July  2017, 37(7): 4053-4069. doi: 10.3934/dcds.2017172

Asymptotic stability and smooth Lyapunov functions for a class of abstract dynamical systems

Institute for Mathematics, University of Würzburg, Emil-Fischer Straβe 40, 97074 Würzburg, Germany

Received  June 2015 Revised  February 2017 Published  April 2017

This paper deals with a characterization of asymptotic stability for a class of dynamical systems in terms of smooth Lyapunov pairs. We point out that well known converse Lyapunov results for differential inclusions cannot be applied to this class of dynamical systems. Following an abstract approach we put an assumption on the trajectories of the dynamical systems which demands for an estimate of the difference between trajectories. Under this assumption, we prove the existence of a $C^∞$-smooth Lyapunov pair. We also show that this assumption is satisfied by differential inclusions defined by Lipschitz continuous set-valued maps taking nonempty, compact and convex values.

Citation: Michael Schönlein. Asymptotic stability and smooth Lyapunov functions for a class of abstract dynamical systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4053-4069. doi: 10.3934/dcds.2017172
References:
[1]

J. -P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory, Springer, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[2]

J. -P. Aubin and H. Frankowska, Set-valued Analysis, Birkhäuser, Boston, 1990.  Google Scholar

[3]

M. Bramson, Stability of queueing networks, Probab. Surv., 5 (2008), 169-345.  doi: 10.1214/08-PS137.  Google Scholar

[4]

A. Bressan and G. Facchi, Trajectories of differential inclusions with state constraints, J. Differ. Equations, 250 (2011), 2267-2281.  doi: 10.1016/j.jde.2010.12.021.  Google Scholar

[5]

H. Chen, Fluid approximations and stability of multiclass queueing networks: Work-conserving disciplines, Ann. Appl. Probab., 5 (1995), 637-665.  doi: 10.1214/aoap/1177004699.  Google Scholar

[6]

F. ClarkeY. Ledyaev and R. Stern, Asymptotic stability and smooth Lyapunov functions, J. Differ. Equations, 149 (1998), 69-114.  doi: 10.1006/jdeq.1998.3476.  Google Scholar

[7]

F. ClarkeR. Stern and P. Wolenski, Subgradient criteria for monotonicity, the {L}ipschitz condition, and convexity, Canad. J. Math, 45 (1993), 1167-1183.  doi: 10.4153/CJM-1993-065-x.  Google Scholar

[8]

J. Dai, On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid limit models, Ann. Appl. Probab., 5 (1995), 49-77.  doi: 10.1214/aoap/1177004828.  Google Scholar

[9]

P. Dupuis and R. J. Williams, Lyapunov functions for semimartingale reflecting Brownian motions, Ann. Probab., 22 (1994), 680-702.  doi: 10.1214/aop/1176988725.  Google Scholar

[10]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 1998. doi: 10.1090/gsm/019.  Google Scholar

[11]

J. K. Hale, Dynamical systems and stability, J. Math. Anal. Appl., 26 (1969), 39-59.  doi: 10.1016/0022-247X(69)90175-9.  Google Scholar

[12]

J. Hale and E. Infante, Extended dynamical systems and stability theory, Proc. Natl. Acad. Sci. USA, 58 (1967), 405-409.  doi: 10.1073/pnas.58.2.405.  Google Scholar

[13]

I. Karafyllis, Lyapunov theorems for systems described by retarded functional differential equations, Nonlinear Anal., 64 (2006), 590-617.  doi: 10.1016/j.na.2005.04.045.  Google Scholar

[14]

C. M. Kellett and A. R. Teel, Smooth Lyapunov functions and robustness of stability for difference inclusions, Syst. Control Lett., 52 (2004), 395-405.  doi: 10.1016/j.sysconle.2004.02.015.  Google Scholar

[15]

C. M. Kellett, Classical converse theorems in Lyapunov's second method, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2333-2360.  doi: 10.3934/dcdsb.2015.20.2333.  Google Scholar

[16]

Y. LinE. D. Sontag and Y. Wang, A smooth converse Lyapunov theorems for robust stability, SIAM J. Control Optim., 34 (1996), 124-160.  doi: 10.1137/S0363012993259981.  Google Scholar

[17]

W. Rudin, Functional Analysis, McGraw-Hill, Inc. , New York, 1991.  Google Scholar

[18]

A. Rybko and A. Stolyar, Ergodicity of stochastic processes describing the operation of open queueing networks, Probl. Inf. Transm., 28 (1992), 199-220.   Google Scholar

[19]

M. Schönlein and F. Wirth, On converse Lyapunov theorems for fluid network models, Queueing Syst., 70 (2012), 339-367.  doi: 10.1007/s11134-012-9279-9.  Google Scholar

[20]

A. Siconolfi and G. Terrone, A metric proof of the converse Lyapunov theorem for semicontinuous multivalued dynamics, Discrete Contin. Dyn. Syst., 32 (2012), 4409-4427.  doi: 10.3934/dcds.2012.32.4409.  Google Scholar

[21]

M. Slemrod, Asymptotic behavior of a class of abstract dynamical systems, J. Differ. Equations, 7 (1970), 584-600.  doi: 10.1016/0022-0396(70)90103-8.  Google Scholar

[22]

A. Stolyar, On the stability of multiclass queueing networks: A relaxed sufficient condition via limiting fluid processes, Markov Process. Relat. Fields, 1 (1995), 491-512.   Google Scholar

[23]

A. Teel and L. Praly, A smooth Lyapunov function from a class-$\mathcal{KL}$ estimate involving two positive semidefinite functions, ESAIM Control Optim. Calc. Var., 5 (2000), 313-367.  doi: 10.1051/cocv:2000113.  Google Scholar

[24]

J. Walker, Dynamical Systems and Evolution Equations. Theory and Applications, Plenum Press, New York, 1980.  Google Scholar

[25]

H. Q. Ye and H. Chen, Lyapunov method for stability of fluid networks, Oper. Res. Lett., 28 (2001), 125-136.  doi: 10.1016/S0167-6377(01)00060-8.  Google Scholar

[26]

V. Zubov, Methods of A. M. Lyapunov and Their Application, Noordhoff Ltd. , Groningen, 1964.  Google Scholar

show all references

References:
[1]

J. -P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory, Springer, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[2]

J. -P. Aubin and H. Frankowska, Set-valued Analysis, Birkhäuser, Boston, 1990.  Google Scholar

[3]

M. Bramson, Stability of queueing networks, Probab. Surv., 5 (2008), 169-345.  doi: 10.1214/08-PS137.  Google Scholar

[4]

A. Bressan and G. Facchi, Trajectories of differential inclusions with state constraints, J. Differ. Equations, 250 (2011), 2267-2281.  doi: 10.1016/j.jde.2010.12.021.  Google Scholar

[5]

H. Chen, Fluid approximations and stability of multiclass queueing networks: Work-conserving disciplines, Ann. Appl. Probab., 5 (1995), 637-665.  doi: 10.1214/aoap/1177004699.  Google Scholar

[6]

F. ClarkeY. Ledyaev and R. Stern, Asymptotic stability and smooth Lyapunov functions, J. Differ. Equations, 149 (1998), 69-114.  doi: 10.1006/jdeq.1998.3476.  Google Scholar

[7]

F. ClarkeR. Stern and P. Wolenski, Subgradient criteria for monotonicity, the {L}ipschitz condition, and convexity, Canad. J. Math, 45 (1993), 1167-1183.  doi: 10.4153/CJM-1993-065-x.  Google Scholar

[8]

J. Dai, On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid limit models, Ann. Appl. Probab., 5 (1995), 49-77.  doi: 10.1214/aoap/1177004828.  Google Scholar

[9]

P. Dupuis and R. J. Williams, Lyapunov functions for semimartingale reflecting Brownian motions, Ann. Probab., 22 (1994), 680-702.  doi: 10.1214/aop/1176988725.  Google Scholar

[10]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 1998. doi: 10.1090/gsm/019.  Google Scholar

[11]

J. K. Hale, Dynamical systems and stability, J. Math. Anal. Appl., 26 (1969), 39-59.  doi: 10.1016/0022-247X(69)90175-9.  Google Scholar

[12]

J. Hale and E. Infante, Extended dynamical systems and stability theory, Proc. Natl. Acad. Sci. USA, 58 (1967), 405-409.  doi: 10.1073/pnas.58.2.405.  Google Scholar

[13]

I. Karafyllis, Lyapunov theorems for systems described by retarded functional differential equations, Nonlinear Anal., 64 (2006), 590-617.  doi: 10.1016/j.na.2005.04.045.  Google Scholar

[14]

C. M. Kellett and A. R. Teel, Smooth Lyapunov functions and robustness of stability for difference inclusions, Syst. Control Lett., 52 (2004), 395-405.  doi: 10.1016/j.sysconle.2004.02.015.  Google Scholar

[15]

C. M. Kellett, Classical converse theorems in Lyapunov's second method, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2333-2360.  doi: 10.3934/dcdsb.2015.20.2333.  Google Scholar

[16]

Y. LinE. D. Sontag and Y. Wang, A smooth converse Lyapunov theorems for robust stability, SIAM J. Control Optim., 34 (1996), 124-160.  doi: 10.1137/S0363012993259981.  Google Scholar

[17]

W. Rudin, Functional Analysis, McGraw-Hill, Inc. , New York, 1991.  Google Scholar

[18]

A. Rybko and A. Stolyar, Ergodicity of stochastic processes describing the operation of open queueing networks, Probl. Inf. Transm., 28 (1992), 199-220.   Google Scholar

[19]

M. Schönlein and F. Wirth, On converse Lyapunov theorems for fluid network models, Queueing Syst., 70 (2012), 339-367.  doi: 10.1007/s11134-012-9279-9.  Google Scholar

[20]

A. Siconolfi and G. Terrone, A metric proof of the converse Lyapunov theorem for semicontinuous multivalued dynamics, Discrete Contin. Dyn. Syst., 32 (2012), 4409-4427.  doi: 10.3934/dcds.2012.32.4409.  Google Scholar

[21]

M. Slemrod, Asymptotic behavior of a class of abstract dynamical systems, J. Differ. Equations, 7 (1970), 584-600.  doi: 10.1016/0022-0396(70)90103-8.  Google Scholar

[22]

A. Stolyar, On the stability of multiclass queueing networks: A relaxed sufficient condition via limiting fluid processes, Markov Process. Relat. Fields, 1 (1995), 491-512.   Google Scholar

[23]

A. Teel and L. Praly, A smooth Lyapunov function from a class-$\mathcal{KL}$ estimate involving two positive semidefinite functions, ESAIM Control Optim. Calc. Var., 5 (2000), 313-367.  doi: 10.1051/cocv:2000113.  Google Scholar

[24]

J. Walker, Dynamical Systems and Evolution Equations. Theory and Applications, Plenum Press, New York, 1980.  Google Scholar

[25]

H. Q. Ye and H. Chen, Lyapunov method for stability of fluid networks, Oper. Res. Lett., 28 (2001), 125-136.  doi: 10.1016/S0167-6377(01)00060-8.  Google Scholar

[26]

V. Zubov, Methods of A. M. Lyapunov and Their Application, Noordhoff Ltd. , Groningen, 1964.  Google Scholar

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