About the Definition of Port Variables for Contact Hamiltonian Systems

  • Bernhard Maschke
  • Arjan van der Schaft
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10589)


Extending the formulation of reversible thermodynamical transformations to the formulation of irreversible transformations of open thermodynamical systems different classes of nonlinear control systems has been defined in terms of control Hamiltonian systems defined on a contact manifold. In this paper we discuss the relation between the definition of variational control contact systems and the input-output contact systems. We have first given an expression of the variational control contact systems in terms of a nonlinear control systems. Secondly we have shown that the conservative input-output contact systems are a subclass of the contact variational systems with integrable output dynamics.


Open irreversible thermodynamic systems Nonlinear control systems Hamiltonian systems on contact manifolds 


  1. 1.
    Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989). 2 edition, ISBN 0-387-96890-3CrossRefGoogle Scholar
  2. 2.
    Bravetti, A., Lopez-Monsalvo, C.S., Nettel, F.: Contact symmetries and Hamiltonian thermodynamics. Ann. Phys. 361, 377–400 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Brockett, R.W.: Geometric control theory, volume 7 of lie groups: history, frontiers and applications. In: Martin, C., Herman, R. (eds.) Control Theory and Analytical Mechanics, pp. 1–46. Math. Sci. Press, Brookline (1977)Google Scholar
  4. 4.
    Eberard, D., Maschke, B.M., van der Schaft, A.J.: An extension of pseudo-Hamiltonian systems to the thermodynamic space: towards a geometry of non-equilibrium thermodynamics. Rep. Math. Phys. 60(2), 175–198 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Favache, A., Dochain, D., Maschke, B.M.: An entropy-based formulation of irreversible processes based on contact structures. Chem. Eng. Sci. 65, 5204–5216 (2010)CrossRefGoogle Scholar
  6. 6.
    Favache, A., Dos Santos Martins, V., Maschke, B., Dochain, D.: Some properties of conservative control systems. IEEE trans. Autom. Control 54(10), 2341–2351 (2009)CrossRefzbMATHGoogle Scholar
  7. 7.
    Grmela, M.: Reciprocity relations in Thermodynamics. Phys. A 309, 304–328 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Libermann, P., Marle, C.-M.: Symplectic Geometry and Analytical Mechanics. D. Reidel Publishing Company, Dordrecht (1987)CrossRefzbMATHGoogle Scholar
  9. 9.
    Maschke, B.M., van der Schaft, A.J.: Port controlled Hamiltonian systems: modeling origins and system theoretic properties. In: Proceedings of 3rd International IFAC Conference on Nonlinear Systems’ Theory and Control, NOLCOS 1992, pp. 282–288, Bordeaux, June 1992Google Scholar
  10. 10.
    Merker, J., Krüger, M.: On a variational principle in Thermodynamics. Continuum Mech. Thermodyn. 25(6), 779–793 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Mrugała, R.: On a special family of thermodynamic processes and their invariants. Rep. Math. Phys. 46(3), 461–468 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Nijmeijer, H., van der Schaft, A.J.: Nonlinear Dynamical Control Systems. Springer, New York (1990). 1st edition, ISBN: 0-387-97234-XCrossRefzbMATHGoogle Scholar
  13. 13.
    Ramirez, H., Maschke, B., Sbarbaro, D.: Feedback equivalence of input-output contact systems. Syst. Control Lett. 62(6), 475–481 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Ramirez, H., Maschke, B., Sbarbaro, D.: Irreversible port-Hamiltonian systems: a general formulation of irreversible processes with application to the CSTR. Chem. Eng. Sci. 89, 223–234 (2013)CrossRefGoogle Scholar
  15. 15.
    Ramirez, H., Maschke, B., Sbarbaro, D.: Partial stabilization of input-output contact systems on a Legendre submanifold. IEEE Trans. Autom. Control 62(3), 1431–1437 (2017)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    van der Schaft, A.: Three decades of mathematical system theory. System Theory and Mechanics. LNCIS, vol. 135, pp. 426–452. Springer, Heidelberg (1989). doi: 10.1007/BFb0008472 Google Scholar
  17. 17.
    van der Schaft, A., Crouch, P.E.: Hamiltonian and self-adjoint control systems. Syst. Control Lett. 8, 289–295 (1987)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Univ Lyon, CNRS, LAGEP UMR 5007Université Claude Bernard Lyon 1VilleurbanneFrance
  2. 2.Johann Bernoulli Institute for Mathematics and Computer ScienceUniversity of GroningenGroningenThe Netherlands

Personalised recommendations