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

Feedback optimal control for stochastic Volterra equations with completely monotone kernels

Fulvia Confortola 1, and  Elisa Mastrogiacomo 2,

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 $x$.
Keywords: optimal control, backward stochastic differential equations, HJB equation, Stochastic Volterra integral equation, feedback law..
Mathematics Subject Classification: 45D05, 93E20, 60H3.
Citation: Fulvia Confortola, Elisa Mastrogiacomo. Feedback optimal control for stochastic Volterra equations with completely monotone kernels. Mathematical Control & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[19]

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

[20]

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

[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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[26]

E. Pardoux, Nonlinear filtering, prediction and smoothing equations,, Stochastics, 6 (): 193.  doi: 10.1080/17442508208833204.  Google Scholar

[27]

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

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[19]

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

[20]

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

[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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[26]

E. Pardoux, Nonlinear filtering, prediction and smoothing equations,, Stochastics, 6 (): 193.  doi: 10.1080/17442508208833204.  Google Scholar

[27]

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

[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.  Google Scholar

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