Asen L. Dontchev

Implicit Functions and Solution Mappings

The purpose of this self-contained work is to provide a reference on the topic and to provide a unified collection of a number of results which are currently scattered throughout the literature. The first chapter of the book treats the classical implicit function theorem in a way that will be useful for students and teachers of undergraduate calculus. The remaining part becomes gradually more advanced, and considers implicit mappings defined by relations other than equations, e.g., variational problems. Applications to numerical analysis and optimization are also provided.

The Euler approximation in state constrained optimal control

We analyze the Euler approximation to a state constrained control problem. We show that if the active constraints satisfy an independence condition and the Lagrangian satisfies a coercivity condition, then locally there exists a solution to the Euler discretization, and the error is bounded by a constant times the mesh size. The proof couples recent stability results for state constrained control problems with results established here on discrete-time regularity. The analysis utilizes mappings of the discrete variables into continuous spaces where classical finite element estimates can be invoked.

Lipschitzian stability in nonlinear control and optimization

This paper studies Lipschitz properties, relative to the parameter p, of the set of solutions to problems of the form\[{\text{find }}z \in \Omega _p {\text{ such that }}T_p (z) \in F_p (z).\] As applications, various problems in control and optimization are examined, focusing in particular on the stability of the feasible set of a control problem, and the stability of solutions of infinite-dimensional mathematical programs and optimal control problems. In another application, an estimate is obtained for the error in the Euler approximation to an optimal control problem.

Error estimates for discretized differential inclusions

We present an estimate for the Hausdorff distance between the set of solutions of a differential inclusion and the set of solutions of its Euler discrete approximation, using an averaged modulus of continuity for multifunctions. A computational procedure to obtain a certain solution of the discretized inclusion is proposed.ZusammenfassungEine Abschätzung der Hausdorff-Distanz zwischen der Menge aller Lösungen einer Differentialeinschließung und der Eulerschen Approximation dieser Einschließung wird vorgelegt, wobei ein Stetigkeitsmodul für Multifunktionen angewendet wird. Eine numerische Prozedur zur Auffindung einer gewissen Lösung der diskretisierten Einschließung wird vorgeschlagen.

Implicit Functions, Lipschitz Maps, and Stability in Optimization

We present an implicit function theorem for set-valued maps associated with the solutions of generalized equations. As corollaries of this theorem, we derive both known and new results. Strong regularity of variational inequalities and Lipschitz stability of optimization problems are discussed.

Lipschitzian Stability for State Constrained Nonlinear Optimal Control

For a nonlinear optimal control problem with state constraints, we give conditions under which the optimal control depends Lipschitz continuously in the L2 norm on a parameter. These conditions involve smoothness of the problem data, uniform independence of active constraint gradients, and a coercivity condition for the integral functional. Under these same conditions, we obtain a new nonoptimal stability result for the optimal control in the

Metric regularity of semi-infinite constraint systems

L^\infty

Implicit function theorems for generalized equations

norm. And under an additional assumption concerning the regularity of the state constraints, a new tight

Error Bounds for Euler Approximation of a State and Control Constrained Optimal Control Problem

L^\infty

Uniform Convergence and Mesh Independence of Newton's Method for Discretized Variational Problems

estimate is obtained. Our approach is based on an abstract implicit function theorem in nonlinear spaces.