Poisson differential inclusion and suboptimality results for convex variational problems

This note deals with a nonsmooth convex problem of calculus of variations in which the cost term has the general form F(x) := G x(0), . . . , x(m−1)(0),x(T ), . . . , x(m−1)(T ) + T 0 L t,x(t), . . ., x(m−1)(t ), x(m)(t ) dt. We derive a necessary and sufficient condition for a trajectory x to be an ε-minimum of F. Optimal trajectories (or exact minima) of F are obtained, of course, by setting the approximation parameter ε at level 0. Our result is specially relevant in a context in which optimal trajectories are not assumed to exist.  2003 Elsevier Inc. All rights reserved