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By Dragutin Debeljkovic

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B): e λ max ( ϒ ) ( t − t0 ) < β , ∀t ∈ ℑ , α (22) where: ( ) ( ) λ max ϒ = λ max ⎛⎜ xT ( t ) P0 A 1 P0 AT1P0 + q 2 P0 x ( t ) : xT ( t ) P0 x ( t ) = 1 ⎞⎟ , ⎝ −1 ⎠ (23) Proof. 1. 38 Time-Delay Systems T τ τ ⎛ ⎞ ⎛ ⎞ V ( x t ,τ ) = ⎜ x ( t ) + ∫ P1 (η ) x ( t − η ) dη ⎟ × ( −Q ) × ⎜ x ( t ) + ∫ P1 (η ) x ( t − η ) dη ⎟ , ⎜ ⎟ ⎜ ⎟ 0 0 ⎝ ⎠ ⎝ ⎠ and since, (25) is negative definite and obviously V ( x t ,τ ) < 0 , time delay system (1) ( −Q ) possesses atractivity property. 8. 2uT ( t ) v ( t − τ ) ≤ uT ( t ) Γ −1 u ( t ) + vT ( t − τ ) Γ v ( t − τ ) , Γ = ΓT > 0 .

The chapter also represents a comprehensive survey on important stability theorems which apply to studied classes of systems. The geometric theory of consistency leads to the natural class of positive definite quadratic forms on the subspace containing all solutions. This fact makes possible the construction of Lyapunov stability theory even for the time delay systems in that sense that asymptotic 28 Time-Delay Systems stability is equivalent to the existence of symmetric, positive definite solutions to a weak form of Lyapunov continuous (discrete) algebraic matrix equation (Owens, Debeljkovic 1985) respectively, incorporating condition which refers to time delay term.

Liu, (1997) Lyapunov stability analysis of linear singular dynamical systems, Proc. Int. Conf. on Intelligent Processing Systems, Beijing, October 28-31, pp. 635-639. Coppel, W. C. Heath. , (1989) Singular control systems, Lecture Notes in Control and Information Sciences, Springer, Berlin, 118. Debeljkovic, D. , (1985) Finite time stability of linear singular discrete time systems, Proc. Conference on Modeling and Simulation, Monastir (Tunisia), November 85, pp. 2-9. Debeljkovic, D. S. (International Symposium Modelling and Simulation for Control of Lumped and Distributed Parameter Systems), Lille, (France), 3 - 6 June, pp.

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