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Well-posedness and regularity of hyperbolic boundary control systemson a one-dimensional spatial domain

Published online by Cambridge University Press:  25 August 2009

Hans Zwart ,
Yann Le Gorrec ,
Bernhard Maschke  and
Javier Villegas
Hans Zwart
Affiliation:
Department of Applied Mathematics, University of Twente, 7500 AE Enschede, The Netherlands. h.j.zwart@math.utwente.nl
Yann Le Gorrec
Affiliation:
FEMTO-ST AS2M, 24 rue Alain Savary, 25000 Besançon, France. Yann.Le.Gorrec@ens2m.fr
Bernhard Maschke
Affiliation:
LAGEP, CNRS UMR 5007, CPE Lyon – Bâtiment 308 G, Université Lyon-1, Université de Lyon, 43 bd. du 11 Novembre 1918, 69622 Villeurbanne Cedex, France. maschke@lagep.univ-lyon1.fr
Javier Villegas
Affiliation:
AVL Powertrain UK, Langdale House, Sable Way, Southfields Business Park, Basildon, SS15 6SR, UK. Javier.Villegas@avl.com
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Abstract

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 C 0-semigroup. Furthermore, we show that the corresponding transfer function is regular, i.e., has a limit for s going to infinity.

Keywords

Infinite-dimensional systemshyperbolic boundary control systemsC 0-semigroupwell-posednessregularity

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 4 , October 2010 , pp. 1077 - 1093
Copyright
© EDP Sciences, SMAI, 2009

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References

Cheng, A. and Morris, K., Well-posedness of boundary control systems. SIAM J. Control Optim. 42 (2003) 1244–1265. CrossRef
R.F. Curtain and H.J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, New York, USA (1995).
R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, Matématiques & Applications 50. Springer-Verlag (2006).
Engel, K.-J., Kramar Fijavž, M., Nagel, R., Sikolya, E., Vertex control of flows in networks. Netw. Heterog. Media 3 (2008) 709–722.
Guo, B.-Z. and Shao, Z.-C., Regularity of a Schödinger equation with Dirichlet control and collocated observation. Syst. Contr. Lett. 54 (2005) 1135–1142. CrossRef
Guo, B.-Z. and Shao, Z.-C., Regularity of an Euler-Bernoulli equation with Neumann control and collocated observation. J. Dyn. Contr. Syst. 12 (2006) 405–418.
Guo, B.-Z. and Zhang, Z.-X., The regularity of the wave equation with partial Dirichlet control and collocated observation. SIAM J. Control Optim. 44 (2005) 1598–1613. CrossRef
Guo, B.-Z. and Zhang, Z.-X., Well-posedness and regularity for an Euler-Bernoulli plate with variable coefficients and boundary control and observation. MCSS 19 (2007) 337–360.
Guo, B.-Z. and Zhang, Z.-X., On the well-posedness and regularity of the wave equation with variable coefficients. ESAIM: COCV 13 (2007) 776–792. CrossRef
Guo, B.-Z. and Zhang, Z.-X., Well-posedness of systems of linear elasticity with Dirichlet boundary control and observation. SIAM J. Control Optim. 48 (2009) 2139–2167. CrossRef
T. Kato, Perturbation Theory for Linear Operators. Corrected printing of the second edition, Springer-Verlag, Berlin, Germany (1980).
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations I, Encyclopedia of Mathematics and its Applications 74. Cambridge University Press (2000).
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations II, Encyclopedia of Mathematics and its Applications 75. Cambridge University Press (2000).
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.
Le Gorrec, Y., Zwart, H. and Maschke, B., 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). CrossRef
Z.-H. Luo, B.-Z. Guo and O. Morgul, Stability and Stabilization of Infinite-Dimensional Systems with Applications. Springer-Verlag (1999).
Macchelli, A. and Melchiorri, C., Modeling and control of the Timoshenko beam, The distributed port Hamiltonian approach. SIAM J. Control Optim. 43 (2004) 743–767. CrossRef
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.
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).
Phillips, R.S., Dissipative operators and hyperbolic systems of partial differential equations. Trans. Amer. Math. Soc. 90 (1959) 193–254.
Russell, D.L., Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Review 20 (1978) 639–739. CrossRef
van der Schaft, A.J. and Maschke, B.M., Hamiltonian formulation of distributed parameter systems with boundary energy flow. J. Geometry Physics 42 (2002) 166–174. CrossRef
O. Staffans, Well-posed Linear Systems, Encyclopedia of Mathematics and its Applications 103. Cambridge University Press (2005).
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts. Basler Lehrbücher (2009).
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.
Weiss, G., Regular linear systems with feedback. Math. Control Signals Syst. 7 (1994) 23–57. CrossRef
Weiss, G. and Curtain, R.F., Exponential stabilization of a Rayleigh beam using collocated control. IEEE Trans. Automat. Contr. 53 (2008) 643–654. CrossRef
Zwart, H., Transfer functions for infinite-dimensional systems. Syst. Contr. Lett. 52 (2004) 247–255. CrossRef
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.