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PDE-constrained optimization of time-dependent 3D electromagnetic induction heating by alternating voltages

Fredi TröltzschIrwin Yousept — 2012

ESAIM: Mathematical Modelling and Numerical Analysis

This paper is concerned with a PDE-constrained optimization problem of induction heating, where the state equations consist of 3D time-dependent heat equations coupled with 3D time-harmonic eddy current equations. The control parameters are given by finite real numbers representing applied alternating voltages which enter the eddy current equations impressed current. The optimization problem is to find optimal voltages so that, under certain constraints on the voltages and the temperature, a desired...

The SQP method for control constrained optimal control of the Burgers equation

Fredi TröltzschStefan Volkwein — 2001

ESAIM: Control, Optimisation and Calculus of Variations

A Lagrange–Newton–SQP method is analyzed for the optimal control of the Burgers equation. Distributed controls are given, which are restricted by pointwise lower and upper bounds. The convergence of the method is proved in appropriate Banach spaces. This proof is based on a weak second-order sufficient optimality condition and the theory of Newton methods for generalized equations in Banach spaces. For the numerical realization a primal-dual active set strategy is applied. Numerical examples are...

Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations

Eduardo CasasFredi Tröltzsch — 2011

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we carry out the numerical analysis of a distributed optimal control problem governed by a quasilinear elliptic equation of non-monotone type. The goal is to prove the strong convergence of the discretization of the problem by finite elements. The main issue is to get error estimates for the discretization of the state equation. One of the difficulties in this analysis is that, in spite of the partial differential equation has a unique solution for any given control, the uniqueness...

Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations

Fredi TröltzschDaniel Wachsmuth — 2006

ESAIM: Control, Optimisation and Calculus of Variations

In this paper sufficient optimality conditions are established for optimal control of both steady-state and instationary 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 .

On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints

Ira NeitzelFredi Tröltzsch — 2009

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we study Lavrentiev-type regularization concepts for linear-quadratic parabolic control problems with pointwise state constraints. In the first part, we apply classical Lavrentiev regularization to a problem with distributed control, whereas in the second part, a Lavrentiev-type regularization method based on the adjoint operator is applied to boundary control problems with state constraints in the whole domain. The analysis for both classes of control problems is investigated and...

Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems

Eduardo CasasFredi Tröltzsch — 2010

ESAIM: Control, Optimisation and Calculus of Variations

Optimal control problems for semilinear elliptic equations with control constraints and pointwise state constraints are studied. Several theoretical results are derived, which are necessary to carry out a numerical analysis for this class of control problems. In particular, sufficient second-order optimality conditions, some new regularity results on optimal controls and a sufficient condition for the uniqueness of the Lagrange multiplier associated with the state constraints are presented.

Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations

Eduardo CasasFredi Tröltzsch — 2011

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we carry out the numerical analysis of a distributed optimal control problem governed by a quasilinear elliptic equation of non-monotone type. The goal is to prove the strong convergence of the discretization of the problem by finite elements. The main issue is to get error estimates for the discretization of the state equation. One of the difficulties in this analysis is that, in spite of the partial differential equation has a unique solution for any given control, the uniqueness...

Optimal Control of Semilinear Parabolic Equations with State-Constraints of Bottleneck Type

Maïtine BergouniouxFredi Tröltzsch — 2010

ESAIM: Control, Optimisation and Calculus of Variations

We consider optimal distributed and boundary control problems for semilinear parabolic equations, where pointwise constraints on the control and pointwise mixed control-state constraints of bottleneck type are given. Our main result states the existence of regular Lagrange multipliers for the state-constraints. Under natural assumptions, we are able to show the existence of bounded and measurable Lagrange multipliers. The method is based on results from the theory of continuous linear programming...

The SQP method for control constrained optimal control of the Burgers equation

Fredi TröltzschStefan Volkwein — 2010

ESAIM: Control, Optimisation and Calculus of Variations

A Lagrange–Newton–SQP method is analyzed for the optimal control of the Burgers equation. Distributed controls are given, which are restricted by pointwise lower and upper bounds. The convergence of the method is proved in appropriate Banach spaces. This proof is based on a weak second-order sufficient optimality condition and the theory of Newton methods for generalized equations in Banach spaces. For the numerical realization a primal-dual active set strategy is applied. Numerical examples are...

On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints

Ira NeitzelFredi Tröltzsch — 2008

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we study Lavrentiev-type regularization concepts for linear-quadratic parabolic control problems with pointwise state constraints. In the first part, we apply classical Lavrentiev regularization to a problem with distributed control, whereas in the second part, a Lavrentiev-type regularization method based on the adjoint operator is applied to boundary control problems with state constraints in the whole domain. The analysis for both classes of control problems is investigated and...

PDE-constrained optimization of time-dependent 3D electromagnetic induction heating by alternating voltages

Fredi TröltzschIrwin Yousept — 2012

ESAIM: Mathematical Modelling and Numerical Analysis

This paper is concerned with a PDE-constrained optimization problem of induction heating, where the state equations consist of 3D time-dependent heat equations coupled with 3D time-harmonic eddy current equations. The control parameters are given by finite real numbers representing applied alternating voltages which enter the eddy current equations impressed current. The optimization problem is to find optimal voltages so that, under certain constraints on the voltages and the temperature, a desired...

Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations

Fredi TröltzschDaniel Wachsmuth — 2005

ESAIM: Control, Optimisation and Calculus of Variations

In this paper sufficient optimality conditions are established for optimal control of both steady-state and instationary 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 -neighborhood, whereby the underlying analysis allows to use weaker norms than .

A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD

Eileen KammannFredi TröltzschStefan Volkwein — 2013

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: let an arbitrary admissible control function be given. How far is it from the next locally optimal control? Under natural assumptions including a second-order sufficient optimality condition for the (unknown) locally optimal control, we estimate the distance between the two controls. To do this, we need some information on the lowest eigenvalue of the reduced Hessian....

Error estimates for the finite element discretization of semi-infinite elliptic optimal control problems

Pedro MerinoIra NeitzelFredi Tröltzsch — 2010

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we derive a priori error estimates for linear-quadratic elliptic optimal control problems with finite dimensional control space and state constraints in the whole domain, which can be written as semi-infinite optimization problems. Numerical experiments are conducted to ilustrate our theory.

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