The following problem looks to be classical, but I fail to find a reference for it. If you know it, please help me.

> Given a bounded measurable function $h:{\mathbb R}_+\to\mathbb R$ and a number $a\ge0$, find a Lipschitz function $u$ satisfying
$$u\ge0,\qquad u'\ge h,\qquad u(u'-h)=0,\qquad u(0)=a.$$
That is, one of both constraints is saturated.

It seems to me that the solution is unique, and that
$$u(T)=\min\{p(T)\,|\,p\ge0,\,p'\ge h\}.$$
Also, there seems to be a *dynamic programming principle* in the following sense: if $0<S<T$, then finding $u(T)$ amounts to finding $u(S)$ and then solving the same problem but replacing $t=0$ by $t=S$ and the data $a$ by $u(S)$.