Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, p. 1077-1093

We study a class of hyperbolic partial differential equations on a one dimensional spatial domain with control and observation at the boundary. Using the idea of feedback we show these systems are well-posed in the sense of Weiss and Salamon if and only if the state operator generates a C0-semigroup. Furthermore, we show that the corresponding transfer function is regular, i.e., has a limit for s going to infinity.

DOI : https://doi.org/10.1051/cocv/2009036
Classification:  93C20,  35L40,  35F15,  37Kxx
Keywords: infinite-dimensional systems, hyperbolic boundary control systems, C0-semigroup, well-posedness, regularity
@article{COCV_2010__16_4_1077_0,
author = {Zwart, Hans and Le Gorrec, Yann and Maschke, Bernhard and Villegas, Javier},
title = {Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {16},
number = {4},
year = {2010},
pages = {1077-1093},
doi = {10.1051/cocv/2009036},
zbl = {1202.93064},
mrnumber = {2744163},
language = {en},
url = {http://www.numdam.org/item/COCV_2010__16_4_1077_0}
}

Zwart, Hans; Le Gorrec, Yann; Maschke, Bernhard; Villegas, Javier. Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, pp. 1077-1093. doi : 10.1051/cocv/2009036. http://www.numdam.org/item/COCV_2010__16_4_1077_0/

[1] A. Cheng and K. Morris, Well-posedness of boundary control systems. SIAM J. Control Optim. 42 (2003) 1244-1265. | Zbl 1049.35041

[2] R.F. Curtain and H.J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, New York, USA (1995). | Zbl 0839.93001

[3] R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, Matématiques & Applications 50. Springer-Verlag (2006). | Zbl 1083.74002

[4] K.-J. Engel, M. Kramar Fijavž, R. Nagel, E. Sikolya, Vertex control of flows in networks. Netw. Heterog. Media 3 (2008) 709-722. | Zbl 1154.93004

[5] B.-Z. Guo and Z.-C. Shao, Regularity of a Schödinger equation with Dirichlet control and collocated observation. Syst. Contr. Lett. 54 (2005) 1135-1142. | Zbl 1129.35447

[6] B.-Z. Guo and Z.-C. Shao, Regularity of an Euler-Bernoulli equation with Neumann control and collocated observation. J. Dyn. Contr. Syst. 12 (2006) 405-418. | Zbl 1111.93033

[7] B.-Z. Guo and Z.-X. Zhang, The regularity of the wave equation with partial Dirichlet control and collocated observation. SIAM J. Control Optim. 44 (2005) 1598-1613. | Zbl 1134.35318

[8] B.-Z. Guo and Z.-X. Zhang, Well-posedness and regularity for an Euler-Bernoulli plate with variable coefficients and boundary control and observation. MCSS 19 (2007) 337-360. | Zbl 1129.93020

[9] B.-Z. Guo and Z.-X. Zhang, On the well-posedness and regularity of the wave equation with variable coefficients. ESAIM: COCV 13 (2007) 776-792. | Zbl 1147.35051

[10] B.-Z. Guo and Z.-X. Zhang, Well-posedness of systems of linear elasticity with Dirichlet boundary control and observation. SIAM J. Control Optim. 48 (2009) 2139-2167. | Zbl 1194.35456

[11] T. Kato, Perturbation Theory for Linear Operators. Corrected printing of the second edition, Springer-Verlag, Berlin, Germany (1980). | Zbl 0435.47001

[12] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations I, Encyclopedia of Mathematics and its Applications 74. Cambridge University Press (2000). | Zbl 0942.93001

[13] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations II, Encyclopedia of Mathematics and its Applications 75. Cambridge University Press (2000). | Zbl 0961.93003

[14] Y. Le Gorrec, B.M. Maschke, H. Zwart and J.A. Villegas, Dissipative boundary control systems with application to distributed parameters reactors, in Proc. IEEE International Conference on Control Applications, Munich, Germany, October 4-6 (2006) 668-673.

[15] Y. Le Gorrec, H. Zwart and B. Maschke, Dirac structures and boundary control systems associated with skew-symmetric differential operators. SIAM J. Control Optim. 44 (2005) 1864-1892. A more detailed version is available at www.math.utwente.nl/publications, Memorandum No. 1730 (2004). | Zbl 1108.93030

[16] Z.-H. Luo, B.-Z. Guo and O. Morgul, Stability and Stabilization of Infinite-Dimensional Systems with Applications. Springer-Verlag (1999). | Zbl 0922.93001

[17] A. Macchelli and C. Melchiorri, Modeling and control of the Timoshenko beam, The distributed port Hamiltonian approach. SIAM J. Control Optim. 43 (2004) 743-767. | Zbl 1073.74040

[18] B. Maschke and A.J. Van Der Schaft,Compositional modelling of distributed-parameter systems, in Advanced Topics in Control Systems Theory - Lecture Notes from FAP 2004, Lecture Notes in Control and Information Sciences, F. Lamnabhi-Lagarrigue, A. Loría and E. Panteley Eds., Springer (2005) 115-154. | Zbl 1119.93044

[19] J. Malinen, Conservatively of time-flow invertible and boundary control systems. Research Report A479, Institute of Mathematics, Helsinki University of Technology, Finland (2004). See also Proceedings of the Joint 44th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC'05).

[20] R.S. Phillips, Dissipative operators and hyperbolic systems of partial differential equations. Trans. Amer. Math. Soc. 90 (1959) 193-254. | Zbl 0093.10001

[21] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Review 20 (1978) 639-739. | Zbl 0397.93001

[22] A.J. Van Der Schaft and B.M. Maschke, Hamiltonian formulation of distributed parameter systems with boundary energy flow. J. Geometry Physics 42 (2002) 166-174. | Zbl 1012.70019

[23] O. Staffans, Well-posed Linear Systems, Encyclopedia of Mathematics and its Applications 103. Cambridge University Press (2005). | Zbl 1057.93001

[24] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts. Basler Lehrbücher (2009). | Zbl 1188.93002

[25] J.A. Villegas, A Port-Hamiltonian Approach to Distributed Parameter Systems. Ph.D. Thesis, University of Twente, The Netherlands (2007). Available at http://doc.utwente.nl.

[26] G. Weiss, Regular linear systems with feedback. Math. Control Signals Syst. 7 (1994) 23-57. | Zbl 0819.93034

[27] G. Weiss and R.F. Curtain, Exponential stabilization of a Rayleigh beam using collocated control. IEEE Trans. Automat. Contr. 53 (2008) 643-654.

[28] H. Zwart, Transfer functions for infinite-dimensional systems. Syst. Contr. Lett. 52 (2004) 247-255. | Zbl 1157.93420

[29] H. Zwart, Y. Le Gorrec, B.M.J. Maschke and J.A. Villegas, Well-posedness and regularity for a class of hyperbolic boundary control systems, in Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan (2006) 1379-1883.