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For a differential inclusion $x'(t)\in h(x(t))$, is there any condition (of course, I don't want the map to be single-valued) under which we can say that for any trajectory $x(.)$ satisfying the differential inclusion there exist a continuous function $F_{x(.)}(.)$ (i.e. depending on the trajectory x(.)) s.t. $F_{x(.)}(x) \in h(x)$ and $x'(t) = F_{x(.)}(x(t))$.

I am aware of continuous selection theorems for set-valued maps, but have not seen anything like above.

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  • $\begingroup$ I can't say that I know much about the subject, but have you looked in the book Differential Inclusions by Aubin & Cellina (1984)? $\endgroup$ Sep 15, 2015 at 13:26
  • $\begingroup$ @IgorKhavkine: Yes, but did not get much. $\endgroup$
    – Sosha
    Sep 15, 2015 at 13:43

1 Answer 1

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There are such results. For example, if $-h$ is a maximal monotone operator on a Hilbert space $H$

$$ A: D(A)\subset H\to 2^H, $$

where $2^H$ denotes the collection of subsets of $H$, then the differential inclusion

$$ x'(t)+A x(t)\ni 0,\;\;x(0)=x_0, \;\;t>0, $$

has a unique (appropriately defined) solution. Moreover,

$$x'(t)+A^0 x(t) =0 $$

for almost all $t$, where $A^0 x(t)$ denotes the point in the closed convex set $A x(t)$ closest to the origin, i.e., the shortest vector in $A x(t)$. For details I refer to Theorem 3.1 in H. Brezis' book

Operateurs maximaux monotone et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co. 1973.

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