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

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  • 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.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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  •   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.
      J. -P. Aubin and H. Frankowska, Set-valued Analysis, Birkhäuser, Boston, 1990.
      M. Bramson , Stability of queueing networks, Probab. Surv., 5 (2008) , 169-345.  doi: 10.1214/08-PS137.
      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.
      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.
      F. Clarke , Y. Ledyaev  and  R. Stern , Asymptotic stability and smooth Lyapunov functions, J. Differ. Equations, 149 (1998) , 69-114.  doi: 10.1006/jdeq.1998.3476.
      F. Clarke , R. 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.
      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.
      P. Dupuis  and  R. J. Williams , Lyapunov functions for semimartingale reflecting Brownian motions, Ann. Probab., 22 (1994) , 680-702.  doi: 10.1214/aop/1176988725.
      L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 1998. doi: 10.1090/gsm/019.
      J. K. Hale , Dynamical systems and stability, J. Math. Anal. Appl., 26 (1969) , 39-59.  doi: 10.1016/0022-247X(69)90175-9.
      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.
      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.
      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.
      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.
      Y. Lin , E. 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.
      W. Rudin, Functional Analysis, McGraw-Hill, Inc. , New York, 1991.
      A. Rybko  and  A. Stolyar , Ergodicity of stochastic processes describing the operation of open queueing networks, Probl. Inf. Transm., 28 (1992) , 199-220. 
      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.
      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.
      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.
      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. 
      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.
      J. Walker, Dynamical Systems and Evolution Equations. Theory and Applications, Plenum Press, New York, 1980.
      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.
      V. Zubov, Methods of A. M. Lyapunov and Their Application, Noordhoff Ltd. , Groningen, 1964.
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