The solution of the global controllability problem for the triangular systems in the singular case

The solution of the global controllability problem is obtained for a class of the triangular systems of O.D.E. that are not feedback linearizable. The introduced class is a generalization of the classes of triangular systems investigated before. The solution of the problem is based on the approach proposed in [18] for the triangular systems of the Volterra equations. This yields the same properties of the considered class of triangular systems as those established in [18] for the Volterra systems. As well as in [18], for the current class of triangular systems, it is proven that there exists a family of continuous controls that solve the global controllability problem for the considered class and continuously depend on the initial and the terminal states. As well as in [18], this implies the global controllability of the bounded perturbations of the current class. In contrast with [18], to prove the existence of the desired family of open-loop controls, we construct a family of closed-loop ones each of which steers the corresponding initial state into an appropriate neighborhood of an appropriate terminal point.