LTI systems in state space representation are systems of the form:
\begin{eqnarray} \dot{\mathbf{x}}(t)=\mathbf{Ax}(t)+\mathbf{Bu}(t) \end{eqnarray} \begin{eqnarray} \mathbf{y}(t)&=&\mathbf{Cx}(t)+\mathbf{Du}(t) \end{eqnarray}
where $\mathbf{x}$ is the state vector, $\mathbf{u}$ is the input and $\mathbf{y}$ is the output. These systems satisfy the superposition principle only if the initial condition $\mathbf{x}=\mathbf{0}$ holds. Let us refer to the class of these systems with the aforementioned initial condition as $\Sigma_{LTI}^0$.
Now a different class of systems, namely $\Sigma_{conv}$ are the ones that are represented by a convolution integral, i.e. the input-output behaviour of the system is described by:
\begin{equation} \mathbf{y}(t)=\int_{-\infty}^\infty h(\tau) \mathbf{u}(t-\tau)d\tau \end{equation}
My question is whether it is true that every system in $\Sigma_{LTI}^0$ admits a representation in $\Sigma_{conv}$? If yes, how?
Note: I understand that my question boils down to finding a function $h$ such that (assume for simplicity that $\mathbf{C}=\mathbf{I}$ and $\mathbf{D}=\mathbf{0}$):
\begin{equation} \int_0^t e^{\mathbf{A}(t-\tau)}\mathbf{Bu}(\tau)d\tau= \int_{-\infty}^\infty h(\tau) \mathbf{u}(t-\tau)d\tau \end{equation}
Update: Let us assume that there exists a $h\in\mathcal{L}^2(\Re;\Re^n)$ such that an LTI system is equivalent to a convolution system and assume for simplicity that the input is one-dimensional. Then, these have identical impulse responses, hence for $u(t)=\delta(t)$ we have:
\begin{equation} \int_0^t e^{\mathbf{A}(t-\tau)}\mathbf{B}\delta(\tau)d\tau= \int_{-\infty}^\infty h(\tau) \delta(t-\tau)d\tau \end{equation}
Therefore:
\begin{equation} \int_0^t e^{\mathbf{A}(t-\tau)}\mathbf{B}\delta(\tau)d\tau= h(t) \implies h(t)=e^{\mathbf{A}t}\mathbf{B} \end{equation}
But then:
\begin{equation} \int_0^t e^{\mathbf{A}(t-\tau)}\mathbf{Bu}(\tau)d\tau= \int_{-\infty}^\infty e^{\mathbf{A}(t-\tau)}\mathbf{Bu}(\tau)d\tau \end{equation}
Doesn't look good! So, if there is no mistake in the above procedure, there is no $h$ that fulfills my requirements. Then, what is the connection between the aforementioned classes of systems.