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Journal of Convex Analysis 13 (2006), No. 2, 353--362
Copyright Heldermann Verlag 2006

Existence and Relaxation Theorems for Unbounded Differential Inclusions

A. Ioffe
Dept. of Mathematics, Technion, Haifa 32000, Israel


We are interested in the existence of solutions of the differential inclusion $${\dot x}\in F(t,x)$$ on the given time interval, say $[0,1]$. Here $F$ is a set-valued mapping from $[0,1]\times \mathbf{R}^n$ into $\mathbf{R}^n$ (we shall write $F: [0,1] \times \mathbf{R}^n \rightrightarrows \mathbf{R}^n$ in what follows) with closed values which will be assumed nonempty whenever necessary. The classical theorems of Filippov and Wazewski theorem uses, as the main assumption characterizing the dependence of $F$ on $x$, the standard Lipschitz condition $$ h(F(t,x),F(t,x'))\le k(t)\| x-x'\|, $$ where $h(P,Q)$ stands for the Hausdorff distance from $P$ to $Q$. This condition, quite reasonable when $F$ is bounded-valued, becomes unacceptably strong if the values of $F$ can be unbounded. Meanwhile unboundedness of the values of the right-hand side set-valued mapping is a fairly natural property of differential inclusions which appear in optimal control problems, e.g. when we deal with a Mayer problem obtained as a result of reformulation of a problem with integral functional. The main purpose of this note is to provide an existence theorem with a weaker version of the Lipschitz condition which is ``more acceptable'' when the values of $F$ are unbounded. This condition which could be characterized as a ``global'' version of Aubin's pseudo-Lipschitz property is very close to that introduced by P. D. Loewen and R. T. Rockafellar [SIAM J. Control Optimization 32 (1994) 442--470].

Keywords: Differential inclusion, relaxation, global pseudo-Lipschitz condition.

MSC: 49K40, 90C20, 90C31

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