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), 50 (2012), 748.  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.  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), 44 (2012), 1562.  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), 9 (2008), 525.  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), 18 (2007), 253.  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), 6 (1997), 441.   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), 29 (1997), 207.   Google Scholar

[8]

Ph. Clément and J. A. Nohel, Abstract linear and nonlinear Volterra equations preserving positivity,, SIAM J. Math. Anal. 10 (1979), 10 (1979), 365.  doi: 10.1137/0510035.  Google Scholar

[9]

Ph. Clément and J. Prüss, Completely positive measures and Feller semigroups,, Math. Ann. 287 (1990), 287 (1990), 73.  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), 27 (2010), 1535.  doi: 10.3934/dcds.2010.27.1535.  Google Scholar

[11]

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

[12]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge Univ. Press, (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.  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.  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.  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.  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, (1971).   Google Scholar

[18]

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

[19]

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

[20]

S. Monniaux and J. Prüss, A theorem of the Dore-Venni type for noncommuting operators,, Trans. Amer. Math. Soc. 349 (1997), 349 (1997), 4787.  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.   Google Scholar

[22]

J. Prüs, Evolutionary Integral Equations and Applications,, Monographs in Mathematics, (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.  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 (2005).   Google Scholar

[25]

I. I. Vrabie, $C_0$-semigroups and Applications,, North-Holland Mathematics Studies, 191 (2003).   Google Scholar

[26]

J. Yong and X. Y. Zhou, Stochastic Controls. Hamiltonian Systems and HJB Equations. Applications of Mathematics,, (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), 50 (2012), 748.  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.  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), 44 (2012), 1562.  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), 9 (2008), 525.  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), 18 (2007), 253.  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), 6 (1997), 441.   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), 29 (1997), 207.   Google Scholar

[8]

Ph. Clément and J. A. Nohel, Abstract linear and nonlinear Volterra equations preserving positivity,, SIAM J. Math. Anal. 10 (1979), 10 (1979), 365.  doi: 10.1137/0510035.  Google Scholar

[9]

Ph. Clément and J. Prüss, Completely positive measures and Feller semigroups,, Math. Ann. 287 (1990), 287 (1990), 73.  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), 27 (2010), 1535.  doi: 10.3934/dcds.2010.27.1535.  Google Scholar

[11]

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

[12]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge Univ. Press, (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.  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.  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.  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.  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, (1971).   Google Scholar

[18]

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

[19]

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

[20]

S. Monniaux and J. Prüss, A theorem of the Dore-Venni type for noncommuting operators,, Trans. Amer. Math. Soc. 349 (1997), 349 (1997), 4787.  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.   Google Scholar

[22]

J. Prüs, Evolutionary Integral Equations and Applications,, Monographs in Mathematics, (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.  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 (2005).   Google Scholar

[25]

I. I. Vrabie, $C_0$-semigroups and Applications,, North-Holland Mathematics Studies, 191 (2003).   Google Scholar

[26]

J. Yong and X. Y. Zhou, Stochastic Controls. Hamiltonian Systems and HJB Equations. Applications of Mathematics,, (New York), (1999).   Google Scholar

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