American Institute of Mathematical Sciences

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, (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. [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. 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. [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). [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. 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. 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, (1990). [10] K. J. Engel, Spectral theory and generator property for one-sided coupled operator matrices,, Semigroup Forum, 58 (1999), 267. 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. doi: 10.1007/BF00251802. [12] W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,", Springer-Verlag, (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. doi: 10.1214/aop/1029867132. [14] J. Keener and J. Sneyd, "Mathematical Physiology,", Springer, (1998). [15] M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks,, Math. Z., 249 (2005), 139. 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. [17] F. Masiero, Stochastic optimal control problems and parabolic equations in Banach spaces,, SIAM J. Control Optim., 47 (2008), 251. doi: 10.1137/050632725. [18] T. Mátrai and E. Sikolya, Asymptotic behavior of flows in networks,, Forum Math., 19 (2007), 429. [19] V. G. Maz'ja, "Sobolev Spaces,", Springer Series in Soviet Mathematics, (1985). [20] M. Métivier, "Semimartingales,", Walter de Gruyter & Co., (1982). [21] D. Mugnolo, Gaussian estimates for a heat equation on a network,, Netw. Heterog. Media, 2 (2007), 55. 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. doi: 10.1002/mma.805. [23] J. D. Murray, "Mathematical Biology. I," third ed.,, Interdisciplinary Applied Mathematics, 17 (2002). [24] R. Nagel, Towards a "matrix theory" for unbounded operator matrices,, Mathematische Zeitschrift, 201 (1989), 57. [25] E. M. Ouhabaz, "Analysis of Heat Equations on Domains,", London Mathematical Society Monographs Series, 31 (2005). [26] D. W. Robinson, "Elliptic Operators and Lie Groups,", Oxford Mathematical Monographs, (1991). [27] C. Rocşoreanu, A. Georgescu and N. Giurgiţeanu, "The FitzHugh-Nagumo Model,", Mathematical Modelling: Theory and Applications, 10 (2000). [28] E. Sikolya, Flows in networks with dynamic ramification nodes,, J. Evol. Equ., 5 (2005), 441. doi: 10.1007/s00028-005-0221-z. [29] Henry C. Tuckwell, "Introduction to Theoretical Neurobiology. Vol. 1,", Cambridge Studies in Mathematical Biology, 8 (1988). [30] D. B. West, "Introduction to Graph Theory - Second Edition,", Prentice Hall Inc., (2001).

show all references

References:
 [1] W. Arendt, Heat kernels,, Manuscript of the 9th Internet Seminar, (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. [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. 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. [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). [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. 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. 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, (1990). [10] K. J. Engel, Spectral theory and generator property for one-sided coupled operator matrices,, Semigroup Forum, 58 (1999), 267. 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. doi: 10.1007/BF00251802. [12] W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,", Springer-Verlag, (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. doi: 10.1214/aop/1029867132. [14] J. Keener and J. Sneyd, "Mathematical Physiology,", Springer, (1998). [15] M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks,, Math. Z., 249 (2005), 139. 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. [17] F. Masiero, Stochastic optimal control problems and parabolic equations in Banach spaces,, SIAM J. Control Optim., 47 (2008), 251. doi: 10.1137/050632725. [18] T. Mátrai and E. Sikolya, Asymptotic behavior of flows in networks,, Forum Math., 19 (2007), 429. [19] V. G. Maz'ja, "Sobolev Spaces,", Springer Series in Soviet Mathematics, (1985). [20] M. Métivier, "Semimartingales,", Walter de Gruyter & Co., (1982). [21] D. Mugnolo, Gaussian estimates for a heat equation on a network,, Netw. Heterog. Media, 2 (2007), 55. 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. doi: 10.1002/mma.805. [23] J. D. Murray, "Mathematical Biology. I," third ed.,, Interdisciplinary Applied Mathematics, 17 (2002). [24] R. Nagel, Towards a "matrix theory" for unbounded operator matrices,, Mathematische Zeitschrift, 201 (1989), 57. [25] E. M. Ouhabaz, "Analysis of Heat Equations on Domains,", London Mathematical Society Monographs Series, 31 (2005). [26] D. W. Robinson, "Elliptic Operators and Lie Groups,", Oxford Mathematical Monographs, (1991). [27] C. Rocşoreanu, A. Georgescu and N. Giurgiţeanu, "The FitzHugh-Nagumo Model,", Mathematical Modelling: Theory and Applications, 10 (2000). [28] E. Sikolya, Flows in networks with dynamic ramification nodes,, J. Evol. Equ., 5 (2005), 441. doi: 10.1007/s00028-005-0221-z. [29] Henry C. Tuckwell, "Introduction to Theoretical Neurobiology. Vol. 1,", Cambridge Studies in Mathematical Biology, 8 (1988). [30] D. B. West, "Introduction to Graph Theory - Second Edition,", Prentice Hall Inc., (2001).
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