On the circle criterion for boundary control systems in factor form : Lyapunov stability and Lur'e equations

Grabowski, Piotr; Callier, Frank M.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 1, pp. 169-197.

A Lur’e feedback control system consisting of a linear, infinite-dimensional system of boundary control in factor form and a nonlinear static sector type controller is considered. A criterion of absolute strong asymptotic stability of the null equilibrium is obtained using a quadratic form Lyapunov functional. The construction of such a functional is reduced to solving a Lur’e system of equations. A sufficient strict circle criterion of solvability of the latter is found, which is based on results by Oostveen and Curtain [Automatica 34 (1998) 953-967]. All the results are illustrated in detail by an electrical transmission line example of the distortionless loaded $\mathrm{\Re 𝔏ℭ𝔊}$-type. The paper uses extensively the philosophy of reciprocal systems with bounded generating operators as recently studied and used by Curtain in (2003) [Syst. Control Lett. 49 (2003) 81-89; SIAM J. Control Optim. 42 (2003) 1671-1702].

Classification : 34G, 35A, 47D, 93B
Mots clés : infinite-dimensional control systems, semigroups, Lyapunov functionals, circle criterion

[1] W. Arendt and C.J.K. Batty, Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc. 306 (1988) 837-852. | Zbl

[2] A.V. Balakrishnan, On a generalization of the Kalman-Yacubovic lemma. Appl. Math. Optim. 31 (1995) 177-187. | Zbl

[3] F. Bucci, Frequency domain stability of nonlinear feedback systems with unbounded input operator. Dynam. Contin. Discrete Impuls. Syst. 7 (2000) 351-368. | Zbl

[4] F.M. Callier and J. Winkin, LQ-optimal control of infinite-dimensional systems by spectral factorization. Automatica 28 (1992) 757-770. | Zbl

[5] R.F. Curtain, Linear operator inequalities for strongly stable weakly regular linear systems. Math. Control Signals Syst. 14 (2001) 299-337. | Zbl

[6] R.F. Curtain, Regular linear systems and their reciprocals: application to Riccati equations. Syst. Control Lett. 49 (2003) 81-89.

[7] R.F. Curtain, Riccati equations for stable well-posed linear systems: The generic case. SIAM J. Control Optim. 42 (2003) 1671-1702. | Zbl

[8] R.F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory. Heidelberg, Springer (1995). | MR | Zbl

[9] R.F. Curtain, H. Logemann and O. Staffans, Stability results of Popov-type for infinite - dimensional systems with applications to integral control. Proc. London Math. Soc. 86 (2003) 779-816. | Zbl

[10] H. Górecki, S. Fuksa, P. Grabowski and A. Korytowski, Analysis and Synthesis of Time-Delay Systems. Warsaw & Chichester: PWN and J. Wiley (1989). | Zbl

[11] P. Grabowski, On the spectral - Lyapunov approach to parametric optimization of DPS. IMA J. Math. Control Inform. 7 (1990) 317-338. | Zbl

[12] P. Grabowski, The LQ controller problem: an example. IMA J. Math. Control Inform. 11 (1994) 355-368. | Zbl

[13] P. Grabowski, On the circle criterion for boundary control systems in factor form. Opuscula Math. 23 (2003) 1-25. | Zbl

[14] P. Grabowski and F.M. Callier, Admissible observation operators. Duality of observation and control using factorizations. Dynamics Continuous, Discrete Impulsive Systems 6 (1999) 87-119. | Zbl

[15] P. Grabowski and F.M. Callier, On the circle criterion for boundary control systems in factor form: Lyapunov approach. Facultés Universitaires Notre-Dame de la Paix à Namur, Publications du Département de Mathématique, Research Report 07 (2000), FUNDP, Namur, Belgium.

[16] P. Grabowski and F.M. Callier, Boundary control systems in factor form: Transfer functions and input-output maps. Integral Equations Operator Theory 41 (2001) 1-37. | Zbl

[17] P. Grabowski and F.M. Callier, Circle criterion and boundary control systems in factor form: Input-output approach. Internat. J. Appl. Math. Comput. Sci. 11 (2001) 1387-1403. | Zbl

[18] P. Grabowski and F.M. Callier, On the circle criterion for boundary control systems in factor form: Lyapunov stability and Lur'e equations. Facultés Universitaires Notre-Dame de la Paix à Namur, Publications du Département de Mathématique, Research Report 05 (2002), FUNDP, Namur, Belgium.

[19] U. Grenander and G. Szegö, Toeplitz Forms and Their Application, Berkeley: University of California Press (1958). | MR | Zbl

[20] K. Hoffman, Banach Spaces of Analytic Functions. Englewood Cliffs: Prentice-Hall (1962). | MR | Zbl

[21] I. Lasiecka and R. Triggiani, Differential and Algebraic Riccati Equations with Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory. Lect. Notes Control Inform. Sci. 164 (1991) 1-160. | Zbl

[22] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Part I: Abstract Parabolic Systems, Cambridge: Cambridge University Press, Encyclopedia Math. Appl. 74 (2000). | MR | Zbl

[23] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Part II: Abstract Hyperbolic-Like Systems over a Finite Time Horizon, Cambridge: Cambridge University Press, Encyclopedia Math. Appl. 75 (2000). | MR | Zbl

[24] A.L. Likhtarnikov and V.A. Yacubovich, The frequency domain theorem for continuous one-parameter semigroups, IZVESTIJA ANSSSR. Seria matematicheskaya. 41 (1977) 895-911 (in Russian). | Zbl

[25] H. Logemann and R.F. Curtain, Absolute stability results for well-posed infinite-dimensional systems with low-gain integral control. ESAIM: COCV 5 (2000) 395-424. | Numdam | Zbl

[26] J.-Cl. Louis and D.Wexler, The Hilbert space regulator problem and operator Riccati equation under stabilizability. Annales de la Société Scientifique de Bruxelles 105 (1991) 137-165. | Zbl

[27] Yu. Lyubich and Vû Quôc Phong, Asymptotic stability of linear differential equations in Banach spaces. Studia Math. 88 (1988) 37-41. | Zbl

[28] E. Noldus, On the stability of systems having several equilibrium points. Appl. Sci. Res. 21 (1969) 218-233. | Zbl

[29] E. Noldus, A. Galle and L. Jasson, The computation of stability regions for systems with many singular points. Intern. J. Control 17 (1973) 641-652. | Zbl

[30] E. Noldus, New direct Lyapunov-type method for studying synchronization problems. Automatica 13 (1977) 139-151. | Zbl

[31] A.A. Nudel'Man and P.A. Schwartzman, On the existence of solution of some operator inequalities. Sibirsk. Mat. Zh. 16 (1975) 563-571 (in Russian). | Zbl

[32] J.C. Oostveen and R.F. Curtain, Riccati equations for strongly stabilizable bounded linear systems. Automatica 34 (1998) 953-967. | Zbl

[33] L. Pandolfi, Kalman-Popov-Yacubovich theorem: an overview and new results for hyperbolic control systems. Nonlinear Anal. Theor. Methods Appl. 30 (1997) 735-745. | Zbl

[34] L. Pandolfi, Dissipativity and Lur'e problem for parabolic boundary control system, Research Report, Dipartamento di Matematica, Politecnico di Torino 1 (1997) 1-27; SIAM J. Control Optim. 36 (1998) 2061-2081. | Zbl

[35] L. Pandolfi, The Kalman-Yacubovich-Popov theorem for stabilizable hyperbolic boundary control systems. Integral Equations Operator Theory 34 (1999) 478-493. | Zbl

[36] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. New York, Springer-Verlag (1983). | MR | Zbl

[37] D. Salamon, Realization theory in Hilbert space. Math. Systems Theory 21 (1989) 147-164. | Zbl

[38] O.J. Staffans, Quadratic optimal control of stable well-posed linear systems through spectral factorization. Math. Control Signals Systems 8 (1995) 167-197. | Zbl

[39] O.J. Staffans and G. Weiss, Transfer functions of regular linear systems, Part II: The system operator and the Lax-Phillips semigroup. Trans. Amer. Math. Soc. 354 (2002) 3229-3262. | Zbl

[40] M. Vidyasagar, Nonlinear Systems Analysis. 2nd Edition, Englewood Cliffs NJ, Prentice-Hall (1993). | Zbl

[41] G. Weiss, Transfer functions of regular linear systems. Part I: Characterization of regularity. Trans. AMS 342 (1994) 827-854. | Zbl

[42] M. Weiss, Riccati Equations in Hilbert Spaces: A Popov function approach. Ph.D. Thesis, Rijksuniversiteit Groningen, The Netherlands (1994).

[43] M. Weiss and G. Weiss, Optimal control of stable weakly regular linear systems. Math. Control Signals Syst. 10 (1997) 287-330. | Zbl

[44] R.M. Young, An Introduction to Nonharmonic Fourier Series. New York, Academic Press (1980). | MR | Zbl