Ursula Felgenhauer

Structural Stability Investigation of Bang-Singular-Bang Optimal Controls

The paper is devoted to parametric optimal control problems with a scalar, partially singular optimal control function. In contrast to the case of pure bang-bang behavior, the investigation of structural stability properties for partially singular controls so far has been rarely addressed in literature. The central result of the paper deals with the case of one first order singular arc under regular concatenation to bang-arcs. Conditions will be provided which ensure the Lipschitz stability of bang-singular junction times positions with respect to small parameter changes. Three examples illustrate the main theorem.

Directional sensitivity differentials for parametric bang-bang control problems

We consider optimal control problems driven by ordinary differential equations depending on a (vector-valued) parameter In case that the state equation is linear w.r.t the control vector function, and the objective functional is of Mayer type, the optimal control is often of bang-bang type The aim of the paper is to consider the structural stability of bang-bang optimal controls with possibly simultaneous switches of two or more components at a time Besides of the local invariance of the number of switches for each component taken separately, existence of their directional parameter–derivatives will be shown.

Lipschitz Stability of Broken Extremals in Bang-Bang Control Problems

Optimal bang-bangcontrols appear in problems where the system dynamics linearly depends on the control input. The principal control structure as well as switching points localization are essential solution characteristics. Under rather strong optimality and regularity conditions, for so-called simpleswitches of (only) one control component, the switching points had been shown being differentiable w.r.t. problem parameters. In case that multiple(or: simultaneous) switches occur, the differentiability is lost but Lipschitz continuous behavior can be observed e.g. for double switches. The proof of local structural stability is based on parametrizations of broken extremals via certain backward shooting approach. In a second step, the Lipschitz property is derived by means of nonsmooth Implicit Function Theorems.

Discretization of semilinear bang-singular-bang control problems

Bang-singular controls may appear in optimal control problems where the control enters the system linearly. We analyze a discretization of the first-order system of necessary optimality conditions written in terms of a variational inequality (or: inclusion) under appropriate assumptions including second-order optimality conditions. For the so-called semilinear case, it is proved that the discrete control has the same principal bang-singular-bang structure as the reference control and, in

On the stable global convergence of particular quasi-newton-methods

L_{1}

Regularity properties of optimal controls with application to discrete approximation

L1 topology, the convergence is of order one w.r.t. the stepsize.

Weak and Strong Optimality Conditions for Constrained Control Problems with Discontinuous Control

The theory of discretization methods to control problems and their convergence under strong stable optimality conditions in recent years has been thoroughly investigated by several authors. A particularly interesting question is to ask for a “natural” smoothness category for the optimal controls as functions of time.In several papers, Hager and Dontchev considered Riemann integrable controls. This smoothness class is characterized by global, averaged criteria. In contrast, we consider strictly local properties of the solution function. As a first step, we introduce tools for the analysis of L∞ elements “at a point”. Using afterwards Robinsons strong regularity theory, under appropriate first and second order optimality conditions we obtain structural as well as certain pseudo-Lipschitz properties with respect to the time variable for the control.Consequences for the behavior of discrete solution approximations are discussed in the concluding section with respect to L∞ as well as L2 topologies.

A Discretization for Control Problems with Optimality Test

In the paper the classical BFGS-method with a Goldstein-Armijo step length rule is considered in the case when the first order information about the function is inexact. There are given accuracy restrictions for which the perturbed method is globally convergent and additional conditions for q-superlinear convergence. For the proofs there were essentially used results of Powell (1976) and of Dennis and Walker (1984, 1985)