Popov condition revisited for other nonlinear systems

The problem of finding an explicit stabilizing control for dynamical systems with sup bounded disturbance and nonlinear terms bounded in norm by a non decreasing function has been addressed. These systems often exhibit from natural bounds a (strong) finite time Lagrange instability due to disturbance adverse effect. Linear PD type control law produces conditional simple stability with weak system robustness to disturbance and shows restricted action to modify complete nonlinear system dynamics, owing to the limited number of gain parameters and the large class of systems satisfying non decreasing law bound. With a nonlinear Lur'ie type control part added to linear PD part, conditional asymptotic stability is obtained from Popov condition, equivalent to application of circle criterium, inside a ball corresponding to a balance between linear and nonlinear terms and reducing to absolute exponential stability result for usual linear bound. Functional robustness is obtained for equivalence class characterized by the same non decreasing function upperbounding nonlinear system part.