June  2011, 6(2): 279-296. doi: 10.3934/nhm.2011.6.279

Gaussian estimates on networks with applications to optimal control

1. 

Department of Mathematics, University of Trento, Povo (TN), 38123, Italy, Italy

Received  April 2010 Revised  April 2011 Published  May 2011

We study a class of reaction-diffusion type equations on a finite network with continuity assumptions and a kind of non-local, stationary Kirchhoff's conditions at the nodes. A multiplicative random Gaussian perturbation acting along the edges is also included. For such a problem we prove Gaussian estimates for the semigroup generated by the evolution operator, hence generalizing similar results previously obtained in [21]. In particular our main goal is to extend known results on Gaussian upper bounds for heat equations on networks with local boundary conditions to those with non-local ones. We conclude showing how our results can be used to apply techniques developed in [13] to solve a class of Stochastic Optimal Control Problems inspired by neurological dynamics.
Citation: Luca Di Persio, Giacomo Ziglio. Gaussian estimates on networks with applications to optimal control. Networks and Heterogeneous Media, 2011, 6 (2) : 279-296. doi: 10.3934/nhm.2011.6.279
References:
[1]

W. Arendt, Heat kernels, Manuscript of the 9th Internet Seminar, Freely available at http://tulka.mathematik.uni-ulm.de/2005/lectures/internetseminar.pdf, 2006.

[2]

W. Arendt and A. F. M. ter Elst, Gaussian estimates for second order elliptic operators with boundary conditions, J. Operator Theory, 38 (1997), 87-130.

[3]

S. Bonaccorsi, F. Confortola and E. Mastrogiacomo, Optimal control of stochastic differential equations with dynamical boundary conditions, J. Math. Anal. Appl., 344 (2008), 667-681. doi: 10.1016/j.jmaa.2008.03.013.

[4]

S. Bonaccorsi, C. Marinelli and G. Ziglio, Stochastic FitzHugh-Nagumo equations on networks with impulsive noise, Electron. J. Probab., 13 (2008), 1362-1379.

[5]

A. J. V. Brandāo, E. Fernández-Cara, P. M. D. Magalhāes and M. A. Rojas-Medar, Theoretical analysis and control results for the FitzHugh-Nagumo equation, Electron. J. Differential Equations, (2008), No. 164, 20.

[6]

V. Casarino, K.-J. Engel, R. Nagel and G. Nickel, A semigroup approach to boundary feedback systems, Integral Equations Operator Theory, 47 (2003), 289-306. doi: 10.1007/s00020-002-1163-2.

[7]

S. Cerrai, Optimal control problems for stochastic reaction-diffusion systems with non-Lipschitz coefficients, SIAM J. Control Optim., 39 (2001), 1779-1816 (electronic). doi: 10.1137/S0363012999356465.

[8]

G. Da Prato and J. Zabczyk, "Ergodicity for Infinite-Dimensional Systems," Cambridge UP, 1996.

[9]

E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1990.

[10]

K. J. Engel, Spectral theory and generator property for one-sided coupled operator matrices, Semigroup Forum, 58 (1999), 267-295. doi: 10.1007/s002339900020.

[11]

E. B. Fabes and D. W. Stroock, A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech.Anal., 96 (1986), 327-338. doi: 10.1007/BF00251802.

[12]

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

[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]

J. Keener and J. Sneyd, "Mathematical Physiology," Springer, New York, 1998.

[15]

M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks, Math. Z., 249 (2005), 139-162. doi: 10.1007/s00209-004-0695-3.

[16]

M. Kramar Fijavž, D. Mugnolo and E. Sikolya, Variational and semigroup methods for waves and diffusion in networks, Appl. Math. Optim., 55 (2007), 219-240.

[17]

F. Masiero, Stochastic optimal control problems and parabolic equations in Banach spaces, SIAM J. Control Optim., 47 (2008), 251-300. doi: 10.1137/050632725.

[18]

T. Mátrai and E. Sikolya, Asymptotic behavior of flows in networks, Forum Math., 19 (2007), 429-461.

[19]

V. G. Maz'ja, "Sobolev Spaces," Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985, Translated from the Russian by T. O. Shaposhnikova.

[20]

M. Métivier, "Semimartingales," Walter de Gruyter & Co., Berlin, 1982.

[21]

D. Mugnolo, Gaussian estimates for a heat equation on a network, Netw. Heterog. Media, 2 (2007), 55-79 (electronic). doi: 10.3934/nhm.2007.2.55.

[22]

D. Mugnolo and S. Romanelli, Dynamic and generalized Wentzell node conditions for network equations, Math. Methods Appl. Sci., 30 (2007), 681-706. doi: 10.1002/mma.805.

[23]

J. D. Murray, "Mathematical Biology. I," third ed., Interdisciplinary Applied Mathematics, vol. 17, Springer-Verlag, New York, 2002, An introduction.

[24]

R. Nagel, Towards a "matrix theory" for unbounded operator matrices, Mathematische Zeitschrift, 201 (1989), 57-68.

[25]

E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Mathematical Society Monographs Series, vol. 31, Princeton University Press, Princeton, NJ, 2005.

[26]

D. W. Robinson, "Elliptic Operators and Lie Groups," Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1991, Oxford Science Publications.

[27]

C. Rocşoreanu, A. Georgescu and N. Giurgiţeanu, "The FitzHugh-Nagumo Model," Mathematical Modelling: Theory and Applications, vol. 10, Kluwer Academic Publishers, Dordrecht, 2000, Bifurcation and dynamics.

[28]

E. Sikolya, Flows in networks with dynamic ramification nodes, J. Evol. Equ., 5 (2005), 441-463. doi: 10.1007/s00028-005-0221-z.

[29]

Henry C. Tuckwell, "Introduction to Theoretical Neurobiology. Vol. 1," Cambridge Studies in Mathematical Biology, vol. 8, Cambridge University Press, Cambridge, 1988, Linear cable theory and dendritic structure.

[30]

D. B. West, "Introduction to Graph Theory - Second Edition," Prentice Hall Inc., Upper Saddle River, NJ, 2001.

show all references

References:
[1]

W. Arendt, Heat kernels, Manuscript of the 9th Internet Seminar, Freely available at http://tulka.mathematik.uni-ulm.de/2005/lectures/internetseminar.pdf, 2006.

[2]

W. Arendt and A. F. M. ter Elst, Gaussian estimates for second order elliptic operators with boundary conditions, J. Operator Theory, 38 (1997), 87-130.

[3]

S. Bonaccorsi, F. Confortola and E. Mastrogiacomo, Optimal control of stochastic differential equations with dynamical boundary conditions, J. Math. Anal. Appl., 344 (2008), 667-681. doi: 10.1016/j.jmaa.2008.03.013.

[4]

S. Bonaccorsi, C. Marinelli and G. Ziglio, Stochastic FitzHugh-Nagumo equations on networks with impulsive noise, Electron. J. Probab., 13 (2008), 1362-1379.

[5]

A. J. V. Brandāo, E. Fernández-Cara, P. M. D. Magalhāes and M. A. Rojas-Medar, Theoretical analysis and control results for the FitzHugh-Nagumo equation, Electron. J. Differential Equations, (2008), No. 164, 20.

[6]

V. Casarino, K.-J. Engel, R. Nagel and G. Nickel, A semigroup approach to boundary feedback systems, Integral Equations Operator Theory, 47 (2003), 289-306. doi: 10.1007/s00020-002-1163-2.

[7]

S. Cerrai, Optimal control problems for stochastic reaction-diffusion systems with non-Lipschitz coefficients, SIAM J. Control Optim., 39 (2001), 1779-1816 (electronic). doi: 10.1137/S0363012999356465.

[8]

G. Da Prato and J. Zabczyk, "Ergodicity for Infinite-Dimensional Systems," Cambridge UP, 1996.

[9]

E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1990.

[10]

K. J. Engel, Spectral theory and generator property for one-sided coupled operator matrices, Semigroup Forum, 58 (1999), 267-295. doi: 10.1007/s002339900020.

[11]

E. B. Fabes and D. W. Stroock, A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech.Anal., 96 (1986), 327-338. doi: 10.1007/BF00251802.

[12]

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

[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]

J. Keener and J. Sneyd, "Mathematical Physiology," Springer, New York, 1998.

[15]

M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks, Math. Z., 249 (2005), 139-162. doi: 10.1007/s00209-004-0695-3.

[16]

M. Kramar Fijavž, D. Mugnolo and E. Sikolya, Variational and semigroup methods for waves and diffusion in networks, Appl. Math. Optim., 55 (2007), 219-240.

[17]

F. Masiero, Stochastic optimal control problems and parabolic equations in Banach spaces, SIAM J. Control Optim., 47 (2008), 251-300. doi: 10.1137/050632725.

[18]

T. Mátrai and E. Sikolya, Asymptotic behavior of flows in networks, Forum Math., 19 (2007), 429-461.

[19]

V. G. Maz'ja, "Sobolev Spaces," Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985, Translated from the Russian by T. O. Shaposhnikova.

[20]

M. Métivier, "Semimartingales," Walter de Gruyter & Co., Berlin, 1982.

[21]

D. Mugnolo, Gaussian estimates for a heat equation on a network, Netw. Heterog. Media, 2 (2007), 55-79 (electronic). doi: 10.3934/nhm.2007.2.55.

[22]

D. Mugnolo and S. Romanelli, Dynamic and generalized Wentzell node conditions for network equations, Math. Methods Appl. Sci., 30 (2007), 681-706. doi: 10.1002/mma.805.

[23]

J. D. Murray, "Mathematical Biology. I," third ed., Interdisciplinary Applied Mathematics, vol. 17, Springer-Verlag, New York, 2002, An introduction.

[24]

R. Nagel, Towards a "matrix theory" for unbounded operator matrices, Mathematische Zeitschrift, 201 (1989), 57-68.

[25]

E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Mathematical Society Monographs Series, vol. 31, Princeton University Press, Princeton, NJ, 2005.

[26]

D. W. Robinson, "Elliptic Operators and Lie Groups," Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1991, Oxford Science Publications.

[27]

C. Rocşoreanu, A. Georgescu and N. Giurgiţeanu, "The FitzHugh-Nagumo Model," Mathematical Modelling: Theory and Applications, vol. 10, Kluwer Academic Publishers, Dordrecht, 2000, Bifurcation and dynamics.

[28]

E. Sikolya, Flows in networks with dynamic ramification nodes, J. Evol. Equ., 5 (2005), 441-463. doi: 10.1007/s00028-005-0221-z.

[29]

Henry C. Tuckwell, "Introduction to Theoretical Neurobiology. Vol. 1," Cambridge Studies in Mathematical Biology, vol. 8, Cambridge University Press, Cambridge, 1988, Linear cable theory and dendritic structure.

[30]

D. B. West, "Introduction to Graph Theory - Second Edition," Prentice Hall Inc., Upper Saddle River, NJ, 2001.

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