Let $J\subset \mathbb{R}$ be any interval, and let $t_{0}$ be an internal point of $J$, (i.e. $t_{0}\in \text{int}(J)$). Let $A:J\to M_{n}(\mathbb{R})$ be continuous. Then $\exists!x\in C^{1}(J;\mathbb{R}^{n})$ that solves $$\begin{cases} \dot{x}(t)&=A(t)x(t) \\ x(t_{0}) & =x_{0} \end{cases}$$ We can then refine this theorem in the case where $J$ is a closed and bounded interval

Let $a,b\in\mathbb{R}$ with $a<b$ and let $A:[a,b]\to M_{n}(\mathbb{R})$ be continuous. Suppose $t_{0}\in(a,b)$, then $\exists!x\in C^{1}([a,b];\mathbb{R}^{n})$ that solves $$\begin{cases} \dot{x}(t)&=A(t)x(t) \\ x(t_{0}) & =x_{0} \end{cases}$$ In class we first proved 2 then used 2 to easily prove 1.

Let $J\subset \mathbb{R}$ be a non-empty interval. There is a unique solution to the following LTVC system $$\begin{cases} \dot{x}(t)&=A(t)x(t)+B(t)u(t) \\ x(t_{0}) & =x_{0} \end{cases}$$given by $$x(t)=\Phi_{A}(t,t_{0})x_{0}+\int\limits _{t_{0}}^{t}\Phi_{A}(t,\tau)B(\tau)u(\tau) \, d\tau$$where $\Phi_{A}(t,t_{0})$ is the Transition Matrix.

Consider a LTVC system $$\begin{cases} \dot{x}(t)=A(t)x(t)+B(t)u(t) \\ x(t_{0})=x_{0} \end{cases}$$Let $t_{0},t_{1}\in J$, $t_{0}<t_{1}$ and let $x_{0},x_{1}\in\mathbb{R}^{n}$. Then, $\exists$ some control $u\in C([t_{0},t_{1}];\mathbb{R}^{m})$ which “steers” the system from $x(t_{0})=x_{0}$ to $x(t_{1})=x_{1}$ if and only if$$\Phi_{A}(t_{0},t_{1})x_{1}-x_{0}\in \text{Image}(W(t_{0},t_{1}))$$holds, where $$W(t_{0},t_{1})=\int\limits _{t_{0}}^{t_{1}}\Phi_{A}(t_{0},\tau)B(\tau)B^{T}(\tau)\Phi^{T}_{A}(t_{0},\tau) \, d\tau$$is our controllability gramian. Therefore, the control input $u(t)$ is defined as $$u(t)=B^{T}(t)\Phi_{A}^{T}(t_{0},t)\eta$$ where $\eta\in\mathbb{R}^{n}$ s.t. $$\Phi_{A}(t_{0},t_{1})x_{1}-x_{0}=W(t_{0},t_{1})\eta$$ gives the desired transfer.

Consider the LTVC System introduced in the Observability Problem: $$\begin{align*} \dot{x}(t)=A(t)x(t)+B(t)u(t)\\ y(t)=C(t)x(t)+D(t)u(t) \end{align*}$$The initial state can be uniquely determined (i.e. our system is Observable) if $$\text{Ker}(M(t_{0},t_{1}))=\{ \vec{0} \}$$