Small-gain theorem
Encyclopedia
In nonlinear systems, the formalism of input-output stability is an important tool in studying the stability of interconnected systems since the gain of a system directly relates to how the norm of a signal increases or decreases as it passes through the system. The small-gain theorem gives a sufficient condition for finite-gain 
stability of the feedback connection. The small gain theorem was proved by George Zames
in 1966. It can be seen as a generalization of the Nyquist criterion to non-linear non-time-invariant MIMO systems.
Theorem. Assume two systems
and 
are connected in a feedback loop, then the closed loop system is input-output stable if 
. (The norm can be the infinity norm, that is the size of the largest singular value of the transfer function over all frequencies. Also any induced Norm will lead to the same results).
stability of the feedback connection. The small gain theorem was proved by George ZamesGeorge Zames
George Zames was a control theorist and professor at McGill University, Montreal, Quebec, Canada. Zames is known for his fundamental contributions to the theory of robust control, and was credited for the development of various well-known results such as small-gain theorem, passivity theorem,...
in 1966. It can be seen as a generalization of the Nyquist criterion to non-linear non-time-invariant MIMO systems.
Theorem. Assume two systems
and are connected in a feedback loop, then the closed loop system is input-output stable if . (The norm can be the infinity norm, that is the size of the largest singular value of the transfer function over all frequencies. Also any induced Norm will lead to the same results).