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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 $u(S)$.

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  • $\begingroup$ You did not specify the role of $a$ - I presume you're imposing $u(0) = a$? I'm guessing you're not happy with the naive approach, namely to first put $v(t) = a + \int_0^t h$, and then defining $u = \max \{ v , 0 \}$, because $u' = h$ is only satisfied at a.e. point of $\{ u \neq 0 \}$? $\endgroup$
    – Leo Moos
    Commented Mar 2, 2023 at 15:40
  • $\begingroup$ @Leo Oh yes, $u(0)=a$. But $u=v_+$ is not the solution, because it does not always satisfy $u'\ge h$ (when $v<0$). $\endgroup$
    – Denis Serre
    Commented Mar 2, 2023 at 16:26
  • $\begingroup$ Oops, how silly! Thanks for pointing this out. $\endgroup$
    – Leo Moos
    Commented Mar 2, 2023 at 16:30
  • $\begingroup$ Have you looked into the optimal control bibliography? The second part of the question seems very similar to the Pontryagin maximum principle with constraints. With the notations of control theory, you would be looking to minimize the objective function $\Psi(x(T)) := x(T)$ where the scalar state satisfies $\dot{x}(t) = u(t)$, $x(0) = a$ with the state constraint $x(t) \geq 0$ and the control constraint $u(t) \geq h(t)$. It looks like the PMP should then imply that, for each $t$, either $u(t) = h(t)$ or $x(t) = 0$. Maybe this doesn't help because you somehow ask for the converse implication. $\endgroup$
    – cs89
    Commented Mar 2, 2023 at 17:24

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