An improved reciprocally convex inequality and an augmented Lyapunov-Krasovskii functional for stability of linear systems with time-varying delay.

*(English)*Zbl 1375.93114Summary: This paper is concerned with stability of a linear system with a time-varying delay. First, an improved reciprocally convex inequality including some existing ones as its special cases is derived. Compared with an extended reciprocally convex inequality recently reported, the improved reciprocally convex inequality can provide a maximum lower bound with less slack matrix variables for some reciprocally convex combinations. Second, an augmented Lyapunov-Krasovskii functional is tailored for the use of a second-order Bessel-Legendre inequality. Third, a stability criterion is derived by employing the proposed reciprocally convex inequality and the augmented Lyapunov-Krasovskii functional. Finally, two well-studied numerical examples are given to show that the obtained stability criterion can produce a larger upper bound of the time-varying delay than some existing criteria.

##### MSC:

93D30 | Lyapunov and storage functions |

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

93C15 | Control/observation systems governed by ordinary differential equations |

93C05 | Linear systems in control theory |

##### Keywords:

time-delay systems; stability; reciprocally convex inequality; Lyapunov-Krasovskii functional
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##### References:

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