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

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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

Abstract
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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 and 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), 12335-12350. doi: 10.1088/1751-8113/40/41/005.
[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), 1-19. doi: 10.3934/mbe.2008.5.1.
[3]	L. Edelstein-Keshet, "Mathematical Models in Biology," Reprint of the 1988 original, Classics in Applied Mathematics, 46, SIAM, Philadelphia, PA, 2005.
[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), 595-610. doi: 10.1137/S0036141094272241.
[5]	D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Nones in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.
[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.
[7]	B. N. Kholodenko, Cell-signalling dynamics in time and space, Nat. Rev. Mol. Cell. Biol., 7 (2006), 165-176. doi: 10.1038/nrm1838.
[8]	Y. Lou, T. Nagylaki and W.-M. Ni, On diffusion-induced blowups in a mutualistic model, Nonlinear Analysis, 45 (2001), 329-342. doi: 10.1016/S0362-546X(99)00346-6.
[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), 287-316. doi: 10.1007/s002200050330.
[10]	J. Morgan, Global existence for semilinear parabolic systems, SIAM J. Math. Anal., 20 (1989), 1128-1144. doi: 10.1137/0520075.
[11]	J. Morgan, Boundedness and decay results for reaction-diffusion systems, SIAM J. Math. Anal., 21 (1990), 1172-1189. doi: 10.1137/0521064.
[12]	P. J. Moylan and D. J. Hill, Stability criteria for large-scale systems, IEEE Trans. Autom. Control, 23 (1978), 143-149. doi: 10.1109/TAC.1978.1101721.
[13]	J. Murray, "Mathematical Biology," Biomathematics, 19, Springer-Verlag, Berlin, 1989.
[14]	H. Othmer and E. Pate, Scale-invariance in reaction-diffusion models of spatial pattern formation, Proc. Natl. Acad. Sci., 77 (1980), 4180-4184. doi: 10.1073/pnas.77.7.4180.
[15]	M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM Review, 42 (2000), 93-106. doi: 10.1137/S0036144599359735.
[16]	R. Redheffer, R. Redlinger and W. Walter, A theorem of La Salle-Lyapunov type for parabolic systems, SIAM J. Math. Anal., 19 (1988), 121-132.
[17]	W. Rudin, "Real and Complex Analysis," Third edition, McGraw-Hill Book Co., New York, 1987.
[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), 324-330. doi: 10.1038/ncb1543.
[19]	H. L. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, AMS, Providence, RI, 1995.
[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), 396-399.
[21]	A. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. B, 273 (1952), 37-72. doi: 10.1098/rstb.1952.0012.
[22]	M. Vidyasagar, "Input-Output Analysis of Large-Scale Interconnected Systems. Decomposition, Well-Posedness and Stability," Lecture Notes in Control and Information Sciences, 29, Springer-Verlag, Berlin-New York, 1981.
[23]	R. L. Wheeden and A. Zygmund, "Measure and Integral: An Introduction to Real Analysis," Pure and Applied Mathematics, Vol. 43, Marcel Dekker, Inc., New York-Basel, 1977.
[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), 321-393.

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), 12335-12350. doi: 10.1088/1751-8113/40/41/005.
[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), 1-19. doi: 10.3934/mbe.2008.5.1.
[3]	L. Edelstein-Keshet, "Mathematical Models in Biology," Reprint of the 1988 original, Classics in Applied Mathematics, 46, SIAM, Philadelphia, PA, 2005.
[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), 595-610. doi: 10.1137/S0036141094272241.
[5]	D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Nones in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.
[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.
[7]	B. N. Kholodenko, Cell-signalling dynamics in time and space, Nat. Rev. Mol. Cell. Biol., 7 (2006), 165-176. doi: 10.1038/nrm1838.
[8]	Y. Lou, T. Nagylaki and W.-M. Ni, On diffusion-induced blowups in a mutualistic model, Nonlinear Analysis, 45 (2001), 329-342. doi: 10.1016/S0362-546X(99)00346-6.
[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), 287-316. doi: 10.1007/s002200050330.
[10]	J. Morgan, Global existence for semilinear parabolic systems, SIAM J. Math. Anal., 20 (1989), 1128-1144. doi: 10.1137/0520075.
[11]	J. Morgan, Boundedness and decay results for reaction-diffusion systems, SIAM J. Math. Anal., 21 (1990), 1172-1189. doi: 10.1137/0521064.
[12]	P. J. Moylan and D. J. Hill, Stability criteria for large-scale systems, IEEE Trans. Autom. Control, 23 (1978), 143-149. doi: 10.1109/TAC.1978.1101721.
[13]	J. Murray, "Mathematical Biology," Biomathematics, 19, Springer-Verlag, Berlin, 1989.
[14]	H. Othmer and E. Pate, Scale-invariance in reaction-diffusion models of spatial pattern formation, Proc. Natl. Acad. Sci., 77 (1980), 4180-4184. doi: 10.1073/pnas.77.7.4180.
[15]	M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM Review, 42 (2000), 93-106. doi: 10.1137/S0036144599359735.
[16]	R. Redheffer, R. Redlinger and W. Walter, A theorem of La Salle-Lyapunov type for parabolic systems, SIAM J. Math. Anal., 19 (1988), 121-132.
[17]	W. Rudin, "Real and Complex Analysis," Third edition, McGraw-Hill Book Co., New York, 1987.
[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), 324-330. doi: 10.1038/ncb1543.
[19]	H. L. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, AMS, Providence, RI, 1995.
[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), 396-399.
[21]	A. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. B, 273 (1952), 37-72. doi: 10.1098/rstb.1952.0012.
[22]	M. Vidyasagar, "Input-Output Analysis of Large-Scale Interconnected Systems. Decomposition, Well-Posedness and Stability," Lecture Notes in Control and Information Sciences, 29, Springer-Verlag, Berlin-New York, 1981.
[23]	R. L. Wheeden and A. Zygmund, "Measure and Integral: An Introduction to Real Analysis," Pure and Applied Mathematics, Vol. 43, Marcel Dekker, Inc., New York-Basel, 1977.
[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), 321-393.

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