Let $u(t)$ be a measurable function on time interval $[0,T]$, consider the "almost everywhere defined" ordinary differential equation:$$\left\{\begin{matrix} \frac{dx(t)}{dt}=b(t,x(t),u(t)) \;\;\;a.e.t\in[0,T]\\ x(0)=x_0\end{matrix}\right.$$ where $b(t,x,u)$ is a measurable function satisfying:
(1) $|b(t,0,u)|<L$ holds for some $L$ and every $(t,u)\in[0,T]\times R$.
(2) $b(t,x,u)$ is Lipschitz continuous in variables $x$ and $u$.
Some literature that I am working with claim that the solution to the above ODE exists and is unique.
I have a few questions though:
(1) Does the solution to the above ODE equivalent to "weak solution"? That is$$\int_0^T x(t)\phi'(t)dt=-\int_0^T b(t,x(t),u(t))\phi(t)dt$$holds for every $\phi\in C_0^\infty([0,T])$?
(2) Does the above ODE equivalent to the integral equation $$x(t)=x(0)+\int_0^tb(s,x(s),u(s))d s$$
(3) Classical theory of ODEs require a continuous $f(t,x)$ to ensure the existence and uniqueness to $dx(t)=f(t,x)dt$ with initial value. Here we don't have any continuity assumed on the time variable, why are the existence and uniqueness still valid?
Please help by leading me to related literatures, thanks.
It is a comment but i can not comment yet:
The first condition needs more information about the function $x$ to be meaningful. We do not know if it is even integrable! However, you can at least define the integrand as a (e.g. tempered) distribution.
The second formula needs $x$ to be absolutely continuous.