• Numerical verification of optimality conditions. (with Arnd Rösch)
  SIAM Journal Control and Optimization 47(5), 2557-2581 (2008). [Abstract]
  [-] Abstract: A class of optimal control problem for a semilinear elliptic partial differential equation with control constraints is considered. It is well known that sufficient second-order conditions ensure the stability of optimal solutions, the convergence of numerical methods. Otherwise, such conditions are very difficult to verify (analytically or numerically). We will propose a new approach: Starting with a numerical solution for a fixed mesh we will show the existence of a local minimizer of the continuous problem. Moreover, we will prove that this minimizer satisfies the sufficient second-order conditions.


• Sensitivity Analysis and the Adjoint Update Strategy for Optimal Control Problems with Mixed Control-State Constraints. (with Roland Griesse)
  To appear in Computational Optimization and Applications. [Abstract]
  [-] Abstract: In this article, an optimal control problem subject to a semilinear elliptic equation and mixed control-state constraints is investigated. The problem data depends on certain parameters. Under an assumption of seperation of the active sets and a second-order sufficient optimality condition, Bouliganddifferentiability (B-differentiability) of the solutions with respect to the parameter is established. Furthermore, an adjoint update strategy is proposed which yields a better approximation of the optimal controls and multipliers than the classical Taylor expansion, with remainder terms vanishing in L^∞.


• Update Strategies for Perturbed Nonsmooth Equations. (with R. Griesse, T. Grund)
  Optimization Methods and Software 23(3), 321-343 (2008). [Abstract]
  [-] Abstract: Nonsmooth operator equations in function spaces are considered, which depend on perturbation parameters. The nonsmoothness arises from a projection onto an admissible interval. Lipschitz stability in L^∞ and Bouligand differentiability in L^p of the parameter-to-solution map are derived. An adjoint problem is introduced for which Lipschitz stability and Bouligand differentiability in L^∞ are obtained. Three different update strategies, which recover a perturbed from an unperturbed solution, are analyzed. They are based on Taylor expansions of the primal and adjoint variables, where the latter admits error estimates in L^∞. Numerical results are provided.


• Analysis of the SQP-method for optimal control problems governed by the instationary Navier-Stokes equations based on L^p-theory.
  SIAM Journal Control and Optimization 46, 1133-1153 (2007). [Abstract]
  [-] Abstract: The aim of this article is to present a convergence theory of the SQP-method applied to optimal control problems for the instationary Navier-Stokes equations. We will employ a second-order sufficient optimality condition, which requires that the second derivative of the Lagrangian is positive definit on a subspace of inactive constraints. Therefore, we have to use L^p-theory of optimal controls of the instationary Navier-Stokes equations rather than Hilbert space methods. We prove local convergence of the SQP-method. This behaviour is confirmed by numerical tests.


• Regularity of solutions for an optimal control problem with mixed control-state constraints. (with Arnd Rösch)
  TOP 14, 263-278 (2006). [ZB]


• Sufficient second-order optimality conditions for convex control constraints.
  Journal of Mathematical Analysis and Applications 319, 228-247 (2006). [ZB] [Abstract]
  [-] Abstract: In this article sufficient optimality conditions are established for optimal control problems with pointwise convex control constraints. Here, the control is a function with values in Rⁿ. The constraint is of the form u(x)∈U(x), where U is an set-valued mapping that is assumed to be measurable with convex and closed images. The second-order condition requires coercivity of the Lagrange function on a suitable subspace, which excludes strongly active constraints, together with first-order necessary conditions. It ensures local optimality of a reference function in a L^∞-neighborhood. The analysis is done for a model problem namely the optimal distributed control of the instationary Navier-Stokes equations.


• Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations. (with Fredi Tröltzsch)
  ESAIM: Control, Optimisation and Calculus of Variations 12, 93-119 (2006). [ZB] [Abstract]
  [-] Abstract: In this paper sufficient optimality conditions are established for optimal control of both steady-state and evolution Navier-Stokes equations. The second-order condition requires coercivity of the Lagrange function on a suitable subspace together with first-order necessary conditions. It ensures local optimality of a reference function in a L^s-neighborhood, whereby the underlying analysis allows to use weaker norms than L^∞.


• Regularity and Stability of optimal controls of instationary Navier-Stokes equations.
  Control and Cybernetics 34, 387-410 (2005). [Abstract]
  [-] Abstract: The regularity and stability of optimal controls of instationary Navier-Stokes equations is investigated. Under suitable assumptions every control satisfying first-order necessary conditions is shown to be a continuous function in both space and time. Moreover, the behaviour of a locally optimal control under certain perturbations of the cost functional and the state equation is investigated. Lipschitz stability is proven provided a second-order sufficient optimality condition holds.


• Regularity of the adjoint state for the instationary Navier-Stokes equations. (with Arnd Rösch)
  Journal for Analysis and its Applications 24, 103-116 (2005). [ZB] [Abstract]
  [-] Abstract: In this article, we are considering imbeddings of abstract functions in spaces of functions being continuous in time. A family of functions depending on certain parameters is discussed in detail. In particular, this example shows that such functions do not belong to the space C([0,T],H). In the second part, we investigate an optimal control problem for the instationary Navier-Stokes equation. We will answer the question, in what sense the initial value problem for the adjoint equation can be solved.


• On instantaneous control for a nonlinear parabolic boundary control problem.
  Numerical Functional Analysis and Optimization 25, 151-181 (2004). [ZB] [Abstract]
  [-] Abstract: A method of instantaneous control type is considered for a nonlinear parabolic boundary control problem with box constraints on the control. It is shown that the method exhibits fixed points. In numerical examples, convergence towards a fixed point state occurs, which might be far away from the desired state. Consequently, a new hybrid method is suggested, which behaves essentially better than the standard method.


• On convergence of a receding horizon method for parabolic boundary control. (with Fredi Tröltzsch)
  Optimization Methods and Software 19, 201-216 (2004). [ZB] [Abstract]
  [-] Abstract: A method of receding horizon type is considered for a simplified linear-quadratic parabolic boundary control problem with bound constraints on the control. The performance of the method is examined numerically and confirmed by an associated analysis. In particular, the method is shown to converge to a unique fixed point. Moreover, a new hybrid method is suggested.

• Optimal control problems with convex control constraints.
  In: Control of Coupled Partial Differential Equations. ISNM Vol. 155. Kunisch, Leugering, Sprekels, Tröltzsch (Eds.), 311-328. Birkhäuser (2007). [Abstract]
  [-] Abstract: We investigate optimal control problems with vector-valued controls. As model problem serve the optimal distributed control of the instationary Navier-Stokes equations. We study pointwise convex control constraints, which is a constraint of the form u(x,t)∈U(x,t) that has to hold on the domain Q. Here, U is an set-valued mapping that is assumed to be measurable with convex and closed images. We establish first-order necessary as well as second-order sufficient optimality conditions. And we prove regularity results for locally optimal controls.


• Numerical Study of the Optimization of Separation Control. (with Angelo Carnarius, Bert Günther, Frank Thiele, Fredi Tröltzsch, Juan Carlos de los Reyes)
  Proceedings of the 45th AIAA Aerospace Sciences Meeting and Exhibit, Reno, 8-11 January 2007, AIAA 2007-58. [Abstract]
  [-] Abstract: The concept of active flow control is applied to the steady flow around a NACA4412 and to the unsteady flow around a generic high-lift configuration in order to delay separation. To the former steady suction upstream of the detachment position is applied. In a series of computations the suction angle is varied and the main flow features are analyzed. A gradient descent method and an adjoint-based method are successfully used to optimize β. For the unsteady case periodic blowing and suction is employed to control the separation. Various calculations are conducted to obtain the dependency of the lift on the amplitude and frequency of the perturbation and the amplitude is optimized with the gradient descent method.


• Numerical solution of optimal control problems with convex control constraints.
  In: Systems, Control, Modeling and Optimization. Ceragioli, Dontchev, Furuta, Marti, Pandolfi (Eds.), 319-327. Springer (2006). For an extended version see also TU Berlin Preprint 31-2005 (2005). [Abstract]
  [-] Abstract: We study optimal control problems with vector-valued controls. As model problem serves the optimal distributed control of the instationary Navier-Stokes equations. In the article, we propose a solution strategy to solve optimal control problems with pointwise convex control constraints. It involves a SQP-like step with an imbedded active-set algorithm. The efficiency of that method is demonstrated in numerical examples and compared to the primal-dual active-set strategy for box-constraints.


• Second-order sufficient optimality conditions for the optimal control of instationary Navier-Stokes equations. (with Fredi Tröltzsch)
  Proceedings in Applied Mathematics and Mechanics 4(1), 628-629 (2004).


• Fast closed loop control of the Navier-Stokes system. (with Michael Hinze)
  In: Modelling, Simulation and Optimization of Complex Processes. Bock, Kostina, Phu, Rannacher (Eds.), 189-202. Springer (2004).

• How to check numerically the sufficient optimality conditions for infinite-dimensional optimization problems. (with Arnd Rösch)
  RICAM Report 2008-23. [Abstract]
  [-] Abstract: We consider general non-convex optimal control problems. Many results for such problems rely on second-order sufficient optimality conditions. We propose a method to verify whether the second-order sufficient optimality conditions hold in a neighborhood of a numerical solution. This method is then applied to abstract optimal control problems. Finally, we consider an optimal control problem subject to a semi-linear elliptic equation that appears to have multiple local minima.


• Optimal control of planar flow of incompressible non-Newtonian fluids. (with Tomáš Roubíček)
  TU Berlin Preprint 11-2008 (2008). [Abstract]
  [-] Abstract: We consider an optimal control problem for the evolutionary flow of incompressible non-Newtonian fluids in a two-dimensional domain. The existence of optimal controls is proven. Furthermore, we investigate first-order necessary as well as second-order sufficient optimality conditions. The analysis relies on new results providing solutions with bounded gradients for the flow equations.


• Semi-smooth Newton's Method for an optimal control problem with control and mixed control-state constraints. (with Arnd Rösch)
  Matheon Preprint 415 (2007). [Abstract]
  [-] Abstract: A class of optimal control problems for a semilinear parabolic partial differential equation with control and mixed control-state constraints is considered. For this problem, a projection formula is derived that is equivalent to the necessary optimality conditions. As main result, the superlinear convergence of a semi-smooth Newton method is shown. Moreover we show the numerical treatment and several numerical experiments.

• Optimal control of the unsteady Navier-Stokes equations.
  TU Berlin, Dissertation (2006).
• Numerische Analysis eines Verfahrens der Momentansteuerung.
  TU Chemnitz, Diplomarbeit (2002).

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