49M05 Methods based on necessary conditions
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Keywords
- necessary optimality conditions (2)
- optimal control (2)
- state constraints (2)
- Evolution Maxwell equations (1)
- adaptivity (1)
- box constraints (1)
- convergence of the central path (1)
- electrical circuits (1)
- elliptic variational inequality, elliptic MPEC, implicit programming, optimal control of variational inequalities, nonsmooth optimization (1)
- exponential moment problem (1)
Using a standard first-order optimality condition for nonsmooth optimization problems, a general framework for a descent method is developed. This setting is applied to a class of mathematical programs with equilibrium constraints in function space from which a new algorithm is derived. Global convergence of the algorithm is demonstrated in function space and the results are then illustrated by numerical experiments.
Two optimal control problems for instationary magnetization
processes are considered in 3D spatial domains that
include electrically conducting and nonconducting regions. The magnetic
fields are generated by induction coils. In the first model, the induction coil
is considered as part of the conducting region and the electrical current is taken
as control. In the second, the coil is viewed as part of the nonconducting region and the
electrical voltage is the control. Here, an integro-differential equation accounts
for the magnetic induction law that couples the given
electrical voltage with the induced electrical current in the
induction coil.
We derive first-order necessary
optimality condition for the optimal controls of both problems. Based on them,
numerical methods of gradient type are applied. Moreover, we report on the application
of model reduction by POD that lead to tremendous savings. Numerical tests are
presented for academic 3D geometries but also for a real-world application.
Some optimal control problems for linear and nonlinear ordinary differential equations related to the optimal switching between
different magnetic fields are considered. The main aim is to move an electrical initial current by a controllable
voltage in shortest time to a desired terminal current and to hold it afterwards. Necessary optimality conditions are derived by
Pontryagin's principle and a Lagrange technique. In the case of a linear system, the principal structure of time-optimal controls is
discussed. The associated optimality systems are solved by a one-shot strategy
using a multigrid software package. Various numerical examples are discussed.
Interior Point Methods in Function Space for State Constraints - Inexact Newton and Adaptivity
(2009)
We consider an interior point method in function space for PDE constrained optimal control problems with state constraints. Our emphasis is on the construction and analysis of an algorithm that integrates a Newton path-following method with adaptive grid refinement. This is done in the framework of inexact Newton methods in function space, where the discretization error of each Newton step is controlled by adaptive grid refinement in the innermost loop. This allows to perform most of the required Newton steps on coarse grids, such that the overall computational time is dominated by the last few steps. For this purpose we propose an a-posteriori error estimator for a problem suited norm.
In this paper we are concerned with the application of interior point methods in function space to gradient constrained optimal control problems, governed by partial differential equations. We will derive existence of solutions together with first order optimality conditions. Afterwards we show continuity of the central path, together with convergence rates depending on the interior point parameter.
Some aspects of reachability for parabolic boundary control problems with control constraints
(2009)
A class of one-dimensional parabolic optimal boundary control problems
is considered. The discussion includes Neumann, Robin, and Dirichlet
boundary conditions. The reachability of a given target state in final
time is discussed under box constraints on the control. As a mathematical
tool, related exponential moment problems are investigated. Moreover,
based on a detailed study of the adjoint state, a technique is presented
to find the location and the number of the switching points of optimal
bang-bang controls. Numerical examples illustrate this procedure.
We propose and analyse an interior point path-following method in function space
for state constrained optimal control. Our emphasis is on proving convergence in
function space and on constructing a practical path-following algorithm. In particular, the introduction of a pointwise damping step leads to a very efficient method, as verified by numerical experiments.