Semidefinite lyapunov functions stability and stabilization

Nontangency-Based Lyapunov Tests for Convergence and Stability in Systems Having a Continuum of Equilibria

“…For Lyapunov stability, the weaker assumptions on the Lyapunov function are supplemented by assuming nontangency of the vector field to invariant or negatively invariant subsets of the level set of the Lyapunov function containing the equilibrium and, for semistability, to invariant or negatively invariant subsets of the zero-level set of the Lyapunov function derivative. These results either extend or complement known results for Lyapunov stability and asymptotic stability involving semidefinite Lyapunov functions and Lyapunov function derivatives as given in [1,16,20,21].…”

“…Theorems 1 and 2 of [16] follow from (i) and (ii) of Corollary 7.1, respectively. However, while Corollary 7.1 requires only that the equilibrium be a local minimizer of the Lyapunov function relative to the set of points at which the Lyapunov function is strictly decreasing, the results of [16] require the equilibrium to be a local minimizer of the Lyapunov function relative to G. Thus Corollary 7.1 is an extension of the main results of [16]. It is shown in [16] that the main result of [1] follows from Theorem 1 in [16].…”

“…The following corollary of Theorem 7.1 is an extension of Theorems 1 and 2 from [16]. (ii) Suppose {x} is an isolated point of N .…”

“…Therefore, with applications to nonnegative dynamics in mind, we consider relative stability of dynamical systems that evolve on (not necessarily open) subsets of R n . Relative stability has been considered previously in [7,16]. We illustrate the main results by applying them to examples from chemical kinetics and adaptive control.…”

On an Eigenflow Equation and its Lie Algebraic Generalization

“…The next result is taken from [20] (Corollary 1) and states, roughly speaking, that an equilibrium point x E in a positively invariant set E, where E itself is assumed to be locally asymptotically stable, is locally asymptotically stable if and only if the equilibrium point x E is locally asymptotically stable on E, which means ∀δ > 0 ∃ǫ > 0 such that if x(0)−x E ≤ ǫ and x(0) ∈ E then x(t)−x E ≤ δ and lim t→∞ x(t) = x E . Lemma 4.…”