Variational problems governed by a multi-valued differential equation (Q800619)

The problem of maximizing the integral functional \((1)\quad J(x)=\int^{T}_{0}u(t,x(t),\dot x(t))dt\) over the set \(K_ a\) of all solutions of the multivalued differential equation (2) \(\dot x(t)\in F(t,x(t))\), \(x(0)=a\) where \(x\in H\), a finite-dimensional Hilbert space, is studied. It is assumed that F(t,x) is non-empty, compact and convex for all (t,x) and satisfies the Carathéodory-type conditions. The solution \(x(\cdot)\) of (2) is understood as an element of the Sobolev space \(W^{1,2}\). In a previous paper of the author (see the paper reviewed above) it was proved that \(K_ a\) is a non-empty, compact subset of \(W^{1,2}\) and the dependence \(a\to K_ a\) is upper hemicontinuous. With respect to (1) it is assumed that u is a normal integrand which is concave in \((x,\dot x)\) for fixed t and satisfies a growth condition. It is proved that J is upper semicontinuous in the weak topology of \(K_ a\); so, the solution of (1), (2) exists. Next, the existence of a maximum of J over \(x\in K_ a\), \(a\in A\) (a compact, convex subset of H) is proved by using of the well-known theorem of Berge.