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January  2012, 17(1): 303-323. doi: 10.3934/dcdsb.2012.17.303

A passivity-based stability criterion for reaction diffusion systems with interconnected structure

Liming Wang 1,

1. 

Department of Mathematics, California State University, 5151 State University Dr., Los Angeles, CA 90032-8204, United States

Received  May 2010 Revised  April 2011 Published  October 2011

In this paper, stability of a class of reaction diffusion systems is studied. Conditions on global asymptotic stability of the homogeneous equilibrium are derived based on the diagonal stability of a dissipativity matrix. This work extends previous result on global asymptotic stability from cyclic systems to general systems with interconnected structure. In addition, it reformulates the approach using an "input-output" formalism that makes the results easier to understand and apply. A biological example from the Mitogen-Activated Protein Kinase (MAPK) system is provided at the end to illustrate the new approach and the main result.
Keywords: Lyapunov stability., Reaction diffusion systems, global stability.
Mathematics Subject Classification: 35K57, 93D20, 93D3.
Citation: Liming Wang. A passivity-based stability criterion for reaction diffusion systems with interconnected structure. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 303-323. doi: 10.3934/dcdsb.2012.17.303
References:
[1]

S. Abdelmalek and S. Kouachi, Proof of existence of global solutions for m-component reaction-diffusion systems with mixed boundary conditions via the Lyapunov functional method,, J. Phys. A, 40 (2007), 12335.  doi: 10.1088/1751-8113/40/41/005.  Google Scholar

[2]

M. Arcak and E. D. Sontag, A passivity-based stability criterion for a class of biochemical reaction networks,, Mathematical Biosciences and Engineering, 5 (2008), 1.  doi: 10.3934/mbe.2008.5.1.  Google Scholar

[3]

L. Edelstein-Keshet, "Mathematical Models in Biology,", Reprint of the 1988 original, 46 (1988).   Google Scholar

[4]

W. B. Fitzgibbon, S. L. Hollis and J. J. Morgan, Stability and Lyapunov functions for reaction-diffusion systems,, SIAM J. Math. Anal., 28 (1997), 595.  doi: 10.1137/S0036141094272241.  Google Scholar

[5]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Nones in Mathematics, 840 (1981).   Google Scholar

[6]

M. R. Jovanović, M. Arcak and E. D. Sontag, A passivity-based approach to stability of spatially distributed systems with a cyclic interconnection structure,, IEEE Transactions on Circuits and Systems I. Regul. Pap., 2008 (): 75.   Google Scholar

[7]

B. N. Kholodenko, Cell-signalling dynamics in time and space,, Nat. Rev. Mol. Cell. Biol., 7 (2006), 165.  doi: 10.1038/nrm1838.  Google Scholar

[8]

Y. Lou, T. Nagylaki and W.-M. Ni, On diffusion-induced blowups in a mutualistic model,, Nonlinear Analysis, 45 (2001), 329.  doi: 10.1016/S0362-546X(99)00346-6.  Google Scholar

[9]

S. Malham and J. Xin, Global solutions to a reactive Boussinesq system with front data on an infinite domain,, Comm. Math. Phys., 193 (1998), 287.  doi: 10.1007/s002200050330.  Google Scholar

[10]

J. Morgan, Global existence for semilinear parabolic systems,, SIAM J. Math. Anal., 20 (1989), 1128.  doi: 10.1137/0520075.  Google Scholar

[11]

J. Morgan, Boundedness and decay results for reaction-diffusion systems,, SIAM J. Math. Anal., 21 (1990), 1172.  doi: 10.1137/0521064.  Google Scholar

[12]

P. J. Moylan and D. J. Hill, Stability criteria for large-scale systems,, IEEE Trans. Autom. Control, 23 (1978), 143.  doi: 10.1109/TAC.1978.1101721.  Google Scholar

[13]

J. Murray, "Mathematical Biology,", Biomathematics, 19 (1989).   Google Scholar

[14]

H. Othmer and E. Pate, Scale-invariance in reaction-diffusion models of spatial pattern formation,, Proc. Natl. Acad. Sci., 77 (1980), 4180.  doi: 10.1073/pnas.77.7.4180.  Google Scholar

[15]

M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass,, SIAM Review, 42 (2000), 93.  doi: 10.1137/S0036144599359735.  Google Scholar

[16]

R. Redheffer, R. Redlinger and W. Walter, A theorem of La Salle-Lyapunov type for parabolic systems,, SIAM J. Math. Anal., 19 (1988), 121.   Google Scholar

[17]

W. Rudin, "Real and Complex Analysis,", Third edition, (1987).   Google Scholar

[18]

S. D. M. Santos, P. J. Verveer and P. I. H. Bastiaens, Growth factor-induced MAPK network topology shapes Erk response determining PC-12 cell fate,, Nature Cell Biology, 9 (2007), 324.  doi: 10.1038/ncb1543.  Google Scholar

[19]

H. L. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,", Mathematical Surveys and Monographs, 41 (1995).   Google Scholar

[20]

M. K. Sundareshan and M. Vidyasagar, $L^2$-stability of large-scale dynamical systems: Criteria via positive operator theory,, IEEE Transactions on Automatic Control, AC-22 (1977), 396.   Google Scholar

[21]

A. Turing, The chemical basis of morphogenesis,, Philos. Trans. Roy. Soc. B, 273 (1952), 37.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[22]

M. Vidyasagar, "Input-Output Analysis of Large-Scale Interconnected Systems. Decomposition, Well-Posedness and Stability,", Lecture Notes in Control and Information Sciences, 29 (1981).   Google Scholar

[23]

R. L. Wheeden and A. Zygmund, "Measure and Integral: An Introduction to Real Analysis,", Pure and Applied Mathematics, (1977).   Google Scholar

[24]

J. C. Willems, Dissipative dynamical systems. I. General Theory; Part II: Linear systems with quadratic supply rates,, Archive for Rational Mechanics and Analysis, 45 (1972), 321.   Google Scholar

show all references

References:
[1]

S. Abdelmalek and S. Kouachi, Proof of existence of global solutions for m-component reaction-diffusion systems with mixed boundary conditions via the Lyapunov functional method,, J. Phys. A, 40 (2007), 12335.  doi: 10.1088/1751-8113/40/41/005.  Google Scholar

[2]

M. Arcak and E. D. Sontag, A passivity-based stability criterion for a class of biochemical reaction networks,, Mathematical Biosciences and Engineering, 5 (2008), 1.  doi: 10.3934/mbe.2008.5.1.  Google Scholar

[3]

L. Edelstein-Keshet, "Mathematical Models in Biology,", Reprint of the 1988 original, 46 (1988).   Google Scholar

[4]

W. B. Fitzgibbon, S. L. Hollis and J. J. Morgan, Stability and Lyapunov functions for reaction-diffusion systems,, SIAM J. Math. Anal., 28 (1997), 595.  doi: 10.1137/S0036141094272241.  Google Scholar

[5]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Nones in Mathematics, 840 (1981).   Google Scholar

[6]

M. R. Jovanović, M. Arcak and E. D. Sontag, A passivity-based approach to stability of spatially distributed systems with a cyclic interconnection structure,, IEEE Transactions on Circuits and Systems I. Regul. Pap., 2008 (): 75.   Google Scholar

[7]

B. N. Kholodenko, Cell-signalling dynamics in time and space,, Nat. Rev. Mol. Cell. Biol., 7 (2006), 165.  doi: 10.1038/nrm1838.  Google Scholar

[8]

Y. Lou, T. Nagylaki and W.-M. Ni, On diffusion-induced blowups in a mutualistic model,, Nonlinear Analysis, 45 (2001), 329.  doi: 10.1016/S0362-546X(99)00346-6.  Google Scholar

[9]

S. Malham and J. Xin, Global solutions to a reactive Boussinesq system with front data on an infinite domain,, Comm. Math. Phys., 193 (1998), 287.  doi: 10.1007/s002200050330.  Google Scholar

[10]

J. Morgan, Global existence for semilinear parabolic systems,, SIAM J. Math. Anal., 20 (1989), 1128.  doi: 10.1137/0520075.  Google Scholar

[11]

J. Morgan, Boundedness and decay results for reaction-diffusion systems,, SIAM J. Math. Anal., 21 (1990), 1172.  doi: 10.1137/0521064.  Google Scholar

[12]

P. J. Moylan and D. J. Hill, Stability criteria for large-scale systems,, IEEE Trans. Autom. Control, 23 (1978), 143.  doi: 10.1109/TAC.1978.1101721.  Google Scholar

[13]

J. Murray, "Mathematical Biology,", Biomathematics, 19 (1989).   Google Scholar

[14]

H. Othmer and E. Pate, Scale-invariance in reaction-diffusion models of spatial pattern formation,, Proc. Natl. Acad. Sci., 77 (1980), 4180.  doi: 10.1073/pnas.77.7.4180.  Google Scholar

[15]

M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass,, SIAM Review, 42 (2000), 93.  doi: 10.1137/S0036144599359735.  Google Scholar

[16]

R. Redheffer, R. Redlinger and W. Walter, A theorem of La Salle-Lyapunov type for parabolic systems,, SIAM J. Math. Anal., 19 (1988), 121.   Google Scholar

[17]

W. Rudin, "Real and Complex Analysis,", Third edition, (1987).   Google Scholar

[18]

S. D. M. Santos, P. J. Verveer and P. I. H. Bastiaens, Growth factor-induced MAPK network topology shapes Erk response determining PC-12 cell fate,, Nature Cell Biology, 9 (2007), 324.  doi: 10.1038/ncb1543.  Google Scholar

[19]

H. L. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,", Mathematical Surveys and Monographs, 41 (1995).   Google Scholar

[20]

M. K. Sundareshan and M. Vidyasagar, $L^2$-stability of large-scale dynamical systems: Criteria via positive operator theory,, IEEE Transactions on Automatic Control, AC-22 (1977), 396.   Google Scholar

[21]

A. Turing, The chemical basis of morphogenesis,, Philos. Trans. Roy. Soc. B, 273 (1952), 37.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[22]

M. Vidyasagar, "Input-Output Analysis of Large-Scale Interconnected Systems. Decomposition, Well-Posedness and Stability,", Lecture Notes in Control and Information Sciences, 29 (1981).   Google Scholar

[23]

R. L. Wheeden and A. Zygmund, "Measure and Integral: An Introduction to Real Analysis,", Pure and Applied Mathematics, (1977).   Google Scholar

[24]

J. C. Willems, Dissipative dynamical systems. I. General Theory; Part II: Linear systems with quadratic supply rates,, Archive for Rational Mechanics and Analysis, 45 (1972), 321.   Google Scholar

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