Nonautonomous Linear-Quadratic Dissipative Control Processes Without Uniform Null Controllability

  • Russell Johnson
  • Sylvia NovoEmail author
  • Carmen Núñez
  • Rafael Obaya


In this paper the dissipativity of a family of linear-quadratic control processes is studied. The application of the Pontryagin Maximum Principle to this problem gives rise to a family of linear Hamiltonian systems for which the existence of an exponential dichotomy is assumed, but no condition of controllability is imposed. As a consequence, some of the systems of this family could be abnormal. Sufficient conditions for the dissipativity of the processes are provided assuming the existence of global positive solutions of the Riccati equation induced by the family of linear Hamiltonian systems or by a convenient disconjugate perturbation of it.


Nonautonomous linear Hamiltonian systems Linear-quadratic control system Dissipativity Abnormal system Rotation number Proper focal point 



Partly supported by MIUR (Italy), by MEC (Spain) under Project MTM2012-30860, and by JCyL (Spain) under Project VA118A12-1.


  1. 1.
    Coppel, W.A.: Disconjugacy. Lecture Notes in Mathematics, vol. 220. Springer, Berlin (1971)zbMATHGoogle Scholar
  2. 2.
    Cornfeld, I.P., Fomin, S.V., Sinai, Ya.G.: Ergodic Theory. Springer, Berlin (1982)Google Scholar
  3. 3.
    Ellis, R.: Lectures on Topological Dynamics. Benjamin, New York (1969)zbMATHGoogle Scholar
  4. 4.
    Fabbri, R., Johnson, R., Núñez, C.: On the Yakubovich frequency theorem for linear non-autonomous control processes. Discret. Contin. Dyn. Syst. Ser. A 9(3), 677–704 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fabbri, R., Johnson, R., Núñez, C.: Disconjugacy and the rotation number for linear, non-autonomous Hamiltonian systems. Ann. Mat. Pura Appl. 185, S3–S21 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fabbri, R., Johnson, R., Novo, S., Núñez, C.: Some remarks concerning weakly disconjugate linear Hamiltonian systems. J. Math. Anal. Appl. 380, 853–864 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fabbri, R., Johnson, R., Novo, S., Núñez, C.: On linear-quadratic dissipative control processes with time-varying coefficients. Discret. Contin. Dyn. Syst. Ser. A 33(1), 193–210 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hill, D.J.: Dissipative nonlinear systems: basic properies and stability analysis. In: Proceedings of the 31st IEEE Conference on Decision and Control, vol. 4, pp. 3259–3264 (1992)Google Scholar
  9. 9.
    Hill, D.J., Moylan, P.J.: Dissipative dynamical systems: basic input–output and state properties. J. Franklin Inst. 309(5), 327–357 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Johnson, R.: \(m\)-Functions and Floquet exponents for linear differential systems. Ann. Mat. Pura Appl. 147, 211–248 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Johnson, R., Nerurkar, M.: Exponential dichotomy and rotation number for linear Hamiltonian systems. J. Differ. Equ. 108, 201–216 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Johnson, R., Nerurkar, M.: Controllability, Stabilization, and the Regulator Problem for Random Differential Systems. Memoirs of the American Mathematical Society, vol. 646. American Mathematical Society, Providence (1998)zbMATHGoogle Scholar
  13. 13.
    Johnson, R., Núñez, C.: Remarks on linear-quadratic dissipative control systems. Discret. Contin. Dyn. Sys. B 20(3), 889–914 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Johnson, R., Novo, S., Obaya, R.: An ergodic and topological approach to disconjugate linear Hamiltonian systems. Ill. J. Math. 45, 1045–1079 (2001)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Johnson, R., Novo, S., Núñez, C., Obaya, R.: Uniform weak disconjugacy and principal solutions for linear Hamiltonian systems, In: Recent Advances in Delay Differential and Difference Equations, Springer Proceedings in Mathematics & Statistics, vol. 94, pp. 131–159. Springer, Berlin (2014)Google Scholar
  16. 16.
    Johnson, R., Núñez, C., Obaya, R.: Dynamical methods for linear Hamiltonian systems with applications to control processes. J. Dyn. Differ. Equ. 25(3), 679–713 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kato, T.: Perturbation Theory for Linear Operators, Corrected Printing of the Second Edition. Springer, Berlin (1995)Google Scholar
  18. 18.
    Kratz, W.: Quadratic Functionals in Variational Analysis and Control Theory. Mathematical Topics, vol. 6. Akademie, Berlin (1995)zbMATHGoogle Scholar
  19. 19.
    Kratz, W.: Definiteness of quadratic functionals. Analysis (Munich) 23(2), 163–183 (2003)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Lidskiĭ, V.B.: Oscillation theorems for canonical systems of differential equations, Dokl. Akad. Nank. SSSR 102 (1955), 877–880. (English translation. In: NASA Technical Translation, P-14, 696.)Google Scholar
  21. 21.
    Matsushima, Y.: Differentiable Manifolds. Marcel Dekker, New York (1972)zbMATHGoogle Scholar
  22. 22.
    Mishchenko, A.S., Shatalov, V.E., Sternin, B.Yu.: Lagrangian Manifolds and the Maslov Operator. Springer, Berlin (1990)Google Scholar
  23. 23.
    Novo, S., Núñez, C., Obaya, R.: Ergodic properties and rotation number for linear Hamiltonian systems. J. Differ. Equ. 148, 148–185 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Polushin, I.G.: Stability results for quasidissipative systems. In: Proceedings of 3rd European Control Conference ECC’95, pp. 681–686 (1995)Google Scholar
  25. 25.
    Reid, W.T.: Principal solutions of nonoscillatory linear differential systems. J. Math. Anal. Appl. 9, 397–423 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Reid, W.T.: Sturmian Theory for Ordinary Differential Equations. Applied Mathematical Sciences, vol. 31. Springer, New York (1980)zbMATHGoogle Scholar
  27. 27.
    Šepitka, P., Šimon Hilscher, R.: Minimal principal solution at infinity for nonoscillatory linear Hamiltonian systems. J. Dynam. Differ. Equ. 26(1), 57–91 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Šepitka, P., Šimon Hilscher, R.: Principal solutions at infinity of given ranks for nonoscillatory linear Hamiltonian systems. J. Dynam. Differ. Equ. 27(1), 137–175 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Savkin, A.V., Petersen, I.R.: Structured dissipativeness and absolute stability of nonlinear systems. Intern. J. Control 62(2), 443–460 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Trentelman, H.L., Willems, J.C.: Storage functions for dissipative linear systems are quadratic state functions. In: Proceedings of 36th IEEE Conference on Decision and Control, pp. 42–49 (1997)Google Scholar
  31. 31.
    Wahrheit, M.: Eigenvalue problems and oscillation of linear Hamiltonian systems. Int. J. Differ. Equ. 2(2), 221–244 (2007)MathSciNetGoogle Scholar
  32. 32.
    Willems, J.C.: Dissipative dynamical systems. Part I: General theory. Part II: Linear systems with quadratic supply rates. Arch. Ration. Mech. Anal. 45, 321–393 (1972)CrossRefGoogle Scholar
  33. 33.
    Yakubovich, V.A.: Oscillatory properties of the solutions of canonical equations. Am. Math. Soc. Transl. Ser. 2(42), 247–288 (1964)zbMATHGoogle Scholar
  34. 34.
    Yakubovich, V.A., Starzhinskii, V.M.: Linear Differential Equations with Periodic Coefficients. Wiley, New York (1975)Google Scholar
  35. 35.
    Yakubovich, V.A., Fradkov, A.L., Hill, D.J., Proskurnikov, A.V.: Dissipativity of \(T\)-periodic linear systems. IEEE Trans. Automat. Control 52(6), 1039–1047 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità di FirenzeFirenzeItaly
  2. 2.Departamento de Matemática AplicadaUniversidad de ValladolidValladolidSpain

Personalised recommendations