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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[9]

Ph. Clément and J. Prüss, Completely positive measures and Feller semigroups, Math. Ann. 287 (1990), 73-105. doi: 10.1007/BF01446879.  Google Scholar

[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.  Google Scholar

[11]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  Google Scholar

[12]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224. doi: 10.1137/0521066.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

J. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204.  Google Scholar

[22]

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

[23]

R. B. Sowers, Multidimensional reaction-diffusion equations with white noise boundary perturbations, Ann. Probab., 22 (1994), 2071-2121. doi: 10.1214/aop/1176988495.  Google Scholar

[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.  Google Scholar

[25]

I. I. Vrabie, $C_0$-semigroups and Applications, North-Holland Mathematics Studies, 191. North-Holland Publishing Co., Amsterdam, 2003, xii+373 pp.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[9]

Ph. Clément and J. Prüss, Completely positive measures and Feller semigroups, Math. Ann. 287 (1990), 73-105. doi: 10.1007/BF01446879.  Google Scholar

[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.  Google Scholar

[11]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  Google Scholar

[12]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224. doi: 10.1137/0521066.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

J. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204.  Google Scholar

[22]

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

[23]

R. B. Sowers, Multidimensional reaction-diffusion equations with white noise boundary perturbations, Ann. Probab., 22 (1994), 2071-2121. doi: 10.1214/aop/1176988495.  Google Scholar

[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.  Google Scholar

[25]

I. I. Vrabie, $C_0$-semigroups and Applications, North-Holland Mathematics Studies, 191. North-Holland Publishing Co., Amsterdam, 2003, xii+373 pp.  Google Scholar

[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.  Google Scholar

[1]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control & Related Fields, 2015, 5 (3) : 613-649. doi: 10.3934/mcrf.2015.5.613

[2]

Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115

[3]

Jiongmin Yong. Forward-backward evolution equations and applications. Mathematical Control & Related Fields, 2016, 6 (4) : 653-704. doi: 10.3934/mcrf.2016019

[4]

Fabio Paronetto. Elliptic approximation of forward-backward parabolic equations. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1017-1036. doi: 10.3934/cpaa.2020047

[5]

Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations & Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035

[6]

Kazuyuki Yagasaki. Optimal control of the SIR epidemic model based on dynamical systems theory. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021144

[7]

G. Bellettini, Giorgio Fusco, Nicola Guglielmi. A concept of solution and numerical experiments for forward-backward diffusion equations. Discrete & Continuous Dynamical Systems, 2006, 16 (4) : 783-842. doi: 10.3934/dcds.2006.16.783

[8]

Flavia Smarrazzo, Alberto Tesei. Entropy solutions of forward-backward parabolic equations with Devonshire free energy. Networks & Heterogeneous Media, 2012, 7 (4) : 941-966. doi: 10.3934/nhm.2012.7.941

[9]

Jie Xiong, Shuaiqi Zhang, Yi Zhuang. A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance. Mathematical Control & Related Fields, 2019, 9 (2) : 257-276. doi: 10.3934/mcrf.2019013

[10]

Chunqiu Li, Desheng Li, Xuewei Ju. On the forward dynamical behavior of nonautonomous systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 473-487. doi: 10.3934/dcdsb.2019190

[11]

Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems & Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002

[12]

Alexandra Rodkina, Henri Schurz, Leonid Shaikhet. Almost sure stability of some stochastic dynamical systems with memory. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 571-593. doi: 10.3934/dcds.2008.21.571

[13]

Elena Goncharova, Maxim Staritsyn. Optimal control of dynamical systems with polynomial impulses. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4367-4384. doi: 10.3934/dcds.2015.35.4367

[14]

Liangquan Zhang, Qing Zhou, Juan Yang. Necessary condition for optimal control of doubly stochastic systems. Mathematical Control & Related Fields, 2020, 10 (2) : 379-403. doi: 10.3934/mcrf.2020002

[15]

Xiao Ding, Deren Han. A modification of the forward-backward splitting method for maximal monotone mappings. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 295-307. doi: 10.3934/naco.2013.3.295

[16]

Andrés Contreras, Juan Peypouquet. Forward-backward approximation of nonlinear semigroups in finite and infinite horizon. Communications on Pure & Applied Analysis, 2021, 20 (5) : 1893-1906. doi: 10.3934/cpaa.2021051

[17]

Hongyong Cui, Peter E. Kloeden, Meihua Yang. Forward omega limit sets of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1103-1114. doi: 10.3934/dcdss.2020065

[18]

Salah-Eldin A. Mohammed, Tusheng Zhang. Large deviations for stochastic systems with memory. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 881-893. doi: 10.3934/dcdsb.2006.6.881

[19]

Ishak Alia. Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach. Mathematical Control & Related Fields, 2020, 10 (4) : 785-826. doi: 10.3934/mcrf.2020020

[20]

Urszula Ledzewicz, Stanislaw Walczak. Optimal control of systems governed by some elliptic equations. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 279-290. doi: 10.3934/dcds.1999.5.279

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (42)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]