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.
A continuity result for Nemyckii Operators and some applications in PDE constrained optimal control
(2008)
This work explores two applications of a classical result on the continuity of Nemyckii operators to optimal control with PDEs. First, we present an alternative approach to the analysis of Newton's method for function space problems involving semi-smooth Nemyckii operators. A concise proof for superlinear convergence is presented, and sharpened bounds on the rate of convergence are derived. Second, we derive second order sufficient conditions for problems, where the underlying PDE has poor regularity properties. We point out that the analytical structure in both topics is essentially the same.
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.
In this paper, we study an optimal control problem for a singular system of partial differential equations that models a nonisothermal phase transition with a nonconserved order parameter. The control acts through a third boundary condition for the absolute temperature and plays the role of the outside temperature. It is shown that the corresponding control-to-state mapping is well defined, and the existence of an optimal control and the first-order optimality conditions for a quadratic cost functional of Bolza type are established.
We study optimal control problems for general unstructured nonlinear differential-algebraic equations of arbitrary index.
In particular, we derive necessary conditions in the case of linear-quadratic control problems and extend them to the general nonlinear case.
We also present a Pontryagin maximum principle for general unstructured nonlinear DAEs in the case of restricted controls.
Moreover, we discuss the numerical solution of the resulting two-point boundary value problems and present a numerical example.
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.
Descriptor systems present a general
mathematical framework for the modelling, simulation and control of complex dynamical systems arising in many areas of mechanical,
electrical and chemical engineering. This
paper presents a survey of the current theory of descriptor systems,concerning
solvability, stability, model reduction, controllability, observability and optimal control.