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December  2020, 12(4): 641-669. doi: 10.3934/jgm.2020030

Symmetry actuated closed-loop Hamiltonian systems

Simon Hochgerner

Österreichische Finanzmarktaufsicht (FMA), Otto-Wagner Platz 5, A-1090 Vienna, Austria

Received  December 2019 Published  December 2020 Early access  November 2020

This paper extends the theory of controlled Hamiltonian systems with symmetries due to [23,9,10,6,7,11] to the case of non-abelian symmetry groups $ G $ and semi-direct product configuration spaces. The notion of symmetry actuating forces is introduced and it is shown, that Hamiltonian systems subject to such forces permit a conservation law, which arises as a controlled perturbation of the $ G $-momentum map. Necessary and sufficient matching conditions are given to relate the closed-loop dynamics, associated to the forced Hamiltonian system, to an unforced Hamiltonian system. These matching conditions are then applied to general Lie-Poisson systems, to the example of ideal charged fluids in the presence of an external magnetic field ([20]), and to the satellite with a rotor example ([9,10]).

Keywords: Hamiltonian mechanics, closed loop control, symmetries.
Mathematics Subject Classification: 70G40, 10H14, 37K45, 76B75.
Citation: Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030
References:
[1]

V. I. Arnold, Sur la géométrie différentielle de groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Annales de l'Institut Fourier, 16 (1966), 319-361.  doi: 10.5802/aif.233.

[2]

V. Arnold and B. Khesin, Topological Methods in Hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, 1998.

[3]

A. Arnaudon, A. L. De Castro and D. D. Holm, Noise and dissipation on coadjoint orbits, J. Nonlinear Sci., 28 (2018), 91-145.  doi: 10.1007/s00332-017-9404-3.

[4]

A. Arnaudon, N. Ganaba and D. Holm, The stochastic energy – Casimir method, Comptes Rendus Mécanique, 346 (2018), 279-290.  doi: 10.1016/j.crme.2018.01.003.

[5]

A. Bloch, D. Chang, N. Leonard and J. E. Marsden, Controlled Lagrangians and the stabilization of mechanical systems II: Potential shaping, IEEE Trans. Automat. Control, 46 (2001), 1556-1571.  doi: 10.1109/9.956051.

[6]

A. Bloch, N. Leonard and J. Marsden, Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem, IEEE Trans. Automat. Control, 45 (2000), 2253-2270.  doi: 10.1109/9.895562.

[7]

A. Bloch, N. Leonard and J. Marsden, Controlled Lagrangians and the stabilization of Euler-Poincaré mechanical systems, Internat. J. Robust Nonlinear Control, 11 (2001), 191-214.  doi: 10.1002/rnc.572.

[8]

A. Bloch and N. Leonard, Symmetries, conservation laws, and control, in Geometry, Mechanics and Dynamics, Springer-Verlag, New York, 2002. doi: 10.1007/b97525.

[9]

A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and G. Sánchez de Alvarez, Stabilization of rigid body dynamics by internal and external torques, Automatica J. IFAC, 28 (1992), 745-756.  doi: 10.1016/0005-1098(92)90034-D.

[10]

A. M. Bloch, J. E. Marsden and G. Sánchez de Alvarez, Feedback stabilization of relative equilibria for mechanical systems with symmetry, in Current and Future Directions in Applied Mathematics, Birkhäuser, Boston, MA, 1997.

[11]

D. E. Chang, A. M. Bloch, N. E. Leonard, J. E. Marsden and C. A. Woolsey, The equivalence of controlled Lagrangian and controlled Hamiltonian systems, ESAIM: Control, Optimisation and Calculus of Variations, 8 (2002), 393-422. 

[12]

D. Crisan, F. Flandoli and D. D. Holm, Solution properties of a 3D stochastic Euler fluid equation, J. Nonlinear Sci., 29 (2019), 813-870.  doi: 10.1007/s00332-018-9506-6.

[13]

T. Drivas, D. Holm and J.-M. Leahy, Lagrangian averaged stochastic advection by Lie transport for fluids, J. Stat. Phys., 179 (2020), 1304-1342.  doi: 10.1007/s10955-020-02493-4.

[14]

F. Gay-Balmaz, C. Tronci and C. Vizman, Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups, J. Geom. Mech., 5 (2013), 39-84.  doi: 10.3934/jgm.2013.5.39.

[15]

F. Gay-Balmaz and T. Ratiu, Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids, J. Symplectic Geom., 6 (2008), 189-237.  doi: 10.4310/JSG.2008.v6.n2.a4.

[16]

J. Gibbons, D. D. Holm and B. Kupershmidt, The Hamiltonian structure of classical chromohydrodynamics, Phys. D, 6 (1982/83), 179-194.  doi: 10.1016/0167-2789(83)90004-0.

[17]

S. Hochgerner, Symmetry reduction of Brownian motion and quantum Calogero-Moser models, Stoch. Dyn., 13 (2013), 1250007, 31 pp. doi: 10.1142/S0219493712500074.

[18]

S. Hochgerner and T. Ratiu, The geometry of non-holonomic diffusion, J. Eur. Math. Soc., 17 (2015), 273-319.  doi: 10.4171/JEMS/504.

[19]

S. Hochgerner, A Hamiltonian mean field system for the Navier–Stokes equation, Proc. A, 474 (2018), 20180178, 20 pp. doi: 10.1098/rspa.2018.0178.

[20]

S. Hochgerner, Feedback control of charged ideal fluids, preprint, 2020, arXiv: 1905.04778.

[21]

D. Holm, J. Marsden, T. Ratiu and A. Weinstein, Nonlinear stability of fluid and plasma equilibria, Phys. Rep., 123 (1985), 1-116.  doi: 10.1016/0370-1573(85)90028-6.

[22]

D. Holm, Variational principles for stochastic fluid dynamics, Proc. A, 471 (2015), 20140963, 19 pp. doi: 10.1098/rspa.2014.0963.

[23]

P. S. Krishnaprasad, Lie-Poisson structures, dual-spinspacecraft and asymptotic stability, Nonlinear Anal., 9 (1985), 1011-1035.  doi: 10.1016/0362-546X(85)90083-5.

[24]

J.-A. Lazaro-Cami and J.-P. Ortega, Stochastic Hamiltonian dynamical systems, Rep. Math. Phys., 61 (2008), 65-122.  doi: 10.1016/S0034-4877(08)80003-1.

[25]

J. E. Marsden, Park City lectures on mechanics, dynamics, and symmetry, in Symplectic Geometry and Topology, IAS/Park City Mathematics Series, No. 7, American Mathematical Society, Providence, RI, 1999,335–430. doi: 10.1090/pcms/007/09.

[26]

J. E. Marsden, A. Weinstein, T. Ratiu, R. Schmid and R. Spencer, Hamiltonian systems with symmetry, coadjoint orbits and plasma physics, Proc. IUTAM-IS1MM Symposium on Modern Developments in Analytical Mechanics (Torino, 1982), Atti Acad. Sci. Torino Cl. Sci. Fis. Math. Natur., 117, (1983), 289–340.

[27]

P. Michor, Topics in Differential Geometry, Graduate Studies in Mathematics, vol. 93, Amer. Math. Soc., Providence, RI, 2008. doi: 10.1090/gsm/093.

[28]

P. Michor, Some geometric evolution equations arising as geodesic equations on groups of diffeomorphism, including the Hamiltonian approach, in Phase space analysis of Partial Differential Equations, Birkhäuser Boston, Boston, 2006,133–215. doi: 10.1007/978-0-8176-4521-2_11.

[29]

M. Puiggalí and A. Bloch, An extension to the theory of controlled Lagrangians using the Helmholtz conditions, J. Nonlinear Sci., 29 (2019), 345-376.  doi: 10.1007/s00332-018-9490-x.

[30]

A. Weinstein, A universal phase space for particles in Yang-Mills fields, Lett. Math. Phys., 2 (1977/78), 417-420.  doi: 10.1007/BF00400169.

show all references

References:
[1]

V. I. Arnold, Sur la géométrie différentielle de groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Annales de l'Institut Fourier, 16 (1966), 319-361.  doi: 10.5802/aif.233.

[2]

V. Arnold and B. Khesin, Topological Methods in Hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, 1998.

[3]

A. Arnaudon, A. L. De Castro and D. D. Holm, Noise and dissipation on coadjoint orbits, J. Nonlinear Sci., 28 (2018), 91-145.  doi: 10.1007/s00332-017-9404-3.

[4]

A. Arnaudon, N. Ganaba and D. Holm, The stochastic energy – Casimir method, Comptes Rendus Mécanique, 346 (2018), 279-290.  doi: 10.1016/j.crme.2018.01.003.

[5]

A. Bloch, D. Chang, N. Leonard and J. E. Marsden, Controlled Lagrangians and the stabilization of mechanical systems II: Potential shaping, IEEE Trans. Automat. Control, 46 (2001), 1556-1571.  doi: 10.1109/9.956051.

[6]

A. Bloch, N. Leonard and J. Marsden, Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem, IEEE Trans. Automat. Control, 45 (2000), 2253-2270.  doi: 10.1109/9.895562.

[7]

A. Bloch, N. Leonard and J. Marsden, Controlled Lagrangians and the stabilization of Euler-Poincaré mechanical systems, Internat. J. Robust Nonlinear Control, 11 (2001), 191-214.  doi: 10.1002/rnc.572.

[8]

A. Bloch and N. Leonard, Symmetries, conservation laws, and control, in Geometry, Mechanics and Dynamics, Springer-Verlag, New York, 2002. doi: 10.1007/b97525.

[9]

A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and G. Sánchez de Alvarez, Stabilization of rigid body dynamics by internal and external torques, Automatica J. IFAC, 28 (1992), 745-756.  doi: 10.1016/0005-1098(92)90034-D.

[10]

A. M. Bloch, J. E. Marsden and G. Sánchez de Alvarez, Feedback stabilization of relative equilibria for mechanical systems with symmetry, in Current and Future Directions in Applied Mathematics, Birkhäuser, Boston, MA, 1997.

[11]

D. E. Chang, A. M. Bloch, N. E. Leonard, J. E. Marsden and C. A. Woolsey, The equivalence of controlled Lagrangian and controlled Hamiltonian systems, ESAIM: Control, Optimisation and Calculus of Variations, 8 (2002), 393-422. 

[12]

D. Crisan, F. Flandoli and D. D. Holm, Solution properties of a 3D stochastic Euler fluid equation, J. Nonlinear Sci., 29 (2019), 813-870.  doi: 10.1007/s00332-018-9506-6.

[13]

T. Drivas, D. Holm and J.-M. Leahy, Lagrangian averaged stochastic advection by Lie transport for fluids, J. Stat. Phys., 179 (2020), 1304-1342.  doi: 10.1007/s10955-020-02493-4.

[14]

F. Gay-Balmaz, C. Tronci and C. Vizman, Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups, J. Geom. Mech., 5 (2013), 39-84.  doi: 10.3934/jgm.2013.5.39.

[15]

F. Gay-Balmaz and T. Ratiu, Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids, J. Symplectic Geom., 6 (2008), 189-237.  doi: 10.4310/JSG.2008.v6.n2.a4.

[16]

J. Gibbons, D. D. Holm and B. Kupershmidt, The Hamiltonian structure of classical chromohydrodynamics, Phys. D, 6 (1982/83), 179-194.  doi: 10.1016/0167-2789(83)90004-0.

[17]

S. Hochgerner, Symmetry reduction of Brownian motion and quantum Calogero-Moser models, Stoch. Dyn., 13 (2013), 1250007, 31 pp. doi: 10.1142/S0219493712500074.

[18]

S. Hochgerner and T. Ratiu, The geometry of non-holonomic diffusion, J. Eur. Math. Soc., 17 (2015), 273-319.  doi: 10.4171/JEMS/504.

[19]

S. Hochgerner, A Hamiltonian mean field system for the Navier–Stokes equation, Proc. A, 474 (2018), 20180178, 20 pp. doi: 10.1098/rspa.2018.0178.

[20]

S. Hochgerner, Feedback control of charged ideal fluids, preprint, 2020, arXiv: 1905.04778.

[21]

D. Holm, J. Marsden, T. Ratiu and A. Weinstein, Nonlinear stability of fluid and plasma equilibria, Phys. Rep., 123 (1985), 1-116.  doi: 10.1016/0370-1573(85)90028-6.

[22]

D. Holm, Variational principles for stochastic fluid dynamics, Proc. A, 471 (2015), 20140963, 19 pp. doi: 10.1098/rspa.2014.0963.

[23]

P. S. Krishnaprasad, Lie-Poisson structures, dual-spinspacecraft and asymptotic stability, Nonlinear Anal., 9 (1985), 1011-1035.  doi: 10.1016/0362-546X(85)90083-5.

[24]

J.-A. Lazaro-Cami and J.-P. Ortega, Stochastic Hamiltonian dynamical systems, Rep. Math. Phys., 61 (2008), 65-122.  doi: 10.1016/S0034-4877(08)80003-1.

[25]

J. E. Marsden, Park City lectures on mechanics, dynamics, and symmetry, in Symplectic Geometry and Topology, IAS/Park City Mathematics Series, No. 7, American Mathematical Society, Providence, RI, 1999,335–430. doi: 10.1090/pcms/007/09.

[26]

J. E. Marsden, A. Weinstein, T. Ratiu, R. Schmid and R. Spencer, Hamiltonian systems with symmetry, coadjoint orbits and plasma physics, Proc. IUTAM-IS1MM Symposium on Modern Developments in Analytical Mechanics (Torino, 1982), Atti Acad. Sci. Torino Cl. Sci. Fis. Math. Natur., 117, (1983), 289–340.

[27]

P. Michor, Topics in Differential Geometry, Graduate Studies in Mathematics, vol. 93, Amer. Math. Soc., Providence, RI, 2008. doi: 10.1090/gsm/093.

[28]

P. Michor, Some geometric evolution equations arising as geodesic equations on groups of diffeomorphism, including the Hamiltonian approach, in Phase space analysis of Partial Differential Equations, Birkhäuser Boston, Boston, 2006,133–215. doi: 10.1007/978-0-8176-4521-2_11.

[29]

M. Puiggalí and A. Bloch, An extension to the theory of controlled Lagrangians using the Helmholtz conditions, J. Nonlinear Sci., 29 (2019), 345-376.  doi: 10.1007/s00332-018-9490-x.

[30]

A. Weinstein, A universal phase space for particles in Yang-Mills fields, Lett. Math. Phys., 2 (1977/78), 417-420.  doi: 10.1007/BF00400169.

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