August  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 and Continuous Dynamical Systems, 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.

[2]

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

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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. 

[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.

[24]

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

[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.

[26]

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

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.

[2]

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

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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. 

[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.

[24]

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

[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.

[26]

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

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