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Mathematics of Complexity and Dynamical Systems pp 1137–1151Cite as

Nonsmooth Analysis in Systems and Control Theory

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Article Outline

Glossary

Definition of the Subject

Introduction

Elements of Nonsmooth Analysis

Necessary Conditions in Optimal Control

Verification Functions

Dynamic Programming and Viscosity Solutions

Lyapunov Functions

Stabilizing Feedback

Future Directions

Bibliography

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Abbreviations

Generalized gradients and subgradients :

These terms refer to various set-valued replacements for the usual derivative which are used in developing differential calculus for functions which are not differentiable in the classical sense. The subject itself is known as nonsmooth analysis . One of the best-known theories of this type is that of generalized gradients. Another basic construct is the subgradient, of which there are several variants. The approach also features generalized tangent and normal vectors which apply to sets which are not classical manifolds. The article contains a summary of the essential definitions.

Pontryagin Maximum Principle :

The main theorem on necessary conditions in optimal control was developed in the 1950s by the Russian mathematician L. Pontryagin and his associates. The Maximum Principle unifies and extends to the control setting the classical necessary conditions of Euler and Weierstrass from the calculus of variations, as well as the transversality conditions. There have been numerous extensions since then, as the need to consider new types of problems continues to arise.

Verification functions :

In attempting to prove that a certain control is indeed the solution to a given optimal control problem, one important approach hinges upon exhibiting a function having certain properties implying the optimality of the given control. Such a function is termed a verification function. The approach becomes widely applicable if one allows nonsmooth verification functions.

Dynamic programming :

A well-known technique in dynamic problems of optimization is to solve (in a discrete context) a backwards recursion for a certain value function related to the problem. This technique, which was developed notably by Bellman, can be applied in particular to optimal control problems. In the continuous setting, the recursion corresponds to the Hamilton–Jacobi Equation . This partial differential equation does not generally admit smooth classical solutions. The theory of viscosity solutions uses subgradients to define generalized solutions, and obtains their existence and uniqueness.

Lyapunov function :

In the classical theory of ordinary differential equations, global asymptotic stability is most often verified by exhibiting a Lyapunov function, a function along which trajectories decrease. In that setting, the existence of a smooth Lyapunov function is both necessary and sufficient for stability. The Lyapunov function concept can be extended to control systems, but in that case it turns out that nonsmooth functions are essential. These generalized control Lyapunov functions play an important role in designing optimal or stabilizing feedback.

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Author information

Authors and Affiliations

  1. Institut universitaire de France et Université de Lyon, Lyon, France

    Francis Clarke

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  1. Francis Clarke
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Clarke, F. (2012). Nonsmooth Analysis in Systems and Control Theory. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_69

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