Pontryagin-type necessary conditions for differential inclusion problems (Q1100422)

Necessary optimality conditions for the finite-dimensional differential inclusion \(\dot x\in F(t,x)\), \(t\in [0,1]\) are studied. It is assumed that the values of F are non-empty, compact and convex sets, F is measurable in t, that F satisfies some semicontinuity-type conditions and admits a Carathéodory selection. The minimum of a locally Lipschitzian function \(l(x(0),x(1))\), is sought where the ends \(x(0)\) and \(x(1)\) belong to some given closed set S. By using some results of Clarke, a short proof of the Pontryagin maximum principle for the above problem is presented.