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Remarks on linear-quadratic dissipative control systems
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 |
References:
[1] |
H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Massob, Paris, 1987. |
[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. |
[3] |
W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics, 220, Springer-Verlag, Berlin, Heidelberg, New York, 1971. |
[4] |
R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969. |
[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. |
[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. |
[7] |
R. Fabbri, R. Johnson, S. Novo, C. Núñez and R. Obaya, Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control,, in preparation., ().
|
[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. |
[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. |
[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. |
[11] |
R. Johnson, Ergodic theory and linear differential equations, J. Differential Equations, 28 (1978), 23-34.
doi: 10.1016/0022-0396(78)90077-3. |
[12] |
R. Johnson, The recurrent Hill's equation, J. Differential Equations, 46 (1982), 165-193.
doi: 10.1016/0022-0396(82)90114-0. |
[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. |
[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. |
[15] |
T. Kato, Perturbation Theory for Linear Operators, Corrected printing of the second edition, Springer-Verlag, Berlin, Heidelberg, 1995. |
[16] |
V. B. Lidskiĭ, Oscillation theorems for canonical systems of differential equations, Dokl. Akad. Nank. SSSR, 102 (1955), 877-880. |
[17] |
R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin, Heidelberg, New York 1987.
doi: 10.1007/978-3-642-70335-5. |
[18] |
Y. Matsushima, Differentiable Manifolds, Marcel Dekker, New York 1972. |
[19] |
A. Mazurov and P. Pakshin, Stochastic dissipativity with risk-sensitive storage function and related control problems, ICIC Express Letters, 3 (2009), 53-60. |
[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. |
[21] |
V. I. Oseledets, A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 179-210. |
[22] |
D. Ruelle, Ergodic theory of differentiable dynamical systems, Publ. I.H.E.S., 50 (1979), 27-58. |
[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. |
[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. |
[25] |
R. E. Vinograd, A problem suggested by N. P. Erugin, Diff. Urav., 11 (1975), 632-638. |
[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. |
[27] |
V. A. Yakubovich, Oscillatory properties of the solutions of canonical equations, Amer. Math. Soc. Transl. Ser., 42 (1964), 247-288. |
[28] |
V. Yakubovich, Contribution to the abstract theory of optimal control I (in Russian), Sib. Mat. Zh., 18 (1977), 685-707. |
[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. |
[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. |
[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. |
show all references
References:
[1] |
H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Massob, Paris, 1987. |
[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. |
[3] |
W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics, 220, Springer-Verlag, Berlin, Heidelberg, New York, 1971. |
[4] |
R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969. |
[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. |
[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. |
[7] |
R. Fabbri, R. Johnson, S. Novo, C. Núñez and R. Obaya, Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control,, in preparation., ().
|
[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. |
[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. |
[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. |
[11] |
R. Johnson, Ergodic theory and linear differential equations, J. Differential Equations, 28 (1978), 23-34.
doi: 10.1016/0022-0396(78)90077-3. |
[12] |
R. Johnson, The recurrent Hill's equation, J. Differential Equations, 46 (1982), 165-193.
doi: 10.1016/0022-0396(82)90114-0. |
[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. |
[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. |
[15] |
T. Kato, Perturbation Theory for Linear Operators, Corrected printing of the second edition, Springer-Verlag, Berlin, Heidelberg, 1995. |
[16] |
V. B. Lidskiĭ, Oscillation theorems for canonical systems of differential equations, Dokl. Akad. Nank. SSSR, 102 (1955), 877-880. |
[17] |
R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin, Heidelberg, New York 1987.
doi: 10.1007/978-3-642-70335-5. |
[18] |
Y. Matsushima, Differentiable Manifolds, Marcel Dekker, New York 1972. |
[19] |
A. Mazurov and P. Pakshin, Stochastic dissipativity with risk-sensitive storage function and related control problems, ICIC Express Letters, 3 (2009), 53-60. |
[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. |
[21] |
V. I. Oseledets, A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 179-210. |
[22] |
D. Ruelle, Ergodic theory of differentiable dynamical systems, Publ. I.H.E.S., 50 (1979), 27-58. |
[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. |
[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. |
[25] |
R. E. Vinograd, A problem suggested by N. P. Erugin, Diff. Urav., 11 (1975), 632-638. |
[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. |
[27] |
V. A. Yakubovich, Oscillatory properties of the solutions of canonical equations, Amer. Math. Soc. Transl. Ser., 42 (1964), 247-288. |
[28] |
V. Yakubovich, Contribution to the abstract theory of optimal control I (in Russian), Sib. Mat. Zh., 18 (1977), 685-707. |
[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. |
[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. |
[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. |
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