In the basic theory of optimal control we must have a unique absolutely continuous function as a solution to a differential system. I will choose the LQR (Linear Quadratic Regulator problem):

$$\begin{cases} x'(t)=Ax+Bu \\ x(t_0)=x_0\end{cases}\ \text{almost everywhere} \ t\in [t_0,T].$$

Here all we know is that the control $u\in L^2(t_0,T;\mathbb{R}^m)$. How can we prove that this Cauchy problem has a **unique absolutely continuous solution** $x:[t_0;T]\to\mathbb{R}^N$, where: $A\in\mathcal{M}_{N}(\mathbb{R}), B\in\mathcal{M}_{N,M}(\mathbb{R})$ and $x_0\in\mathbb{R}^N$ are fixed and are constants.

Practically, we have to prove that:

$$x(t)=e^{(t-t_0)A}\cdot x_0+\int_{t_0}^t e^{(t-s)A}\cdot B\cdot u(s)\ ds,\ \forall\ t\in [t_0,T]$$ is

**1. absolutely continuous**

**2. solution of the Cauchy problem stated below**

**3. it is the unique absolutely continuous solution of our Cauchy problem**

I cannot show rigorously any of them (I hardly made 1., just for $N=M=1$). Any advice will be great received by me.

I am interested in a more general existence result in that sense (for $L^2$ controls, and $A.C.$ solutions), but taking into account the controls. I know that all is good if we work with $u$ as a piecewise continuous functions, but that's not the context.