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 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}, pages = {1077--1093}, publisher = {EDP-Sciences}, volume = {16}, number = {4}, year = {2010}, doi = {10.1051/cocv/2009036}, mrnumber = {2744163}, zbl = {1202.93064}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2009036/} }

TY - JOUR AU - Zwart, Hans AU - Le Gorrec, Yann AU - Maschke, Bernhard AU - Villegas, Javier TI - Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2010 SP - 1077 EP - 1093 VL - 16 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2009036/ DO - 10.1051/cocv/2009036 LA - en ID - COCV_2010__16_4_1077_0 ER -

%0 Journal Article %A Zwart, Hans %A Le Gorrec, Yann %A Maschke, Bernhard %A Villegas, Javier %T Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 1077-1093 %V 16 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2009036/ %R 10.1051/cocv/2009036 %G en %F 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://archive.numdam.org/articles/10.1051/cocv/2009036/

[1] Well-posedness of boundary control systems. SIAM J. Control Optim. 42 (2003) 1244-1265. | Zbl

and ,[2] An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, New York, USA (1995). | Zbl

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

and ,[4] Vertex control of flows in networks. Netw. Heterog. Media 3 (2008) 709-722. | Zbl

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

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

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

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

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

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

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

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

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

and ,[14] 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.

, , and ,[15] 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

, and ,[16] Stability and Stabilization of Infinite-Dimensional Systems with Applications. Springer-Verlag (1999). | Zbl

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

and ,[18] 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

and ,[19] 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] Dissipative operators and hyperbolic systems of partial differential equations. Trans. Amer. Math. Soc. 90 (1959) 193-254. | Zbl

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

,[22] Hamiltonian formulation of distributed parameter systems with boundary energy flow. J. Geometry Physics 42 (2002) 166-174. | Zbl

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

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

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

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

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

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

,[29] 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.

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