Jet space extensions of infinite-dimensional Hamiltonian systems (2401.15096v1)
Abstract: We analyze infinite-dimensional Hamiltonian systems corresponding to partial differential equations on one-dimensional spatial domains formulated with formally skew-adjoint Hamiltonian operators and nonlinear Hamiltonian density. In various applications, the Hamiltonian density can depend on spatial derivatives of the state such that these systems can not straightforwardly be formulated as boundary port-Hamiltonian system using a Stokes-Dirac structure. In this work, we show that any Hamiltonian system of the above class can be reformulated as a Hamiltonian system on the jet space, in which the Hamiltonian density only depends on the extended state variable itself and not on its derivatives. Consequently, well-known geometric formulations with Stokes- Dirac structures are applicable. Additionally, we provide a similar result for dissipative systems. We illustrate the developed theory by means of the the Boussinesq equation, the dynamics of an elastic rod and the Allen-Cahn equation.
- Structure-preserving discretization of a coupled Allen-Cahn and heat equation system. IFAC-PapersOnLine, 55(18):99–104, 2022.
- Theodore James Courant. Dirac manifolds. Transactions of the American Mathematical Society, 319(2):631–661, 1990.
- T.J. Courant and A. Weinstein. Beyond Poisson structures. In Séminaire Sud-Rhodanien de Géométrie, volume 8 of Travaux en cours, Paris, 1988. Hermann.
- B. Jacob and H. Zwart. Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces, volume 223 of Operator Theory: Advances and Applications. Birkhäuser Basel, 2012.
- Dirac structures and boundary control systems associated with skew-symmetric differential operators. SIAM J. Control Optim., 44(5):1864, 2005.
- A. Macchelli and B.M. Maschke. Modeling and Control of Complex Physical Systems - The Port-Hamiltonian Approach, chapter Infinite-dimensional Port-Hamiltonian Systems, pages 211–272. Springer, Sept. 2009.
- Bernhard Maschke and Arjan J van der Schaft. On alternative poisson brackets for fluid dynamical systems and their extension to Stokes-Dirac structures. IFAC Proceedings Volumes, 46(26):109–114, 2013.
- Bernhard M Maschke and Arjan J van der Schaft. Port-controlled Hamiltonian systems: modelling origins and systemtheoretic properties. In Nonlinear Control Systems Design 1992, pages 359–365. Elsevier, 1993.
- An intrinsic Hamiltonian formulation of network dynamics: Non-standard Poisson structures and gyrators. J. Frank. Inst., 329(5):923–966, 1992.
- Peter J. Olver. Applications of Lie groups to differential equations. Graduate texts in mathematics 107. Springer, New York, 2nd edition, 1993.
- Jet bundle formulation of infinite-dimensional port-Hamiltonian systems using differential operators. Automatica, 50(2):607–613, 2014.
- Linear boundary port-Hamiltonian systems with implicitly defined energy. arXiv:2305.13772, 2023.
- A.J. van der Schaft and B.M. Maschke. Hamiltonian formulation of distributed parameter systems with boundary energy flow. J. Geom. Phys., 42:166–174, 2002.
- Arjan J van der Schaft and Bernhard M Maschke. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and physics, 42(1-2):166–194, 2002.
- J.A. Villegas. A port-Hamiltonian approach to distributed parameter systems. PhD thesis, U Twente, 2007.
- Port Hamiltonian systems with moving interface: a phase field approach. IFAC-PapersOnLine, 53(2):7569–7574, 2020.
- Port Hamiltonian formulation of the solidification process for a pure substance: A phase field approach. IFAC-PapersOnLine, 55(18):93–98, 2022.
- Building systems from simple hyperbolic ones. Systems & Control Letters, 91:1 – 6, 2016.