# Nonsmooth Analysis in Control Theory: A Survey

@article{Clarke2001NonsmoothAI, title={Nonsmooth Analysis in Control Theory: A Survey}, author={Frank H. Clarke}, journal={Eur. J. Control}, year={2001}, volume={7}, pages={145-159} }

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.

## 42 Citations

### Nonsmooth Analysis in Systems and Control Theory

- MathematicsEncyclopedia 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

- Mathematics
- 2004

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

- Mathematics
- 2009

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

- MathematicsIEEE Trans. Autom. Control.
- 2003

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

- Mathematics
- 2014

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

- MathematicsMath. Program.
- 2003

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

- MathematicsProceedings of the 32nd Chinese Control Conference
- 2013

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

- Mathematics
- 2010

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

- Mathematics2009 European Control Conference (ECC)
- 2009

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

- Mathematics
- 2003

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…

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