Second method of Lyapunov for stability of linear impulsive differential-difference equations with variable impulsive perturbations
Using the second method of Lyapunov, the authors study the stability of a linear system of impulsive differential-difference equations \[ \dot{x} (t) = A(t)x(t) +B (t)x(t-h), \quad t\neq \tau_{k}(x(t)), \;t>t_0, \tag{1} \] \[ \Delta x(t)|_{t= \tau_{k}(x(t))} = C_{k}x(t), \quad t > t_0,\;k = 1,2,\dots,\tag{2} \] where \(x\in{\mathbb{R}^{n}}, A(t)\) and \(B(t)\) are \(n \times n\)-matrix-valued functions; \(C_{k}, k=1,2,\cdots,\) are \(n \times n\)-matrices; \(\tau_{k}:{\mathbb{R}^{n}} \rightarrow (t_{0}, \infty)\); \(\Delta x(t)= x (t+0)-x(t-0)\). If the impulses are realized at fixed moments of time, the results can be easily derived by virtue of the corresponding result in the continuous case. Studies of impulsive differential-difference equations with variable impulsive perturbations carry lots of difficulties due to the presence of phenomena such as ''beating'' of the solutions, bifurcation, loss of property of autonomy, etc. The importance of these equations in mathematical modeling necessitates to prove criteria for the stability of their solutions. The uniform stability and uniform asymptotical stability of the zero solution to systems (1-2) are established with the aid of comparison techniques, piecewise continuous Lyapunov functions and a technique that uses minimal subsets of suitable spaces of piecewise continuous functions. The elements of these subsets help to estimate the derivatives of piecewise continuous auxiliary functions.
scientific article; zbMATH DE number 1214057