Some kinds of Lyapunov functional in stability theory of RFDE
By means of Lyapunov functionals the uniform asymptotic stability of the zero solution of one of the following types of retarded functional differential equations is studied: \[ \dot x(t) = A_0 x(t) + \sum^N_{k = 1} A_k x(t - T_k), \tag{1} \] \[ \dot x(t) = A_0 x(t) + Bx \bigl( t - \tau (t) \bigr) + \int^t_{t - \tau (t)} C(t,s) x(s) ds, \tag{2} \] \[ \dot x(t) = - f \bigl( t,x(t) \bigr) + g \bigl( ,x(t - \tau) \bigr), \tag{3} \] \[ \dot x(t) = - f \bigl( t,x(t) \bigr) + g \bigl( t,x (t - \tau) \bigr) + \int^t_{t - \tau (t)} h \bigl( t,s,x(s) \bigr) ds. \tag{4} \] In these equations \(x \in R^n\) and in particular (1) can be written as \[ \dot x(t) = \sum^n_{j = 1} a^{(0)}_{ij} x_j(t) + \sum^N_{k = 1} \sum^n_{j = 1} a_{ij}^{(k)} x_j (t - T_k), \quad i = 1,2, \dots, n. \tag{5} \] Some easily checked sufficient conditions for the uniform asymptotic stability of the zero solution are obtained. The following Theorem being one of these results gives an idea of what type they are: Theorem. If there exist positive constants \(\alpha_k\) \((k = 1,2, \dots, N)\), \(\mu, \nu\) and \(d_i\) \((i = 1,2, \dots, n)\) satisfying the inequalities \[ d_i a_{ii}^{(0)} + \sum^n_{j = 1,j \neq j} d_j |a^{(0)}_{ji} |, \quad \mu - \sum^N_{k = 1} \alpha_k \geq \nu, \] \[ \alpha_k - \beta \max_{1 \leq j \leq n} \sum^n_{i = 1} |a_{ij}^{(k)} |\geq 0 \quad \left( \text{here } \beta = \max_{1 \leq j \leq n} \{d_j\} \right) \] for \(i = 1,2, \dots, n\) and \(k = 1,2, \dots, N\), then the zero solution of (5) is uniformly asymptotically stable.
retarded functional differential equations
scientific article; zbMATH DE number 801091