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May  2015, 20(3): 889-914. doi: 10.3934/dcdsb.2015.20.889

Remarks on linear-quadratic dissipative control systems

Russell Johnson 1, and  Carmen Núñez 2,

1. 

Dipartimento di Matematica e Informatica, Università di Firenze, Via di Santa Marta 3, 50139 Firenze, Italy
2. 

Departamento de Matemática Aplicada, E. Ingenierías Industriales, Universidad de Valladolid, Paseo del Cauce 59, 47011 Valladolid

Received  August 2013 Revised  May 2014 Published  January 2015

We study the concept of dissipativity in the sense of Willems for nonautonomous linear-quadratic (LQ) control systems. A nonautonomous system of Hamiltonian ODEs is associated with such an LQ system by way of the Pontryagin Maximum Principle. We relate the concepts of exponential dichotomy and weak disconjugacy for this Hamiltonian ODE to that of dissipativity for the LQ system.
Keywords: Linear-quadratic control systems, Hamiltonian systems, dissipativity, weak disconjugacy..
Mathematics Subject Classification: 37B55, 34C10, 34L0.
Citation: Russell Johnson, Carmen Núñez. Remarks on linear-quadratic dissipative control systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 889-914. doi: 10.3934/dcdsb.2015.20.889
References:
[1]

H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Massob, Paris, 1987. Google Scholar

[2]

N. D. Cong, A generic bounded linear cocycle has simple Lyapunov spectrum, Ergod. Th. Dynam. Sys., 25 (2005), 1775-1797. doi: 10.1017/S0143385705000337.  Google Scholar

[3]

W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics, 220, Springer-Verlag, Berlin, Heidelberg, New York, 1971.  Google Scholar

[4]

R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.  Google Scholar

[5]

R. Fabbri, R. Johnson, S. Novo and C. Núñez, Some remarks concerning weakly disconjugate linear Hamiltonian systems, J. Math. Anal. Appl., 380 (2011), 853-864. doi: 10.1016/j.jmaa.2010.11.036.  Google Scholar

[6]

R. Fabbri, R. Johnson, S. Novo and C. Núñez, On linear-quadratic dissipative control processes with time-varying coefficients, Discrete Contin. Dynam. Systems, Ser. A, 33 (2013), 193-210. doi: 10.3934/dcds.2013.33.193.  Google Scholar

[7]

R. Fabbri, R. Johnson, S. Novo, C. Núñez and R. Obaya, Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control,, in preparation., ().   Google Scholar

[8]

R. Fabbri, R. Johnson and C. Núñez, On the Yakubovich Frequency Theorem for linear non-autonomous control processes, Discrete Contin. Dynam. Systems, Ser. A, 9 (2003), 677-704. doi: 10.3934/dcds.2003.9.677.  Google Scholar

[9]

R. Fabbri, R. Johnson and C. Núñez, Disconjugacy and the rotation number for linear, nonautonomus linear Hamiltonian systems, Ann. Mat. Pura App., 185 (2006), S3-S21. Google Scholar

[10]

R. Johnson and M. Nerurkar, Controllability, stabilization, and the regulator problem for random differential systems, Mem. Amer. Math. Soc., 136 (1998), viii+48 pp. doi: 10.1090/memo/0646.  Google Scholar

[11]

R. Johnson, Ergodic theory and linear differential equations, J. Differential Equations, 28 (1978), 23-34. doi: 10.1016/0022-0396(78)90077-3.  Google Scholar

[12]

R. Johnson, The recurrent Hill's equation, J. Differential Equations, 46 (1982), 165-193. doi: 10.1016/0022-0396(82)90114-0.  Google Scholar

[13]

R. Johnson, S. Novo and R. Obaya, An ergodic and topological approach to disconjugate linear Hamiltonian systems, Illinois J. Math., 45 (2001), 803-822.  Google Scholar

[14]

R. Johnson, C. Núñez and R. Obaya, Dynamical methods for linear Hamiltonian systems with applications to control processes, J. Dynam. Differential Equations, 25 (2013), 679-713. doi: 10.1007/s10884-013-9300-y.  Google Scholar

[15]

T. Kato, Perturbation Theory for Linear Operators, Corrected printing of the second edition, Springer-Verlag, Berlin, Heidelberg, 1995.  Google Scholar

[16]

V. B. Lidskiĭ, Oscillation theorems for canonical systems of differential equations, Dokl. Akad. Nank. SSSR, 102 (1955), 877-880.  Google Scholar

[17]

R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin, Heidelberg, New York 1987. doi: 10.1007/978-3-642-70335-5.  Google Scholar

[18]

Y. Matsushima, Differentiable Manifolds, Marcel Dekker, New York 1972.  Google Scholar

[19]

A. Mazurov and P. Pakshin, Stochastic dissipativity with risk-sensitive storage function and related control problems, ICIC Express Letters, 3 (2009), 53-60. Google Scholar

[20]

V. M. Millionščikov, Proof of the existence of irregular systems of linear differential equations with almost periodic coefficients, Diff. Urav., 4 (1968), 391-396.  Google Scholar

[21]

V. I. Oseledets, A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 179-210.  Google Scholar

[22]

D. Ruelle, Ergodic theory of differentiable dynamical systems, Publ. I.H.E.S., 50 (1979), 27-58.  Google Scholar

[23]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8.  Google Scholar

[24]

H. L. Trentelman and J. C. Willems, Dissipative linear differential systems and the state-space H-infinity control problem, Int. Jour. Robust Nonlin. Control, 10 (2000), 1039-1057. doi: 10.1002/1099-1239(200009/10)10:11/12<1039::AID-RNC538>3.0.CO;2-5.  Google Scholar

[25]

R. E. Vinograd, A problem suggested by N. P. Erugin, Diff. Urav., 11 (1975), 632-638.  Google Scholar

[26]

J. C. Willems, Dissipative dynamical systems. Part I: General theory. Part II: Linear systems with quadratic supply rates, Arch. Rational Mech. Anal., 45 (1972), 321-351. doi: 10.1007/BF00276493.  Google Scholar

[27]

V. A. Yakubovich, Oscillatory properties of the solutions of canonical equations, Amer. Math. Soc. Transl. Ser., 42 (1964), 247-288. Google Scholar

[28]

V. Yakubovich, Contribution to the abstract theory of optimal control I (in Russian), Sib. Mat. Zh., 18 (1977), 685-707. Google Scholar

[29]

V. A. Yakubovich, Linear-quadratic optimization problem and the frequency theorem for periodic systems I (in Russian), Sib. Mat. Zh., 27 (1986), 181-200.  Google Scholar

[30]

V. A. Yakubovich, Linear-quadratic optimization problem and the frequency theorem for periodic systems II, Siberian Math. J., 31 (1990), 1027-1039. doi: 10.1007/BF00970068.  Google Scholar

[31]

V. A. Yakubovich, A. L. Fradkov, D. J. Hill and A. V. Proskurnikov, Dissipativity of $T$-periodic linear systems, IEEE Trans. Automat. Control, 52 (2007), 1039-1047. doi: 10.1109/TAC.2007.899013.  Google Scholar

show all references

References:
[1]

H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Massob, Paris, 1987. Google Scholar

[2]

N. D. Cong, A generic bounded linear cocycle has simple Lyapunov spectrum, Ergod. Th. Dynam. Sys., 25 (2005), 1775-1797. doi: 10.1017/S0143385705000337.  Google Scholar

[3]

W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics, 220, Springer-Verlag, Berlin, Heidelberg, New York, 1971.  Google Scholar

[4]

R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.  Google Scholar

[5]

R. Fabbri, R. Johnson, S. Novo and C. Núñez, Some remarks concerning weakly disconjugate linear Hamiltonian systems, J. Math. Anal. Appl., 380 (2011), 853-864. doi: 10.1016/j.jmaa.2010.11.036.  Google Scholar

[6]

R. Fabbri, R. Johnson, S. Novo and C. Núñez, On linear-quadratic dissipative control processes with time-varying coefficients, Discrete Contin. Dynam. Systems, Ser. A, 33 (2013), 193-210. doi: 10.3934/dcds.2013.33.193.  Google Scholar

[7]

R. Fabbri, R. Johnson, S. Novo, C. Núñez and R. Obaya, Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control,, in preparation., ().   Google Scholar

[8]

R. Fabbri, R. Johnson and C. Núñez, On the Yakubovich Frequency Theorem for linear non-autonomous control processes, Discrete Contin. Dynam. Systems, Ser. A, 9 (2003), 677-704. doi: 10.3934/dcds.2003.9.677.  Google Scholar

[9]

R. Fabbri, R. Johnson and C. Núñez, Disconjugacy and the rotation number for linear, nonautonomus linear Hamiltonian systems, Ann. Mat. Pura App., 185 (2006), S3-S21. Google Scholar

[10]

R. Johnson and M. Nerurkar, Controllability, stabilization, and the regulator problem for random differential systems, Mem. Amer. Math. Soc., 136 (1998), viii+48 pp. doi: 10.1090/memo/0646.  Google Scholar

[11]

R. Johnson, Ergodic theory and linear differential equations, J. Differential Equations, 28 (1978), 23-34. doi: 10.1016/0022-0396(78)90077-3.  Google Scholar

[12]

R. Johnson, The recurrent Hill's equation, J. Differential Equations, 46 (1982), 165-193. doi: 10.1016/0022-0396(82)90114-0.  Google Scholar

[13]

R. Johnson, S. Novo and R. Obaya, An ergodic and topological approach to disconjugate linear Hamiltonian systems, Illinois J. Math., 45 (2001), 803-822.  Google Scholar

[14]

R. Johnson, C. Núñez and R. Obaya, Dynamical methods for linear Hamiltonian systems with applications to control processes, J. Dynam. Differential Equations, 25 (2013), 679-713. doi: 10.1007/s10884-013-9300-y.  Google Scholar

[15]

T. Kato, Perturbation Theory for Linear Operators, Corrected printing of the second edition, Springer-Verlag, Berlin, Heidelberg, 1995.  Google Scholar

[16]

V. B. Lidskiĭ, Oscillation theorems for canonical systems of differential equations, Dokl. Akad. Nank. SSSR, 102 (1955), 877-880.  Google Scholar

[17]

R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin, Heidelberg, New York 1987. doi: 10.1007/978-3-642-70335-5.  Google Scholar

[18]

Y. Matsushima, Differentiable Manifolds, Marcel Dekker, New York 1972.  Google Scholar

[19]

A. Mazurov and P. Pakshin, Stochastic dissipativity with risk-sensitive storage function and related control problems, ICIC Express Letters, 3 (2009), 53-60. Google Scholar

[20]

V. M. Millionščikov, Proof of the existence of irregular systems of linear differential equations with almost periodic coefficients, Diff. Urav., 4 (1968), 391-396.  Google Scholar

[21]

V. I. Oseledets, A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 179-210.  Google Scholar

[22]

D. Ruelle, Ergodic theory of differentiable dynamical systems, Publ. I.H.E.S., 50 (1979), 27-58.  Google Scholar

[23]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8.  Google Scholar

[24]

H. L. Trentelman and J. C. Willems, Dissipative linear differential systems and the state-space H-infinity control problem, Int. Jour. Robust Nonlin. Control, 10 (2000), 1039-1057. doi: 10.1002/1099-1239(200009/10)10:11/12<1039::AID-RNC538>3.0.CO;2-5.  Google Scholar

[25]

R. E. Vinograd, A problem suggested by N. P. Erugin, Diff. Urav., 11 (1975), 632-638.  Google Scholar

[26]

J. C. Willems, Dissipative dynamical systems. Part I: General theory. Part II: Linear systems with quadratic supply rates, Arch. Rational Mech. Anal., 45 (1972), 321-351. doi: 10.1007/BF00276493.  Google Scholar

[27]

V. A. Yakubovich, Oscillatory properties of the solutions of canonical equations, Amer. Math. Soc. Transl. Ser., 42 (1964), 247-288. Google Scholar

[28]

V. Yakubovich, Contribution to the abstract theory of optimal control I (in Russian), Sib. Mat. Zh., 18 (1977), 685-707. Google Scholar

[29]

V. A. Yakubovich, Linear-quadratic optimization problem and the frequency theorem for periodic systems I (in Russian), Sib. Mat. Zh., 27 (1986), 181-200.  Google Scholar

[30]

V. A. Yakubovich, Linear-quadratic optimization problem and the frequency theorem for periodic systems II, Siberian Math. J., 31 (1990), 1027-1039. doi: 10.1007/BF00970068.  Google Scholar

[31]

V. A. Yakubovich, A. L. Fradkov, D. J. Hill and A. V. Proskurnikov, Dissipativity of $T$-periodic linear systems, IEEE Trans. Automat. Control, 52 (2007), 1039-1047. doi: 10.1109/TAC.2007.899013.  Google Scholar

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