On "uniformity" in definitions of global asymptotic stability for time-varying nonlinear systems

We discuss the lack of ''uniformity'' in definitions of uniform global asymptotic stability (UGAS) that have been used in various textbooks, monographs, and papers over the years. Sometimes UGAS is taken to be the combination of uniform local stability (ULS) and uniform global attractivity (UGA). Other times it also encompasses uniform global boundedness (UGB). This paper contains an explicit, smooth scalar example that shows that these definitions do not agree in general, even when the right-hand side is locally Lipschitz in the state uniformly in time (and thus bounded in time). We also discuss various notions of global asymptotic stability with relaxed uniformity (with respect to the initial time) requirements for the behavior of the solutions. In particular, we consider class-KL estimates and Lyapunov characterizations.

[1]  C. A. Desoer,et al.  Nonlinear Systems Analysis , 1978 .

[2]  N. Rouche,et al.  Stability Theory by Liapunov's Direct Method , 1977 .

[3]  A. Teel,et al.  A Smooth Lyapunov Function from a Class-kl Estimate Involving Two Positive Semideenite Functions , 1999 .

[4]  A. Bacciotti,et al.  Liapunov functions and stability in control theory , 2001 .

[5]  Yuandan Lin,et al.  A Smooth Converse Lyapunov Theorem for Robust Stability , 1996 .

[6]  G. P. Szegö,et al.  Stability theory of dynamical systems , 1970 .

[7]  吉沢 太郎.  Stability theory and the existence of periodic solutions and almost periodic solutions , 1975 .

[8]  Eduardo D. Sontag,et al.  Lyapunov Characterizations of Input to Output Stability , 2000, SIAM J. Control. Optim..

[9]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[10]  Yuan Wang,et al.  Stabilization in spite of matched unmodeled dynamics and an equivalent definition of input-to-state stability , 1996, Math. Control. Signals Syst..

[11]  Iasson Karafyllis,et al.  A Converse Lyapunov Theorem for Nonuniform in Time Global Asymptotic Stability and Its Application to Feedback Stabilization , 2003, SIAM J. Control. Optim..

[12]  Eduardo Sontag,et al.  Notions of input to output stability , 1999, Systems & Control Letters.

[13]  Luca Zaccarian,et al.  On the Literature's two different definitions of uniform global asymptotic stability for nonlinear systems , 2006 .

[14]  A. Teel,et al.  A smooth Lyapunov function from a class- ${\mathcal{KL}}$ estimate involving two positive semidefinite functions , 2000 .

[15]  Elena Panteley,et al.  Advanced topics in control systems theory : lecture notes from FAP 2004 , 2005 .

[16]  Shouchuan Hu.  Differential equations with discontinuous right-hand sides☆ , 1991 .

[17]  V. Lakshmikantham,et al.  Differential and integral inequalities : theory and applications , 1969 .

[18]  W. Walter.  Differential and Integral Inequalities , 1970 .

[19]  M. Vidyasagar,et al.  Nonlinear systems analysis (2nd ed.) , 1993 .

[20]  J. L. Massera.  Contributions to Stability Theory , 1956 .

[21]  Eduardo Sontag.  Comments on integral variants of ISS , 1998 .

[22]  Wolfgang Hahn,et al.  Stability of Motion , 1967 .