# American Institute of Mathematical Sciences

March  2013, 3(1): 1-19. doi: 10.3934/mcrf.2013.3.1

## Compositions of passive boundary control systems

 1 Department of Mathematics and Systems Analysis, Aalto University School of Science, PB 11100, 00076-Aalto, Finland, Finland

Received  December 2011 Revised  May 2012 Published  February 2013

We show under mild assumptions that a composition of internally well-posed, impedance passive (or conservative) boundary control systems through Kirchhoff type connections is also an internally well-posed, impedance passive (resp., conservative) boundary control system. The proof is based on results of Malinen and Staffans [21]. We also present an example of such composition involving Webster's equation on a Y-shaped graph.
Citation: Atte Aalto, Jarmo Malinen. Compositions of passive boundary control systems. Mathematical Control & Related Fields, 2013, 3 (1) : 1-19. doi: 10.3934/mcrf.2013.3.1
##### References:
 [1] A. Aalto and J. Malinen, Wave propagation in networks: A system theoretic approach, in "Proceedings of the $18^{th}$ IFAC World Congress" (eds. S. Bittanti, A. Cenedese and S. Zampieri), (2011), 8854-8859. Google Scholar [2] W. Arendt, C. Batty, M. Hieber and F. Neubrander, "Vector-valued Laplace Transforms and Cauchy Problems," Monographs in Mathematics, 96, Birkhäuser Verlag, Basel, 2001.  Google Scholar [3] J. Cervera, A. J. van der Schaft and A. Baños, Interconnection of port-Hamiltonian systems and composition of Dirac structures, Automatica J. of IFAC, 43 (2007), 212-225. doi: 10.1016/j.automatica.2006.08.014.  Google Scholar [4] R. F. Curtain and H. Zwart, "An Introduction to Infinite-Dimensional Linear Systems Theory," Texts in Applied Mathematics, 21, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar [5] V. Derkach, S. Hassi, M. Malamud and H. de Snoo, Boundary relations and their Weyl families, Transactions of the American Mathematical Society, 358 (2006), 5351-5400. doi: 10.1090/S0002-9947-06-04033-5.  Google Scholar [6] M. Gugat, G. Leugering, K. Schittkowski and E. J. P. Georg Schmidt, Modelling, stabilization, and control of flow in networks of open channels, in "Online Optimization of Large Scale Systems," Springer, Berlin, (2001), 251-270.  Google Scholar [7] K.-J. Engel, M. Kramar Fijavž, R. Nagel and E. Sikolya, Vertex control of flows in networks, Networks and Heterogeneous Media, 3 (2008), 709-722. doi: 10.3934/nhm.2008.3.709.  Google Scholar [8] H. Fattorini, Boundary control systems, SIAM Journal of Control, 6 (1968), 349-385.  Google Scholar [9] V. I. Gorbachuk and M. L. Gorbachuk, "Boundary Value Problems for Operator Differential Equations," Mathematics and its Applications (Soviet Series), 48, Kluwer Academic Publishers Group, Dordrecht, 1991.  Google Scholar [10] G. Greiner, Perturbing the boundary conditions of a generator, Houston Journal of Mathematics, 13 (1987), 213-229.  Google Scholar [11] A. Hannukainen, T. Lukkari, J. Malinen and P. Palo, Vowel formants from the wave equation, Journal of Acoustical Society of America Express Letters, 122 (2007). Google Scholar [12] R. Hundhammer and G. Leugering, Instantaneous control of vibrating string networks, in "Online Optimization of Large Scale Systems," Springer, Berlin, (2001), 229-249.  Google Scholar [13] P. Kuchment and H. Zeng, Convergence of spectra of mesoscopic systems collapsing onto a graph, Journal of Mathematical Analysis and Applications, 258 (2001), 671-700. doi: 10.1006/jmaa.2000.7415.  Google Scholar [14] Mikael Kurula, "Towards Input/Output-Free Modelling of Linear Infinite-Dimensional Systems in Continuous Time," Ph.D thesis, Å bo Akademi, 2010. Google Scholar [15] M. Kurula, H. Zwart, A. van der Schaft and J. Behrndt, Dirac structures and their composition on Hilbert spaces, Journal of Mathematical Analysis and Applications, 372 (2010), 402-422. doi: 10.1016/j.jmaa.2010.07.004.  Google Scholar [16] Y. Latushkin and V. Pivovarchik, Scattering in a forked-shaped waveguide, Integral Equations and Operator Theory, 61 (2008), 365-399. doi: 10.1007/s00020-008-1597-2.  Google Scholar [17] M. S. Livšic, "Operators, Oscillations, Waves (Open Systems)," Translations of Mathematical Monographs, Vol. 34, American Mathematical Society, Providence, Rhode Island, 1973.  Google Scholar [18] T. Lukkari and J. Malinen, Webster's equation with curvature and dissipation, preprint, arXiv:1204.4075, 2012. Google Scholar [19] J. Malinen, Conservativivity of time-flow invertible and boundary control systems, Helsinki University of Technology Institute of Mathematics Research Reports, A479, (2004). Google Scholar [20] J. Malinen and O. Staffans, Conservative boundary control systems, Journal of Differential Equations, 231 (2006), 290-312. doi: 10.1016/j.jde.2006.05.012.  Google Scholar [21] J. Malinen and O. Staffans, Impedance passive and conservative boundary control systems, Complex Analysis and Operator Theory, 1 (2007), 279-300. doi: 10.1007/s11785-006-0009-3.  Google Scholar [22] J. Malinen, O. Staffans and G. Weiss, When is a linear system conservative, Quarterly of Applied Mathematics, 64 (2006), 61-91.  Google Scholar [23] J. Rubinstein and M. Schatzman, Variational problems on multiply connected thin strips. I. Basic estimates and convergence of the Laplacian spectrum, Archive for Rational Mechanics and Analysis, 160 (2001), 271-308. doi: 10.1007/s002050100164.  Google Scholar [24] D. Salamon, Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach, Transactions of the American Mathematical Society, 300 (1987), 383-431. doi: 10.2307/2000351.  Google Scholar [25] D. Salamon, Realization theory in Hilbert space, Mathematical Systems Theory, 21 (1989), 147-164. doi: 10.1007/BF02088011.  Google Scholar [26] O. Staffans, "Well-Posed Linear Systems," Encyclopedia of Mathematics and its Applications, 103, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197.  Google Scholar [27] Javier Villegas, "A Port-Hamiltonian Approach to Distributed Parameter Systems," Ph.D thesis, University of Twente, 2007. Google Scholar [28] G. Weiss, Regular linear systems with feedback, Mathematics of Control, Signals, and Systems, 7 (1994), 23-57. doi: 10.1007/BF01211484.  Google Scholar [29] G. Weiss and X. Zhao, Well-posedness and controllability of a class of coupled linear systems, SIAM Journal of Control and Optimization, 48 (2009), 2719-2750. doi: 10.1137/090752833.  Google Scholar [30] H. Zwart, Y. Le Gorrec, B. Maschke and J. Villegas, Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain, ESAIM: Control, Optimisation and Calculus of Variations, 16 (2010), 1077-1093. doi: 10.1051/cocv/2009036.  Google Scholar

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##### References:
 [1] A. Aalto and J. Malinen, Wave propagation in networks: A system theoretic approach, in "Proceedings of the $18^{th}$ IFAC World Congress" (eds. S. Bittanti, A. Cenedese and S. Zampieri), (2011), 8854-8859. Google Scholar [2] W. Arendt, C. Batty, M. Hieber and F. Neubrander, "Vector-valued Laplace Transforms and Cauchy Problems," Monographs in Mathematics, 96, Birkhäuser Verlag, Basel, 2001.  Google Scholar [3] J. Cervera, A. J. van der Schaft and A. Baños, Interconnection of port-Hamiltonian systems and composition of Dirac structures, Automatica J. of IFAC, 43 (2007), 212-225. doi: 10.1016/j.automatica.2006.08.014.  Google Scholar [4] R. F. Curtain and H. Zwart, "An Introduction to Infinite-Dimensional Linear Systems Theory," Texts in Applied Mathematics, 21, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar [5] V. Derkach, S. Hassi, M. Malamud and H. de Snoo, Boundary relations and their Weyl families, Transactions of the American Mathematical Society, 358 (2006), 5351-5400. doi: 10.1090/S0002-9947-06-04033-5.  Google Scholar [6] M. Gugat, G. Leugering, K. Schittkowski and E. J. P. Georg Schmidt, Modelling, stabilization, and control of flow in networks of open channels, in "Online Optimization of Large Scale Systems," Springer, Berlin, (2001), 251-270.  Google Scholar [7] K.-J. Engel, M. Kramar Fijavž, R. Nagel and E. Sikolya, Vertex control of flows in networks, Networks and Heterogeneous Media, 3 (2008), 709-722. doi: 10.3934/nhm.2008.3.709.  Google Scholar [8] H. Fattorini, Boundary control systems, SIAM Journal of Control, 6 (1968), 349-385.  Google Scholar [9] V. I. Gorbachuk and M. L. Gorbachuk, "Boundary Value Problems for Operator Differential Equations," Mathematics and its Applications (Soviet Series), 48, Kluwer Academic Publishers Group, Dordrecht, 1991.  Google Scholar [10] G. Greiner, Perturbing the boundary conditions of a generator, Houston Journal of Mathematics, 13 (1987), 213-229.  Google Scholar [11] A. Hannukainen, T. Lukkari, J. Malinen and P. Palo, Vowel formants from the wave equation, Journal of Acoustical Society of America Express Letters, 122 (2007). Google Scholar [12] R. Hundhammer and G. Leugering, Instantaneous control of vibrating string networks, in "Online Optimization of Large Scale Systems," Springer, Berlin, (2001), 229-249.  Google Scholar [13] P. Kuchment and H. Zeng, Convergence of spectra of mesoscopic systems collapsing onto a graph, Journal of Mathematical Analysis and Applications, 258 (2001), 671-700. doi: 10.1006/jmaa.2000.7415.  Google Scholar [14] Mikael Kurula, "Towards Input/Output-Free Modelling of Linear Infinite-Dimensional Systems in Continuous Time," Ph.D thesis, Å bo Akademi, 2010. Google Scholar [15] M. Kurula, H. Zwart, A. van der Schaft and J. Behrndt, Dirac structures and their composition on Hilbert spaces, Journal of Mathematical Analysis and Applications, 372 (2010), 402-422. doi: 10.1016/j.jmaa.2010.07.004.  Google Scholar [16] Y. Latushkin and V. Pivovarchik, Scattering in a forked-shaped waveguide, Integral Equations and Operator Theory, 61 (2008), 365-399. doi: 10.1007/s00020-008-1597-2.  Google Scholar [17] M. S. Livšic, "Operators, Oscillations, Waves (Open Systems)," Translations of Mathematical Monographs, Vol. 34, American Mathematical Society, Providence, Rhode Island, 1973.  Google Scholar [18] T. Lukkari and J. Malinen, Webster's equation with curvature and dissipation, preprint, arXiv:1204.4075, 2012. Google Scholar [19] J. Malinen, Conservativivity of time-flow invertible and boundary control systems, Helsinki University of Technology Institute of Mathematics Research Reports, A479, (2004). Google Scholar [20] J. Malinen and O. Staffans, Conservative boundary control systems, Journal of Differential Equations, 231 (2006), 290-312. doi: 10.1016/j.jde.2006.05.012.  Google Scholar [21] J. Malinen and O. Staffans, Impedance passive and conservative boundary control systems, Complex Analysis and Operator Theory, 1 (2007), 279-300. doi: 10.1007/s11785-006-0009-3.  Google Scholar [22] J. Malinen, O. Staffans and G. Weiss, When is a linear system conservative, Quarterly of Applied Mathematics, 64 (2006), 61-91.  Google Scholar [23] J. Rubinstein and M. Schatzman, Variational problems on multiply connected thin strips. I. Basic estimates and convergence of the Laplacian spectrum, Archive for Rational Mechanics and Analysis, 160 (2001), 271-308. doi: 10.1007/s002050100164.  Google Scholar [24] D. Salamon, Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach, Transactions of the American Mathematical Society, 300 (1987), 383-431. doi: 10.2307/2000351.  Google Scholar [25] D. Salamon, Realization theory in Hilbert space, Mathematical Systems Theory, 21 (1989), 147-164. doi: 10.1007/BF02088011.  Google Scholar [26] O. Staffans, "Well-Posed Linear Systems," Encyclopedia of Mathematics and its Applications, 103, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197.  Google Scholar [27] Javier Villegas, "A Port-Hamiltonian Approach to Distributed Parameter Systems," Ph.D thesis, University of Twente, 2007. Google Scholar [28] G. Weiss, Regular linear systems with feedback, Mathematics of Control, Signals, and Systems, 7 (1994), 23-57. doi: 10.1007/BF01211484.  Google Scholar [29] G. Weiss and X. Zhao, Well-posedness and controllability of a class of coupled linear systems, SIAM Journal of Control and Optimization, 48 (2009), 2719-2750. doi: 10.1137/090752833.  Google Scholar [30] H. Zwart, Y. Le Gorrec, B. Maschke and J. Villegas, Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain, ESAIM: Control, Optimisation and Calculus of Variations, 16 (2010), 1077-1093. doi: 10.1051/cocv/2009036.  Google Scholar
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