## Small gain problem in coupled differential-difference equations, time-varying delays, and direct Lyapunov method.(English)Zbl 1213.93085

Summary: This article presents a Lyapunov-Krasovskii formulation of scaled small gain problem for systems described by coupled differential-difference equations. This problem includes $$H_\infty$$ problem with block-diagonal uncertainty as a special case. A discretization may be applied to reduce the conditions into linear matrix inequalities. As an application, the stability problem of systems with time-varying delays is transformed into the scaled small gain problem through a process of either one-term approximation or two-term approximation. The cases of time-varying delays with and without derivative upper-bound are compared. Finally, it is shown that similar conditions can also be obtained by a direct Lyapunov-Krasovskii functional method for coupled differential-functional equations. Numerical examples are presented to illustrate the effectiveness of the method in tackling systems with time-varying delays.

### MSC:

 93C15 Control/observation systems governed by ordinary differential equations 93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 93B18 Linearizations
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