Small-Gain Results

Modern stability theory, i.e., stability theory for systems with inputs (Input-to-State Stability, Input-to-Output Stability, etc.), has enabled researchers to give a new way of proving stability: small-gain theorems. Everything started with this paper: 

Z.-P. Jiang, A. R. Teel and L. Praly, Small-gain theorem for ISS systems and applications, Mathematics of Control, Signals, and Systems, 7, 1994, pp. 95–120. 

but it should be noted that the "small-gain" idea existed earlier in the classical input-output operator framework. 

I have also produced some small-gain results, focusing mainly on abstract control systems, i.e., a wide class of systems that contains systems described by ODEs, PDEs, delay systems and most hybrid systems (using and extending the classical abstract formulation of a control system given by Kalman and Sontag; see Chapter 2 in Sontag's book). Here is a list of papers that provide small-gain results. 

• I. Karafyllis and J. Tsinias, “Non-Uniform in Time Input-to-State Stability and the Small-Gain Theorem”, IEEE Transactions Automatic Control, 49(2), 2004, pp. 196-216.

• I. Karafyllis, “The Non-Uniform in Time Small-Gain Theorem for a Wide Class of Control Systems with Outputs”, European Journal of Control, 10(4), 2004, pp. 307-323.

• I. Karafyllis, Z.-P. Jiang and G. Athanasiou, “Nash Equilibrium and Robust Stability in Dynamic Games: A Small-Gain Perspective”, Computers and Mathematics with Applications, 60(11), 2010, pp. 2936-2952.

• I. Karafyllis and Z.-P. Jiang, “A Vector Small-Gain Theorem for General Nonlinear Control Systems”, IMA Journal of Mathematical Control and Information, 28(3), 2011, pp. 309-344.

• I. Karafyllis and Z.-P. Jiang, “A New Small-Gain Theorem with an Application to the Stabilization of the Chemostat”, International Journal of Robust and Nonlinear Control, 22(14), 2012, pp. 1602–1630.

Small-Gain results are also available in the book:

• I. Karafyllis and Z.-P. Jiang, Stability and Stabilization of Nonlinear Systems, Springer-Verlag, London (Series: Communications and Control Engineering), 2011.

Chapter 7 of the book 

• I. Karafyllis and M. Krstic, Input-to-State Stability for PDEs, Springer-Verlag, London (Series: Communications and Control Engineering), 2019.

provides small-gain results based on the so-called "exp-ISS" property, that can allow the derivation of exponential-type stability estimates. These small-gain results are used in an instrumental way in the above book for the proof of various stability results in systems with PDEs.