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...
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...
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...
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 .
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...
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.
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...
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...
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...
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 .
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...
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...
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....
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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