Necessary optimality conditions for nonconvex differential inclusions with endpoint constraints (Q1906999)

The problem \(\min g_0(x(1))\) subject to \[ x'(t)\in F(t, x(t)),\quad x(0)= x_0,\quad g_1(x(1))= 0 \] is considered, where \(x\in\mathbb{R}^n\) and values of \(F\) may be nonconvex. A necessary condition (a kind of maximum principle) is derived in terms of Lojasiewicz parametrization of \(\overline{\text{co}} F(t,x)\). When \(F\) admits a \(C^1\) Lojasiewicz representation, this result implies an intrinsic necessary condition (in terms of the Hamiltonian of \(F\) with no explicit use of a parametrization). Comparisons of other known necessary conditions and examples are given. At the end, a theorem is formulated where more general constraints of the form \(g_2(t, x(t))\in A\) are included.