American Institute of Mathematical Sciences

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

Gaussian estimates on networks with applications to optimal control

Luca Di Persio 1, and  Giacomo Ziglio 1,

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.
Keywords: Optimal stochastic control., Stochastic partial differential equations on networks, Gaussian estimates.
Mathematics Subject Classification: Primary: 35R02, 60H15, 93E20; Secondary: 90B1.
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).   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.   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.  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.   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).   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.  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.  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, (1990).   Google Scholar

[10]

K. J. Engel, Spectral theory and generator property for one-sided coupled operator matrices,, Semigroup Forum, 58 (1999), 267.  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.  doi: 10.1007/BF00251802.  Google Scholar

[12]

W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,", Springer-Verlag, (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.  doi: 10.1214/aop/1029867132.  Google Scholar

[14]

J. Keener and J. Sneyd, "Mathematical Physiology,", Springer, (1998).   Google Scholar

[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.  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.   Google Scholar

[17]

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

[18]

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

[19]

V. G. Maz'ja, "Sobolev Spaces,", Springer Series in Soviet Mathematics, (1985).   Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

J. D. Murray, "Mathematical Biology. I," third ed.,, Interdisciplinary Applied Mathematics, 17 (2002).   Google Scholar

[24]

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

[25]

E. M. Ouhabaz, "Analysis of Heat Equations on Domains,", London Mathematical Society Monographs Series, 31 (2005).   Google Scholar

[26]

D. W. Robinson, "Elliptic Operators and Lie Groups,", Oxford Mathematical Monographs, (1991).   Google Scholar

[27]

C. Rocşoreanu, A. Georgescu and N. Giurgiţeanu, "The FitzHugh-Nagumo Model,", Mathematical Modelling: Theory and Applications, 10 (2000).   Google Scholar

[28]

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

[29]

Henry C. Tuckwell, "Introduction to Theoretical Neurobiology. Vol. 1,", Cambridge Studies in Mathematical Biology, 8 (1988).   Google Scholar

[30]

D. B. West, "Introduction to Graph Theory - Second Edition,", Prentice Hall Inc., (2001).   Google Scholar

show all references

References:
[1]

W. Arendt, Heat kernels,, Manuscript of the 9th Internet Seminar, (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.   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.  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.   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).   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.  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.  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, (1990).   Google Scholar

[10]

K. J. Engel, Spectral theory and generator property for one-sided coupled operator matrices,, Semigroup Forum, 58 (1999), 267.  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.  doi: 10.1007/BF00251802.  Google Scholar

[12]

W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,", Springer-Verlag, (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.  doi: 10.1214/aop/1029867132.  Google Scholar

[14]

J. Keener and J. Sneyd, "Mathematical Physiology,", Springer, (1998).   Google Scholar

[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.  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.   Google Scholar

[17]

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

[18]

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

[19]

V. G. Maz'ja, "Sobolev Spaces,", Springer Series in Soviet Mathematics, (1985).   Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

J. D. Murray, "Mathematical Biology. I," third ed.,, Interdisciplinary Applied Mathematics, 17 (2002).   Google Scholar

[24]

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

[25]

E. M. Ouhabaz, "Analysis of Heat Equations on Domains,", London Mathematical Society Monographs Series, 31 (2005).   Google Scholar

[26]

D. W. Robinson, "Elliptic Operators and Lie Groups,", Oxford Mathematical Monographs, (1991).   Google Scholar

[27]

C. Rocşoreanu, A. Georgescu and N. Giurgiţeanu, "The FitzHugh-Nagumo Model,", Mathematical Modelling: Theory and Applications, 10 (2000).   Google Scholar

[28]

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

[29]

Henry C. Tuckwell, "Introduction to Theoretical Neurobiology. Vol. 1,", Cambridge Studies in Mathematical Biology, 8 (1988).   Google Scholar

[30]

D. B. West, "Introduction to Graph Theory - Second Edition,", Prentice Hall Inc., (2001).   Google Scholar

[1]

Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199
[2]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[3]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[4]

Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203
[5]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221
[6]

Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks & Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014
[7]

Sergio Albeverio, Sonia Mazzucchi. Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena. Journal of Geometric Mechanics, 2019, 11 (2) : 123-137. doi: 10.3934/jgm.2019006
[8]

Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061
[9]

Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control & Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97
[10]

Weidong Zhao, Jinlei Wang, Shige Peng. Error estimates of the $\theta$-scheme for backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 905-924. doi: 10.3934/dcdsb.2009.12.905
[11]

Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4099-4116. doi: 10.3934/dcdsb.2019052
[12]

Fulvia Confortola, Elisa Mastrogiacomo. Feedback optimal control for stochastic Volterra equations with completely monotone kernels. Mathematical Control & Related Fields, 2015, 5 (2) : 191-235. doi: 10.3934/mcrf.2015.5.191
[13]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control & Related Fields, 2015, 5 (3) : 613-649. doi: 10.3934/mcrf.2015.5.613
[14]

N. U. Ahmed. Weak solutions of stochastic reaction diffusion equations and their optimal control. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1011-1029. doi: 10.3934/dcdss.2018059
[15]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473
[16]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
[17]

Jiahui Zhu, Zdzisław Brzeźniak. Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3269-3299. doi: 10.3934/dcdsb.2016097
[18]

Kai Liu. Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3915-3934. doi: 10.3934/dcdsb.2018117
[19]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019213
[20]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences