The authors deal with optimal control with states belonging to a finite dimensional Riemann manifold \(M\): Minimize \(J(\xi,v)=c_0(\xi(0))+c_T(\xi(T))+\int_0^T f^0_t(\xi(t),v(t))dt\) over all \(\xi\), \(v\) satisfying the following control equation with end-point and control constraints \(\dot\xi(t)=f_t(\xi(t),v(t))\), a.e. \(t\in I\); \(\xi(0)\in N_0\), \(\xi(T)\in N_T\); \(v(t)\in \Omega_t \subseteq U\), a.e. \(t\in I,\) where \(N_0\), \(N_T\) are submanifolds of \(M\). They give a sufficient condition of geometric type on a manifold assuming that the maximized Hamiltonian is sufficiently smooth. The condition extends the classical condition of nonexistence of conjugate points in the calculus of variations.
For the entire collection see [Zbl 0903.00046].