A class of absolutely stable distributed systems (Q1820112)

The author studies a class of distributed systems which can be described by the equation \[ (1)\quad du/dt=Au+B\phi (Cu,t),\quad t\geq 0 \] where A is a closed linear operator which acts in a Hilbert space H, B and C are linear bounded operators which act from a Hilbert space E to H and from H to E respectively, \(\phi\) : \(E\times [0,\infty)\to E\) is a function that ensures the existence and uniqueness of a solution of equation (1) for any initial vector u(0), belonging to the domain of definition D(A) of the operator A, and that satisfies the inequality \[ | \phi (u,t)|_ E\leq q| u|_ E\quad (q=const;\quad t\geq 0) \] in which \(| \cdot |_ E\) denotes a norm on E. The author proves that if the linear part of (1) has an impulse function being in some sense positive, then the system (1) is absolutely stable in the Hurwitz' angle in the sense that the system satisfies the generalized hypothesis of absolute stability of dynamical systems in the Hurwitz' angle.