Stability for impulsive control systems

In this paper we extend the notion of the control Lyapounov pair of functions and derive a stability theory for impulsive control systems. The control system is a measure driven differential inclusion that is partly absolutely continuous and partly singular. Some examples illustrating the features of Lyapounov stability are provided.

[1]  V. Borkar,et al.  A unified framework for hybrid control: model and optimal control theory , 1998, IEEE Trans. Autom. Control..

[2]  A. Bressan,et al.  Impulsive control systems without commutativity assumptions , 1994 .

[3]  J. Warga.  Relaxed variational problems , 1962 .

[4]  F. Lobo Pereira,et al.  Some questions about hybrid systems , 2001, 2001 European Control Conference (ECC).

[5]  J. Warga.  Optimal control of differential and functional equations , 1972 .

[6]  F. Clarke,et al.  The Optimal Exploitation of Renewable Resource Stocks: Problems of Irreversible Investment , 1979 .

[7]  Raymond W. Rishel,et al.  An Extended Pontryagin Principle for Control Systems whose Control Laws Contain Measures , 1965 .

[8]  H. Witsenhausen.  A class of hybrid-state continuous-time dynamic systems , 1966 .

[9]  R. Vinter,et al.  Necessary conditions for optimal impulsive control problems , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[10]  Yu. S. Ledyaev,et al.  Asymptotic controllability implies feedback stabilization , 1997, IEEE Trans. Autom. Control..

[11]  R. Vinter,et al.  A maximum principle for optimal processes with discontinuous trajectories , 1988 .

[12]  G. N. Silva,et al.  Measure Driven Differential Inclusions , 1996 .

[13]  H. Komatsu.  Ultradifferentiability of solutions of ordinary differential equations , 1980 .

[14]  Fernando Lobo Pereira,et al.  Second Order Necessary Conditions for Optimal Impulsive Control Problems , 2003 .

[15]  Jean Pierre Marec,et al.  Optimal Space Trajectories , 1979 .

[16]  Kolokolnikova Galina.  Discontinuous Trajectories Optimality in the Nonlinear Optimal Control Problems , 1996 .

[17]  E. Ryan.  On Brockett's Condition for Smooth Stabilizability and its Necessity in a Context of Nonsmooth Feedback , 1994 .

[18]  Yu. S. Ledyaev,et al.  A Lyapunov characterization of robust stabilization , 1999 .

[19]  A. Bressan,et al.  Impulsive control systems with commutative vector fields , 1991 .

[20]  B. Brogliato.  Nonsmooth Impact Mechanics: Models, Dynamics and Control , 1996 .

[21]  Ludovic Rifford,et al.  Existence of Lipschitz and Semiconcave Control-Lyapunov Functions , 2000, SIAM J. Control. Optim..

[22]  B. Miller.  The generalized solutions of nonlinear optimization problems with impulse control , 1996 .

[23]  A. L. Fradkov,et al.  Nonlinear control systems 2001 (NOLCOS 2001) : a proceedings volume from the 5th IFAC Symposium, St. Petersburg, Russia, 4-6 July 2001 , 2002 .

[24]  Yu. S. Ledyaev,et al.  Nonsmooth analysis and control theory , 1998 .

[25]  Francis H. Clarke,et al.  Feedback Stabilization and Lyapunov Functions , 2000, SIAM J. Control. Optim..

[26]  Eduardo Sontag.  A Lyapunov-Like Characterization of Asymptotic Controllability , 1983, SIAM Journal on Control and Optimization.

[27]  Geraldo Nunes Silva,et al.  Necessary conditions of optimality for vector-valued impulsive control problems , 2000 .

[28]  F. Rampazzo,et al.  Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls , 1995, Differential and Integral Equations.

[29]  J. Warga,et al.  Variational Problems with Unbounded Controls , 1965 .

[30]  Eduardo Sontag,et al.  Nonsmooth control-Lyapunov functions , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[31]  Aram V. Arutyunov,et al.  Second order necessary conditions of optimality for impulsive control problems , 2001, 2001 European Control Conference (ECC).