  This paper provides a necessary and sufficient condition for guaranteeing
exponential stability of the linear difference equation $x(t)=Ax(t-a)+Bx(t-b)$
where $a>0,b>0$ are constants and $A,B$ are $n\times n$ square matrices, in
terms of a linear matrix inequality (LMI) of size $\left( k+1\right) n\times
\left( k+1\right) n$ where $k\geq1$ is some integer. Different from an existing
condition where the coefficients $\left( A,B\right) $ appear as highly
nonlinear functions, the proposed LMI condition involves matrices that are
linear functions of $\left( A,B\right) .$ Such a property is further used to
deal with the robust stability problem in case of norm bounded uncertainty and
polytopic uncertainty, and the state feedback stabilization problem. Solutions
to these two problems are expressed by LMIs. A time domain interpretation of
the proposed LMI condition in terms of Lyapunov-Krasovskii functional is given,
which helps to reveal the relationships among the existing methods. Numerical
example demonstrates the effectiveness of the proposed method.
