The comparison method in asymptotic stability problems

Stabilization of the motions of mechanical systems with variable masses

“…We will use the stability theorem in Ref. 10. For the function V 1 = ||p − p 0 (t)|| 2 , we findV 1 ≡ 0 or, in the case of condition (2.5),…”

On stability of discontinuous systems via vector Lyapunov functions

“…Theorem 2.2 Consider the nonlinear system (12). Assume that the function w : Q×V → R q is continuous and w(·, y) is of class W. Let z(t) (t ∈ I z0 ) be the solution of Eq.…”

“…Assume that the function w : Q×V → R q is continuous and w(·, y) is of class W. Let z(t) (t ∈ I z0 ) be the solution of Eq. (12) and [t 0 , t 0 +τ ] ⊆ I z0 . If there exist locally Lipschitz continuous and regular functions v i , i = 1, · · · , q.…”

Stabilization of the motions of non-autonomous mechanical systems

“…It follows from the precompact nature of system (1.5) that there is a family of limit reference systems 11 that differ from system (1.5) by the replacement of the functions (t) and b(t) by the limit functions *(t) and b*(t). It is easy to see here that the solutions (u * 1 (t), u * 2 (t)) of the limit reference system satisfy the relation 11 Then, on the basis of the principle of quasi-invariance of the positive limit set of the perturbed motion, 11 it can be stated that if the set (1.6)…”

“…11 This fact is proved as follows: on the set (1.6) the solutions of the limit system for (1.4) satisfy the relation 12 (1.7) which, by virtue of the non-degeneracy of the matrix P*(t), implies the condition x * 1 (t) ≡ 0, i.e., the solutions of the limit system for (1.4) do not belong to set (1.6). Therefore, the unperturbed motion (1.2) of system (1.1) will be stabilized up to uniform asymptotic stability.…”