Stability of linear functional differential systems with a multivalued feedback with respect to impulse disturbances (Q619482)

The paper studies a linear controlled functional differential system with a linear feedback channel, under the assumption that the system is exponentially stable in the closed state. Explicitly, the authors consider the following functional differential inclusion: \[ \dot{x}(t) \in \int_0^t d_s A(t,s) x(s) \;+ B \;\overline{\text{co}}\; x[{\max}\;(0,t-\delta),t] + F'(t), \] where the feedback term is nonideal in the sense that it involves a multivalued delay; \(A\) is a matrix valued function satisfying certain suitable assumptions; \(F\) is a function of locally bounded variation and and its generalized derivative is \(F'(t)\), representing an uncontrolled impulsive effect. Each solution (in the sense that the authors define) \(x\) to the above inclusion is shown to satisfy the linear functional differential equation with a measure: \[ \dot{x}(t) = \int_0^t d_s A(t,s) x(s) \;+ B \int_{\max(t-\delta,0)}^t d_s r(t,s) x(s) + F'(t) + \phi(t), \] where \(\delta\) is a small number and \(\phi\) is am arbitrarily small function. Finally, the exponential stability of the above inclusion is established for sufficiently small \(\delta.\)