An optimal control problem governed by a bilinear elliptic equation is considered. This problem is solved by the sequential quadratic programming (SQP) method in an infinite-dimensional framework. In each level of this iterative method the solution of linear-quadratic subproblem is computed by a Galerkin projection using proper orthogonal decomposition (POD). Thus, an approximate (inexact) solution of the subproblem is determined. Based on a POD error estimator developed by Tröltzsch and Volkwein...
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...
Proper orthogonal decomposition (POD) is a powerful technique for model reduction of linear and non-linear systems. It is based on a Galerkin type discretization with basis elements created from the system itself. In this work, error estimates for Galerkin POD methods for linear elliptic, parameter-dependent systems are proved. The resulting error bounds depend on the number of POD basis functions and on the parameter grid that is used to generate the snapshots and to compute the POD basis. The...
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...
An optimal control problem governed by a bilinear elliptic equation is considered. This
problem is solved by the sequential quadratic programming (SQP) method in an
infinite-dimensional framework. In each level of this iterative method the solution of
linear-quadratic subproblem is computed by a Galerkin projection using proper orthogonal
decomposition (POD). Thus, an approximate (inexact) solution of the subproblem is
determined. Based on a POD...
The construction of reduced order models for dynamical systems using
proper orthogonal decomposition (POD) is based on the information
contained in so-called snapshots. These provide the spatial
distribution of the dynamical system at discrete time instances.
This work is devoted to optimizing the choice of these time
instances in such a manner that the error between the POD-solution
and the trajectory of the dynamical system is minimized. First and
second order optimality systems are given. Numerical...
Proper orthogonal decomposition (POD) is a
powerful technique for model reduction of non-linear systems. It
is based on a Galerkin type discretization with basis elements
created from the dynamical system itself. In the context of
optimal control this approach may suffer from the fact that the
basis elements are computed from a reference trajectory containing
features which are quite different from those of the optimally
controlled trajectory. A method is proposed which avoids this
problem of unmodelled...
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....
Optimal control problems for the heat equation with pointwise bilateral control-state constraints are considered. A locally superlinearly convergent numerical solution algorithm is proposed and its mesh independence is established. Further, for the efficient numerical solution reduced space and Schur complement based preconditioners are proposed which take into account the active and inactive set structure of the problem. The paper ends by numerical tests illustrating our theoretical findings and...
Optimal control problems for the heat equation with pointwise
bilateral control-state constraints are considered. A locally
superlinearly convergent numerical solution algorithm is proposed
and its mesh independence is established. Further, for the
efficient numerical solution reduced space and Schur complement
based preconditioners are proposed which take into account the
active and inactive set structure of the problem. The paper ends
by numerical tests illustrating our theoretical findings and
comparing...
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