# American Institute of Mathematical Sciences

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.

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##### 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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