Global robust controllability of the triangular integro-differential Volterra systems

Abstract

A solution of the global controllability problem for a class of nonlinear control systems of the Volterra integro-differential equations is presented. It is proven that there exists a family of continuous controls that solve the global controllability problem for this class. The constructed controls depend continuously on the initial and the terminal states. It makes possible to prove the global controllability of the uniformly bounded perturbations of these systems under the global Lipschitz condition for the unperturbed system with respect to the states and the controls.  2004 Elsevier Inc. All rights reserved.

Global properties of the triangular systems in the singular case

“…We omit this argument, which is the same as in [20,Section 4]. It is clear that Theorem 3.3 is a corollary of Theorem 3.1 as well.…”

“…however, in this case, we can treat our system as a bounded perturbation of another smooth system around the points x 1 = x 2 + l 0 , x 1 = x 2 − l 0 , and apply Theorem 3.2 instead of Theorems 3.1, 3.3, and our theory will work anyway (see also Example 3.3 from [20]). )…”

“…Later, especially in 90th, and 2000th, TF became very popular [1,3,4,7,[10][11][12]15,[19][20][21]23,[26][27][28][29]31,35,37]. There were several reasons for that.…”

“…To prove our main results, we reduce the problem to our version of the "back-stepping" process (Section 4, Theorem 4.1), which we applied to the Volterra triangular systems in [20], and which differs widely from the well-known back-stepping algorithms [7,[10][11][12]15,23,28,29,37]. We modify essentially our previous technique proposed in [20], which works now locally, around a regular point only (Section 5), and combine it with constructing a family of discontinuous time-varying feedbacks (44) which solve globally a certain problem of "practical stabilization" of the curve constructed at the previous step of our back-stepping procedure (Lemma 5.1 and its proof in Section 6).…”

“…We modify essentially our previous technique proposed in [20], which works now locally, around a regular point only (Section 5), and combine it with constructing a family of discontinuous time-varying feedbacks (44) which solve globally a certain problem of "practical stabilization" of the curve constructed at the previous step of our back-stepping procedure (Lemma 5.1 and its proof in Section 6).…”