Francis Clarke

Feedback Stabilization and Lyapunov Functions

Given a locally defined, nondifferentiable but Lipschitz Lyapunov function, we employ it in order to construct a (discontinuous) feedback law which stabilizes the underlying system to any given tolerance. A converse result shows that suitable Lyapunov functions of this type exist under mild assumptions. We also establish that the feedback in question possesses a robustness property relative to measurement error, despite the fact that it may not be continuous.

Necessary conditions in dynamic optimization

Introduction Boundary trajectories Differential inclusions The calculus of variations Optimal control of vector fields The Hamiltonian inclusion Bibliography Index.

Optimal Control Problems with Mixed Constraints

We develop necessary conditions of broad applicability for optimal control problems in which the state and control are subject to mixed constraints. We unify, subsume, and significantly extend most of the results on this subject, notably in the three special cases that comprise the bulk of the literature: calculus of variations, differential-algebraic systems, and mixed constraints specified by equalities and inequalities. Our approach also provides a new and unified calibrated formulation of the appropriate constraint qualifications, and shows how to extend them to nonsmooth data. Other features include a very weak hypothesis concerning the type of local minimum, nonrestrictive hypotheses on the data, and stronger conclusions, notably as regards the maximum (or Weierstrass) condition. The necessary conditions are stratified, in the sense that they are asserted on precisely the domain upon which the hypotheses (and the optimality) are assumed to hold. This leads to local, intermediate, and global versions of the necessary conditions, according to how the hypotheses are formulated.

Discontinuous Feedback and Nonlinear Systems

This tutorial paper is devoted to the controllability and stability of control systems that are nonlinear, and for which, for whatever reason, linearization fails. We begin by motivating the need for two seemingly exotic tools: nonsmooth control-Lyapunov functions, and discontinuous feedbacks. Then, after a (very) short course on nonsmooth analysis, we build a theory around these tools. We proceed to apply it in various contexts, focusing principally on the design of discontinuous stabilizing feedbacks.

Nonsmooth Analysis in Control Theory: A Survey

In the classical calculus of variations, the question of regularity (smoothness or otherwise of certain functions) plays a dominant role. This same issue, although it emerges in different guises, has turned out to be crucial in nonlinear control theory, in contexts as various as necessary conditions for optimal control, the existence of Lyapunov functions, and the construction of stabilizing feedbacks. In this report we give an overview of the subject, and of some recent developments.

State Constrained Feedback Stabilization

A standard finite dimensional nonlinear control system is considered, along with a state constraint set S and a target set

Lyapunov functions and discontinuous stabilizing feedback

\Sigma

Lyapunov and feedback characterizations of state constrained controllability and stabilization

. It is proven that open loop S-constrained controllability to

On Global Optimality Conditions for Nonlinear Optimal Control Problems

\Sigma

Nonconvex Duality in Optimal Control

implies closed loop S-constrained controllability to the closed