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**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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\textit{K. Gu} et al., Int. J. Robust Nonlinear Control 21, No. 4, 429--451 (2011; Zbl 1213.93085)

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