A journal of IEEE and CAA , publishes high-quality papers in English on original theoretical/experimental research and development in all areas of automation
Advanced Search
Volume 8 Issue 12
Dec.  2021

IEEE/CAA Journal of Automatica Sinica

  • JCR Impact Factor: 15.3, Top 1 (SCI Q1)
    CiteScore: 23.5, Top 2% (Q1)
    Google Scholar h5-index: 77， TOP 5
Turn off MathJax
Article Contents
Article Navigation >  IEEE/CAA Journal of Automatica Sinica  > 2021  >  8(12): 1956-1963
Z. Q. Ge, "Exact Controllability and Exact Observability of Descriptor Infinite Dimensional Systems," IEEE/CAA J. Autom. Sinica, vol. 8, no. 12, pp. 1956-1963, Dec. 2021. doi: 10.1109/JAS.2020.1003411
Citation: Z. Q. Ge, "Exact Controllability and Exact Observability of Descriptor Infinite Dimensional Systems," IEEE/CAA J. Autom. Sinica, vol. 8, no. 12, pp. 1956-1963, Dec. 2021. doi: 10.1109/JAS.2020.1003411
shu

Exact Controllability and Exact Observability of Descriptor Infinite Dimensional Systems

doi: 10.1109/JAS.2020.1003411

  • School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China

Funds:  This work was supported by the National Natural Science Foundation of China (11926402, 61973338)
More Information
    Abstract
  • Necessary and sufficient conditions for the exact controllability and exact observability of a descriptor infinite dimensional system are obtained in the sense of distributional solution. These general results are used to examine the exact controllability and exact observability of the Dzektser equation in the theory of seepage and the exact controllability of wave equation.

     

    • FullText(HTML)
  • loading
    • References(30)
  • [1]
    H. D. Gou and B. L. Li, “Existence of mild solutions for Sobolev-type Hilfer fractional evolution equations with boundary conditions,” Boundary Value Problems, vol. 2018, no. 1, pp. 1–25, 2018. doi: 10.1186/s13661-017-0918-2
    [2]
    Z. Q. Ge, F. Liu, and D. X. Feng, “Pulse controllability of singular distributed parameter systems,” SCI China Inf. Sci., vol. 62, no. 4, pp. 049201:1–049201:3, Apr. 2019.
    [3]
    Y. K. Chang, Y. T. Pei, and R. Ponce, “Existence and optimal Controls for fractional stochastic evolution equations of Sobolev type via Fractional resolvent operators,” Journal of Optimization Theory and Applications, vol. 182, no. 2, pp. 558–572, Aug. 2019. doi: 10.1007/s10957-018-1314-5
    [4]
    J. Liang, Y. Y. Mu, and T. J. Xiao, “Solutions to fractional Sobolev type integral-differential equations in Banach spaces with operator pairs and impulsive conditions,” Banach Journal of Mathematical Analysis, vol. 13, no. 4, pp. 745–768, 2019. doi: 10.1215/17358787-2019-0017
    [5]
    Z. Q. Ge and X. C. Ge, “An exact null controllability of stochastic singular systems,” SCI China Inf. Sci., vol. 64, no. 7, pp. 179202:1–179202:3, Jul. 2021. doi: 10.1007/s11432-019-9902-y
    [6]
    J. Mohamed and B. Samet, “Instantaneous blow-up for a fractional in time equation of Sobolev type,” Mathematical Methods in the Applied Sciences, vol. 43, no. 8, pp. 5645–5652, May 2020. doi: 10.1002/mma.6290
    [7]
    Z. Q. Ge, X. C. Ge, and J. F. Zhang, “Approximate controllability and approximate observability of singular distributed parameter systems,” IEEE Trans. Autom. Contr., vol. 65, no. 5, pp. 2294–2299, May 2020. doi: 10.1109/TAC.2019.2920215
    [8]
    A. Favini, “Controllability conditions of linear degenerate evolution systems,” Appl. Math. Optim., vol. 6, no. 1, pp. 153–167, Mar. 1980. doi: 10.1007/BF01442890
    [9]
    Z. Q. Ge, G. T. Zhu, and Y. H. Ma, “Spectrum distribution of the second order generalized distributed parameter systems,” Appl. Math. Mech., vol. 24, no. 10, pp. 1224–1232, Oct. 2003. doi: 10.1007/BF02438111
    [10]
    T. Reis, “Controllability and observability of infinite dimensional descriptor systems,” IEEE Trans. Autom. Contr., vol. 53, no. 4, pp. 929–940, Apr. 2008. doi: 10.1109/TAC.2008.920236
    [11]
    W. Marszalek, “On solving of some heat exchangers problems via image processing equations,” Archiwum Termodynamiki, vol. 8, no. 1–2, pp. 55–72, Jan. 1987.
    [12]
    G. Chen and G. Zhang, “Initial boundary value problem for a system of generalized IMBq equations,” Math. Methods Appl. Sci., vol. 27, no. 5, pp. 497–518, 2004. doi: 10.1002/mma.444
    [13]
    A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces. New York, USA: Marcel Dekker, 1999.
    [14]
    G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, Inverse and Ill-Posed Problems Series. Utrecht, Holland: VSP, 2003.
    [15]
    D. Doleckv and D. L. Russell, “A general theory of observation and control,” SIAM J. Control and Optimization, vol. 15, no. 2, pp. 185–220, Feb. 1977. doi: 10.1137/0315015
    [16]
    R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory. New York, USA: Springer-Verlag, 1995.
    [17]
    M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups. Berlin, Germany: Birkhauser, 2009.
    [18]
    E. L. Yip and R. F. Sincovec, “Solvability, controllability, and observability of continuous descriptor systems,” IEEE Trans. Autom. Contr., vol. 26, no. 3, pp. 702–707, Mar. 1981. doi: 10.1109/TAC.1981.1102699
    [19]
    D. Cobb, “Controllability, observability and duality in singular systems,” IEEE Trans. Autom. Contr., vol. 29, no. 12, pp. 1076–1082, Dec. 1984. doi: 10.1109/TAC.1984.1103451
    [20]
    L. Dai, Singular Control Systems. New York, USA: Springer-Verlag, 1989.
    [21]
    G. R. Duan, Analysis and Design of Descriptor Linear Systems. New York, USA: Springer-Verlag, 2010.
    [22]
    V. Barbu, A. Favini, “Control of degenerate differential systems,” Contr. Cybern., vol. 28, no. 3, pp. 397–420, 1999.
    [23]
    M. R. A. Moubarak, “Exact controllability and spectrum assignment for infinite dimensional singular systems,” Publ. Math. Debrecen, vol. 54, no. 1–2, pp. 11–19, 1999.
    [24]
    Z. Q. Ge, G. T. Zhu, and D. X. Feng, “Exact controllability for singular distributed parameter system in Hilbert space,” Sci. China Ser. F-Inf. Sci., vol. 52, no. 11, pp. 2045–2052, Nov. 2009. doi: 10.1007/s11432-009-0204-8
    [25]
    V. E. Fedorov and B. Shklyar, “Exact null controllability of degenerate evolution equations with scalar control,” Sbornik:Mathematics, vol. 203, no. 12, pp. 1817–1836, 2012. doi: 10.1070/SM2012v203n12ABEH004289
    [26]
    E. S. Dzektser, “Generalization of the equation of motion of ground waters with a free surface,” English Transl. in Soviety Phys. Dokl, vol. 17, no. 8, pp. 108–110, Aug. 1972.
    [27]
    J. S. Respondek, “Numerical analysis of controllability of diffusiveconvective system with limited manipulating variables,” Int. Communications in Heat and Mass Transfer, vol. 34, no. 8, pp. 934–944, Oct. 2007. doi: 10.1016/j.icheatmasstransfer.2007.04.005
    [28]
    J. S. Respondek, “Numerical simulation in the partial differential equation controllability analysis with physically meaningful constraints,” Mathematics and Computers in Simulation, vol. 81, no. 1, pp. 120–132, Sep. 2010. doi: 10.1016/j.matcom.2010.07.016
    [29]
    I. M. Gelfand and G. E. Shilov, Generalized Function. New York, USA: Academic, 1964, vol.I.
    [30]
    S. P. Sethe, Optimal Control Theory. New York, USA: Springer, 2019.
    • Related Articles
    • Supplements(0)

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索
    Get Citation
    PDF
    XML

    Article Metrics

    Article views (1125) PDF downloads(63) Cited by()

    Highlights

    • The difference between the impulsive solution and the mild solution is explained, which shows the difference between the exact controllability in the sense of the two solutions.
    • The dual principle holds in the sense of impulsive solution but not in the sense of mild solution.
    • The example shows that the approximate controllable system is not necessarily exactly controllable.
    • For a specific descriptor infinite dimensional system, appropriate controllability can be defined according to the needs of various optimal control problems.

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return