The global asymptotic stability and stabilization in nonlinear cascade systems with delay

Exponential Stability of Impulsive Cascaded Systems and its Application in Robot Control

“…To illustrate our theoretical results, we consider the problem of tracking a wheeled robot that moves in a plane. The motion of this tracking robot is given by the following system of differential equations [26], [49];…”

Sufficient stability conditions and stabilizing control design for delay differential systems of a special type

“…Some comparison equations can be conveniently used to obtain this estimate. For example, the following consequence of Theorem 1 of [9] holds: Lemma 2 [6]. Assume that the function V ∈ C 1 (R + × R n , R + ) satisfies (2) in the definition domain, and the function W (t, r) ∈ C(R + × R + , R + ) is such that the derivative of V along the solution curve of (1) satisfies…”

“…Some additional conditions providing the global character of the results are discussed. In the general case such a character is not guaranteed when using semidefinite functions [6,7].…”

“…Therefore, in the case of a nonlinear system, the asymptotic stability of the zero solution does not generally mean the global stability when a local semidefinite function (the functional) is used. Appropriate counterexamples are presented in [6,7].…”

“…We will specify for the sequel some sufficient conditions for stability and asymptotic stability of the zero solution of (1) in terms of auxiliary semidefinite functions and functionals: Lemma 1 [6]. Assume that there is a Lyapunov function V (t, x) such that…”