Symplectic geometry in hybrid and impulsive optimal control

William Clark; Maria Oprea; William Clark; Maria Oprea

doi:10.3934/cam.2025037

Communications in Analysis and Mechanics

2025, Volume 17, Issue 4: 910-943. doi: 10.3934/cam.2025037

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Symplectic geometry in hybrid and impulsive optimal control

William Clark ^{1
,
,},
Maria Oprea ²

1.
Department of Mathematics, Ohio University, Athens, OH 45701, USA
2.
Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, USA

Received: 22 April 2025 Revised: 17 September 2025 Accepted: 24 September 2025 Published: 03 November 2025
49N25, 34A38, 37J39

Hybrid dynamical systems are systems which undergo both continuous and discrete transitions. The Bolza problem from optimal control theory was applied to these systems and a hybrid version of Pontryagin's maximum principle was presented. This hybrid maximum principle was presented to emphasize its geometric nature which made its study amenable to the tools of geometric mechanics and symplectic geometry. One explicit benefit of this geometric approach was that the symplectic structure (and hence the induced volume) was preserved. This allowed for a hybrid analog of caustics and conjugate points. Additionally, an introductory analysis of singular solutions (beating and Zeno) was discussed geometrically. This work concluded on a biological example where beating can occur.
- hybrid systems,
- optimal control,
- symplectic geometry
Citation: William Clark, Maria Oprea. Symplectic geometry in hybrid and impulsive optimal control[J]. Communications in Analysis and Mechanics, 2025, 17(4): 910-943. doi: 10.3934/cam.2025037

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Abstract

Hybrid dynamical systems are systems which undergo both continuous and discrete transitions. The Bolza problem from optimal control theory was applied to these systems and a hybrid version of Pontryagin's maximum principle was presented. This hybrid maximum principle was presented to emphasize its geometric nature which made its study amenable to the tools of geometric mechanics and symplectic geometry. One explicit benefit of this geometric approach was that the symplectic structure (and hence the induced volume) was preserved. This allowed for a hybrid analog of caustics and conjugate points. Additionally, an introductory analysis of singular solutions (beating and Zeno) was discussed geometrically. This work concluded on a biological example where beating can occur.

References

[1]	K. Aihara, H. Suzuki, Theory of hybrid dynamical systems and its applications to biological and medical systems, Philos Trans A Math Phys Eng Sci., 368 (2010), 4893–4914 https://doi.org/10.1098/rsta.2010.0237 doi: 10.1098/rsta.2010.0237
[2]	V. Azhmyakov, S. A. Attia, D. Gromov, J. Raisch, Necessary optimality conditions for a class of hybrid optimal control problems, in Hybrid Systems: Computation and Control (eds. A. Bemporad, A. Bicchi, and G. Buttazzo), Springer Berlin Heidelberg, 2007,637–640. https://doi.org/10.1007/978-3-540-71493-4_50
[3]	A. Pakniyat, P. E. Caines, On the minimum principle and dynamic programming for hybrid systems, IFAC Proceedings Volumes, 47 (2014), 9629–9634. https://doi.org/10.3182/20140824-6-ZA-1003.02700 doi: 10.3182/20140824-6-ZA-1003.02700
[4]	T. Westenbroek, X. Xiong, A. Ames, S. Sastry, Optimal control of piecewise-smooth control systems via singular perturbations, in 2019 IEEE Conference on Decision and Control (CDC), 2019, 3046–3053. https://doi.org/10.1109/CDC40024.2019.9029232
[5]	K. Yunt, C. Glocker, Modeling and optimal control of hybrid rigidbody mechanical systems, in Hybrid Systems: Computation and Control (eds. A. Bemporad, A. Bicchi, and G. Buttazzo), Springer Berlin Heidelberg, 2007,614–627. https://doi.org/10.1007/978-3-540-71493-4_47
[6]	V. Azhmyakov, S. A. Attia, J. Raisch, On the maximum principle for impulsive hybrid systems, in Hybrid Systems: Computation and Control (eds. M. Egerstedt and B. Mishra), Springer Berlin Heidelberg, 2008, 30–42. https://doi.org/10.1007/978-3-540-78929-1_3
[7]	A. Cristofaro, C. Possieri, M. Sassano, Time-optimal control for the hybrid double integrator with state-driven jumps, in 2019 IEEE Conference on Decision and Control (CDC), 2019, 6301–6306. https://doi.org/10.1109/CDC40024.2019.9030262
[8]	A. Pakniyat, P. E. Caines, On the minimum principle and dynamic programming for hybrid systems with low dimensional switching manifolds, in 2015 IEEE Conference on Decision and Control (CDC), 2015, 2567–2573. https://doi.org/10.1109/CDC.2015.7402603
[9]	D. N. Pekarek, J. E. Marsden, Variational collision integrators and optimal control, in Proc. of the 18th International Symposium on Mathematical Theory of Networks and Systems (MTNS), 2008.
[10]	D. Liberzon, Calculus of variations and optimal control theory: A concise introduction, Princeton University Press, 2012.
[11]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The mathematical theory of optimal processes, New York: Interscience Publishers, Inc., 1962.
[12]	M. Barbero-Liñán, M.C. Muñoz-Lecanda, Geometric approach to pontryagin's maximum principle, Acta Appl Math, 108 (2009), 429–485. https://doi.org/10.1007/s10440-008-9320-5 doi: 10.1007/s10440-008-9320-5
[13]	B. Piernicola, C. Franco, Lagrangian submanifold landscapes of necessary conditions for maxima in optimal control: global parameterizations and generalized solutions, J Math Sci, 135 (2006), 3125–3144. https://doi.org/10.1007/s10958-006-0149-z doi: 10.1007/s10958-006-0149-z
[14]	W. Clark, A. Bloch, Invariant forms in hybrid and impact systems and a taming of Zeno, Arch Rational Mech Anal, 247 (2023), 13. https://doi.org/10.1007/s00205-023-01844-1 doi: 10.1007/s00205-023-01844-1
[15]	S. N. Simic, K. H. Johansson, J. Lygeros, S. Sastry, Structural stability of hybrid systems, in 2001 European Control Conference (ECC), 2001, 3858–3863. https://doi.org/10.23919/ECC.2001.7076536
[16]	M. Barbero-Liñán, J. Cortés, D. Martín de Diego, S. Martínez, M. C. Muñoz Lecanda, Global controllability tests for geometric hybrid control systems, Nonlinear Analysis: Hybrid Systems, 38 (2020), 100935. https://doi.org/10.1016/j.nahs.2020.100935 doi: 10.1016/j.nahs.2020.100935
[17]	R. Geobel, R. G. Sanfelice, A. R. Teel, Hybrid dynamical systems, Princeton University Press, 2012. https://doi.org/10.23943/princeton/9780691153896.001.0001
[18]	W. M. Haddad, V. Chellaboina, S. G. Nersesov, Impulsive and hybrid dynamical systems: stability, dissipativity, and control, Princeton University Press, 2006. https://doi.org/10.1515/9781400865246
[19]	N. J. Kong, J. J. Payne, J. Zhu, A. M. Johnson, Saltation matrices: The essential tool for linearizing hybrid dynamical systems, Proceedings of the IEEE, 112 (2024), 585–608. https://doi.org/10.1109/JPROC.2024.3440211 doi: 10.1109/JPROC.2024.3440211
[20]	A. D. Ames, H. Zheng, R. D. Gregg, S. Sastry, Is there life after Zeno? Taking executions past the breaking (Zeno) point, Proceedings of the 25th American Control Conference, 2006, 2652–2657. https://doi.org/10.1109/ACC.2006.1656623
[21]	B. Brogliato, Nonsmooth mechanics, In: Communications and Control Engineering, Springer International Publishing, 2016. https://doi.org/10.1007/978-3-319-28664-8
[22]	I. Gelfand, S. Fomin, Calculus of variations, In: Dover Books on Mathematics, Dover Publications, 2000.
[23]	J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, In: Applied Mathematical Sciences, Springer-Verlag New York, 1983. https://doi.org/10.1007/978-1-4612-1140-2
[24]	R. C. Fetecau, J. E. Marsden, M. Oritz, M. West, Nonsmooth Lagrangian mechanics and variational collision integrators, SIAM J. Appl. Dyn. Syst., 2 (2003), 381–416. https://doi.org/10.1137/S1111111102406038 doi: 10.1137/S1111111102406038
[25]	W. A. Clark, M. Oprea, A. Shaw, Optimal control of hybrid systems with submersive resets, IFAC-PapersOnLine, 58 (2024), 89–94. https://doi.org/10.1016/j.ifacol.2024.08.262 doi: 10.1016/j.ifacol.2024.08.262
[26]	L. Colombo, M. de León, M. E. E. Irazú, A. López-Gordón, Hamilton-Jacobi theory for nonholonomic and forced hybrid mechanical systems, Geom. Mech., 1 (2024), 123–152. https://doi.org/10.1142/S2972458924500059 doi: 10.1142/S2972458924500059
[27]	A. Pakniyat, Optimal control of deterministic and stochastic hybrid systems: Theory and applications, Ph.D thesis, McGill University, 2016.
[28]	W. Clark, A. Bloch, Stable orbits for a simple passive walker experiencing foot slip, in 2018 IEEE Conference on Decision and Control (CDC), 2018, 2366–2371. https://doi.org/10.1109/CDC.2018.8618880
[29]	R. Abraham, J. Marsden, Foundations of mechanics, AMS Chelsea Pub./American Mathematical Society, 2008.
[30]	W. Clark, M. Oprea, Optimal control of hybrid systems via hybrid Lagrangian submanifolds, in 7th IFAC Workshop on Lagrangian and Hamiltonian Methods for Optimal Control, 2021, 88–93. https://doi.org/10.1016/j.ifacol.2021.11.060
[31]	A. A. Agrachev, Y. L. Sachkov, Control theory from the geometric viewpoint, Springer Berlin Heidelberg, 2004. https://doi.org/10.1007/978-3-662-06404-7
[32]	W. Clark, M. Oprea, Optimality of Zeno executions in hybrid systems, in 2023 American Control Conference (ACC), 2023, 3983–3988. https://doi.org/10.23919/ACC55779.2023.10155928
[33]	S. Phillips, R. G. Sanfelice, A Framework for modeling and analysis of dynamical properties of spiking neurons, in 2014 American Control Conference, 2014, 1414–1419. https://doi.org/10.1109/ACC.2014.6859494
[34]	W. Gerstner, W. M. Kistler, R. Naud, L. Paninski, Neuronal dynamics: From single neurons to networks and models of cognition, Cambridge University Press, 2014.
[35]	O. V. Popovych, P. A. Tass, Control of abnormal synchronization in neurological disorders, Front. Neurol., 5 (2014), 268. https://doi.org/10.3389/fneur.2014.00268 doi: 10.3389/fneur.2014.00268
[36]	C. J. Wilson, B. Beverlin, T. Netoff, Chaotic desynchronization as the therapeutic mechanism of deep brain stimulation, Front. Syst. Neurosci., 5 (2011), 50. https://doi.org/10.3389/fnsys.2011.00050 doi: 10.3389/fnsys.2011.00050
[37]	B. Duchet, G. Weerasinghe, H. Cagnan, P. Brown, C. Bick, R. Bogacz, Phase-dependence of response curves to deep brain stimulation and their relationship: from essential tremor patient data to a Wilson–Cowan model, J. Math. Neurosci., 10 (2020), 4. https://doi.org/10.1186/s13408-020-00081-0 doi: 10.1186/s13408-020-00081-0
[38]	T. Kreuz, F. Mormann, R. Andrezejak, A. Kraskov, K. Lehertz, P. Grassberger, Measuring synchronization in coupled model systems: A comparison of different approaches, Physica D, 225 (2007), 29–42. https://doi.org/10.1016/j.physd.2006.09.039 doi: 10.1016/j.physd.2006.09.039
[39]	M. Rungger, O. Stursberg, Optimal control for deterministic hybrid systems using dynamic programming, IFAC Proceedings Volumes, 42 (2009), 316–321. https://doi.org/10.3182/20090916-3-ES-3003.00055 doi: 10.3182/20090916-3-ES-3003.00055
[40]	E. Gutkin, A. Katok, Caustics for inner and outer billiards, Commun. Math. Phys., 173 (1995), 101–133. https://doi.org/10.1007/BF02100183 doi: 10.1007/BF02100183
[41]	X. Wang, J. Liu, H. Peng, Symplectic pseudospectral methods for optimal control, Springer Singapore, Singapore, 2021. https://doi.org/10.1007/978-981-15-3438-6

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