Nonsmooth Analysis in Control Theory: A Survey

  title={Nonsmooth Analysis in Control Theory: A Survey},
  author={Frank H. Clarke},
  journal={Eur. J. Control},
  • F. Clarke
  • Published 2001
  • Mathematics
  • Eur. J. Control
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. 

Nonsmooth Analysis in Systems and Control Theory

  • F. Clarke
  • Mathematics
    Encyclopedia of Complexity and Systems Science
  • 2009
Generalized gradients and subgradients These terms refer to various setvalued replacements for the usual derivative which are used in developing differential calculus for functions which are not

Lyapunov Functions and Feedback in Nonlinear Control

This article synthesizes in this article a number of recent developments bearing upon the regularity properties of Lyapunov functions, and discusses the equivalence between open-loop controllability, feedback stabilizability, and the existence of Lyapsinov functions with appropriateregularity properties.

Optimal control: Nonlocal conditions, computational methods, and the variational principle of maximum

This paper surveys theoretical results on the Pontryagin maximum principle (together with its conversion) and nonlocal optimality conditions based on the use of the Lyapunov-type functions (solutions

Nonregular feedback linearization: a nonsmooth approach

A criterion of nonregular static state feedback linearizability is presented for a class of nonlinear affine systems with two control inputs, and its application to nonholonomic systems is briefly discussed.

Sliding mode control design via piecewise smooth Lipschitz surfaces based on contingent cone criteria

In order to improve flexibility of sliding mode control (SMC) for a class of nonlinear systems, a new control design method is proposed in this paper. The sliding surface is extended to be a generic

Necessary conditions in optimal control: a new approach

The principal focus here is the development of structural criteria on the problem which guarantee a priori that the necessary conditions apply to any local minimizer.

Sliding mode control for a class of second-order systems via linear Lipschitz surfaces

In order to improve flexibility of sliding mode control for a class of second-order systems, a new control design method is proposed in this paper. The sliding surface is designed to be a linear

Regularization and bang-bang conjugate times in optimal control

In this thesis we consider a minimal time control problem for single-input control-affine systems in finite dimension with fixed initial and final conditions, where the scalar control take values on

On a small gain theorem for ISS networks in dissipative Lyapunov form

This paper considers several interconnected ISS systems supplied with ISS Lyapunov functions defined in the dissipative form and provides a condition of a small gain type under which this construction is possible and describes a method of an explicit construction of such an ISS LyAPunov function.

Discontinuous feedbacks, discontinuous optimal controls, and continuous-time model predictive control

It is known that there is a class of nonlinear systems that cannot be stabilized by a continuous time-invariant feedback. This class includes systems with interest in practice, such as nonholonomic



Semiconcave Control-Lyapunov Functions and Stabilizing Feedbacks

This work constructs discontinuous feedback laws and makes it possible to choose these continuous outside a small set (closed with measure zero) of discontinuity in the case of control systems which are affine in the control.

Nonsmooth control-Lyapunov functions

It is shown that the existence of a continuous control-Lyapunov function (CLF) is necessary and sufficient for null asymptotic controllability of nonlinear finite-dimensional control systems. The CLF

Liapunov and lagrange stability: Inverse theorems for discontinuous systems

A converse Liapunov theorem is proved which applies to systems of ordinary differential equations with a discontinuous righthand side and it is shown that in order to achieve a necessary and sufficient condition in terms of continuous Liap unov functions, the classical definitions need to be strengthened in a convenient way.

Asymptotic controllability implies feedback stabilization

It is shown that every asymptotically controllable system can be globally stabilized by means of some (discontinuous) feedback law. The stabilizing strategy is based on pointwise optimization of a

A Smooth Converse Lyapunov Theorem for Robust Stability

This paper presents a converse Lyapunov function theorem motivated by robust control analysis and design. Our result is based upon, but generalizes, various aspects of well-known classical theorems.

Some characterizations of optimal trajectories in control theory

Several characterizations of optimal trajectories for the classical Mayer problem in optimal control are presented. The regularity of directional derivatives of the value function is studied. For

The value function in optimal control: Sensitivity, controllability, and time-optimality

We consider a general optimal control problem in which the constraints depend on a parameter $\alpha $, and the resulting value function $V(\alpha )$. A formula for the generalized gradient of V is

Existence of Lipschitz and Semiconcave Control-Lyapunov Functions

Given a locally Lipschitz control system which is globally asymptotically controllable at the origin, a control-Lyapunov function is constructed and the existence of another one which is semiconcave outside the origin is deduced.

A Lyapunov-Like Characterization of Asymptotic Controllability

It is shown that a control system in ${\bf R}^n $ is asymptotically controllable to the origin if and only if there exists a positive definite continuous functional of the states whose derivative can