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  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 & Control Theory, 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.  Google Scholar

[2]

A. BrugnoliD. AlazardV. 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.  Google Scholar

[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.  Google Scholar

[4]

R. Cross, Multivalued Linear Operators, volume 213 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1998.  Google Scholar

[5]

R. Dautray and J. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3, Springer-Verlag, Berlin, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. van der Schaft and D. Jeltsema, Port-Hamiltonian systems theory: An introductory overview, Found. Trends Syst. Control, 1 (2014), 173-378.   Google Scholar

[15]

J. A. Villegas, A Port-Hamiltonian Approach to Distributed Parameter Systems, Ph.D thesis, University of Twente, Netherlands, 2007. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

A. BrugnoliD. AlazardV. 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.  Google Scholar

[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.  Google Scholar

[4]

R. Cross, Multivalued Linear Operators, volume 213 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1998.  Google Scholar

[5]

R. Dautray and J. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3, Springer-Verlag, Berlin, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. van der Schaft and D. Jeltsema, Port-Hamiltonian systems theory: An introductory overview, Found. Trends Syst. Control, 1 (2014), 173-378.   Google Scholar

[15]

J. A. Villegas, A Port-Hamiltonian Approach to Distributed Parameter Systems, Ph.D thesis, University of Twente, Netherlands, 2007. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Illustration of a quasi Gelfand triple, where $ D_{+} = \text{dom} \ \iota_{+} $ and $ D_{-} = \text{dom} \ \iota_{-} $
[1]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

[2]

Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, 2021, 20 (1) : 359-368. doi: 10.3934/cpaa.2020270

[3]

Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030

[4]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[5]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336

[6]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[7]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073

[8]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351

[9]

Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258

[10]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

[11]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[12]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240

[13]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[14]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[15]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052

[16]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[17]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267

[18]

Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107

[19]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

2019 Impact Factor: 0.953

Article outline

Figures and Tables

[Back to Top]