On the condition $c^T A^{-1} b + r > 0$, in the Lurie problem

Abstract:

The problem of Lurie consists in finding NASC’s for all solutions of the system $\{ x’ = Ax + bf(\sigma ),\sigma ’ = {c^T}x - rf(\sigma )\}$ to tend to zero as $t \to \infty$ under appropriate conditions on the functions involved. When $f(\sigma )/\sigma < M$, for all $\sigma$ and a certain $M$, we obtain NASC’s for the system to be absolutely stable. When $f(\sigma )/\sigma < M$ as $|\sigma | \to \infty$, we obtain conditions for ultimate uniform boundedness of the solutions of the system.