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Stability and dissipativity theory for nonnegative dynamical systems: a unified analysis framework for biological and physiological systems. (English) Zbl 1074.93030

Summary: Nonnegative dynamical system models are derived from mass and energy balance considerations that involve dynamic states whose values are nonnegative. These models are widespread in biological, physiological, and ecological sciences and play a key role in the understanding of these processes. In this paper we develop several results on stability, dissipativity, and stability of feedback interconnections of linear and nonlinear nonnegative dynamical systems. Specifically, using linear Lyapunov functions we develop necessary and sufficient conditions for Lyapunov stability, semistability, that is, system trajectory convergence to Lyapunov stable equilibrium points, and asymptotic stability for nonnegative dynamical systems. In addition, using linear and nonlinear storage functions with linear supply rates we develop new notions of dissipativity theory for nonnegative dynamical systems. Finally, these results are used to develop general stability criteria for feedback interconnections of nonnegative dynamical systems.

MSC:

93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
92B05 General biology and biomathematics
93C10 Nonlinear systems in control theory
34A30 Linear ordinary differential equations and systems
34D20 Stability of solutions to ordinary differential equations
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[1] Abraham, R.; Marsden, J. E.; Ratin, T., Manifolds, Tensor Analysis, and Applications (1983), Addison-Wesley: Addison-Wesley Reading, MA
[2] Anderson, D. H., Compartmental Modeling and Tracer Kinetics (1983), Springer-Verlag: Springer-Verlag Berlin, New York · Zbl 0509.92001
[3] Bellman, R., Topics in pharmacokinetics I. Concentration dependent rates, Math. Biosci., 6, 13-17 (1970)
[4] Berman, A.; Neumann, M.; Stern, R. J., Nonnegative Matrices in Dynamic Systems (1989), Wiley and Sons: Wiley and Sons New York · Zbl 0723.93013
[5] Berman, A.; Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences (1979), Academic Press: Academic Press New York · Zbl 0484.15016
[6] Berman, A.; Varga, R. S.; Ward, R. C., ALPSMatrices with nonpositive off-diagonal entries, Linear Algebra Appl., 21, 233-244 (1978) · Zbl 0401.15018
[7] D.S. Bernstein, S.P. Bhat, Nonnegativity, reducibilty and semistability of mass action kinetics, in: Proc. IEEE Conf. Dec. Contr., Phoenix, AZ, 1999, pp. 2206-2211.; D.S. Bernstein, S.P. Bhat, Nonnegativity, reducibilty and semistability of mass action kinetics, in: Proc. IEEE Conf. Dec. Contr., Phoenix, AZ, 1999, pp. 2206-2211.
[8] Bernstein, D. S.; Hyland, D. C., Compartmental modeling and second-moment analysis of state space systems, SIAM J. Matrix Anal. Appl., 14, 880-901 (1993) · Zbl 0782.93004
[9] S.P. Bhat, D.S. Bernstein, Lyapunov analysis of semistability, in Amer. Contr. Conference San Diego, CA, 1999, pp. 1608-1612.; S.P. Bhat, D.S. Bernstein, Lyapunov analysis of semistability, in Amer. Contr. Conference San Diego, CA, 1999, pp. 1608-1612.
[10] Boyd, S. P.; Ghaoui, L. E.; Feron, E.; Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory (1994), SIAM: SIAM Philadelphia, PA
[11] Brayton, R. K.; Tong, C. H., Stability dynamical systemsa constructive approach, IEEE Trans. Circ. System, 26, 224-234 (1979) · Zbl 0413.93048
[12] Campbell, S. L.; Rose, N. J., Singular perturbation of autonomous linear systems, SIAM J. Math. Anal.,, 10, 542-551 (1979) · Zbl 0413.34055
[13] Carson, E. R.; Cobelli, C.; Finkelstein, L., The Mathematical Modeling of Metabolic and Endocrine Systems (1983), Wiley: Wiley New York
[14] Chellabonia, V.; Haddad, W. M., Exponentially dissipative nonlinear dynamical systemsA nonlinear extension of strict positive realness, J. Math. Prob. Eng., 2003, 25-45 (2003) · Zbl 1075.93034
[15] Cherruault, Y., Mathematical modelling in biomedicine (1986), Dordrecht: Dordrecht Reidel
[16] Farina, L.; Rinaldi, S., Positive Linear SystemsTheory and Applications (2000), Wiley: Wiley New York
[17] Feinberg, M., The existence of uniqueness of states for a class of chemical reaction networks, Arch. Rat. Mech. Anal., 132, 311-370 (1995) · Zbl 0853.92024
[18] Fiedler, M.; Ptak, V. P., On matrices with non-positive off-diagonal elements and positive principal minors, Czechoslovak Math. J., 12, 36-48 (1962)
[19] Fife, D., Which linear compartmental systems contain traps?, Math. Biosci., 14, 311-315 (1972) · Zbl 0249.92011
[20] Foster, D. M.; Jacquez, J. A., Multiple zeros for eigenvalues and the multiplicity of traps of a linear compartmental system, Math. Biosci., 26, 89-97 (1975) · Zbl 0311.93002
[21] Funderlic, R. E.; Mankin, J. B., Solution of homogeneous systems of linear equations arising from compartmental models, SIAM J. Sci. Statist. Comp., 2, 375-383 (1981) · Zbl 0468.65042
[22] Godfrey, K., Compartmental Models and their Applications (1983), Academic Press: Academic Press New York
[23] Haddad, W. M.; Bernstein, D. S., Explicit construction of quadratic Lyapunov functions for small gain, positivity, circle, and Popov theorems and their application to robust stability. Part IContinuous-time theory, Int. J. Robust. Nonlin. Contr., 3, 313-339 (1993) · Zbl 0794.93095
[24] Hill, D. J.; Moylan, P. J., The stability of nonlinear dissipative systems, IEEE Trans. Autom. Contr., 21, 708-711 (1976) · Zbl 0339.93014
[25] Hill, D. J.; Moylan, P. J., Stability results of nonlinear feedback systems, Automatica, 13, 377-382 (1977) · Zbl 0356.93025
[26] Hirsch, M. W., Systems of differential equations which are competitive or cooperative. ILimit sets, SIAM J. Math. Anal., 13, 167-179 (1982) · Zbl 0494.34017
[27] Horn, R. A.; Johnson, R. C., Topics in Matrix Analysis (1995), Cambridge University Press: Cambridge University Press Cambridge
[28] Jacquez, J. A., Compartmental Analysis in Biology and Medicine (1985), University of Michigan Press: University of Michigan Press Ann Arbor · Zbl 0703.92001
[29] Jacquez, J. A.; Simon, C. P., Qualitative theory of compartmental systems, SIAM Rev., 35, 43-79 (1993) · Zbl 0776.92001
[30] Kaszkurewicz, E.; Bhaya, A., Matrix Diagonal Stability in Systems and Computation (2000), Birkhauser: Birkhauser Boston · Zbl 0951.93058
[31] Kaczorek, T., Positive 1D and 2D Systems (2002), Springer-Verlag: Springer-Verlag London · Zbl 1005.68175
[32] Keener, J.; Sneyd, J., Mathematical Physiology (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0913.92009
[33] Khalil, H. K., Nonlinear systems (1996), Prentice-Hall: Prentice-Hall Upper Saddle River, NJ
[34] LaSalle, J. P., Some extensions to Lyapunov’s second method, IRE Trans. Circ. Theory, 7, 520-527 (1960)
[35] Lozano, R.; Brogliato, B.; Egeland, O.; Maschke, B., Dissipative Systems Analysis and Control (2000), Springer-Verlag: Springer-Verlag London
[36] Maeda, H.; Kodama, S.; Kajiya, F., Compartmental system analysisRealization of a class of linear systems with physical constraints, IEEE Trans. Circ. System, 24, 8-14 (1977) · Zbl 0351.93005
[37] Maeda, H.; Kodama, S.; Ohta, Y., Asymptotic behavior of nonlinear compartmental systemsNonoscillation and stability, IEEE Trans. Circ. System, 25, 372-378 (1978) · Zbl 0382.93041
[38] Meyer, C. D.; Stadelmaier, M. W., Singular M-matrices and inverse positivity, Linear Algebra Appl., 22, 139-156 (1978) · Zbl 0411.15006
[39] Mohler, R. R., Biological modeling with variable compartmental structure, IEEE Trans. Autom. Contr.,, 1974, 922-926 (1974)
[40] Nieuwenhuis, J. W., About nonnegative realizations, System Contr. Lett., 1, 283-287 (1982) · Zbl 0484.93018
[41] Ohta, Y., Stability criteria for off-diagonal monotone nonlinear dynamical systems, IEEE Trans. Circ. System, 27, 956-962 (1980) · Zbl 0461.93051
[42] Ohta, Y.; Imanishi, H.; Gong, L.; Haneda, H., Computer generated Lyapunov functions for a class of nonlinear systems, IEEE Trans. Circ. System, 40, 343-354 (1993) · Zbl 0790.93120
[43] Ohta, Y.; Maeda, H.; Kodama, S., observability and realizability of continuous-time positive systems, SIAM J. Contr. Optimiz., 22, 171-180 (1984) · Zbl 0539.93005
[44] Sandberg, W., On the mathematical foundations of compartmental analysis in biology medicine and ecology, IEEE Trans. Circ. System, 25, 273-279 (1978) · Zbl 0377.93034
[45] Sheppard, C. W.; Martin, W. R., Cation exchange between cells and plasma of mammalian blood, J. Gen. Physiol., 33, 703-722 (1950)
[46] Siljak, D. D., Large-Scale Dynamic Systems (1978), North-Holland: North-Holland Amsterdam · Zbl 0384.93002
[47] Siljak, D. D., Complex dynamical systemsDimensionality, structure and uncertainty, Large Scale Systems, 4, 279-294 (1983) · Zbl 0512.93002
[48] Tsokos, J. O.; Tsokos, C. P., Statistical modeling of pharmacokinetic systems, ASME J. Dyn. System Meas. Contr., 98, 37-43 (1976)
[49] Willems, J. C., Dissipative dynamical systems part IGeneral theory, Arch. Rat. Mech. Anal., 45, 321-351 (1972) · Zbl 0252.93002
[50] Willems, J. C., Dissipative dynamical systems part IIQuadratic supply rates, Arch. Rat. Mech. Anal., 45, 359-393 (1972)
[51] Willems, J. C., Lyapunov functions for diagonally dominant systems, Automatica, 12, 519-523 (1976) · Zbl 0345.93040
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