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

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Stability of Linear Systems With Time-Varying Delays Using Bessel–Legendre Inequalities (2018, background): “Remark 1: As noticed in [28], the additional variables X1 and X2 can be removed by taking X1 = R1 − Y1R−1 2 Y T 1 and X2 = R2 − Y T 2 R−1 1 Y2 , which obviously verifies conditions (10).”
Overview of recent advances in stability of linear systems with time‐varying delays (2019, methodology): “By employing the improved reciprocally convex inequality (21), a less conservative stability criterion is presented in [72].”
Observer-Based Event-Triggered Control for Networked Linear Systems Subject to Denial-of-Service Attacks (2020, methodology): “…L 1 , n , ∀ k ∈ K ( n ) , n ∈ N , taking the derivative of V ( t ) with respect to t along the system (18) yields Using the inequality in [46] and [47], we obtain Using (46) and (47) to deal with integral terms in (45) by [46, Lemma 3], we have Substituting (48) into (45), adding the terms e Tk ,…”