Problem

Consider an LTVC System: $$\begin{cases} \dot{x}(t)=A(t)x(t)+B(t)u(t)\quad & x(t)\in\mathbb{R}^{n}, \, u(t)\in\mathbb{R}^{m}\\ y(t)=C(t)x(t)+D(t)u(t)\quad & \ \end{cases}$$where $A(t)\in\mathbb{R}^{n\times n},B(t)\in\mathbb{R}^{n\times m},C(t)\in\mathbb{R}^{p\times n},D(t)\in\mathbb{R}^{p\times m}$.

Let’s assume we know $A,B,C,D$ and $u(t)|_{[t_{0},t_{1}]},y(t)|_{[t_{0},t_{1}]}$. Now we recall by Linear Time Varying Control System that $$x(t)=\Phi_{A}(t,t_{0})x_{0}+\int\limits _{t_{0}}^{t}\Phi_{A}(t,\tau)B(\tau)u(\tau) \, d\tau$$ By our assumption, if we know the initial state, we can find $x(t)$. Hence $x_{0}$ is the unknown in the observability problem, and it is what we want to recover. We can express $y$ as follows: $$\begin{align*} y(t)&=C(t)x(t)+D(t)u(t)\\ y(t)&=C(t)\left( \Phi_{A}(t,t_{0})x_{0}+\int\limits _{t_{0}}^{t}\Phi_{A}(t,\tau)B(\tau)u(\tau) \, d\tau \right)+D(t)u(t)\\ y(t)&=\underbrace{ C(t)\Phi_{A}(t,t_{0})x_{0} }_{ \text{unknown} } +\underbrace{ \int\limits _{t_{0}}^{t_{1}}C(t)\Phi_{A}(t,\tau)B(\tau)u(\tau) \, d\tau+D(t)u(t) }_{ \text{known} } \end{align*}$$hence we can rearrange and write $$C(t)\Phi_{A}(t,t_{0})x_{0}=y(t)-\int\limits _{t_{0}}^{t_{1}}C(t)\Phi_{A}(t,\tau)B(\tau)u(\tau) \, d\tau+D(t)u(t)$$Then we define the linear map $$\begin{align*} \hat{L}:&\mathbb{R}^{n}\to C([t_{0},t_{1};\mathbb{R}^{p}])\\ &x\mapsto H(t)x=C(t)\Phi_{A}(t,t_{0})x \end{align*}$$