American Institute of Mathematical Sciences

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

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

Roberta Fabbri 1, , Russell Johnson 2, , Sylvia Novo 3, and  Carmen Núñez 3,

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.
Keywords: dissipativity, Linear-quadratic control system, supply rate, storage function..
Mathematics Subject Classification: 37B55, 93D20, 34D2.
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

show all references

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

[1]

Shigeaki Koike, Hiroaki Morimoto, Shigeru Sakaguchi. A linear-quadratic control problem with discretionary stopping. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 261-277. doi: 10.3934/dcdsb.2007.8.261
[2]

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
[3]

Galina Kurina, Sahlar Meherrem. Decomposition of discrete linear-quadratic optimal control problems for switching systems. Conference Publications, 2015, 2015 (special) : 764-774. doi: 10.3934/proc.2015.0764
[4]

Mohamed Aliane, Mohand Bentobache, Nacima Moussouni, Philippe Marthon. Direct method to solve linear-quadratic optimal control problems. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021002
[5]

Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields, 2021, 11 (1) : 169-188. doi: 10.3934/mcrf.2020032
[6]

Yadong Shu, Bo Li. Linear-quadratic optimal control for discrete-time stochastic descriptor systems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021034
[7]

Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control & Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97
[8]

Hongyan Yan, Yun Sun, Yuanguo Zhu. A linear-quadratic control problem of uncertain discrete-time switched systems. Journal of Industrial & Management Optimization, 2017, 13 (1) : 267-282. doi: 10.3934/jimo.2016016
[9]

Hanxiao Wang, Jingrui Sun, Jiongmin Yong. Weak closed-loop solvability of stochastic linear-quadratic optimal control problems. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2785-2805. doi: 10.3934/dcds.2019117
[10]

Georg Vossen, Stefan Volkwein. Model reduction techniques with a-posteriori error analysis for linear-quadratic optimal control problems. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 465-485. doi: 10.3934/naco.2012.2.465
[11]

Walter Alt, Robert Baier, Matthias Gerdts, Frank Lempio. Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 547-570. doi: 10.3934/naco.2012.2.547
[12]

Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026
[13]

Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020040
[14]

Henri Bonnel, Ngoc Sang Pham. Nonsmooth optimization over the (weakly or properly) Pareto set of a linear-quadratic multi-objective control problem: Explicit optimality conditions. Journal of Industrial & Management Optimization, 2011, 7 (4) : 789-809. doi: 10.3934/jimo.2011.7.789
[15]

Martino Bardi. Explicit solutions of some linear-quadratic mean field games. Networks & Heterogeneous Media, 2012, 7 (2) : 243-261. doi: 10.3934/nhm.2012.7.243
[16]

Yulin Zhao. On the monotonicity of the period function of a quadratic system. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 795-810. doi: 10.3934/dcds.2005.13.795
[17]

Xingwu Chen, Jaume Llibre, Weinian Zhang. Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3953-3965. doi: 10.3934/dcdsb.2017203
[18]

Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001
[19]

Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021026
[20]

Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences