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

Roberta Fabbri; Russell Johnson; Sylvia Novo; Carmen Núñez

doi:10.3934/dcds.2013.33.193

Article Contents

2013, Volume 33, Issue 1: 193-210. Doi: 10.3934/dcds.2013.33.193

This issue Previous Article A class of singular first order differential equations with applications in reaction-diffusion Next Article Boundedness and stability for the damped and forced single well Duffing equation

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

Received: August 2011

Revised: January 2012

Published: January 2013

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: 37B55, 93D20, 34D20.

Citation:

Related Papers

Cited by

References

[1]	F. Colonius and W. Kliemann, "The Dynamics of Control," Birkhäuser, Basel, 2000.
[2]	W. A. Coppel, "Dichotomies in Stability Theory," Lecture Notes in Math., Springer-Verlag, Berlin, Heidelberg, New York, 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., Ser. A, 9 (2003), 677-704.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), 853-864.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, Vol., 4 (1992), 3259-3264.
[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.
[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), 1045-1079.doi: 10.1017/S0308210500000573.
[9]	E. Lee and B. Markus, "Foundation of Optimal Control Theory," John Wiley & Sons, New York, 1967.
[10]	I. G. Polushin, Stability results for quasidissipative systems, Proc. 3rd Eur. Control Conference ECC'95, (1995), 681-686.
[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.
[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.
[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.
[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.
[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.
[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.
[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.