Advanced Search
Article Contents
Article Contents

Feedback optimal control for stochastic Volterra equations with completely monotone kernels

Abstract Related Papers Cited by
  • In this paper we are concerned with a class of stochastic Volterra integro-differential problems with completely monotone kernels, where we assume that the noise enters the system when we introduce a control. We start by reformulating the state equation into a semilinear evolution equation which can be treated by semigroup methods. The application to optimal control provides other interesting results and requires a precise description of the properties of the generated semigroup.
        The first main result of the paper is the proof of existence and uniqueness of a mild solution for the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The main technical point consists in the differentiability of the BSDE associated with the reformulated equation with respect to its initial datum $x$.
    Mathematics Subject Classification: 45D05, 93E20, 60H30.


    \begin{equation} \\ \end{equation}
  • [1]

    O. P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynamics, 29 (2002), 145-155.doi: 10.1023/A:1016539022492.


    S. A. Belbas, A new method for optimal control of Volterra integral equations, Appl. Math. Comput., 189 (2007), 1902-1915.doi: 10.1016/j.amc.2006.12.077.


    S. Bonaccorsi, Some Applications in Malliavin Calculus, Ph.D thesis, Dept. Mathematics, Univ. Trento., 1998.


    S. Bonaccorsi and G. Desch, Volterra equations in Banach spaces with completely monotone kernels, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 557-594.doi: 10.1007/s00030-012-0167-0.


    S. Bonaccorsi and E. Mastrogiacomo, An analytic approach to stochastic Volterra equations with completely monotone kernels, J. Evol. Equ., 9 (2009), 315-339.doi: 10.1007/s00028-009-0010-1.


    S. Bonaccorsi, F. Confortola and E. Mastrogiacomo, Optimal control for stochastic Volterra equations with completely monotone kernels, SIAM J. Control Optim., 50 (2012), 748-789.doi: 10.1137/100782875.


    P. Briand and F. Confortola, SDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces, Stochastic Process. Appl., 118 (2008), 818-838.doi: 10.1016/j.spa.2007.06.006.


    F. Confortola and E. Mastrogiacomo, Optimal control for stochastic heat equation with memory, Evolution Equations and Control Theory, 3 (2014), 35-58.doi: 10.3934/eect.2014.3.35.


    G. Da Prato and J. Zabczyk, Evolution Equations with White-Noise Boundary Conditions, Cambridge Univ. Press, Cambridge, 1993.doi: 10.1080/17442509308833817.


    G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992.doi: 10.1017/CBO9780511666223.


    A. Debussche, M. Fuhrman and G. Tessitore, Optimal control of a stochastic heat equation with boundary-noise and boundary-control, ESAIM Control Optim. Calc. Var., 13 (2007), 178-205 (electronic).doi: 10.1051/cocv:2007001.


    W. Desch and R. K. Miller, Exponential stabilization of Volterra integral equations with singular kernels, J. Integral Equations Appl., 1 (1988), 397-433.doi: 10.1216/JIE-1988-1-3-397.


    K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math., {194}, Springer-Verlag, Berlin, 2000.


    W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993.


    M. Fuhrman and G. Tessitore, Nonlinear Kolmogorov equations in infinite dimensional spaces: The backward stochastic differential equations approach and applications to optimal control, Ann. Probab., 30 (2002), 1397-1465.doi: 10.1214/aop/1029867132.


    G. Gripenberg, S. O. Londen and O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, 1990.doi: 10.1017/CBO9780511662805.


    A. Grorud and E. Pardoux, Intégrales Hilbertiennes anticipantes par rapport à un processus de Wienner cylindrique et calcul stochastique associé, Appl. Math. Optim., 25 (1992), 31-49.doi: 10.1007/BF01184155.


    K. W. Homan, An Analytic Semigroup Approach to Convolution Volterra Equations, Ph.D. Thesis, Delft University Press, 2003.


    G. E. Ladas and V. Lakshmikantham, Differential Equations in Abstract Spaces, Mathematics in Science and Engineering, 85, Academic Press, New York, 1972,


    A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, 2013 reprint of the 1995 original, Modern Birkhäuser Classics, Basel, 1995.


    F. Masiero, A stochastic optimal control problem for the heat equation on the halfline with dirichlet boundary-noise and boundary-control, Appl. Math. Optim., 62 (2010), 253-294.doi: 10.1007/s00245-010-9103-z.


    R. K. Miller, Linear Volterra integrodifferential equations as semigroups, Funkcial. Ekvac., 17 (1974), 39-55.


    D. Nualart, Malliavin Calculus and Related Topics, Probability and its Applications (New York), Springer-Verlag, Berlin, 2006.


    B. Øksendal and T. Zhang, The stochastic Volterra equation, in Barcelona Seminar on Stochastic Analysis (St. Feliu de Guíxols, 1991), Progr. Probab., 32, Birkhäuser, Basel, 1993, 168-202.


    B. Øksendal and T. Zhang, The general linear stochastic Volterra equations with anticipating coefficients, in Stochastic Analysis and Applications (Powys, 1995), World Sci. Publ., River Edge, NJ, 1996, 343-366.


    E. Pardoux, Nonlinear filtering, prediction and smoothing equations, Stochastics, 6 (1981/82), 193-231. doi: 10.1080/17442508208833204.


    J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87, Birkhäuser Verlag, Basel, 1993.doi: 10.1007/978-3-0348-8570-6.


    W. R. Schneider and W. Wyss, Fractional diffusion and wave equations, Journal of Mathematical Physics, 30 (1989), 134-144.doi: 10.1063/1.528578.

  • 加载中

Article Metrics

HTML views() PDF downloads(83) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint