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December  2021, 10(4): 965-1006. doi: 10.3934/eect.2020098

## Well-posedness of linear first order port-Hamiltonian Systems on multidimensional spatial domains

 University of Wuppertal, School of Mathematics and Natural Science, Gaußstraße 20, D-42119 Wuppertal, Germany

* Corresponding author: skrepek@uni-wuppertal.de

Received  October 2019 Revised  June 2020 Published  December 2021 Early access  October 2020

Fund Project: This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 765579

We consider a port-Hamiltonian system on an open spatial domain $\Omega \subseteq \mathbb{R}^n$ with bounded Lipschitz boundary. We show that there is a boundary triple associated to this system. Hence, we can characterize all boundary conditions that provide unique solutions that are non-increasing in the Hamiltonian. As a by-product we develop the theory of quasi Gelfand triples. Adding "natural" boundary controls and boundary observations yields scattering/impedance passive boundary control systems. This framework will be applied to the wave equation, Maxwell's equations and Mindlin plate model. Probably, there are even more applications.

Citation: Nathanael Skrepek. Well-posedness of linear first order port-Hamiltonian Systems on multidimensional spatial domains. Evolution Equations and Control Theory, 2021, 10 (4) : 965-1006. doi: 10.3934/eect.2020098
##### References:
 [1] J. Behrndt and M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal., 243 (2007), 536-565.  doi: 10.1016/j.jfa.2006.10.009. [2] A. Brugnoli, D. Alazard, V. Pommier-Budinger and D. Matignon, Port-Hamiltonian formulation and symplectic discretization of plate models. Part Ⅰ: Mindlin model for thick plates, Appl. Math. Model., 75 (2019), 940-960.  doi: 10.1016/j.apm.2019.04.035. [3] L. Carbone and R. De Arcangelis, Unbounded Functionals in the Calculus of Variations, volume 125 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2002. [4] R. Cross, Multivalued Linear Operators, volume 213 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1998. [5] R. Dautray and J. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3, Springer-Verlag, Berlin, 1990. [6] V. I. Gorbachuk and M. L. Gorbachuk, Boundary Value Problems for Operator Differential Equations, volume 48 of Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3714-0. [7] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Classics in Mathematics, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-61497-2. [8] B. Jacob and H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces, volume 223 of Operator Theory: Advances and Applications, Birkhäuser/Springer Basel AG, Basel, 2012. doi: 10.1007/978-3-0348-0399-1. [9] M. Kurula and H. Zwart, Linear wave systems on $n$-D spatial domains, Internat. J. Control, 88 (2015), 1063-1077.  doi: 10.1080/00207179.2014.993337. [10] A. Macchelli, C. Melchiorri and L. Bassi, Port-based modelling and control of the Mindlin plate, in Decision and Control, 2005 and 2005 European Control Conference, IEEE, (2005), 5989-5994. doi: 10.1109/CDC.2005.1583120. [11] J. Malinen and O. J. Staffans, Conservative boundary control systems, J. Differential Equations, 231 (2006), 290-312.  doi: 10.1016/j.jde.2006.05.012. [12] J. Malinen and O. J. Staffans, Impedance passive and conservative boundary control systems, Complex Anal. Oper. Theory, 1 (2007), 279-300.  doi: 10.1007/s11785-006-0009-3. [13] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9. [14] A. van der Schaft and D. Jeltsema, Port-Hamiltonian systems theory: An introductory overview, Found. Trends Syst. Control, 1 (2014), 173-378. [15] J. A. Villegas, A Port-Hamiltonian Approach to Distributed Parameter Systems, Ph.D thesis, University of Twente, Netherlands, 2007. [16] G. Weiss and O. J. Staffans, Maxwell's equations as a scattering passive linear system, SIAM J. Control Optim., 51 (2013), 3722-3756.  doi: 10.1137/120869444. [17] K. Yosida, Functional Analysis, volume 123 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin-New York, sixth edition, 1980.

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##### References:
 [1] J. Behrndt and M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal., 243 (2007), 536-565.  doi: 10.1016/j.jfa.2006.10.009. [2] A. Brugnoli, D. Alazard, V. Pommier-Budinger and D. Matignon, Port-Hamiltonian formulation and symplectic discretization of plate models. Part Ⅰ: Mindlin model for thick plates, Appl. Math. Model., 75 (2019), 940-960.  doi: 10.1016/j.apm.2019.04.035. [3] L. Carbone and R. De Arcangelis, Unbounded Functionals in the Calculus of Variations, volume 125 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2002. [4] R. Cross, Multivalued Linear Operators, volume 213 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1998. [5] R. Dautray and J. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3, Springer-Verlag, Berlin, 1990. [6] V. I. Gorbachuk and M. L. Gorbachuk, Boundary Value Problems for Operator Differential Equations, volume 48 of Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3714-0. [7] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Classics in Mathematics, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-61497-2. [8] B. Jacob and H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces, volume 223 of Operator Theory: Advances and Applications, Birkhäuser/Springer Basel AG, Basel, 2012. doi: 10.1007/978-3-0348-0399-1. [9] M. Kurula and H. Zwart, Linear wave systems on $n$-D spatial domains, Internat. J. Control, 88 (2015), 1063-1077.  doi: 10.1080/00207179.2014.993337. [10] A. Macchelli, C. Melchiorri and L. Bassi, Port-based modelling and control of the Mindlin plate, in Decision and Control, 2005 and 2005 European Control Conference, IEEE, (2005), 5989-5994. doi: 10.1109/CDC.2005.1583120. [11] J. Malinen and O. J. Staffans, Conservative boundary control systems, J. Differential Equations, 231 (2006), 290-312.  doi: 10.1016/j.jde.2006.05.012. [12] J. Malinen and O. J. Staffans, Impedance passive and conservative boundary control systems, Complex Anal. Oper. Theory, 1 (2007), 279-300.  doi: 10.1007/s11785-006-0009-3. [13] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9. [14] A. van der Schaft and D. Jeltsema, Port-Hamiltonian systems theory: An introductory overview, Found. Trends Syst. Control, 1 (2014), 173-378. [15] J. A. Villegas, A Port-Hamiltonian Approach to Distributed Parameter Systems, Ph.D thesis, University of Twente, Netherlands, 2007. [16] G. Weiss and O. J. Staffans, Maxwell's equations as a scattering passive linear system, SIAM J. Control Optim., 51 (2013), 3722-3756.  doi: 10.1137/120869444. [17] K. Yosida, Functional Analysis, volume 123 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin-New York, sixth edition, 1980.
Illustration of a quasi Gelfand triple, where $D_{+} = \text{dom} \ \iota_{+}$ and $D_{-} = \text{dom} \ \iota_{-}$
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