• Previous Article
    Delay-induced instabilities of stationary solutions in a single species nonlocal hyperbolic-parabolic population model
  • DCDS-S Home
  • This Issue
  • Next Article
    Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations
September  2020, 13(9): 2489-2508. doi: 10.3934/dcdss.2020119

Minimum energy compensation for discrete delayed systems with disturbances

Faculty of Sciences Ain Chock, University Hassan II, B.P.5366, Maȃrif, Casablanca, Morocco

* Corresponding author

Received  October 2018 Revised  January 2019 Published  October 2019

This work is devoted to the remediability problem for a class of discrete delayed systems. We investigate the possibility of reducing the disturbance effect with a convenient choice of the control operator. We give the main properties and characterization results of this concept, according to the delay and the observation. Then, under an appropriate hypothesis, we demonstrate how to find the optimal control which ensures the compensation of a disturbance measured through the observation (measurements, signals, ...). The discrete version of the wave equation, as well as the usual actuators and sensors, are examined. Numerical results are also presented.

Citation: Salma Souhaile, Larbi Afifi. Minimum energy compensation for discrete delayed systems with disturbances. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2489-2508. doi: 10.3934/dcdss.2020119
References:
[1]

L. AfifiM. BahadiA. Chafiai and A. El Mizane, Asymptotic compensation in discrete distributed systems: Analysis, approximations and simulations, Applied Mathematical Sciences, 2 (2008), 99-137.   Google Scholar

[2]

L. Afifi and A. El Jai, Systémes Distribués Perturbés, Presses Universitaires de Perpignan (frensh), 2015. Google Scholar

[3]

R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences, 8. Springer-Verlag, Berlin-New York, 1978.  Google Scholar

[4]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Applied Mathematical Sciences, 110. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.  Google Scholar

[5]

A. El Jaï and A. J. Pritchard, Sensors and actuators in distributed systems, International Journal of Control, 46 (1987), 1139-1153.  doi: 10.1080/00207178708933956.  Google Scholar

[6]

E. Fridman, Introduction to Time-Delay Systems: Analysis and Control, Systems & Control: Foundations & Applications, Birkhäuser/Springer, Cham, 2014. doi: 10.1007/978-3-319-09393-2.  Google Scholar

[7]

S. Hadd and A. Idrissi, Regular linear systems governed by systems with state, input and output delays, IMA Journal of Mathematical Control and Information, 22 (2005), 423-439.  doi: 10.1093/imamci/dni035.  Google Scholar

[8]

S. Hadd, An evolution equation approach to nonautonomous linear systems with state, input, and output delays, SIAM Journal on Control and Optimization, 45 (2006), 246-272.  doi: 10.1137/040612178.  Google Scholar

[9]

S. Hadd and Q.-C. Zhong, On feedback stabilizability of linear systems with state and input delays in Banach spaces, IEEE Transactions on Automatic Control, 54 (2009), 438-451.  doi: 10.1109/TAC.2009.2012969.  Google Scholar

[10]

H. Shi, G. M. Xie and W. G. Luo, Controllability of linear discrete time systems with both delayed states and delayed inputs, Abstract and Applied Analysis, (2013), Art. ID 975461, 5 pp. doi: 10.1155/2013/975461.  Google Scholar

[11]

V. Isakov, Inverse Source Problems, Mathematical Surveys and Monographs, 34. American Mathematical Society, Providence, RI, 1990. doi: 10.1090/surv/034.  Google Scholar

[12]

H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Wiley-Interscience, New York-London-Sydney, 1972.  Google Scholar

[13]

J.-L. Lions, Contrȏle Optimal de Systémes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris, Gauthier-Villars, Paris, 1968.  Google Scholar

[14]

M. Naim, F. Lahmidi, A. Namir and M. Rachik, On the output controllability of positive discrete linear delay systems, Abstract and Applied Analysis, (2017), Art. ID 3651271, 12 pp. doi: 10.1155/2017/3651271.  Google Scholar

[15]

V. N. Phat and T. C. Dieu, Constrained controllability of linear discrete nonstationary systems in banach spaces, SIAM J. Control Optim., 30 (1992), 1311-1318.  doi: 10.1137/0330069.  Google Scholar

[16]

V. N. Phat, Controllability of discrete-time systems with multiple delays on controls and states, International Journal of Control, 49 (1989), 1645-1654.  doi: 10.1080/00207178908559731.  Google Scholar

[17]

R. Rabah and M. Malabare, Structure at infinity revisited for delay systems, IEEE-SMC-IMACS Multiconference, Symposium on Robotics and Cybernetics (CESA'96). Symposium in Modelling, Analysis and Simulation, (1996), 87–90. https://hal.archives-ouvertes.fr/hal-01466183 Google Scholar

[18]

R. Rabah and M. Malabare, Weak structure at infinity and row-by-row decoupling for linear delay systems, Kybernetika, 40 (2004), 181-195.   Google Scholar

[19]

M. RachikM. Lhous and A. Tridane, Controllability and Optimal Control Problem for Linear Time-varying Discrete Distributed Systems, Systems Analysis Modelling Simulation, 43 (2003), 137-164.   Google Scholar

[20]

J.-P. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica J. IFAC, 39 (2003), 1667-1694.  doi: 10.1016/S0005-1098(03)00167-5.  Google Scholar

[21]

S. Souhaile and L. Afifi, Cheap compensation in disturbed linear dynamical systems with multi-input delays, International Journal of Dynamics and Control. https://doi.org/10.1007/s40435-018-00505-6. doi: 10.1007/s40435-018-00505-6.  Google Scholar

[22]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

show all references

References:
[1]

L. AfifiM. BahadiA. Chafiai and A. El Mizane, Asymptotic compensation in discrete distributed systems: Analysis, approximations and simulations, Applied Mathematical Sciences, 2 (2008), 99-137.   Google Scholar

[2]

L. Afifi and A. El Jai, Systémes Distribués Perturbés, Presses Universitaires de Perpignan (frensh), 2015. Google Scholar

[3]

R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences, 8. Springer-Verlag, Berlin-New York, 1978.  Google Scholar

[4]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Applied Mathematical Sciences, 110. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.  Google Scholar

[5]

A. El Jaï and A. J. Pritchard, Sensors and actuators in distributed systems, International Journal of Control, 46 (1987), 1139-1153.  doi: 10.1080/00207178708933956.  Google Scholar

[6]

E. Fridman, Introduction to Time-Delay Systems: Analysis and Control, Systems & Control: Foundations & Applications, Birkhäuser/Springer, Cham, 2014. doi: 10.1007/978-3-319-09393-2.  Google Scholar

[7]

S. Hadd and A. Idrissi, Regular linear systems governed by systems with state, input and output delays, IMA Journal of Mathematical Control and Information, 22 (2005), 423-439.  doi: 10.1093/imamci/dni035.  Google Scholar

[8]

S. Hadd, An evolution equation approach to nonautonomous linear systems with state, input, and output delays, SIAM Journal on Control and Optimization, 45 (2006), 246-272.  doi: 10.1137/040612178.  Google Scholar

[9]

S. Hadd and Q.-C. Zhong, On feedback stabilizability of linear systems with state and input delays in Banach spaces, IEEE Transactions on Automatic Control, 54 (2009), 438-451.  doi: 10.1109/TAC.2009.2012969.  Google Scholar

[10]

H. Shi, G. M. Xie and W. G. Luo, Controllability of linear discrete time systems with both delayed states and delayed inputs, Abstract and Applied Analysis, (2013), Art. ID 975461, 5 pp. doi: 10.1155/2013/975461.  Google Scholar

[11]

V. Isakov, Inverse Source Problems, Mathematical Surveys and Monographs, 34. American Mathematical Society, Providence, RI, 1990. doi: 10.1090/surv/034.  Google Scholar

[12]

H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Wiley-Interscience, New York-London-Sydney, 1972.  Google Scholar

[13]

J.-L. Lions, Contrȏle Optimal de Systémes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris, Gauthier-Villars, Paris, 1968.  Google Scholar

[14]

M. Naim, F. Lahmidi, A. Namir and M. Rachik, On the output controllability of positive discrete linear delay systems, Abstract and Applied Analysis, (2017), Art. ID 3651271, 12 pp. doi: 10.1155/2017/3651271.  Google Scholar

[15]

V. N. Phat and T. C. Dieu, Constrained controllability of linear discrete nonstationary systems in banach spaces, SIAM J. Control Optim., 30 (1992), 1311-1318.  doi: 10.1137/0330069.  Google Scholar

[16]

V. N. Phat, Controllability of discrete-time systems with multiple delays on controls and states, International Journal of Control, 49 (1989), 1645-1654.  doi: 10.1080/00207178908559731.  Google Scholar

[17]

R. Rabah and M. Malabare, Structure at infinity revisited for delay systems, IEEE-SMC-IMACS Multiconference, Symposium on Robotics and Cybernetics (CESA'96). Symposium in Modelling, Analysis and Simulation, (1996), 87–90. https://hal.archives-ouvertes.fr/hal-01466183 Google Scholar

[18]

R. Rabah and M. Malabare, Weak structure at infinity and row-by-row decoupling for linear delay systems, Kybernetika, 40 (2004), 181-195.   Google Scholar

[19]

M. RachikM. Lhous and A. Tridane, Controllability and Optimal Control Problem for Linear Time-varying Discrete Distributed Systems, Systems Analysis Modelling Simulation, 43 (2003), 137-164.   Google Scholar

[20]

J.-P. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica J. IFAC, 39 (2003), 1667-1694.  doi: 10.1016/S0005-1098(03)00167-5.  Google Scholar

[21]

S. Souhaile and L. Afifi, Cheap compensation in disturbed linear dynamical systems with multi-input delays, International Journal of Dynamics and Control. https://doi.org/10.1007/s40435-018-00505-6. doi: 10.1007/s40435-018-00505-6.  Google Scholar

[22]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

Figure 1.  Control observation for $ N = 10 $
Figure 2.  Control observation for $ N = 20 $
[1]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

[2]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[3]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

[4]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[5]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[6]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[7]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

[8]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[9]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375

[10]

Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166

[11]

Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020269

[12]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[13]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[14]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273

[15]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[16]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467

[17]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[18]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[19]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[20]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (92)
  • HTML views (288)
  • Cited by (0)

Other articles
by authors

[Back to Top]