Figure 1.
Illustration of a quasi Gelfand triple, where $ D_{+} = \text{dom} \ \iota_{+} $ and $ D_{-} = \text{dom} \ \iota_{-} $
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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 |
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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.
Keywords:
port-Hamiltonian,
multidimensional spatial domain,
scattering passive,
boundary triple,
quasi Gelfand triple.
Mathematics Subject Classification:
47D06, 93C20, 35Q93, 47F05.
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:
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Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal., 243 (2007), 536-565.
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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.
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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. |
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R. Cross, Multivalued Linear Operators, volume 213 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1998. |
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R. Dautray and J. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3, Springer-Verlag, Berlin, 1990. |
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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.
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L. Hörmander, The Analysis of Linear Partial Differential Operators I, Classics in Mathematics, Springer-Verlag, Berlin, 2003.
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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. |
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M. Kurula and H. Zwart,
Linear wave systems on $n$-D spatial domains, Internat. J. Control, 88 (2015), 1063-1077.
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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. |
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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. |
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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. |
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K. Yosida, Functional Analysis, volume 123 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin-New York, sixth edition, 1980. |
show all references
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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