I'm now approaching for the first time to first order differential equations with discontinuous right hand side.

Let $A\subset \mathbb{R}\times\mathbb{R}^n$ and $f=f(t, x):A\to \mathbb{R}^n$. Fix $(t_0,x_0)\in A$ and consider the initial value problem $$\begin{cases} y'(t)=f(t,y(t))\\y(t_0)=x_0\end{cases}.\qquad[A]$$ It is well known that if $f$ is continuous then Peano's Theorem guarantee the existence of a local classical solution to $[A]$, which is also a $C^1$ function.

The notion of Carathéodory solution for an initial problem like $[A]$ arises when $f$ is a general function, not necessarily continuous.

Def. An absolutely continuous function $y:[t_0,t_0+a]\to \mathbb{R}^n$ ($a \in \mathbb{R},a>0$), such that $(t,y(t))\in A \,\forall t \in [t_0,t_0+a]$, is said to be a Carathedory solution for $[A]$ if $$y(t)=x_0+\int_{t_0}^{t}f(s,y(s)ds\qquad t \in [t_0,t_0+a].$$ By equivalent definitions of absolute continuity for functions on closed and bounded intervals it follow immediately that if $y:[t_0,t_0+a]\to \mathbb{R}^n$, such that $(t,y(t))\in A \,\forall t \in [t_0,t_0+a]$, is an absolutely continuous function, then $$\text{y is a Carathédory solution of}\, [A] \iff \,\text y(t_0)=x_0\,\text{and}\, y'(t)=f(t,y(t))\, \text {for a.e.}\, t \in [t_0,t_0+a].$$

By Carathédory existence theorem we know that, under suitable conditions on $f$, there is at least a solution of $[A]$, in the weak sense as above.

Now let's suppose we are given a function $f:[0,+\infty)\times \mathbb{R}^n\to \mathbb{R}^n$ and a point $x_0 \in \mathbb{R}^n$. Consider the initial value problem $$\begin{cases} y'(t)=f(t,y(t))\\y(0)=x_0\end{cases}.\qquad[B]$$ In many books I read that under the hypothesis:

(i)$f$ is bounded

(ii) the map $t\to f(x, t)$ is measurable for each $x$

(iii) Exists $L>0$ such that $$|f(t,x_1)-f(t,x_2)|\leq L|x_1-x_2| \qquad \forall (t,x_1)(t,x_2) \in [0,+\infty)\times \mathbb{R}^n $$ then there exist a unique global solution $y_{x_0}:[0,+\infty)\to \mathbb{R}^n$ to $[B]$.

The problem is that I'm missing what "solution" in this case means (the interval of definition of $y_{x_0}$ is not closed and it is also unbounded). I think that it could be understood like a function $y:[0,+\infty)\to \mathbb{R}^n$ such that $y$ is absolutely continuous on every compact subinterval $[a,b]\subset [0,+\infty)$ and $$y(t)=x_0+\int_{t_0}^{t}f(s,y(s))ds \qquad t>0,$$ but I'm not sure about it. For instance, this problem arises in studying optimal control theory.

Can anyone give me some help or some good references?

Thanks a lot.


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