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 & 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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[30]

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

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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[30]

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

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