In this paper, we address the (uniform) finite-time input-to-state stability problem for switched nonlinear systems. We prove that a switched nonlinear system has a useful finite-time input-to-state stability property under average dwell-time switching signals if each constituent subsystem has finite-time input-to-state stability. Moreover, we prove the equivalence between the optimal costs for the switched nonlinear systems and for the relaxed differential inclusion.

References

References
1.
Bengea
,
S.
, and
DeCarlo
,
R.
,
2005
, “
Optimal Control of Switching Systems
,”
Automatica
,
41
(
1
), pp.
11
27
.10.1016/j.automatica.2004.08.003
2.
Mancilla-Aguilar
,
J. L.
, and
Garcia
,
R. A.
,
2000
, “
A Converse Lyapunov Theorem for Nonlinear Switched Systems
,”
Syst. Control Lett.
,
41
, pp.
67
71
.10.1016/S0167-6911(00)00040-2
3.
Mancilla-Aguilar
,
J. L.
, and
Garcia
,
R. A.
,
2001
, “
A Converse Lyapunov Theorems for ISS and iISS Switched Nonlinear Systems
,”
Syst. Control Lett.
,
42
, pp.
47
53
.10.1016/S0167-6911(00)00079-7
4.
Zhang
,
L.
,
Chen
,
Y.
, and
Cui
,
P.
,
2005
, “
Stabilization for a Class of Second-Order Switched Systems
,”
Nonlinear Anal.
,
62
, pp.
1527
1535
.10.1016/j.na.2005.03.082
5.
Sun
,
Z.
,
2006
, “
Stabilization and Optimization of Switched Linear Systems
,”
Automatica
,
42
, pp.
783
788
.10.1016/j.automatica.2005.12.022
6.
Bhat
,
S.
, and
Bernstein
,
D.
,
2000
, “
Finite-Time Stability of Continuous Autonomous Systems
,”
SIAM J. Control Optim.
,
38
, pp.
751
766
.10.1137/S0363012997321358
7.
Orlov
,
Y.
,
2005
, “
Finite-Time Stability and Robust Control Synthesis of Uncertain Switched Systems
,”
SIAM J. Control Optim.
,
43
, pp.
1253
1271
.10.1137/S0363012903425593
8.
Wang
,
X.
, and
Hong
,
Y.
,
2008
, “
Finite-Time Consensus for Multi-Agent Networks With Second-Order Agent Dynamics
,”
Proceedings of the
IFAC
World Congress, pp.
15185
15190
.10.3182/20080706-5-KR-1001.02568
9.
Ryan
,
E.
,
1991
, “
Finite-Time Stabilization of Uncertain Nonlinear Planar Systems
,”
Dyn. Control
,
1
, pp.
83
94
.10.1007/BF02169426
10.
Xu
,
J.
,
Sun
,
J.
, and
Yue
,
D.
,
2012
, “
Stochastic Finite-Time Stability of Nonlinear Markovian Switching Systems With Impulsive Effects
,”
ASME J. Dyn. Sys., Meas., Control
,
134
(
1
), p.
011011
.10.1115/1.4005359
11.
Sontag
,
E. D.
, and
Wang
,
Y.
,
1995
, “
On Characterizations of the Input to State Stability Property
,”
Syst. Control Lett.
,
24
, pp.
351
359
.10.1016/0167-6911(94)00050-6
12.
Sontag
,
E. D.
,
2000
, “
The ISS Philosophy as a Unifying Framework for Stability-Like Behavior
,”
Nonlinear Control in the Year 2000, Lecture Notes in Control and Information Sciences
,
A.
Isidori
,
F.
Lamnabhi-Lagarrigue
, and
W.
Respondek
, eds.,
Springer-Verlag
,
Berlin
, pp.
443
468
.
13.
Lin
,
Y.
,
Sontag
,
E. D.
, and
Wang
,
Y.
,
1996
, “
A Smooth Converse Lyapunov Theorem for Robust Stability
,”
SIAM J. Control Optim.
,
34
(
1
), pp.
124
160
.10.1137/S0363012993259981
14.
Hong
,
Y.
,
Jiang
,
Z. P.
, and
Feng
,
G.
,
2010
, “
Finite-Time Input-to-State Stability and Applications to Finite-Time Control Design
,”
SIAM J. Control Optim.
48
, pp.
4395
4418
.10.1137/070712043
15.
Xie
,
W.
,
Wen
,
C.
, and
Li
,
Z.
,
2001
, “
Input-to-State Stabilization of Switched Nonlinear Systems
,”
IEEE Trans. Autom. Control
,
46
, pp.
1111
1116
.10.1109/9.975483
16.
Vu
,
L.
,
Chatterjee
,
D.
, and
Liberzon
,
D.
,
2007
, “
Input-to-State Stability of Switched Systems and Switching Adaptive Control
,”
Automatica
,
43
, pp.
639
646
.10.1016/j.automatica.2006.10.007
17.
Kartsatos
,
A.
,
1980
,
Advanced Ordinary Differential Equations
,
Mariner
,
Tampa, FL
.
18.
Yoshizawa
,
T.
,
1966
, “
Stability Theory by Lyapunov Second Method
.”
J. Math. Soci. Jpn.
,
9
, pp.
521
535
.
19.
Aubin
,
J. P.
, and
Cellina
,
A.
,
1984
,
Differential Inclusions
,
Springer-Verlag
,
Berlin
.
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