  An example of a nonunique solution of the Cauchy problem of
Hamilton-Jacobi-Bellman (HJB) equation with surprisingly regular Hamiltonian is
presented. The Hamiltonian H(t,x,p) is locally Lipschitz continuous with
respect to all variables, convex in p and with linear growth with respect to p
and x. The HJB equation possesses two distinct lower semicontinuous solutions
with the same final conditions; moreover, one of them is the value function of
the corresponding Bolza problem. The definition of lower semicontinuous
solution was proposed by Barron-Jensen (1990) and Frankowska (1993). Using the
example an analysis and comparison of assumptions in some uniqueness results in
HJB equations is provided.
