# American Institute of Mathematical Sciences

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

## 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

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.
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, (1987). [2] N. D. Cong, A generic bounded linear cocycle has simple Lyapunov spectrum,, Ergod. Th. Dynam. Sys., 25 (2005), 1775. doi: 10.1017/S0143385705000337. [3] W. A. Coppel, Disconjugacy,, Lecture Notes in Mathematics, (1971). [4] R. Ellis, Lectures on Topological Dynamics,, Benjamin, (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. 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, 33 (2013), 193. 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, 9 (2003), 677. 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). [10] R. Johnson and M. Nerurkar, Controllability, stabilization, and the regulator problem for random differential systems,, Mem. Amer. Math. Soc., 136 (1998). doi: 10.1090/memo/0646. [11] R. Johnson, Ergodic theory and linear differential equations,, J. Differential Equations, 28 (1978), 23. doi: 10.1016/0022-0396(78)90077-3. [12] R. Johnson, The recurrent Hill's equation,, J. Differential Equations, 46 (1982), 165. 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. [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. doi: 10.1007/s10884-013-9300-y. [15] T. Kato, Perturbation Theory for Linear Operators,, Corrected printing of the second edition, (1995). [16] V. B. Lidskiĭ, Oscillation theorems for canonical systems of differential equations,, Dokl. Akad. Nank. SSSR, 102 (1955), 877. [17] R. Mañé, Ergodic Theory and Differentiable Dynamics,, Springer-Verlag, (1987). doi: 10.1007/978-3-642-70335-5. [18] Y. Matsushima, Differentiable Manifolds,, Marcel Dekker, (1972). [19] A. Mazurov and P. Pakshin, Stochastic dissipativity with risk-sensitive storage function and related control problems,, ICIC Express Letters, 3 (2009), 53. [20] V. M. Millionščikov, Proof of the existence of irregular systems of linear differential equations with almost periodic coefficients,, Diff. Urav., 4 (1968), 391. [21] V. I. Oseledets, A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 179. [22] D. Ruelle, Ergodic theory of differentiable dynamical systems,, Publ. I.H.E.S., 50 (1979), 27. [23] R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems,, J. Differential Equations, 27 (1978), 320. 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. 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. [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. doi: 10.1007/BF00276493. [27] V. A. Yakubovich, Oscillatory properties of the solutions of canonical equations,, Amer. Math. Soc. Transl. Ser., 42 (1964), 247. [28] V. Yakubovich, Contribution to the abstract theory of optimal control I (in Russian),, Sib. Mat. Zh., 18 (1977), 685. [29] V. A. Yakubovich, Linear-quadratic optimization problem and the frequency theorem for periodic systems I (in Russian),, Sib. Mat. Zh., 27 (1986), 181. [30] V. A. Yakubovich, Linear-quadratic optimization problem and the frequency theorem for periodic systems II,, Siberian Math. J., 31 (1990), 1027. 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. doi: 10.1109/TAC.2007.899013.

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##### References:
 [1] H. Brezis, Analyse Fonctionnelle. Théorie et Applications,, Massob, (1987). [2] N. D. Cong, A generic bounded linear cocycle has simple Lyapunov spectrum,, Ergod. Th. Dynam. Sys., 25 (2005), 1775. doi: 10.1017/S0143385705000337. [3] W. A. Coppel, Disconjugacy,, Lecture Notes in Mathematics, (1971). [4] R. Ellis, Lectures on Topological Dynamics,, Benjamin, (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. 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, 33 (2013), 193. 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, 9 (2003), 677. 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). [10] R. Johnson and M. Nerurkar, Controllability, stabilization, and the regulator problem for random differential systems,, Mem. Amer. Math. Soc., 136 (1998). doi: 10.1090/memo/0646. [11] R. Johnson, Ergodic theory and linear differential equations,, J. Differential Equations, 28 (1978), 23. doi: 10.1016/0022-0396(78)90077-3. [12] R. Johnson, The recurrent Hill's equation,, J. Differential Equations, 46 (1982), 165. 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. [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. doi: 10.1007/s10884-013-9300-y. [15] T. Kato, Perturbation Theory for Linear Operators,, Corrected printing of the second edition, (1995). [16] V. B. Lidskiĭ, Oscillation theorems for canonical systems of differential equations,, Dokl. Akad. Nank. SSSR, 102 (1955), 877. [17] R. Mañé, Ergodic Theory and Differentiable Dynamics,, Springer-Verlag, (1987). doi: 10.1007/978-3-642-70335-5. [18] Y. Matsushima, Differentiable Manifolds,, Marcel Dekker, (1972). [19] A. Mazurov and P. Pakshin, Stochastic dissipativity with risk-sensitive storage function and related control problems,, ICIC Express Letters, 3 (2009), 53. [20] V. M. Millionščikov, Proof of the existence of irregular systems of linear differential equations with almost periodic coefficients,, Diff. Urav., 4 (1968), 391. [21] V. I. Oseledets, A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 179. [22] D. Ruelle, Ergodic theory of differentiable dynamical systems,, Publ. I.H.E.S., 50 (1979), 27. [23] R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems,, J. Differential Equations, 27 (1978), 320. 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. 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. [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. doi: 10.1007/BF00276493. [27] V. A. Yakubovich, Oscillatory properties of the solutions of canonical equations,, Amer. Math. Soc. Transl. Ser., 42 (1964), 247. [28] V. Yakubovich, Contribution to the abstract theory of optimal control I (in Russian),, Sib. Mat. Zh., 18 (1977), 685. [29] V. A. Yakubovich, Linear-quadratic optimization problem and the frequency theorem for periodic systems I (in Russian),, Sib. Mat. Zh., 27 (1986), 181. [30] V. A. Yakubovich, Linear-quadratic optimization problem and the frequency theorem for periodic systems II,, Siberian Math. J., 31 (1990), 1027. 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. doi: 10.1109/TAC.2007.899013.

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