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Volume 7 Issue 2
Mar.  2020

IEEE/CAA Journal of Automatica Sinica

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Khac Duc Do, "Stability in Probability and Inverse Optimal Control of Evolution Systems Driven by Lévy Processes," IEEE/CAA J. Autom. Sinica, vol. 7, no. 2, pp. 405-419, Mar. 2020. doi: 10.1109/JAS.2020.1003036
Citation: Khac Duc Do, "Stability in Probability and Inverse Optimal Control of Evolution Systems Driven by Lévy Processes," IEEE/CAA J. Autom. Sinica, vol. 7, no. 2, pp. 405-419, Mar. 2020. doi: 10.1109/JAS.2020.1003036

Stability in Probability and Inverse Optimal Control of Evolution Systems Driven by Lévy Processes

doi: 10.1109/JAS.2020.1003036
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  • This paper first develops a Lyapunov-type theorem to study global well-posedness (existence and uniqueness of the strong variational solution) and asymptotic stability in probability of nonlinear stochastic evolution systems (SESs) driven by a special class of Lévy processes, which consist of Wiener and compensated Poisson processes. This theorem is then utilized to develop an approach to solve an inverse optimal stabilization problem for SESs driven by Lévy processes. The inverse optimal control design achieves global well-posedness and global asymptotic stability of the closed-loop system, and minimizes a meaningful cost functional that penalizes both states and control. The approach does not require to solve a Hamilton-Jacobi-Bellman equation (HJBE). An optimal stabilization of the evolution of the frequency of a certain genetic character from the population is included to illustrate the theoretical developments.

     

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  • [1]
    D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, 2nd ed., 2009.
    [2]
    E. Pardoux, “Stochastic partial differential equations and filtering of diffusion processes,” Stochastics, vol. 3, pp. 127–167, 1979.
    [3]
    K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations With Applications, Boca Raton, FL: Chapman and Hall/CRC, 2006.
    [4]
    P. L. Chow, Stochastic Partial Differential Equations. Boca Raton: Chapman & Hall/CRC, 2007.
    [5]
    G. D. Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge, UK: Cambridge University Press, 1992.
    [6]
    R. Liu and V. Mandrekar, “Stochastic semilinear evolution equations: Lyapunov function, stability and ultimate boundedness,” J. Mathematical Analysis and Applications, vol. 212, no. 2, pp. 537–553, 1997. doi: 10.1006/jmaa.1997.5534
    [7]
    L. Gawarecki and V. Mandrekar, Stochastic Differential Equations in Infinite Dimensions With Applications to Stochastic Partial Differential Equations, Berlin: Springer, 2011.
    [8]
    K. D. Do and A. D. Lucey, “Stochastic stabilization of slender beams in space: modeling and boundary control,” Automatica, vol. 91, pp. 279–293, 2018. doi: 10.1016/j.automatica.2018.01.017
    [9]
    K. D. Do, “Stability of nonlinear stochastic distributed parameter systems and its applications,” J. Dynamic Systems,Measurement,and Control, vol. 138, pp. 101010: 1–12, 1010.
    [10]
    K. D. Do, “Stochastic boundary control design for extensible marine risers in three dimensional space,” Automatica, vol. 77, pp. 184–197, 2017. doi: 10.1016/j.automatica.2016.11.032
    [11]
    Z. Brzézniak, W. Liu, and J. Zhu, “Strong solutions for spde with locally monotone coefficients driven by Lévy noise,” Nonlinear Analysis:Real World Applications, vol. 17, pp. 283–310, 2014. doi: 10.1016/j.nonrwa.2013.12.005
    [12]
    Z. Dong and T. Xu, “One-dimensional stochastic burgers equation driven by Lévy processes,” J. Functional Analysis, vol. 243, pp. 631–678, 2007. doi: 10.1016/j.jfa.2006.09.010
    [13]
    M. Röckner and T. Zhang, “Stochastic evolution equation of jump type: existence, uniqueness and large deviation principles,” Potential Analysis, vol. 26, pp. 255–279, 2007. doi: 10.1007/s11118-006-9035-z
    [14]
    S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations With Levy Noise: An Evolution Equation Approach, Cambridge: Cambridge University Press, 2007.
    [15]
    H. Li and Q. Zhu, “The pth moment exponential stability and almost surely exponential stability of stochastic differential delay equations with poisson jump,” J. Mathematical Analysis and Applications, vol. 471, pp. 197–210, 2019. doi: 10.1016/j.jmaa.2018.10.072
    [16]
    Q. Zhu, “Stability analysis of stochastic delay differential equations with Lévy noise,” Systems &Control Letters, vol. 118, pp. 62–68, 2018.
    [17]
    Q. Zhu, “Razumikhin-type theorem for stochastic functional differential equations with Lévy noise and Markov switching,” Int. J. Control, vol. 90, pp. 1703–1712, 2017. doi: 10.1080/00207179.2016.1219069
    [18]
    Q. Zhu, “Asymptotic stability in the pth moment for stochastic differential equations with Lévy noise,” J. Mathematical Analysis and Applications, vol. 416, pp. 126–142, 2014. doi: 10.1016/j.jmaa.2014.02.016
    [19]
    H. Khalil, Nonlinear Systems, Prentice Hall, 2002.
    [20]
    P. Kokotovic and M. Arcak, “Constructive nonlinear control: a history perspective,” Automatica, vol. 37, no. 5, pp. 637–662, 2001. doi: 10.1016/S0005-1098(01)00002-4
    [21]
    R. Sepulchre, M. Jankovic, and P. Kokotovic, Constructive Nonlinear Control, New York: Springer, 1997.
    [22]
    M. Krstic and H. Deng, Stabilization of Nonlinear Uncertain Systems, London: Springer, 1998.
    [23]
    K. D. Do and A. D. Lucey, “Inverse optimal control of evolution systems and its application to extensible and shearable slender beams,” IEEE/CAA J. Autom. Sinica, vol. 6, pp. 395–409, 2019. doi: 10.1109/JAS.2019.1911381
    [24]
    E. D. Sontag, “Smooth stabilization implies comprime factorization,” IEEE Trans. Automatic Control, vol. 34, no. 4, pp. 435–443, 1989. doi: 10.1109/9.28018
    [25]
    K. D. Do, “Global inverse optimal stabilization of stochastic nonholonomic systems,” Systems &Control Letters, vol. 75, pp. 41–55, 2015.
    [26]
    H. Deng, M. Krstic, and R. Williams, “Stabilization of stochastic nonlinear systems driven by noise of unknown covariance,” IEEE Trans. Automatic Control, vol. 46, no. 8, pp. 1237–1253, 2001. doi: 10.1109/9.940927
    [27]
    M. Krstic, “On global stabilization of Burgers’ equation by boundary control,” Systems &Control Letters, vol. 37, pp. 123–141, 1999.
    [28]
    K. D. Do, “Inverse optimal gain assignment control of evolution systems and its application to boundary control of marine risers,” Automatica, vol. 106, pp. 242–256, 2019. doi: 10.1016/j.automatica.2019.05.020
    [29]
    F. B. Hanson, Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis, and Computation, Society for Industrial and Applied Mathematics, 2007.
    [30]
    C. Prevot and M. Rockner, A Concise Course on Stochastic Partial Differential Equations, Berlin: Springer, 2007.
    [31]
    G. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge: Cambridge University Press, 2nd ed., 1989.
    [32]
    W. H. Fleming, Distributed Parameter Stochastic Systems in Population Biology, Berlin-New York: Lecture Notes in Economics and Mathematical Systems, vol. 107, Springer, 1975.
    [33]
    K. D. Do, “Inverse optimal control of stochastic systems driven by Lévy processes,” Automatica, vol. 107, pp. 539–550, Sept. 2019.
    [34]
    X. Mao, Stochastic Differential Equations and Applications, Cambridge: Woodhead Publishing, 2nd ed., 2007.
    [35]
    I. Gyongy and N. V. Krylov, “On stochastic equations with respect to semimartingales I,” Stochastics, vol. 4, pp. 1–21, 1980. doi: 10.1080/03610918008833154
    [36]
    R. Khasminskii, Stochastic Stability of Differential Equations, Rockville MD: S&N Int., 1980.
    [37]
    R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Oxford, UK: Academic Press, 2nd ed., 2003.
    [38]
    P. E. Protter, Stochastic Integration and Differential Equations, Berlin: Springer, 2nd ed., 2005.
    [39]
    P. Protter, Stochastic Integration and Differential Equations. Springer Verlag, 2nd ed., 2005.

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    Highlights

    • Global well-posedness and stability in probability of evolution systems driven by Levy processes.
    • Inverse optimal stabilizers for evolution systems driven by Levy processes.
    • No need to solve Hamilton-Jacobi-Bellman equations.

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