it is proved here that if $g$ is a locally graded group satisfying the minimal condition on subgroups which are not locally supersoluble, then $g$ is either locally supersoluble or a vcernikov group. the same conclusion holds for locally finite groups satisfying the weak minimal condition on non-(locally supersoluble) subgroups. as a consequence, it is shown that any infinite locally graded gro...

We derive necessary optimality conditions for the time of crisis problem under a more general hypothesis than usual one encountered in hybrid setting, which requires that any optimal solution should cross boundary constraint set transversely. Doing so, we apply Pontryagin Maximum Principle to sequence regular control problems whose integral cost approximates crisis. Optimality are derived by pa...