Necessary and sufficient conditions for bounded global stability of certain nonlinear systems

Nonlinear systems having the form \[ x ˙ = − A x + B y \dot x = - Ax + By \] \[ y ˙ = C x − f ( y ) \dot y = Cx - f\left ( y \right ) \] , where ∂ f / ∂ y \partial f/\partial y is a symmetric matrix, are considered. Such systems include the class of nonlinear reciprocal networks where the nonlinearity is voltage (or current) controlled. Also included, provided c T b ≠ 0 {c^T}b \ne 0 , are the equations of nonlinear feedback systems, \[ x ˙ = A x + b f ( c T x ) \dot x = Ax + bf\left ( {{c^T}x} \right ) \] , considered by Aizerman [1], A type of stability called bounded global stability is considered which requires that all bounded solutions decay as t → ∞ t \to \infty to the set of equilibrium points. A necessary and sufficient condition on the linear parts of these systems for their bounded global stability is given. It is also shown that this condition insures the existence of at least one stable equilibrium point.