1.4 Some Main Definitions and Lemmas
11
holds:
lim
t→∞
Ex(t)
2
= 0
( 1 . 4 )
Definition 1.3 The multi-model jumping system (1.2) is mean exponentially stable
for any t ≥ 0 and any modes, if there exists two positive scalars α and β, the following
relation holds for any initial state and mode (x 0 , r 0 ):
E
∞
0
x(t)
2 dt|x 0 , r 0
≤ αx(t)
2 e
−βt
(1.5)
In general, the above three definitions are equivalent [9, 16]. When we consider the
uncertainties and state-feedback schemes of system (1.2), we still have the following
equivalent definitions.
Definition 1.4 Consider the following multi-model jumping system with uncertainties:
˙
x(t) = [A(r t ) + A(r t )]x(t) + [B(r t ) + B(r t )]u(t) + [B d (r t ) + B d (r t )]ω(t)
x(t) = x 0 , r t = r 0 , t = 0.
(1.6)
where A(r t ), B(r t ) and B d (r t ) are the uncertain time-varying matrix of multimodel jumping system and we have the following equivalent definitions.
(1) The multi-model jumping system (1.6) is robustly stochastically stable for any
t ≥ 0 and any modes, if there exists a positive scalar M(x 0 , r 0 ), we have relation
(1.4) for any initial state and mode (x 0 , r 0 ).
(2) The multi-model jumping system (1.6) is robustly mean square stable for any
t ≥ 0 and any modes, if for any initial state and mode (x 0 , r 0 ), we have relation
(1.5).
(3) The multi-model jumping system (1.6) is robustly mean exponentially stable for
any t ≥ 0 and any modes, if there exists two positive scalars α and β, we have relation
(1.6) for any initial state and mode (x 0 , r 0 ).
Definition 1.5 For the state feedback law u(t) = K (r t )x(t), we have the following
equivalent definitions.
(1) The multi-model jumping system (1.2) is stochastically stabilizable for any t ≥ 0
and any modes, if there exists a positive scalar M(x 0 , r 0 ), we have relation (1.4) for
any initial state and mode (x 0 , r 0 ).
(2) The multi-model jumping system (1.2) is mean square stabilizable for any t ≥ 0
and any modes, if for any initial state and mode (x 0 , r 0 ), we have relation (1.5).
(3) The multi-model jumping system (1.2) is mean exponentially stabilizable for any
t ≥ 0 and any modes, if there exists two positive scalars α and β, we have relation
(1.6) for any initial state and mode (x 0 , r 0 ).
Definition 1.6 In the Euclidean space R
n
× M × R + , we introduce a Lyapunov–
Krasovskii V [x(t), r t = i, t > 0] = V [x(t), i], and define the weak infinitesimal
operator [5, 9, 15, 16] as:
11
holds:
lim
t→∞
Ex(t)
2
= 0
( 1 . 4 )
Definition 1.3 The multi-model jumping system (1.2) is mean exponentially stable
for any t ≥ 0 and any modes, if there exists two positive scalars α and β, the following
relation holds for any initial state and mode (x 0 , r 0 ):
E
∞
0
x(t)
2 dt|x 0 , r 0
≤ αx(t)
2 e
−βt
(1.5)
In general, the above three definitions are equivalent [9, 16]. When we consider the
uncertainties and state-feedback schemes of system (1.2), we still have the following
equivalent definitions.
Definition 1.4 Consider the following multi-model jumping system with uncertainties:
˙
x(t) = [A(r t ) + A(r t )]x(t) + [B(r t ) + B(r t )]u(t) + [B d (r t ) + B d (r t )]ω(t)
x(t) = x 0 , r t = r 0 , t = 0.
(1.6)
where A(r t ), B(r t ) and B d (r t ) are the uncertain time-varying matrix of multimodel jumping system and we have the following equivalent definitions.
(1) The multi-model jumping system (1.6) is robustly stochastically stable for any
t ≥ 0 and any modes, if there exists a positive scalar M(x 0 , r 0 ), we have relation
(1.4) for any initial state and mode (x 0 , r 0 ).
(2) The multi-model jumping system (1.6) is robustly mean square stable for any
t ≥ 0 and any modes, if for any initial state and mode (x 0 , r 0 ), we have relation
(1.5).
(3) The multi-model jumping system (1.6) is robustly mean exponentially stable for
any t ≥ 0 and any modes, if there exists two positive scalars α and β, we have relation
(1.6) for any initial state and mode (x 0 , r 0 ).
Definition 1.5 For the state feedback law u(t) = K (r t )x(t), we have the following
equivalent definitions.
(1) The multi-model jumping system (1.2) is stochastically stabilizable for any t ≥ 0
and any modes, if there exists a positive scalar M(x 0 , r 0 ), we have relation (1.4) for
any initial state and mode (x 0 , r 0 ).
(2) The multi-model jumping system (1.2) is mean square stabilizable for any t ≥ 0
and any modes, if for any initial state and mode (x 0 , r 0 ), we have relation (1.5).
(3) The multi-model jumping system (1.2) is mean exponentially stabilizable for any
t ≥ 0 and any modes, if there exists two positive scalars α and β, we have relation
(1.6) for any initial state and mode (x 0 , r 0 ).
Definition 1.6 In the Euclidean space R
n
× M × R + , we introduce a Lyapunov–
Krasovskii V [x(t), r t = i, t > 0] = V [x(t), i], and define the weak infinitesimal
operator [5, 9, 15, 16] as: