Abstract
This chapter is devoted to the stability of nonlinear autonomous systems with distributed parameters and delay, governed by functional-differential equations in a Banach space with nonlinear causal mappings and bounded operators acting on the delayed state. These equations include partial differential, integro-differential and other traditional equations. Estimates for the norms of solutions are established. They give us explicit conditions for the delay-dependent Lyapunov and exponential stabilities of the considered systems. These conditions are formulated in terms of the spectra of the operator coefficients of the equations. In addition, the obtained solution estimates provide us bounds for the regions of attraction of steady states. The global exponential stability conditions are also derived. As particular cases we consider systems with discrete and distributed delays. The illustrative examples with the Dirichlet and Neumann boundary conditions are also presented. These examples show that the obtained stability conditions allow us to avoid the construction of the Lyapunov type functionals in appropriate situations. Our approach is based on a combined usage of properties of operator semigroups with estimates for fundamental solutions of the considered equations.
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References
Corduneanu C (2002) Functional equations with causal operators. Taylor and Francis, London
Datko R (1988) Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J Control Optim 26:697–713
Engel K-J, Nagel R (2006) A short course on operator semigroups. Springer, New York
Fridmana E, Orlov Y (2009) Exponential stability of linear distributed parameter systems with time-varying delays. Automatica 45:194–201
Gil’ MI (1998) On global stability of parabolic systems with delay. Appl Anal 69:57–71
Gil’ MI (1998) On the generalized Wazewski and Lozinskii inequalities for semilinear abstract differential-delay equations. J Inequal Appl 2:255–267
Gil’ MI (2000) Stability of linear time-variant functional differential equations in a Hilbert space. Funcialaj Ekvasioj 43:31–38
Gil’ MI (2002) Solution estimates for abstract nonlinear time-variant differential delay equations. J Math Anal Appl 270:51–65
Gil’ MI (2002) Stability of solutions of nonlinear nonautonomous differential-delay equations in a Hilbert space. Electron J Differ Equ 2002(94):1–15
Gil’ MI (2003) The generalized Aizerman - Myshkis problem for abstract differential-delay equations. Nonlinear Anal TMA 55:771–784
Gil’ MI (2003) On the generalized Aizerman - Myshkis problem for retarded distributed parameter systems. IMA J Math Control 20:129–136
Gil’ MI (2005) Stability of abstract nonlinear nonautonomous differential-delay equations with unbounded history-responsive operators. J Math Anal Appl 308:140–158
Gil’ MI (2013) Stability of vector differential delay equations. Birkhäuser Verlag, Basel
Gil’ MI (2014) Stability of neutral functional differential equations. Atlantis Press, Amsterdam
Hale JK (1977) Theory of functional differential equations. Springer, New York
He Y, Wang Q-G, Lin C, Wu M (2007) Delay-range-dependent stability for systems with time-varying delay. Automatica 43(2):371–376
Henry D (1981) Geometric theory of semilinear parabolic equations. In: Lectures notes in mathematics, vol 840. Springer, New York
Ikeda K, Azuma T, Uchida K (2001) Infinite-dimensional LMI approach to analysis and synthesis for linear time-delay systems. Special issue on advances in analysis and control of time-delay systems. Kybernetika (Prague) 37(4):505–520
Lakshmikantham V, Leela S, Drici Z, McRae FA (2009) Theory of causal differential equations, Atlantis studies in mathematics for engineering and science, vol 5. Atlantis Press, Paris
Logemann H, Rebarber R, Weiss G (1996) Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop. SIAM J Control Optim 34(2):572–600
Luo YP, Deng FQ (2006) LMI-based approach of robust control for uncertain distributed parameter control systems with time-delay. Control Theory Appl 23:318–324
Nicaise S, Pignotti C (2006) Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J Control Optim 45(5):1561–1585
Tanabe H (1997) Functional analytic methods for partial differential equations. Marcel Dekker, Inc., New York
Vrabie II (1987) Compactness methods for nonlinear evolutions. Pitman, New York
Wang L, Yangfan W (2009) LMI-based approach of global exponential robust stability for a class of uncertain distributed parameter control. Systems with time-varying delays. J Vib Control 15(8):1173–1185
Wu J (1996) Theory and applications of partial functional differential equations. Springer, New York
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Gil’, M. (2016). Explicit Delay-Dependent Stability Criteria for Nonlinear Distributed Parameter Systems. In: Vaidyanathan, S., Volos, C. (eds) Advances and Applications in Nonlinear Control Systems. Studies in Computational Intelligence, vol 635. Springer, Cham. https://doi.org/10.1007/978-3-319-30169-3_14
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DOI: https://doi.org/10.1007/978-3-319-30169-3_14
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