Input-to-state stability is an L infinity stability property for systems with disturbances that was introduced by Sontag in 1989. Roughly speaking a system is ISS if bounded inputs imply that the states are bounded and, moreover, the ultimate bound on trajectories depends only on the disturbance size. This property turned out to be very useful and natural to use in many context and has helped generate numerous new analysis and design methods for nonlinear systems. There are many equivalent characterizations of ISS each of which may be important in certain situations. In particular, its Lyapunov characterization makes a nice link to Lyapunov theory and provides a tool to check ISS via the so called Lyapunov ISS functions. Another notion that is very important in this context is the nonlinear gain function that allows one to state small gain theorems that are very useful in robustness analysis, as well as nonlinear controller design. Many other variants of the ISS property are available, including the integral ISS (iISS), input-output-to-state stability (IOSS), and so on.
A large majority of my work uses the ISS methodology in different contexts, such as sampled-data nonlinear systems or averaging. However, there several results in which I was exclusively dealing with analysis of the ISS gains or design of input-to-state stabilizing controllers. In particular, we have presented a framework for computation of "smallest" ISS gains via the technique of dynamic programming. Moreover, we provided a unifying framework for design of controllers to achieve various ISS like properties using the method of dynamic programming. I also developed a controller for a class of nonlinear systems with positive outputs that achieve ISS with respect to measurement disturbances, which is known to be a hard problem for general nonlinear systems.