Synthesis of optimal trajectories on representation spaces of Lie groups

Let $X$ be a representation space of a Lie group $G$, $\omega$ a differential form of the first degree on $X$ and $K(x)$ a field of closed convex cones on $X$. Problem 1 is the minimization of the integral of the differential form $\omega$ along curves which satisfy certain boundary conditions and are solutions of the differential inclusion $\dot x(t)\in K(x(t))$. This problem is assumed to be equivariant in the sense that the field of cones $K(x)$ and the differential form $\omega$ are invariant under the action of $G$. For Problem 1 the author introduces the concept of a totally extremal manifold, which is an analogue of the concept of a totally geodesic manifold in a Riemannian or Finsler space. Some theorems are proved on totally extremal manifolds for fundamental representations of Lie groups. These theorems are used along with techniques developed in previous papers by the author to construct a synthesis of optimal trajectories for some multidimensional equivariant problems.
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