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2 Control and Cybernetics

2 2009

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The nonsmooth maximum principle

Clarke F. H., Pinho M. R. de

Control and Cybernetics

2009

Vol. 38, no 4A

1151-1167

We present a brief survey of the nonsmooth maximum principle of optimal control, focusing, in particular, upon the alternative forms of the adjoint equation. We obtain a new version of the theorem that asserts for the first time the full Weierstrass condition together with the Euler form of the adjoint equation, thereby extending a result of de Pinho and Vinter. The new theorem also features stratified hypotheses and conclusions. Two examples illustrate its use.

Stability analysis of sliding-mode feedback control

Clarke F. H., Vinter R. B.

Control and Cybernetics

2009

Vol. 38, no 4A

1169-1192

This paper provides new analytic tools leading to the first rigorous stability and robustness analysis of sliding-mode feedback controllers. Unrestrictive conditions are given, under which these controllers are stabilizing in the presence of large disturbances, conditions invoke the existence of two Lyapunov-type functions, the first associated with passage to the sliding set in finite time, and the second with convergence to the desired state. In this approach, account is taken, from the outset, of implementational constraints. We provide a framework for establishing stability and robustness of the closed-loop system, for a variety of implementation schemes. We illustrate our results by means of two examples of the type frequently encountered in the sliding-mode literature.