The nonsmooth maximum principle

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.