Existence and Relaxation Theorems for Unbounded Differential Inclusions

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].