# American Institute of Mathematical Sciences

2013, 33(1): 193-210. doi: 10.3934/dcds.2013.33.193

## On linear-quadratic dissipative control processes with time-varying coefficients

 1 Dipartimento di Sistemi e Informatica, Università di Firenze, Via di Santa Marta 3, 50139 Firenze, Italy 2 Dipartimento di Sistemi e Informatica, Università di Firenze, Facolta' di Ingegneria, Via di Santa Marta 3, 50139 Firenze, Italy 3 Departamento de Matemática Aplicada, E. Ingenierías Industriales, Universidad de Valladolid, Paseo del Cauce 59, 47011 Valladolid, Spain, Spain

Received  August 2011 Revised  January 2012 Published  September 2012

Yakubovich, Fradkov, Hill and Proskurnikov have used the Yaku-bovich Frequency Theorem to prove that a strictly dissipative linear-quadratic control process with periodic coefficients admits a storage function, and various related results. We extend their analysis to the case when the coefficients are bounded uniformly continuous functions.
Citation: Roberta Fabbri, Russell Johnson, Sylvia Novo, Carmen Núñez. On linear-quadratic dissipative control processes with time-varying coefficients. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 193-210. doi: 10.3934/dcds.2013.33.193
##### References:
 [1] F. Colonius and W. Kliemann, "The Dynamics of Control,", Birkhäuser, (2000). [2] W. A. Coppel, "Dichotomies in Stability Theory,", Lecture Notes in Math., 629 (1978). [3] R. Fabbri, R. Johnson and C. Núñez, On the Yakubovich Frequency Theorem for linear non autonomous control processes,, Discrete Contin. Dyn. Syst., 9 (2003), 677. doi: 10.3934/dcds.2003.9.677. [4] 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. [5] D. J. Hill, Dissipative nonlinear systems: basic properies and stability analysis,, Proc. 31st IEEE Conference on Decision and Control, 4 (1992), 3259. [6] D. J. Hill and P. J. Moylan, Dissipative dynamical systems: Basic input-output and state properties,, J. Franklin Inst., 309 (1980), 327. doi: 10.1016/0016-0032(80)90026-5. [7] R. Johnson and M. Nerurkar, "Controllability, Stabilization and the Regulator Problem for Random Differential Systems,", Mem. Amer. Math. Soc., 646 (1998). [8] R. Johnson, S. Novo and R. Obaya, Ergodic properties and Weyl $M$-functions for linear Hamiltonian systems,, Proc. Roy. Soc. Edinburgh, 130A (2000), 1045. doi: 10.1017/S0308210500000573. [9] E. Lee and B. Markus, "Foundation of Optimal Control Theory,", John Wiley & Sons, (1967). [10] I. G. Polushin, Stability results for quasidissipative systems,, Proc. 3rd Eur. Control Conference ECC'95, (1995), 681. [11] 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. [12] A. V. Savkin and I. R. Petersen, Structured dissipativeness and absolute stability of nonlinear systems,, Internat. J. Control, 62 (1995), 443. doi: 10.1080/00207179508921550. [13] H. L. Trentelman and J. C. Willems, "Storage Functions for Dissipative Linear Systems Are Quadratic State Functions,", Proc. 36th IEEE Conf. Decision and Control, (1997), 42. [14] 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. [15] V. A. Yakubovich, A linear-quadratic problem of optimization and the frequency theorem for periodic systems. I,, Siberian Math. J., 27 (1986), 614. doi: 10.1007/BF00969175. [16] V. A. Yakubovich, A linear-quadratic problem of optimization and the frequency theorem for periodic systems. II,, Siberian Math. J., 31 (1990), 1027. doi: 10.1007/BF00970068. [17] 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] F. Colonius and W. Kliemann, "The Dynamics of Control,", Birkhäuser, (2000). [2] W. A. Coppel, "Dichotomies in Stability Theory,", Lecture Notes in Math., 629 (1978). [3] R. Fabbri, R. Johnson and C. Núñez, On the Yakubovich Frequency Theorem for linear non autonomous control processes,, Discrete Contin. Dyn. Syst., 9 (2003), 677. doi: 10.3934/dcds.2003.9.677. [4] 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. [5] D. J. Hill, Dissipative nonlinear systems: basic properies and stability analysis,, Proc. 31st IEEE Conference on Decision and Control, 4 (1992), 3259. [6] D. J. Hill and P. J. Moylan, Dissipative dynamical systems: Basic input-output and state properties,, J. Franklin Inst., 309 (1980), 327. doi: 10.1016/0016-0032(80)90026-5. [7] R. Johnson and M. Nerurkar, "Controllability, Stabilization and the Regulator Problem for Random Differential Systems,", Mem. Amer. Math. Soc., 646 (1998). [8] R. Johnson, S. Novo and R. Obaya, Ergodic properties and Weyl $M$-functions for linear Hamiltonian systems,, Proc. Roy. Soc. Edinburgh, 130A (2000), 1045. doi: 10.1017/S0308210500000573. [9] E. Lee and B. Markus, "Foundation of Optimal Control Theory,", John Wiley & Sons, (1967). [10] I. G. Polushin, Stability results for quasidissipative systems,, Proc. 3rd Eur. Control Conference ECC'95, (1995), 681. [11] 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. [12] A. V. Savkin and I. R. Petersen, Structured dissipativeness and absolute stability of nonlinear systems,, Internat. J. Control, 62 (1995), 443. doi: 10.1080/00207179508921550. [13] H. L. Trentelman and J. C. Willems, "Storage Functions for Dissipative Linear Systems Are Quadratic State Functions,", Proc. 36th IEEE Conf. Decision and Control, (1997), 42. [14] 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. [15] V. A. Yakubovich, A linear-quadratic problem of optimization and the frequency theorem for periodic systems. I,, Siberian Math. J., 27 (1986), 614. doi: 10.1007/BF00969175. [16] V. A. Yakubovich, A linear-quadratic problem of optimization and the frequency theorem for periodic systems. II,, Siberian Math. J., 31 (1990), 1027. doi: 10.1007/BF00970068. [17] 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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