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 another work [18] devoted to the triangular systems of the Volterra equations and written jointly with W.H. Schmidt by the current authors. 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.