are shown to be directionally differentiable functions of perturbation parameters such as the Reynolds number, the desired trajectory, or the initial conditions. The derivative is characterized as the solution of an auxiliary linearquadratic optimal control problem. Thus, it can be computed at relatively low cost. Taylor expansions of the minimum value function are provided as well. 1
Control and Cybernetics
"... Regularity and stability of optimal controls of nonstationary NavierStokes equations 1 by ..."
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Regularity and stability of optimal controls of nonstationary NavierStokes equations 1 by
Stability and Sensitivity Analysis in Optimal Control of Partial Differential Equations
, 2007
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ERROR ESTIMATES FOR THE NUMERICAL APPROXIMATION OF A DISTRIBUTED CONTROL PROBLEM FOR THE
"... Abstract. We obtain error estimates for the numerical approximation of a distributed control problem governed by the stationary Navier–Stokes equations, with pointwise control constraints. We show that the L2norm of the error for the control is of order h2 if the control set is not discretized, whi ..."
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Abstract. We obtain error estimates for the numerical approximation of a distributed control problem governed by the stationary Navier–Stokes equations, with pointwise control constraints. We show that the L2norm of the error for the control is of order h2 if the control set is not discretized, while it is of order h if it is discretized by piecewise constant functions. These error estimates are obtained for local solutions of the control problem, which are nonsingular in the sense that the linearized Navier–Stokes equations around these solutions define some isomorphisms, and which satisfy a second order sufficient optimality condition. We establish a second order necessary optimality condition. The gap between the necessary and sufficient second order optimality conditions is the usual gap known for finite dimensional optimization problems.
Regularity and stability of optimal controls of nonstationary NavierStokes equations
 CONTROL AND CYBERNETICS
, 2005
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