Timeline for Differential inequality with convex constraint

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Mar 2, 2023 at 17:24	comment	added	cs89		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.
Mar 2, 2023 at 16:30	comment	added	Leo Moos		Oops, how silly! Thanks for pointing this out.
Mar 2, 2023 at 16:26	history	edited	Denis Serre	CC BY-SA 4.0	added 14 characters in body
Mar 2, 2023 at 16:26	comment	added	Denis Serre		@Leo Oh yes, $u(0)=a$. But $u=v_+$ is not the solution, because it does not always satisfy $u'\ge h$ (when $v<0$).
Mar 2, 2023 at 15:40	comment	added	Leo Moos		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 $\{ u \neq 0 \}$?
Mar 2, 2023 at 15:23	history	asked	Denis Serre	CC BY-SA 4.0