# American Institute of Mathematical Sciences

October  2015, 20(8): 2497-2526. doi: 10.3934/dcdsb.2015.20.2497

## Separable Lyapunov functions for monotone systems: Constructions and limitations

 1 Institut für Mathematik, Universität Würzburg, Campus Hubland Nord, Emil-Fischer-Str. 40, 97074 Würzburg, Germany 2 Department of Systems Design and Informatics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka 820-8502, Japan 3 Automatic Control LTH, Lund University, Box 118, SE-221 00 Lund, Sweden 4 School of Mathematical & Physical Sciences, Faculty of Science & IT, The University of Newcastle (UON), University Drive, Callaghan NSW 2308, Australia

Received  July 2014 Revised  January 2015 Published  August 2015

For monotone systems evolving on the positive orthant of $\mathbb{R}^n_+$ two types of Lyapunov functions are considered: Sum- and max-separable Lyapunov functions. One can be written as a sum, the other as a maximum of functions of scalar arguments. Several constructive existence results for both types are given. Notably, one construction provides a max-separable Lyapunov function that is defined at least on an arbitrarily large compact set, based on little more than the knowledge about one trajectory. Another construction for a class of planar systems yields a global sum-separable Lyapunov function, provided the right hand side satisfies a small-gain type condition. A number of examples demonstrate these methods and shed light on the relation between the shape of sublevel sets and the right hand side of the system equation. Negative examples show that there are indeed globally asymptotically stable systems that do not admit either type of Lyapunov function.
Citation: Gunther Dirr, Hiroshi Ito, Anders Rantzer, Björn S. Rüffer. Separable Lyapunov functions for monotone systems: Constructions and limitations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2497-2526. doi: 10.3934/dcdsb.2015.20.2497
##### References:
 [1] D. Angeli and A. Astolfi, A tight small-gain theorem for not necessarily ISS systems, Systems Control Lett., 56 (2007), 87-91. doi: 10.1016/j.sysconle.2006.08.003. [2] D. Angeli, E. D. Sontag and Y. Wang, A characterization of integral input-to-state stability, IEEE Trans. Autom. Control, 45 (2000), 1082-1097. doi: 10.1109/9.863594. [3] A. Bacciotti, F. Ceragioli and L. Mazzi, Differential inclusions and monotonicity conditions for nonsmooth Lyapunov functions, Set-Valued Analysis, 8 (2000), 299-309. doi: 10.1023/A:1008763931789. [4] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. [5] N. P. Bhatia and G. P. Szegö, Stability Theory of Dynamical Systems, Die Grundlehren der mathematischen Wissenschaften, Band 161, Springer-Verlag, New York, 1970. [6] F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, 1998. [7] G. Como, E. Lovisari and K. Savla, Throughput optimality and overload behavior of dynamical flow networks under monotone distributed routing, IEEE Trans. Contr. Network Systems, 2 (2014), 57-67. doi: 10.1109/TCNS.2014.2367361. [8] S. N. Dashkovskiy, B. S. Rüffer and F. R. Wirth, Small gain theorems for large scale systems and construction of ISS Lyapunov functions, SIAM J. Control Optim., 48 (2010), 4089-4118. doi: 10.1137/090746483. [9] P. De Leenheer, D. Angeli and E. D. Sontag, Monotone chemical reaction networks, Journal of Mathematical Chemistry, 41 (2007), 295-314. doi: 10.1007/s10910-006-9075-z. [10] D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness, Springer, Berlin, 2005. doi: 10.1007/b137541. [11] M. W. Hirsch and H. Smith, Monotone dynamical systems, in Handbook of Differential Equations: Ordinary Differential Equations, II, Elsevier B. V., Amsterdam, 2005, 239-357. [12] M. W. Hirsch, Systems of differential equations which are competitive or cooperative. I. Limit sets, SIAM J. Math. Anal., 13 (1982), 167-179. doi: 10.1137/0513013. [13] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. II. Convergence almost everywhere, SIAM J. Math. Anal., 16 (1985), 423-439. doi: 10.1137/0516030. [14] M. W. Hirsch, Systems of differential equations which are competitive or cooperative. III. Competing species, Nonlinearity, 1 (1988), 51-71. doi: 10.1088/0951-7715/1/1/003. [15] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. V. Convergence in $3$-dimensional systems, J. Diff. Eqns., 80 (1989), 94-106. doi: 10.1016/0022-0396(89)90097-1. [16] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. IV. Structural stability in three-dimensional systems, SIAM J. Math. Anal., 21 (1990), 1225-1234. doi: 10.1137/0521067. [17] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. VI. A local $C^r$ closing lemma for 3-dimensional systems, Ergodic Theory Dynam. Systems, 11 (1991), 443-454. doi: 10.1017/S014338570000626X. [18] M. W. Hirsch and H. L. Smith, Competitive and cooperative systems: Mini-review, in Positive systems (Rome, 2003), Lecture Notes in Control and Inform. Sci., 294, Springer, Berlin, 2003, 183-190. doi: 10.1007/978-3-540-44928-7_25. [19] H. Ito, A Lyapunov approach to cascade interconnection of integral input-to-state stable systems, IEEE Trans. Autom. Control, 55 (2010), 702-708. doi: 10.1109/TAC.2009.2037457. [20] H. Ito and Z.-P. Jiang, Necessary and sufficient small gain conditions for integral input-to-state stable systems: A Lyapunov perspective, IEEE Trans. Autom. Control, 54 (2009), 2389-2404. doi: 10.1109/TAC.2009.2028980. [21] H. Ito, State-dependent scaling problems and stability of interconnected iISS and ISS systems, IEEE Trans. Autom. Control, 51 (2006), 1626-1643. doi: 10.1109/TAC.2006.882930. [22] H. Ito, S. Dashkovskiy and F. Wirth, Capability and limitation of max- and sum-type construction of Lyapunov functions for networks of iISS systems, Automatica J. IFAC, 48 (2012), 1197-1204. doi: 10.1016/j.automatica.2012.03.018. [23] H. Ito, Z.-P. Jiang, S. Dashkovskiy and B. S. Rüffer, Robust stability of networks of iISS systems: Construction of sum-type Lyapunov functions, IEEE Trans. Autom. Control, 58 (2013), 1192-1207. doi: 10.1109/TAC.2012.2231552. [24] H. Ito, B. S. Rüffer and A. Rantzer, Max- and sum-separable Lyapunov functions for monotone systems and their level sets, in Proc. 53rd IEEE Conf. Decis. Control, 2014, 2371-2377. doi: 10.1109/CDC.2014.7039750. [25] Z.-P. Jiang, I. M. Y. Mareels and Y. Wang, A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems, Automatica J. IFAC, 32 (1996), 1211-1215. doi: 10.1016/0005-1098(96)00051-9. [26] H. K. Khalil, Nonlinear systems, 3rd edition, Prentice-Hall, Upper Saddle River, NJ, USA, 2002. [27] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications, Vol. I: Ordinary Differential Equations, Academic Press, New York, 1969. [28] M. Margaliot and T. Tuller, Stability analysis of the ribosome flow model, IEEE/ACM Trans. Comp. Biology & Bioinformatics, 9 (2012), 1545-1552. doi: 10.1109/TCBB.2012.88. [29] A. Rantzer, Distributed control of positive systems, in Proc. 50th IEEE Conf. Decis. Control and Europ. Contr. Conf., Orlando, FL, USA, 2011, 6608-6611. doi: 10.1109/CDC.2011.6161293. [30] A. Rantzer, Optimizing positively dominated systems, in Proc. 51st IEEE Conf. Decis. Control, Maui, Hawaii, USA, 2012, 272-277. doi: 10.1109/CDC.2012.6426312. [31] A. Rantzer, B. S. Rüffer and G. Dirr, Separable Lyapunov functions for monotone systems, in Proc. 52nd IEEE Conf. Decis. Control, 2013, 4590-4594. doi: 10.1109/CDC.2013.6760604. [32] B. S. Rüffer, Monotone inequalities, dynamical systems, and paths in the positive orthant of Euclidean $n$-space, Positivity, 14 (2010), 257-283. doi: 10.1007/s11117-009-0016-5. [33] B. S. Rüffer, Small-gain conditions and the comparison principle, IEEE Trans. Autom. Control, 55 (2010), 1732-1736. doi: 10.1109/TAC.2010.2048053. [34] B. S. Rüffer, P. M. Dower and H. Ito, Computational comparison principles for large-scale system stability analysis, in Proc. of the 10th SICE Annual Conference on Control Systems, Kumamoto, Japan, 2010 (electronic). [35] B. S. Rüffer, H. Ito and P. M. Dower, Computing asymptotic gains of large-scale interconnections, in Proc. 49th IEEE Conf. Decis. Control, 2010, 7413-7418. [36] B. S. Rüffer, C. M. Kellett and S. R. Weller, Connection between cooperative positive systems and integral input-to-state stability of large-scale systems, Automatica J. IFAC, 46 (2010), 1019-1027. doi: 10.1016/j.automatica.2010.03.012. [37] H. L. Smith, Monotone Dynamical Systems, Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995. [38] E. D. Sontag, Input to state stability, in The Control Systems Handbook: Control System Advanced Methods (ed. W. S. Levine), 2nd edition, CRC Press, Boca Raton, 2010, 45.1-45.21 (1034-1054). [39] E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Autom. Control, 34 (1989), 435-443. doi: 10.1109/9.28018. [40] E. D. Sontag, Comments on integral variants of ISS, Systems Control Lett., 34 (1998), 93-100. doi: 10.1016/S0167-6911(98)00003-6. [41] E. D. Sontag, Monotone and near-monotone biochemical networks, Systems and Synthetic Biology, 1 (2007), 59-87. doi: 10.1007/s11693-007-9005-9. [42] E. D. Sontag and Y. Wang, On characterizations of the input-to-state stability property, Systems Control Lett., 24 (1995), 351-359. doi: 10.1016/0167-6911(94)00050-6. [43] J. A. Yorke, Extending Liapunov's second method to non-Lipschitz Liapunov functions, Bull. Amer. Math. Soc., 74 (1968), 322-325. doi: 10.1090/S0002-9904-1968-11940-8. [44] T. Yoshizawa, Stability Theory by Liapunov's Second Method, Publications of the Mathematical Society of Japan, No. 9, The Mathematical Society of Japan, Tokyo, 1966.

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##### References:
 [1] D. Angeli and A. Astolfi, A tight small-gain theorem for not necessarily ISS systems, Systems Control Lett., 56 (2007), 87-91. doi: 10.1016/j.sysconle.2006.08.003. [2] D. Angeli, E. D. Sontag and Y. Wang, A characterization of integral input-to-state stability, IEEE Trans. Autom. Control, 45 (2000), 1082-1097. doi: 10.1109/9.863594. [3] A. Bacciotti, F. Ceragioli and L. Mazzi, Differential inclusions and monotonicity conditions for nonsmooth Lyapunov functions, Set-Valued Analysis, 8 (2000), 299-309. doi: 10.1023/A:1008763931789. [4] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. [5] N. P. Bhatia and G. P. Szegö, Stability Theory of Dynamical Systems, Die Grundlehren der mathematischen Wissenschaften, Band 161, Springer-Verlag, New York, 1970. [6] F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, 1998. [7] G. Como, E. Lovisari and K. Savla, Throughput optimality and overload behavior of dynamical flow networks under monotone distributed routing, IEEE Trans. Contr. Network Systems, 2 (2014), 57-67. doi: 10.1109/TCNS.2014.2367361. [8] S. N. Dashkovskiy, B. S. Rüffer and F. R. Wirth, Small gain theorems for large scale systems and construction of ISS Lyapunov functions, SIAM J. Control Optim., 48 (2010), 4089-4118. doi: 10.1137/090746483. [9] P. De Leenheer, D. Angeli and E. D. Sontag, Monotone chemical reaction networks, Journal of Mathematical Chemistry, 41 (2007), 295-314. doi: 10.1007/s10910-006-9075-z. [10] D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness, Springer, Berlin, 2005. doi: 10.1007/b137541. [11] M. W. Hirsch and H. Smith, Monotone dynamical systems, in Handbook of Differential Equations: Ordinary Differential Equations, II, Elsevier B. V., Amsterdam, 2005, 239-357. [12] M. W. Hirsch, Systems of differential equations which are competitive or cooperative. I. Limit sets, SIAM J. Math. Anal., 13 (1982), 167-179. doi: 10.1137/0513013. [13] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. II. Convergence almost everywhere, SIAM J. Math. Anal., 16 (1985), 423-439. doi: 10.1137/0516030. [14] M. W. Hirsch, Systems of differential equations which are competitive or cooperative. III. Competing species, Nonlinearity, 1 (1988), 51-71. doi: 10.1088/0951-7715/1/1/003. [15] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. V. Convergence in $3$-dimensional systems, J. Diff. Eqns., 80 (1989), 94-106. doi: 10.1016/0022-0396(89)90097-1. [16] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. IV. Structural stability in three-dimensional systems, SIAM J. Math. Anal., 21 (1990), 1225-1234. doi: 10.1137/0521067. [17] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. VI. A local $C^r$ closing lemma for 3-dimensional systems, Ergodic Theory Dynam. Systems, 11 (1991), 443-454. doi: 10.1017/S014338570000626X. [18] M. W. Hirsch and H. L. Smith, Competitive and cooperative systems: Mini-review, in Positive systems (Rome, 2003), Lecture Notes in Control and Inform. Sci., 294, Springer, Berlin, 2003, 183-190. doi: 10.1007/978-3-540-44928-7_25. [19] H. Ito, A Lyapunov approach to cascade interconnection of integral input-to-state stable systems, IEEE Trans. Autom. Control, 55 (2010), 702-708. doi: 10.1109/TAC.2009.2037457. [20] H. Ito and Z.-P. Jiang, Necessary and sufficient small gain conditions for integral input-to-state stable systems: A Lyapunov perspective, IEEE Trans. Autom. Control, 54 (2009), 2389-2404. doi: 10.1109/TAC.2009.2028980. [21] H. Ito, State-dependent scaling problems and stability of interconnected iISS and ISS systems, IEEE Trans. Autom. Control, 51 (2006), 1626-1643. doi: 10.1109/TAC.2006.882930. [22] H. Ito, S. Dashkovskiy and F. Wirth, Capability and limitation of max- and sum-type construction of Lyapunov functions for networks of iISS systems, Automatica J. IFAC, 48 (2012), 1197-1204. doi: 10.1016/j.automatica.2012.03.018. [23] H. Ito, Z.-P. Jiang, S. Dashkovskiy and B. S. Rüffer, Robust stability of networks of iISS systems: Construction of sum-type Lyapunov functions, IEEE Trans. Autom. Control, 58 (2013), 1192-1207. doi: 10.1109/TAC.2012.2231552. [24] H. Ito, B. S. Rüffer and A. Rantzer, Max- and sum-separable Lyapunov functions for monotone systems and their level sets, in Proc. 53rd IEEE Conf. Decis. Control, 2014, 2371-2377. doi: 10.1109/CDC.2014.7039750. [25] Z.-P. Jiang, I. M. Y. Mareels and Y. Wang, A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems, Automatica J. IFAC, 32 (1996), 1211-1215. doi: 10.1016/0005-1098(96)00051-9. [26] H. K. Khalil, Nonlinear systems, 3rd edition, Prentice-Hall, Upper Saddle River, NJ, USA, 2002. [27] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications, Vol. I: Ordinary Differential Equations, Academic Press, New York, 1969. [28] M. Margaliot and T. Tuller, Stability analysis of the ribosome flow model, IEEE/ACM Trans. Comp. Biology & Bioinformatics, 9 (2012), 1545-1552. doi: 10.1109/TCBB.2012.88. [29] A. Rantzer, Distributed control of positive systems, in Proc. 50th IEEE Conf. Decis. Control and Europ. Contr. Conf., Orlando, FL, USA, 2011, 6608-6611. doi: 10.1109/CDC.2011.6161293. [30] A. Rantzer, Optimizing positively dominated systems, in Proc. 51st IEEE Conf. Decis. Control, Maui, Hawaii, USA, 2012, 272-277. doi: 10.1109/CDC.2012.6426312. [31] A. Rantzer, B. S. Rüffer and G. Dirr, Separable Lyapunov functions for monotone systems, in Proc. 52nd IEEE Conf. Decis. Control, 2013, 4590-4594. doi: 10.1109/CDC.2013.6760604. [32] B. S. Rüffer, Monotone inequalities, dynamical systems, and paths in the positive orthant of Euclidean $n$-space, Positivity, 14 (2010), 257-283. doi: 10.1007/s11117-009-0016-5. [33] B. S. Rüffer, Small-gain conditions and the comparison principle, IEEE Trans. Autom. Control, 55 (2010), 1732-1736. doi: 10.1109/TAC.2010.2048053. [34] B. S. Rüffer, P. M. Dower and H. Ito, Computational comparison principles for large-scale system stability analysis, in Proc. of the 10th SICE Annual Conference on Control Systems, Kumamoto, Japan, 2010 (electronic). [35] B. S. Rüffer, H. Ito and P. M. Dower, Computing asymptotic gains of large-scale interconnections, in Proc. 49th IEEE Conf. Decis. Control, 2010, 7413-7418. [36] B. S. Rüffer, C. M. Kellett and S. R. Weller, Connection between cooperative positive systems and integral input-to-state stability of large-scale systems, Automatica J. IFAC, 46 (2010), 1019-1027. doi: 10.1016/j.automatica.2010.03.012. [37] H. L. Smith, Monotone Dynamical Systems, Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995. [38] E. D. Sontag, Input to state stability, in The Control Systems Handbook: Control System Advanced Methods (ed. W. S. Levine), 2nd edition, CRC Press, Boca Raton, 2010, 45.1-45.21 (1034-1054). [39] E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Autom. Control, 34 (1989), 435-443. doi: 10.1109/9.28018. [40] E. D. Sontag, Comments on integral variants of ISS, Systems Control Lett., 34 (1998), 93-100. doi: 10.1016/S0167-6911(98)00003-6. [41] E. D. Sontag, Monotone and near-monotone biochemical networks, Systems and Synthetic Biology, 1 (2007), 59-87. doi: 10.1007/s11693-007-9005-9. [42] E. D. Sontag and Y. Wang, On characterizations of the input-to-state stability property, Systems Control Lett., 24 (1995), 351-359. doi: 10.1016/0167-6911(94)00050-6. [43] J. A. Yorke, Extending Liapunov's second method to non-Lipschitz Liapunov functions, Bull. Amer. Math. Soc., 74 (1968), 322-325. doi: 10.1090/S0002-9904-1968-11940-8. [44] T. Yoshizawa, Stability Theory by Liapunov's Second Method, Publications of the Mathematical Society of Japan, No. 9, The Mathematical Society of Japan, Tokyo, 1966.
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