Problem

Consider a special case of LTVC Systems: $$\begin{cases} \dot{x}(t)=A(t)x(t)+B(t)u(t) \\ y(t)=C(t)x(t) \end{cases}$$where $D=0$ and $x(t_{0})=0$. Recall from Linear Time Varying Control System that $$x(t)=\Phi_{A}(t,t_{0})x_{0}+\int\limits _{t_{0}}^{t}\Phi_{A}(t,\tau)B(\tau)u(\tau) \, d\tau$$hence, $$\begin{align*} y(t)=C(t)x(t)&=C(t)\Phi_{A}(t,t_{0})x_{0}+\int\limits _{t_{0}}^{t}\underbrace{ C(t)\Phi_{A}(t,\tau)B(\tau) }_{ T(t,\tau) }u(\tau) \, d\tau \\ &=\int\limits _{t_{0}}^{t_{1}}T(t,\tau)u(\tau) \, d\tau \end{align*}$$We call $T:J\times J\to M_{p\times n}(\mathbb{R})$ the Weighting Pattern of the LTVC system. The question we pose is: > Given a mapping $T:J\times J\to M_{p\times n}(\mathbb{R})$ does $\exists A,B,C$ continuous maps s.t. $$T(t,\tau)=C(t)\Phi_{A}(t,\tau)B(\tau)$$?

If yes, we say $T$ is Realizable and $(A,B,C)$ is the Realization of $T$.