Page 21 - Read Online

Sun et al. Complex Eng Syst 2022;2:17 I http://dx.doi.org/10.20517/ces.2022.48 Page 5 of 15

where = ( − ) and

( , , )
( , , ) = ,

( )
( , , ) ( , , )
( , , ) = , · · · , ,
1
( 2 )
( , , )
( , , ) = .

×
Moreover, assume that there exists a constant > 0, such that

3 − 4 ¯ − 2 = 0, (10)
1 − 2 ¯ > 0. (11)
Remark 2 The system studied in this research has the property of high nonlinearity. In other words, the LGC is
removed from the SSS (1), which makes the considered system more general. Without the LGC, the solution of a
stochastic system may explode in a finite time. To ensure the existence of a global solution, a PGC (i.e., condition
(6)) is imposed on the SSS (1) (see, e.g., [13,27,28] ). Therefore, the system (1) we studied obeys the LLC (i.e., condition
(5)) and the PGC. By combining the MLF approach and ADT method, we then prove the existence and uniqueness
of the global solution.
Before presenting the main results, the definition of ADT is revisited.

Definition 1 [28] For a switching signal ( ) and any ≥ ≥ 0, ( , ) and ( , ) denote the whole running
time and the switching number of the i-th subsystem over the interval [ , ], respectively, ∈ Γ. Then, the following
inequality holds:

( , )
( , ) ≤ + 0 ,
J
where J > 0 is called the mode-dependent ADT and 0 > 0 is the mode-dependent chatter bound.

3. MAIN RESULTS
In this section, we prove the existence of a unique global solution for a highly nonlinear SSS (1) by using the
ADT and MLF approaches. Then, both the th moment exponential stability and almost surely exponential
stability are provided for a highly nonlinear SSS (1).

Theorem 1 Under Assumptions 1-3, if there exists a constant > 0 such that

ln
J > . (12)

Then, foranyinitialdata(2), thereexistsauniqueglobalsolution ( ) fortheSSS(1)on [− , ∞), andthesolution
satisfies

sup E| ( )| < ∞. (13)
− ≤ <∞
Proof. We divide the whole proof into two steps. In step 1, for all ∈ S, we prove that the -th subsystem with
the initial value (0) has a unique global solution ( ). In step 2, when each subsystem has a unique global
solution, the SSS (1) with a deterministic switching signal has a unique global solution ( ) on [− , ∞).
Step 1. For all ∈ S, the control system becomes

( ) = ( , ( ), ( )) + ( , ( ), ( )) ( ), ≥ − , (14)