Proof. For system (3), by Lemma 2, linear matrix inequalities (LMIs) (5) and (6) imply that (*E*,*A*) is admissible, and then there exist two nonsingular matrices , such that

where . By [23], we can always assume that . Let , by (5), we have

Let

where . Premultiplying (3) by *M*, we have

- (7)

where

Now system (7) is

- (8)

- (9)

By inequality (6), we have

- (10)

Inequality (10) implies that

- (11)

By inequality (11), we have

- (12)

which implies that

- (13)

Hence,

which is equivalent to

which implies that . Therefore, for any given and ,

- (14)

It follows from the fixed-point principle that (14) implies that there exists unique solution for any given and *u* from the second equation of (9). Next, we show that exists and is unique; by means of the Picard Lindelof theorem for the existence and uniqueness of solution to ordinary differential equations, we need to show that is Lipschitz with respect to , that is, we only need to show that is Lipschitz with respect to . In fact, for any given and ,

- (15)

Therefore,

- (16)

By (4) and (16), we have is Lipschitz with respect to .

Next premultiplying and postmultiplying (13) by and its transposition, respectively, we have

and hence,

- (17)

Notice that ; then from (9) and (17), we have

where *f* = *f*(*x*,*u*). By (4), we have

- (18)

Therefore,

which implies that

thereby

By inequality (12), we have , note that *P*_{1} *⩾*0, and we can prove that *P*_{1} > 0. In fact, if min{*λ*(*P*_{1})} = 0, then there exists *ξ* such that *P*_{1}*ξ* = 0; thus , which is impossible.

Now let and *K*_{1}(*ξ*) = min{*λ*(*P*_{1})}*ξ*^{2} and *K*_{2}(*ξ*) = max{*λ*(*P*_{1})} *ξ*^{2}; then, *K*_{1},*K*_{2} are *K*_{ ∞ }-functions, and

From (3) and (5),

By (6), choose *λ* = max{*λ*(Θ)} < 0,*λ*_{N} = min{*λ*(*N*^{T}*N*)} > 0; we have

Then, there exist *K*_{ ∞ }-function *K*_{3}(*ξ*) = − *λλ*_{N}*ξ*^{2} and *K*-function *K*_{4}(*ξ*) = *βξ*^{2}, such that

By Lemma 1, there exist *KL*-function *γ*_{1} and *K*-function *χ*_{1}, such that

Denote , and then

Hence,

Then, there exist *KL*-function *γ*_{2} and *K*-function *χ*_{2}, such that

which implies that there exist *KL*-function *γ*_{3} and *K*-function *χ* such that

Note that

Hence,

Consequently, there exist *KL*-function *γ* and *K*-function *χ* such that

thus, system (3) is ISS.□

Remark 6. The results in this paper is more general than that in [4]. When considering ISS, one important issue is to find a proper condition on the function *f*(*x*,*u*) such that it is easy to check and compatible with ISS. In this paper, condition (4) is less strict than that used in [4]. In [4], the bound of the norm of *f*(*x*,*u*) only depends on the state *x*; it does not depend on the input *u*. In our results, the norm of *f*(*x*,*u*) is assumed to be bounded by the sum of a function on *x* and another function on *u*. Thus, the model used in this paper is more general and will have more applications than that in [4].

Next, we consider the following special descriptor system:

- (19)

where , matrix , and .