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Stability of impulsive time-varying systems and compactness of the operators mapping the input space into the state and output spaces. (English) Zbl 1111.93072
Summary: Concerned with time-varying systems with non-necessarily bounded everywhere continuous time-differentiable time-varying point delays. The delay-free and delayed dynamics are assumed to be time-varying and impulsive, in general, and the external input may also be impulsive. For given bounded initial conditions, the (unique) homogeneous state-trajectory and output trajectory are equivalently constructed from three different auxiliary homogeneous systems, the first one being delay-free and time-invariant, the second one possessing the delay-free dynamics of the current delayed system and the third one being the homogeneous part of the system under study. In this way, the constructed solution trajectories of both the unforced and forced systems are obtained from different (input-state space/output space and state space to output space) operators. The system stability and the compactness of the operators describing the solution trajectories are investigated.

MSC:
93D25Input-output approaches to stability of control systems
34K20Stability theory of functional-differential equations
34K45Functional-differential equations with impulses
47N70Applications of operator theory in systems theory, circuits, and control theory
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References:
[1] Burton, T. A.: Stability and periodic solutions of ordinary and differential equations. Ser. math. Sci. tech. (1985) · Zbl 0635.34001
[2] Nakagiri, S. I.: Structural properties of functional differential equations in Banach spaces. Osaka J. Math. 25, 353-398 (1988) · Zbl 0713.34069
[3] Phat, V. Ngoc: Stabilization of linear continuous time-varying systems with state delays in Hilbert spaces. Electron. J. Differential equations 67, 1-13 (2001) · Zbl 1020.93017
[4] Zheng, F.; Frank, P. M.: Finite-dimensional variable structure control design for distributed delay systems. Internat. J. Control 74, 398-408 (2001) · Zbl 1033.93008
[5] Jugo, J.; De La Sen, M.: Input -- output based pole-placement controller for a class of time-delay systems. IEE proc. Ser. D control theory appl. 149, 323-330 (2002)
[6] Faria, T.; Huang, W.; Wu, J.: Smoothness of center manifolds for maps and formal adjoints for semilinear FDES in generic Banach spaces. SIAM J. Math. anal. 34, 173-203 (2002) · Zbl 1085.34064
[7] Oucheriah, S.: Exponential stabilization of linear delayed systems. IEEE trans. Circuits systems I. Fund. theory appl. 50, 826-830 (2003)
[8] Aziz, B.: Nonlinear robust control problems of parabolic type equations with time-varying delays given in the integral form. J. dynam. Control systems 9, 489-512 (2003) · Zbl 1026.49005
[9] Richard, J. P.: Time-delay systems: an overview of some recent advances and open problems. Automatica 39, 1667-1694 (2003) · Zbl 1145.93302
[10] Zeng, Z. Gang; Wang, J.; Liao, X.: Global exponential stability of neural networks with time-varying delays. IEEE trans. Circuits systems I. Fund. theory appl. 50, 1353-1358 (2003)
[11] De La Sen, M.; Luo, N.: A note on the stability of linear time-delay systems with impulsive inputs. IEEE trans. Circuits systems I. Fund. theory appl. 50, 149-152 (2003)
[12] De La Sen, M.; Luo, N.: On the uniform exponential stability of a wide class of linear time-delay systems. J. math. Anal. appl. 289, 456-476 (2004) · Zbl 1046.34086
[13] Ioannou, P.; Datta, A.: Robust adaptive control: design, analysis and robustness bounds. Lecture notes in control and inform. Sci. 160 (1991) · Zbl 0787.93055
[14] Akhiezer, N. I.; Glazman, I. M.: Theory of linear operators in Hilbert space. (1963) · Zbl 0098.30702
[15] Franks, L. E.: Signal theory. (1975) · Zbl 0296.62084