Manifold turnpikes of nonlinear port-Hamiltonian descriptor systems under minimal energy supply

Authors

Attila Karsai

Abstract

Turnpike phenomena of nonlinear port-Hamiltonian descriptor systems under minimal energy supply are studied. Under assumptions on the smoothness of the system nonlinearities, it is shown that the optimal control problem is dissipative with respect to a manifold. Then, under controllability assumptions, it is shown that the optimal control problem exhibits a manifold turnpike property.

Keywords

Turnpike phenomenon; Nonlinear systems; Port-Hamiltonian systems

Citation

• Journal: Mathematics of Control, Signals, and Systems
• Year: 2024
• Volume: 36
• Issue: 3
• Pages: 707–728
• Publisher: Springer Science and Business Media LLC
• DOI: 10.1007/s00498-024-00384-7

BibTeX

@article{Karsai_2024,
  title={{Manifold turnpikes of nonlinear port-Hamiltonian descriptor systems under minimal energy supply}},
  volume={36},
  ISSN={1435-568X},
  DOI={10.1007/s00498-024-00384-7},
  number={3},
  journal={Mathematics of Control, Signals, and Systems},
  publisher={Springer Science and Business Media LLC},
  author={Karsai, Attila},
  year={2024},
  pages={707--728}
}

Download the bib file

References

• Dorfman R, Samuelson P, Solow R (1987) Linear programming and economic analysis. Dover books on advanced mathematics. Dover Publications, New York. https://books.google.de/books?id=k5_vzaCNQP4C
• McKenzie, L. W. Turnpike Theorems for a Generalized Leontief Model. Econometrica vol. 31 165 (1963) – 10.2307/1910955
• Carlson, D. A., Haurie, A. B. & Leizarowitz, A. Infinite Horizon Optimal Control. (Springer Berlin Heidelberg, 1991). doi:10.1007/978-3-642-76755-5 – 10.1007/978-3-642-76755-5
• Turnpike Properties in the Calculus of Variations and Optimal Control. Nonconvex Optimization and Its Applications (Springer-Verlag, 2006). doi:10.1007/0-387-28154-1 – 10.1007/0-387-28154-1
• Trélat, E. & Zhang, C. Integral and measure-turnpike properties for infinite-dimensional optimal control systems. Mathematics of Control, Signals, and Systems vol. 30 (2018) – 10.1007/s00498-018-0209-1
• Porretta, A. & Zuazua, E. Long Time versus Steady State Optimal Control. SIAM Journal on Control and Optimization vol. 51 4242–4273 (2013) – 10.1137/130907239
• Porretta, A. & Zuazua, E. Remarks on Long Time Versus Steady State Optimal Control. Springer INdAM Series 67–89 (2016) doi:10.1007/978-3-319-39092-5_5 – 10.1007/978-3-319-39092-5_5
• Trélat, E. & Zuazua, E. The turnpike property in finite-dimensional nonlinear optimal control. Journal of Differential Equations vol. 258 81–114 (2015) – 10.1016/j.jde.2014.09.005
• Rapaport, A. & Cartigny, P. Turnpike theorems by a value function approach. ESAIM: Control, Optimisation and Calculus of Variations vol. 10 123–141 (2004) – 10.1051/cocv:2003039
• Faulwasser, T., Flaßkamp, K., Ober-Blöbaum, S., Schaller, M. & Worthmann, K. Manifold turnpikes, trims, and symmetries. Mathematics of Control, Signals, and Systems vol. 34 759–788 (2022) – 10.1007/s00498-022-00321-6
• Paynter H (1961) Analysis and Design of Engineering Systems: Class Notes for M.I.T. Course 2,751. M.I.T. Press, Cambridge. https://books.google.de/books?id=FcdhAAAAMAAJ
• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002
• Moser T, Lohmann B (2022) Rosenbrock framework for tangential interpolation of port-Hamiltonian descriptor systems. https://arxiv.org/abs/2210.16071
• Lamoline F, Hastir A (2022) On Dirac structure of infinite-dimensional stochastic port-Hamiltonian systems. https://arxiv.org/abs/2210.06358
• van der Schaft, A. & Mehrmann, V. Linear Port-Hamiltonian DAE Systems Revisited. SSRN Electronic Journal (2022) doi:10.2139/ssrn.4292998 – 10.2139/ssrn.4292998
• Breiten T, Hinsen D, Unger B (2022) Towards a modeling class for port-Hamiltonian systems with time-delay. https://arxiv.org/abs/2211.10687
• Gernandt H, Hinsen D (2022) Stability and passivity for a class of infinite dimensional port-Hamiltonian networks with application to power grids. https://arxiv.org/abs/2212.02792
• Mehrmann, V. & Unger, B. Control of port-Hamiltonian differential-algebraic systems and applications. Acta Numerica vol. 32 395–515 (2023) – 10.1017/s0962492922000083
• Schaller, M., Philipp, F., Faulwasser, T., Worthmann, K. & Maschke, B. Control of port-Hamiltonian systems with minimal energy supply. European Journal of Control vol. 62 33–40 (2021) – 10.1016/j.ejcon.2021.06.017
• Faulwasser, T., Maschke, B., Philipp, F., Schaller, M. & Worthmann, K. Optimal Control of Port-Hamiltonian Descriptor Systems with Minimal Energy Supply. SIAM Journal on Control and Optimization vol. 60 2132–2158 (2022) – 10.1137/21m1427723
• Philipp, F., Schaller, M., Faulwasser, T., Maschke, B. & Worthmann, K. Minimizing the energy supply of infinite-dimensional linear port-Hamiltonian systems. IFAC-PapersOnLine vol. 54 155–160 (2021) – 10.1016/j.ifacol.2021.11.071
• Staffans, O. J. Passive and Conservative Infinite-Dimensional Impedance and Scattering Systems (From a Personal Point of View). The IMA Volumes in Mathematics and its Applications 375–413 (2003) doi:10.1007/978-0-387-21696-6_14 – 10.1007/978-0-387-21696-6_14
• Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica vol. 38 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
• Willems, J. C. Dissipative dynamical systems part I: General theory. Archive for Rational Mechanics and Analysis vol. 45 321–351 (1972) – 10.1007/bf00276493
• Dudek, E. & Holly, K. Nonlinear orthogonal projection. Annales Polonici Mathematici vol. 59 1–31 (1994) – 10.4064/ap-59-1-1-31
• Leobacher, G. & Steinicke, A. Existence, uniqueness and regularity of the projection onto differentiable manifolds. Annals of Global Analysis and Geometry vol. 60 559–587 (2021) – 10.1007/s10455-021-09788-z
• Walter, R. Introducing Analysis 3. (Walter de Gruyter, 2009). doi:10.1515/9783110214840 – 10.1515/9783110214840
• Łojasiewicz, S. Sur le problème de la division. Studia Mathematica vol. 18 87–136 (1959) – 10.4064/sm-18-1-87-136
• Ji, S., Kollar, J. & Shiffman, B. A Global Lojasiewicz Inequality for Algebraic Varieties. Transactions of the American Mathematical Society vol. 329 813 (1992) – 10.2307/2153965
• Faulwasser, T., Korda, M., Jones, C. N. & Bonvin, D. On turnpike and dissipativity properties of continuous-time optimal control problems. Automatica vol. 81 297–304 (2017) – 10.1016/j.automatica.2017.03.012
• Grüne, L. & Guglielmi, R. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields vol. 11 169–188 (2021) – 10.3934/mcrf.2020032
• Domschke P, Hiller B, Lang J, Mehrmann V, Morandin R, Tischendorf C (2021) Gas network modeling: an overview. https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/411
• Andersson, J. A. E., Gillis, J., Horn, G., Rawlings, J. B. & Diehl, M. CasADi: a software framework for nonlinear optimization and optimal control. Mathematical Programming Computation vol. 11 1–36 (2018) – 10.1007/s12532-018-0139-4
• Wächter, A. & Biegler, L. T. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming vol. 106 25–57 (2005) – 10.1007/s10107-004-0559-y
• Faulwasser, T. et al. Hidden regularity in singular optimal control of port-Hamiltonian systems. Preprint at https://doi.org/10.48550/ARXIV.2305.03790 (2023) – 10.48550/arxiv.2305.03790
• Trélat, E., Zhang, C. & Zuazua, E. Steady-State and Periodic Exponential Turnpike Property for Optimal Control Problems in Hilbert Spaces. SIAM Journal on Control and Optimization vol. 56 1222–1252 (2018) – 10.1137/16m1097638
• Kunkel, P. & Mehrmann, V. Differential-Algebraic Equations. EMS Textbooks in Mathematics (2006) doi:10.4171/017 – 10.4171/017
• Macki, J. & Strauss, A. Introduction to Optimal Control Theory. Undergraduate Texts in Mathematics (Springer New York, 1982). doi:10.1007/978-1-4612-5671-7 – 10.1007/978-1-4612-5671-7
• Kunkel, P. & Mehrmann, V. Optimal control for unstructured nonlinear differential-algebraic equations of arbitrary index. Mathematics of Control, Signals, and Systems vol. 20 227–269 (2008) – 10.1007/s00498-008-0032-1