# American Institute of Mathematical Sciences

March  2014, 3(1): 35-58. doi: 10.3934/eect.2014.3.35

## Optimal control for stochastic heat equation with memory

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

Received  December 2012 Revised  July 2013 Published  February 2014

In this paper, we investigate the existence and uniqueness of solutions for a class of evolutionary integral equations perturbed by a noise arising in the theory of heat conduction. As a motivation of our results, we study an optimal control problem when the control enters the system together with the noise.
Citation: Fulvia Confortola, Elisa Mastrogiacomo. Optimal control for stochastic heat equation with memory. Evolution Equations and Control Theory, 2014, 3 (1) : 35-58. doi: 10.3934/eect.2014.3.35
##### References:
 [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. [2] S. Bonaccorsi and W. Desch, Volterra equations perturbed by noise, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 557-594. doi: 10.1007/s00030-012-0167-0. [3] S. Bonaccorsi, G. Da Prato and L. Tubaro, Asymptotic behavior of a class of nonlinear heat conduction problems with memory effects, SIAM J. Math. Anal. 44 (2012), 1562-1587. doi: 10.1137/110841795. [4] T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B 9 (2008), 525-539. doi: 10.3934/dcdsb.2008.9.525. [5] T. Caraballo, I. D. Chueshov, P. Marín-Rubio and J. Real, Existence and asymptotic behavior for stochastic heat equations with multiplicative noise in materials with memory, Discrete Contin. Dyn. Syst. 18 (2007), 253-270. doi: 10.3934/dcds.2007.18.253. [6] Ph. Clément and G. Da Prato, White noise perturbation of the heat equation in materials with memory, Dynam. Systems Appl. 6 (1997), 441-460. [7] Ph. Clément and G. Da Prato and J. Prüss, White noise perturbation of the equations of linear parabolic viscoelasticity, Rend. Istit. Mat. Univ. Trieste 29 (1997), 207-220 (1998). [8] Ph. Clément and J. A. Nohel, Abstract linear and nonlinear Volterra equations preserving positivity, SIAM J. Math. Anal. 10 (1979), 365-388. doi: 10.1137/0510035. [9] Ph. Clément and J. Prüss, Completely positive measures and Feller semigroups, Math. Ann. 287 (1990), 73-105. doi: 10.1007/BF01446879. [10] M. Conti, E. M. Marchini and V. Pata, Semilinear wave equations of viscoelasticity in the minimal state framework, Discrete Contin. Dyn. Syst. 27 (2010), 1535-1552. doi: 10.3934/dcds.2010.27.1535. [11] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. [12] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. [13] 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. [14] M. Grasselli and V. Pata, Upper semicontinuous attractor for a hyperbolic phase-field model with memory, Indiana Univ. Math. J., 50 (2001), 1281-1308. doi: 10.1512/iumj.2001.50.2122. [15] M. Grasselli and V. Pata, A reaction-diffusion equation with memory, Discrete Contin. Dyn. Syst., 15 (2006), 1079-1088. doi: 10.3934/dcds.2006.15.1079. [16] M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373. [17] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170 Springer-Verlag, New York-Berlin, 1971. xi+396 pp. [18] A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224. doi: 10.1137/0521066. [19] R. K. Miller, Linear Volterra integrodifferential equations as semigroups, Funkcial. Ekvac., 17 (1974), 39-55. [20] S. Monniaux and J. Prüss, A theorem of the Dore-Venni type for noncommuting operators, Trans. Amer. Math. Soc. 349 (1997), 4787-4814. doi: 10.1090/S0002-9947-97-01997-1. [21] J. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204. [22] J. Prüs, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87. Birkhüser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6. [23] R. B. Sowers, Multidimensional reaction-diffusion equations with white noise boundary perturbations, Ann. Probab., 22 (1994), 2071-2121. doi: 10.1214/aop/1176988495. [24] F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Translated from the 2005 German original by Jorgen Sprekels. Graduate Studies in Mathematics, 112. American Mathematical Society, Providence, RI, 2010. xvi+399 pp. [25] I. I. Vrabie, $C_0$-semigroups and Applications, North-Holland Mathematics Studies, 191. North-Holland Publishing Co., Amsterdam, 2003, xii+373 pp. [26] J. Yong and X. Y. Zhou, Stochastic Controls. Hamiltonian Systems and HJB Equations. Applications of Mathematics, (New York), 43. Springer-Verlag, New York, 1999.

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##### References:
 [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. [2] S. Bonaccorsi and W. Desch, Volterra equations perturbed by noise, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 557-594. doi: 10.1007/s00030-012-0167-0. [3] S. Bonaccorsi, G. Da Prato and L. Tubaro, Asymptotic behavior of a class of nonlinear heat conduction problems with memory effects, SIAM J. Math. Anal. 44 (2012), 1562-1587. doi: 10.1137/110841795. [4] T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B 9 (2008), 525-539. doi: 10.3934/dcdsb.2008.9.525. [5] T. Caraballo, I. D. Chueshov, P. Marín-Rubio and J. Real, Existence and asymptotic behavior for stochastic heat equations with multiplicative noise in materials with memory, Discrete Contin. Dyn. Syst. 18 (2007), 253-270. doi: 10.3934/dcds.2007.18.253. [6] Ph. Clément and G. Da Prato, White noise perturbation of the heat equation in materials with memory, Dynam. Systems Appl. 6 (1997), 441-460. [7] Ph. Clément and G. Da Prato and J. Prüss, White noise perturbation of the equations of linear parabolic viscoelasticity, Rend. Istit. Mat. Univ. Trieste 29 (1997), 207-220 (1998). [8] Ph. Clément and J. A. Nohel, Abstract linear and nonlinear Volterra equations preserving positivity, SIAM J. Math. Anal. 10 (1979), 365-388. doi: 10.1137/0510035. [9] Ph. Clément and J. Prüss, Completely positive measures and Feller semigroups, Math. Ann. 287 (1990), 73-105. doi: 10.1007/BF01446879. [10] M. Conti, E. M. Marchini and V. Pata, Semilinear wave equations of viscoelasticity in the minimal state framework, Discrete Contin. Dyn. Syst. 27 (2010), 1535-1552. doi: 10.3934/dcds.2010.27.1535. [11] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. [12] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. [13] 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. [14] M. Grasselli and V. Pata, Upper semicontinuous attractor for a hyperbolic phase-field model with memory, Indiana Univ. Math. J., 50 (2001), 1281-1308. doi: 10.1512/iumj.2001.50.2122. [15] M. Grasselli and V. Pata, A reaction-diffusion equation with memory, Discrete Contin. Dyn. Syst., 15 (2006), 1079-1088. doi: 10.3934/dcds.2006.15.1079. [16] M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373. [17] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170 Springer-Verlag, New York-Berlin, 1971. xi+396 pp. [18] A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224. doi: 10.1137/0521066. [19] R. K. Miller, Linear Volterra integrodifferential equations as semigroups, Funkcial. Ekvac., 17 (1974), 39-55. [20] S. Monniaux and J. Prüss, A theorem of the Dore-Venni type for noncommuting operators, Trans. Amer. Math. Soc. 349 (1997), 4787-4814. doi: 10.1090/S0002-9947-97-01997-1. [21] J. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204. [22] J. Prüs, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87. Birkhüser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6. [23] R. B. Sowers, Multidimensional reaction-diffusion equations with white noise boundary perturbations, Ann. Probab., 22 (1994), 2071-2121. doi: 10.1214/aop/1176988495. [24] F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Translated from the 2005 German original by Jorgen Sprekels. Graduate Studies in Mathematics, 112. American Mathematical Society, Providence, RI, 2010. xvi+399 pp. [25] I. I. Vrabie, $C_0$-semigroups and Applications, North-Holland Mathematics Studies, 191. North-Holland Publishing Co., Amsterdam, 2003, xii+373 pp. [26] J. Yong and X. Y. Zhou, Stochastic Controls. Hamiltonian Systems and HJB Equations. Applications of Mathematics, (New York), 43. Springer-Verlag, New York, 1999.
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