On the asymptotic stability by nondecrescent Lyapunov function. (English) Zbl 0534.34054
As is known, Lyapunov’s classical theorem on the asymptotic stability of the zero solution of the non-autonomous system \(\dot x=X(x,t) (x\in R^ k\), \(t\in R_+)\) requires the existence of a scalar function (Lyapunov function) V(x,t) which is, among others, decrescent (admits an infinitely small upper bound). This means that V(x,t)\(\to 0\) uniformly in \(t\in R_+\) as \(| x| \to 0\). In practice to construct such a Lyapunov function is rather difficult. In the main theorem of this paper this condition is replaced by the existence of a further auxiliary vector function, which is positive definite, decrescent and ”cannot change too fast along the solutions”. The results concern partial stability properties and are applied to the study of both partial and concerning all variables stability properties of the equilibrium of the mathematical plain pendulum of changing length.
MSC:
| 34D20 | Stability of solutions to ordinary differential equations |
| 70J99 | Linear vibration theory |
| 34A40 | Differential inequalities involving functions of a single real variable |
Keywords:
decrescent Lyapunov function; differential inequalities; partial asymptotic stability; partial stability properties; mathematical plain pendulumReferences:
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