Abstract
A preview of the material contained in the book is given in this chapter. The technical difficulties for the extension of Input-to-State Stability (ISS) to systems containing at least one PDE are illustrated by means of various examples. The chapter also offers an overview of the topics covered in the book as well as a brief presentation of the contents of all subsequent chapters. A list of all applications contained in the book is provided. The applications include mathematical models arising in various scientific disciplines. Finally, the required background for a reader of the book is detailed.
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References
Angeli D, Sontag ED, Wang Y (2003) Input-to-state stability with respect to inputs and their derivatives. Int J Robust Nonlinear Control 13:1035–1056
Balakrishnan AV (1981) Applied functional analysis, 2nd edn. Springer, New York
Bensoussan A, Da Prato G, Delfour MC, Mitter SK (2007) Representation and control of infinite dimensional systems, 2nd edn. Series: Mathematics, systems & control: foundations & applications. Birkhäuser, Boston
Bribiesca Argomedo F, Witrant E, Prieur C (2012) D1-input-to-state stability of a time-varying nonhomogeneous diffusive equation subject to boundary disturbances. In: Proceedings of the American control conference, Montreal, QC, pp 2978–2983
Bribiesca Argomedo F, Prieur C, Witrant E, Bremond S (2013) A strict control Lyapunov function for a diffusion equation with time-varying distributed coefficients. IEEE Trans Autom Control 58:290–303
Cai C, Teel A (2009) Characterizations of input-to-state stability for hybrid systems. Syst Control Lett 58(1):47–53
Dashkovskiy S, Mironchenko A (2010) On the uniform input-to-state stability of reaction-diffusion systems. In: Proceedings of the 49th conference on decision and control, Atlanta, GA, USA, pp 6547–6552
Dashkovskiy S, Mironchenko A (2011) Local ISS of reaction-diffusion systems. In: Proceedings of the 18th IFAC world congress, Milano, Italy, pp 11018–11023
Dashkovskiy S, Mironchenko A (2013) Input-to-state stability of infinite-dimensional control systems. Math Control Signals Syst 25:1–35
Dashkovskiy S, Mironchenko A (2013) Input-to-state stability of nonlinear impulsive systems. SIAM J Control Optim 51:1962–1987
Day WA (1982) Extension of a property of the heat equation to linear thermoelasticity and other theories. Q Appl Math 40:319–330
Day WA (1983) A decreasing property of solutions of parabolic equations with applications to thermoelasticity. Q Appl Math 40:468–475
Desch W, Lasiecka I, Schappacher W (1985) Feedback boundary control problems for linear semigroups. Isr J Math 51:177–207
Ekolin G (1991) Finite difference methods for a nonlocal boundary value problem for the heat equation. BIT 31:245–261
Fairweather G, Lopez-Marcos JC (1996) Galerkin methods for a semilinear parabolic problem with nonlocal boundary conditions. Adv Comput Math 6:243–262
Friedman A (1986) Monotone decay of solutions of parabolic equations with nonlocal boundary conditions. Q Appl Math 44:401–407
Hespanha JP, Liberzon D, Teel AR (2008) Lyapunov conditions for input-to-state stability of impulsive systems. Automatica 44(11):2735–2744
Jacob B, Nabiullin R, Partington JR, Schwenninger F. Infinite-dimensional input-to-state stability and Orlicz spaces. arXiv:1609.09741 [math.OC]
Jacob B, Nabiullin R, Partington JR, Schwenninger F (2016) On input-to-state-stability and integral input-to-state-stability for parabolic boundary control systems. In: Proceedings of MTNS 2016
Jayawardhana B, Logemann H, Ryan EP (2008) Infinite-dimensional feedback systems: the circle criterion and input-to-state stability. Commun Inf Syst 8:403–434
Jiang Z-P, Teel A, Praly L (1994) Small-gain theorem for ISS systems and applications. Math Control Signals Syst 7:95–120
Karafyllis I, Pepe P, Jiang Z-P (2008) Global output stability for systems described by retarded functional differential equations: Lyapunov characterizations. Eur J Control 14(6):516–536
Karafyllis I, Pepe P, Jiang Z-P (2008) Input-to-output stability for systems described by retarded functional differential equations. Eur J Control 14(6):539–555
Karafyllis I, Jiang Z-P (2011) Stability and stabilization of nonlinear systems. Series: Communications and control engineering. Springer, London
Karafyllis I, Krstic M (2014) On the relation of delay equations to first-order hyperbolic partial differential equations. ESAIM Control Optim Calc Var 20:894–923
Karafyllis I, Pepe P (2015) A note on converse Lyapunov results for neutral systems. In: Karafyllis I, Malisoff M, Mazenc F, Pepe P (eds) Recent results on nonlinear time delayed systems. Advances in delays and dynamics, vol 4. Springer, Berlin
Karafyllis I, Krstic M (2016) Input-to state stability with respect to boundary disturbances for the 1-D heat equation. In: Proceedings of the 55th IEEE conference on decision and control, pp 2247–2252
Karafyllis I, Krstic M (2016) ISS with respect to boundary disturbances for 1-D parabolic PDEs. IEEE Trans Autom Control 61:3712–3724
Karafyllis I, Krstic M (2017) ISS in different norms for 1-D parabolic PDEs with boundary disturbances. SIAM J Control Optim 55:1716–1751
Karafyllis I, Krstic M (2017) Predictor feedback for delay systems: implementations and approximations. Series: Mathematics, systems & control: foundations & applications. Birkhäuser, Boston
Karafyllis I, Krstic M. Decay estimates for 1-D parabolic PDEs with boundary disturbances. Submitted to ESAIM Control Optim Calc Var (see also arXiv:1706.01410 [math.OC])
Krstic M, Kanellakopoulos I, Kokotovic PV (1995) Nonlinear and adaptive control design. Wiley, New York
Krstic M, Smyshlyaev A (2008) Boundary control of PDEs: a course on backstepping designs. SIAM, USA
Lasiecka I (1980) Unified theory for abstract parabolic boundary problems—a semigroup approach. Appl Math Optim 6:287–333
Lasiecka I, Triggiani R (1991) Differential and algebraic Riccati equations with application to boundary point control problems: continuous theory and approximation theory. Lecture notes in control and information sciences. Springer, Berlin
Lasiecka I, Tataru D (1993) Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differ Integr Equ 6:507–533
Lasiecka I, Triggiani R (2000) Control theory for partial differential equations: continuous and approximation theories I. Abstract parabolic systems. Cambridge University Press, Cambridge
Lasiecka I, Triggiani R (2000) Control theory for partial differential equations: continuous and approximation theories II. Abstract hyperbolic-like systems over a finite time horizon. Cambridge University Press, Cambridge
Lin Y, Sontag ED, Wang Y (1996) A smooth converse Lyapunov theorem for robust stability. SIAM J Control Optim 34:124–160
Liu Y (1999) Numerical solution of the heat equation with nonlocal boundary conditions. J Comput Appl Math 110:115–127
Logemann H (2014) Infinite-dimensional Lur’e systems: the circle criterion, input-to-state stability and the converging-input-converging-state property. In: Proceedings of the 21st international symposium on mathematical theory of networks and systems, Groningen, The Netherlands, pp 1624–1627
Mazenc F, Prieur C (2011) Strict Lyapunov functionals for nonlinear parabolic partial differential equations. In: Proceedings of the 18th IFAC world congress, Milan, Italy, vol 44, pp 12550–12555
Mazenc F, Prieur C (2011) Strict Lyapunov functions for semilinear parabolic partial differential equations. Math Control Relat Fields AIMS 1:231–250
Mironchenko A, Ito H (2014) Integral input-to-state stability of bilinear infinite-dimensional systems. In: Proceedings of the 53rd IEEE conference on decision and control, Los Angeles, California, USA, pp 3155–3160
Mironchenko A, Ito H (2015) Construction of Lyapunov functions for interconnected parabolic systems: an iISS approach. SIAM J Control Optim 53:3364–3382
Mironchenko A (2016) Local input-to-state stability: characterizations and counterexamples. Syst Control Lett 87:23–28
Mironchenko A, Wirth F (2016) Restatements of input-to-state stability in infinite dimensions: what goes wrong. In: Proceedings of the 22nd international symposium on mathematical theory of systems and networks, pp 667–674
Mironchenko A, Wirth F (2016) Global converse Lyapunov theorems for infinite-dimensional systems. In: Proceedings of the 10th IFAC symposium on nonlinear control systems, pp 909–914
Mironchenko A, Wirth F. Characterizations of input-to-state stability for infinite-dimensional systems. To appear in IEEE transactions on automatic control
Mironchenko A, Wirth F (2017) A non-coercive Lyapunov framework for stability of distributed parameter systems. Proceedings of the 56th IEEE conference on decision and control, pp 1900–1905
Mironchenko A, Wirth F (2017) Input-to-state stability of time-delay systems: criteria and open problems. Proceedings of the 56th IEEE conference on decision and control, pp 3719–3724
Mironchenko A, Karafyllis I, Krstic M. Monotonicity methods for input-to-state stability of nonlinear parabolic PDEs with boundary disturbances. Submitted to SIAM J Control Optim (see also arXiv:1706.07224 [math.OC])
Pao CV (1998) Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions. J Comput Appl Math 88:225–238
Pao CV (2001) Numerical solutions of reaction-diffusion equations with nonlocal boundary conditions. J Comput Appl Math 136:227–243
Pazy A (1983) Semigroups of linear operators and applications to partial differential equations. Springer, New York
Pepe P, Jiang Z-P (2006) A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems. Syst Control Lett 55(12):1006–1014
Pepe P, Karafyllis I, Jiang Z-P (2017) Lyapunov-Krasovskii characterization of the input-to-state stability for neutral systems in Hale’s form. Syst Control Lett 102:48–56
Prieur C, Mazenc F (2012) ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws. Math Control Signals Syst 24:111–134
Smyshlyaev A, Krstic M (2004) Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations. IEEE Trans Autom Control 49:2185–2202
Smyshlyaev A, Krstic M (2010) Adaptive control of parabolic PDEs. Princeton University Press, Princeton
Sontag ED (1989) Smooth stabilization implies coprime factorization. IEEE Trans Autom Control 34:435–443
Sontag ED, Wang Y (1995) On characterizations of the input-to-state stability property. Syst Control Lett 24:351–359
Sontag ED, Wang Y (1996) New characterizations of input to state stability. IEEE Trans Autom Control 41:1283–1294
Sontag ED (1998) Comments on integral variants of ISS. Syst Control Lett 34:93–100
Sontag ED, Wang Y (1999) Notions of input to output stability. Syst Control Lett 38:235–248
Sontag ED, Wang Y (2001) Lyapunov characterizations of input to output stability. SIAM J Control Optim 39:226–249
Sontag ED (2008) Input-to-state stability: basic concepts and results. In: Nistri P, Stefani G (eds) Nonlinear and optimal control theory. Lectures given at the C.I.M.E. Summer School Held in Cetraro, Italy, June 19–29 2004, vol 1932. Lecture notes in mathematics, pp 163–220. Springer, Berlin
Vu L, Chatterjee D, Liberzon D (2007) Input-to-state stability of switched systems and switching adaptive control. Automatica 43(4):639–646
Zheng J, Zhu G. Input-to state stability with respect to boundary disturbances for a class of semi-linear parabolic equations. arXiv:1709.01880 [math.OC]
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Karafyllis, I., Krstic, M. (2019). Preview. In: Input-to-State Stability for PDEs. Communications and Control Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-91011-6_1
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DOI: https://doi.org/10.1007/978-3-319-91011-6_1
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