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Nonautonomous Linear-Quadratic Dissipative Control Processes Without Uniform Null Controllability

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Abstract

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.

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References

  1. Coppel, W.A.: Disconjugacy. Lecture Notes in Mathematics, vol. 220. Springer, Berlin (1971)

    MATH  Google Scholar 

  2. Cornfeld, I.P., Fomin, S.V., Sinai, Ya.G.: Ergodic Theory. Springer, Berlin (1982)

  3. Ellis, R.: Lectures on Topological Dynamics. Benjamin, New York (1969)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  9. Hill, D.J., Moylan, P.J.: Dissipative dynamical systems: basic input–output and state properties. J. Franklin Inst. 309(5), 327–357 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  10. Johnson, R.: \(m\)-Functions and Floquet exponents for linear differential systems. Ann. Mat. Pura Appl. 147, 211–248 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  11. Johnson, R., Nerurkar, M.: Exponential dichotomy and rotation number for linear Hamiltonian systems. J. Differ. Equ. 108, 201–216 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  13. Johnson, R., Núñez, C.: Remarks on linear-quadratic dissipative control systems. Discret. Contin. Dyn. Sys. B 20(3), 889–914 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Johnson, R., Novo, S., Obaya, R.: An ergodic and topological approach to disconjugate linear Hamiltonian systems. Ill. J. Math. 45, 1045–1079 (2001)

    MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kato, T.: Perturbation Theory for Linear Operators, Corrected Printing of the Second Edition. Springer, Berlin (1995)

    Google Scholar 

  18. Kratz, W.: Quadratic Functionals in Variational Analysis and Control Theory. Mathematical Topics, vol. 6. Akademie, Berlin (1995)

    MATH  Google Scholar 

  19. Kratz, W.: Definiteness of quadratic functionals. Analysis (Munich) 23(2), 163–183 (2003)

    MathSciNet  MATH  Google Scholar 

  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.)

  21. Matsushima, Y.: Differentiable Manifolds. Marcel Dekker, New York (1972)

    MATH  Google Scholar 

  22. Mishchenko, A.S., Shatalov, V.E., Sternin, B.Yu.: Lagrangian Manifolds and the Maslov Operator. Springer, Berlin (1990)

  23. Novo, S., Núñez, C., Obaya, R.: Ergodic properties and rotation number for linear Hamiltonian systems. J. Differ. Equ. 148, 148–185 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Polushin, I.G.: Stability results for quasidissipative systems. In: Proceedings of 3rd European Control Conference ECC’95, pp. 681–686 (1995)

  25. Reid, W.T.: Principal solutions of nonoscillatory linear differential systems. J. Math. Anal. Appl. 9, 397–423 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  26. Reid, W.T.: Sturmian Theory for Ordinary Differential Equations. Applied Mathematical Sciences, vol. 31. Springer, New York (1980)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  29. Savkin, A.V., Petersen, I.R.: Structured dissipativeness and absolute stability of nonlinear systems. Intern. J. Control 62(2), 443–460 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  31. Wahrheit, M.: Eigenvalue problems and oscillation of linear Hamiltonian systems. Int. J. Differ. Equ. 2(2), 221–244 (2007)

    MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  33. Yakubovich, V.A.: Oscillatory properties of the solutions of canonical equations. Am. Math. Soc. Transl. Ser. 2(42), 247–288 (1964)

    MATH  Google Scholar 

  34. Yakubovich, V.A., Starzhinskii, V.M.: Linear Differential Equations with Periodic Coefficients. Wiley, New York (1975)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

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

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Firenze, Via di Santa Marta 3, 50139, Firenze, Italy

    Russell Johnson

  2. Departamento de Matemática Aplicada, Universidad de Valladolid, Paseo del Cauce 59, 47011, Valladolid, Spain

    Sylvia Novo, Carmen Núñez & Rafael Obaya

Authors
  1. Russell Johnson
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  2. Sylvia Novo
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  3. Carmen Núñez
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  4. Rafael Obaya
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Correspondence to Sylvia Novo.

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Johnson, R., Novo, S., Núñez, C. et al. Nonautonomous Linear-Quadratic Dissipative Control Processes Without Uniform Null Controllability. J Dyn Diff Equat 29, 355–383 (2017). https://doi.org/10.1007/s10884-015-9495-1

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  • DOI: https://doi.org/10.1007/s10884-015-9495-1

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