On stabilisability of nonlinear systems on time scales

In this article, stabilisability of nonlinear finite-dimensional control systems on arbitrary time scales is studied. The classical results on stabilisation of nonlinear continuous-time and discrete-time systems are extended to systems on arbitrary time scales with bounded graininess function. It is shown that uniform exponential stability of the linear approximation of a nonlinear system implies uniform exponential stability of the nonlinear system. Then this result is used to show a similar implication for uniform exponential stabilisability.

[1]  V. Jurdjevic,et al.  Controllability and stability , 1978 .

[2]  G. Guseinov.  Integration on time scales , 2003 .

[3]  Robert J. Marks,et al.  CONTROLLABILITY, OBSERVABILITY, REALIZABILITY, AND STABILITY OF DYNAMIC LINEAR SYSTEMS , 2009, 0901.3764.

[4]  R. J. Stern,et al.  Brockett's stabilization condition under state constraints , 2002, Syst. Control. Lett..

[5]  Desheng Li,et al.  On the dynamics of nonautonomous periodic general dynamical systems and differential inclusions , 2006 .

[6]  A. Peterson,et al.  Dynamic Equations on Time Scales , 2001 .

[7]  Dirk Aey Els.  Stabilization of a class of nonlinear systems by a smooth feedback control , 1985 .

[8]  E B Lee,et al.  Foundations of optimal control theory , 1967 .

[9]  Fabian R. Wirth,et al.  A spectral characterization of exponential stability for linear time-invariant systems on time scales , 2003 .

[10]  J. Zabczyk.  Some comments on stabilizability , 1989 .

[11]  J. J. Dacunha.  Stability for time varying linear dynamic systems on time scales , 2005 .

[12]  Ai-Lian Liu,et al.  Boundedness and exponential stability of solutions to dynamic equations on time scales. , 2007 .

[13]  Robert J. Marks,et al.  Linear state feedback stabilisation on time scales , 2009, 0910.3034.

[14]  Zbigniew Bartosiewicz,et al.  Stability, stabilization and observers of linear control systems on time scales , 2007, 2007 46th IEEE Conference on Decision and Control.

[15]  S. Siegmund,et al.  Exponential Stability of Linear Time-Invariant Systems on Time Scales , 2009 .

[16]  M. Rashford,et al.  Exponential Stability of Dynamic Equations on Time Scales , 2022 .

[17]  Zbigniew Bartosiewicz,et al.  Lyapunov functions in stability of nonlinear systems on time scales , 2011 .

[18]  Nicholas Kalouptsidis,et al.  Output Feedback Stabilization of Discrete-Time Control Systems , 1990 .

[19]  Wei Lin,et al.  Stabilization of discrete-time nonlinear systems by smooth state feedback , 1993 .

[20]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[21]  Ewa Pawluszewicz,et al.  Realizations of linear control systems on time scales , 2006 .

[22]  John Tsinias,et al.  Stabilizability of Discrete-Time Nonlinear Systems , 1989 .