The following notations will be used throughout this article. denotes the set of all real non-negative numbers; denotes the n-dimensional space with the scalar product and the vector norm ; denotes the space of all matrices of -dimension. denotes the transpose of A; a matrix A is symmetric if , a matrix I is the identity matrix of appropriate dimension.
Matrix A is semi-positive definite () if , for all ; A is positive definite () if for all ; means .
Consider a nonlinear uncertain delay-difference systems with polytopic uncertainties of the form where
is the state, the system matrices are subjected to uncertainties and belong to the polytope Ω given by
, are given constant matrices with appropriate dimensions.
The nonlinear perturbations
satisfies the following condition
where β is positive constants. For simplicity, we denote by f, respectively.
The time-varying uncertain matrices
are defined by
are known constant real matrices with appropriate dimensions.
are unknown uncertain matrices satisfying
where I is the identity matrix of appropriate dimension.
The time-varying function
satisfies the condition:
Remark 2.1 It is worth noting that the time delay is a time-varying function belonging to a given interval, which allows the time-delay to be a fast time-varying function and the lower bound is not restricted to being zero as considered in [2, 6, 9–11].
Definition 2.1 The nonlinear uncertain system () is robustly stable if the zero solution of the system is asymptotically stable for all uncertainties which satisfy (2.1), (2.3), and (2.4).
Proposition 2.1 For real numbers
, the following inequality hold
The proof is followed from the completing the square:
Proposition 2.2 (Cauchy inequality)
For any symmetric positive definite matrix
Proposition 2.3 ()
, H and F be any constant matrices of appropriate dimensions and
. For any
, we have