# American Institute of Mathematical Sciences

January  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, 2013, 33 (1) : 193-210. doi: 10.3934/dcds.2013.33.193
##### References:
 [1] F. Colonius and W. Kliemann, "The Dynamics of Control," Birkhäuser, Basel, 2000.  Google Scholar [2] W. A. Coppel, "Dichotomies in Stability Theory," Lecture Notes in Math., Springer-Verlag, Berlin, Heidelberg, New York, 629 (1978).  Google Scholar [3] R. Fabbri, R. Johnson and C. Núñez, On the Yakubovich Frequency Theorem for linear non autonomous control processes, Discrete Contin. Dyn. Syst., Ser. A, 9 (2003), 677-704. doi: 10.3934/dcds.2003.9.677.  Google Scholar [4] R. Fabbri, R. Johnson, S. Novo and C. Núñez, Some remarks concerning weakly disconjugate linear Hamiltonian systems, J. Math. Anal. Appl., 380 (2011), 853-864. doi: 10.1016/j.jmaa.2010.11.036.  Google Scholar [5] D. J. Hill, Dissipative nonlinear systems: basic properies and stability analysis, Proc. 31st IEEE Conference on Decision and Control, Vol., 4 (1992), 3259-3264. Google Scholar [6] D. J. Hill and P. J. Moylan, Dissipative dynamical systems: Basic input-output and state properties, J. Franklin Inst., 309 (1980), 327-357. doi: 10.1016/0016-0032(80)90026-5.  Google Scholar [7] R. Johnson and M. Nerurkar, "Controllability, Stabilization and the Regulator Problem for Random Differential Systems," Mem. Amer. Math. Soc., 646 (1998).  Google Scholar [8] R. Johnson, S. Novo and R. Obaya, Ergodic properties and Weyl $M$-functions for linear Hamiltonian systems, Proc. Roy. Soc. Edinburgh, 130A (2000), 1045-1079. doi: 10.1017/S0308210500000573.  Google Scholar [9] E. Lee and B. Markus, "Foundation of Optimal Control Theory," John Wiley & Sons, New York, 1967.  Google Scholar [10] I. G. Polushin, Stability results for quasidissipative systems, Proc. 3rd Eur. Control Conference ECC'95, (1995), 681-686. Google Scholar [11] 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 [12] A. V. Savkin and I. R. Petersen, Structured dissipativeness and absolute stability of nonlinear systems, Internat. J. Control, 62 (1995), 443-460. doi: 10.1080/00207179508921550.  Google Scholar [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), 42-49. Google Scholar [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), 321-393. doi: 10.1007/BF00276493.  Google Scholar [15] V. A. Yakubovich, A linear-quadratic problem of optimization and the frequency theorem for periodic systems. I, Siberian Math. J., 27 (1986), 614-630. doi: 10.1007/BF00969175.  Google Scholar [16] V. A. Yakubovich, A linear-quadratic problem of optimization and the frequency theorem for periodic systems. II, Siberian Math. J., 31 (1990), 1027-1039. doi: 10.1007/BF00970068.  Google Scholar [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), 1039-1047. doi: 10.1109/TAC.2007.899013.  Google Scholar

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##### References:
 [1] F. Colonius and W. Kliemann, "The Dynamics of Control," Birkhäuser, Basel, 2000.  Google Scholar [2] W. A. Coppel, "Dichotomies in Stability Theory," Lecture Notes in Math., Springer-Verlag, Berlin, Heidelberg, New York, 629 (1978).  Google Scholar [3] R. Fabbri, R. Johnson and C. Núñez, On the Yakubovich Frequency Theorem for linear non autonomous control processes, Discrete Contin. Dyn. Syst., Ser. A, 9 (2003), 677-704. doi: 10.3934/dcds.2003.9.677.  Google Scholar [4] R. Fabbri, R. Johnson, S. Novo and C. Núñez, Some remarks concerning weakly disconjugate linear Hamiltonian systems, J. Math. Anal. Appl., 380 (2011), 853-864. doi: 10.1016/j.jmaa.2010.11.036.  Google Scholar [5] D. J. Hill, Dissipative nonlinear systems: basic properies and stability analysis, Proc. 31st IEEE Conference on Decision and Control, Vol., 4 (1992), 3259-3264. Google Scholar [6] D. J. Hill and P. J. Moylan, Dissipative dynamical systems: Basic input-output and state properties, J. Franklin Inst., 309 (1980), 327-357. doi: 10.1016/0016-0032(80)90026-5.  Google Scholar [7] R. Johnson and M. Nerurkar, "Controllability, Stabilization and the Regulator Problem for Random Differential Systems," Mem. Amer. Math. Soc., 646 (1998).  Google Scholar [8] R. Johnson, S. Novo and R. Obaya, Ergodic properties and Weyl $M$-functions for linear Hamiltonian systems, Proc. Roy. Soc. Edinburgh, 130A (2000), 1045-1079. doi: 10.1017/S0308210500000573.  Google Scholar [9] E. Lee and B. Markus, "Foundation of Optimal Control Theory," John Wiley & Sons, New York, 1967.  Google Scholar [10] I. G. Polushin, Stability results for quasidissipative systems, Proc. 3rd Eur. Control Conference ECC'95, (1995), 681-686. Google Scholar [11] 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 [12] A. V. Savkin and I. R. Petersen, Structured dissipativeness and absolute stability of nonlinear systems, Internat. J. Control, 62 (1995), 443-460. doi: 10.1080/00207179508921550.  Google Scholar [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), 42-49. Google Scholar [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), 321-393. doi: 10.1007/BF00276493.  Google Scholar [15] V. A. Yakubovich, A linear-quadratic problem of optimization and the frequency theorem for periodic systems. I, Siberian Math. J., 27 (1986), 614-630. doi: 10.1007/BF00969175.  Google Scholar [16] V. A. Yakubovich, A linear-quadratic problem of optimization and the frequency theorem for periodic systems. II, Siberian Math. J., 31 (1990), 1027-1039. doi: 10.1007/BF00970068.  Google Scholar [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), 1039-1047. doi: 10.1109/TAC.2007.899013.  Google Scholar

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