Stability of difference equations with impulsive actions at the instants of time dependent on the state vector
A dynamic system with jumps of the state vector is given by the difference equations \[ x(n+1)= F(x(n)),\quad n+1\neq n_i, \] \[ x(n_i)= x(n_i+ 0)= x(n_i- 0)+ \varphi(x(n_i- 0)),\;x(n_0)= x_0, \] where \(x(n)\in S\subset R^k\), \(n\in \{0,1,\dots\}\); \(n_i\), \(i= 0,1,\dots\) are instants of impulsive inputs and \(F\), \(\varphi: S\to R^k\) are some continuous functions. The instants of impulsive inputs are defined by the transitions of solutions through some hypersurfaces in an extended phase space. Under some additional assumptions, the author establishes the conditions for uniform asymptotic stability of the zero solution of linear and nonlinear systems, instability of nonlinear systems, uniform asymptotic stability of the zero solution of nonperturbed systems under impulsive perturbations and the Chetaev \((\lambda,A, n_0, T)\)-stability [cf. the author, Autom. Remote Control 56, No. 1, Pt. 2, 88-95 (1995); translation from Avtom. Telemekh. 1995, No. 1, 107-116 (1995; Zbl 0932.93060)].
scientific article; zbMATH DE number 1263103