# American Institute of Mathematical Sciences

June  2015, 5(2): 191-235. doi: 10.3934/mcrf.2015.5.191

## Feedback optimal control for stochastic Volterra equations with completely monotone kernels

 1 Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32 - 20133 Milano 2 Dipartimento di statistica e metodi quantitativi, Universitá degli studi di Milano Bicocca, Piazza dell'Ateneo Nuovo, 1 - 20126, Milano

Received  September 2013 Revised  August 2014 Published  April 2015

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 $x$.
Citation: Fulvia Confortola, Elisa Mastrogiacomo. Feedback optimal control for stochastic Volterra equations with completely monotone kernels. Mathematical Control and Related Fields, 2015, 5 (2) : 191-235. doi: 10.3934/mcrf.2015.5.191
##### References:
 [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. [2] 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. [3] S. Bonaccorsi, Some Applications in Malliavin Calculus, Ph.D thesis, Dept. Mathematics, Univ. Trento., 1998. [4] 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. [5] 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. [6] 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. [7] 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. [8] 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. [9] G. Da Prato and J. Zabczyk, Evolution Equations with White-Noise Boundary Conditions, Cambridge Univ. Press, Cambridge, 1993. doi: 10.1080/17442509308833817. [10] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223. [11] 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. [12] 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. [13] K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math., {194}, Springer-Verlag, Berlin, 2000. [14] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993. [15] 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. [16] G. Gripenberg, S. O. Londen and O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, 1990. doi: 10.1017/CBO9780511662805. [17] 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. [18] K. W. Homan, An Analytic Semigroup Approach to Convolution Volterra Equations, Ph.D. Thesis, Delft University Press, 2003. [19] G. E. Ladas and V. Lakshmikantham, Differential Equations in Abstract Spaces, Mathematics in Science and Engineering, 85, Academic Press, New York, 1972, [20] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, 2013 reprint of the 1995 original, Modern Birkhäuser Classics, Basel, 1995. [21] 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. [22] R. K. Miller, Linear Volterra integrodifferential equations as semigroups, Funkcial. Ekvac., 17 (1974), 39-55. [23] D. Nualart, Malliavin Calculus and Related Topics, Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. [24] 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. [25] 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. [26] E. Pardoux, Nonlinear filtering, prediction and smoothing equations, Stochastics, 6 (1981/82), 193-231. doi: 10.1080/17442508208833204. [27] J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6. [28] W. R. Schneider and W. Wyss, Fractional diffusion and wave equations, Journal of Mathematical Physics, 30 (1989), 134-144. doi: 10.1063/1.528578.

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##### References:
 [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. [2] 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. [3] S. Bonaccorsi, Some Applications in Malliavin Calculus, Ph.D thesis, Dept. Mathematics, Univ. Trento., 1998. [4] 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. [5] 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. [6] 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. [7] 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. [8] 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. [9] G. Da Prato and J. Zabczyk, Evolution Equations with White-Noise Boundary Conditions, Cambridge Univ. Press, Cambridge, 1993. doi: 10.1080/17442509308833817. [10] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223. [11] 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. [12] 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. [13] K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math., {194}, Springer-Verlag, Berlin, 2000. [14] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993. [15] 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. [16] G. Gripenberg, S. O. Londen and O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, 1990. doi: 10.1017/CBO9780511662805. [17] 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. [18] K. W. Homan, An Analytic Semigroup Approach to Convolution Volterra Equations, Ph.D. Thesis, Delft University Press, 2003. [19] G. E. Ladas and V. Lakshmikantham, Differential Equations in Abstract Spaces, Mathematics in Science and Engineering, 85, Academic Press, New York, 1972, [20] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, 2013 reprint of the 1995 original, Modern Birkhäuser Classics, Basel, 1995. [21] 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. [22] R. K. Miller, Linear Volterra integrodifferential equations as semigroups, Funkcial. Ekvac., 17 (1974), 39-55. [23] D. Nualart, Malliavin Calculus and Related Topics, Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. [24] 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. [25] 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. [26] E. Pardoux, Nonlinear filtering, prediction and smoothing equations, Stochastics, 6 (1981/82), 193-231. doi: 10.1080/17442508208833204. [27] J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6. [28] W. R. Schneider and W. Wyss, Fractional diffusion and wave equations, Journal of Mathematical Physics, 30 (1989), 134-144. doi: 10.1063/1.528578.
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