Let $A_i$ be an $2\times 2$ matrix for every $i\in\mathbb{N}$. Also, let $U:\ell^2\to\ell^2$ be an arbitrary weakly contraction operator. Define $A:\mathbb R^\infty\to \mathbb R^\infty$ as
$A=\text{diag}(A_1,A_2,...)+U$.
We consider a differential game described by the following system of differential equations
\begin{equation}
\dot{x}=Ax, \ \ x(0)=x_0\in\ell^p, \ p\in[0,\infty].
\end{equation}
Suppose that $\{A_i\}$ is a uniformly normalizable family.
Let $\{\lambda_{i,j}\}_{j = 1}^{d_i}$ denote the set of eigenvalues of $A_i$, $i\ge 2$.
Assertion. If $\sup_{i, j}Re(\lambda_{i,j}) < 0$ the above system is exponentially stable for $p\in[0,\infty]$.
By assumption the spectrum of $e^U$ is contained in $\{z\,:\,|z|<1\}.$ Thus the spectral radius $r$ of $e^U$ is strictly less than $1.$ Since $$r=\lim_n\|e^{nU}\|^{1/n}$$ we get $\|e^{nU}\|\underset{n}{\to} 0.$ Hence $$\lim_{t\to \infty}\|e^{tU}\|=0$$