Computation of variations for almost-periodic trajectories.

Summary We present here a variational approach to almost-periodic trajectories of Euler-Lagrange differential equations. The computation of mean variations consists in studying nonlinear time-averaged functionals defined on spaces of almost-periodic functions. The critical points of these functionals are then exactly the almost-periodic solutions of Euler-Lagrange equations. We thus obtain results on the structure of the set of almost-periodic and periodic solutions of autonomous equations with convex Lagrangians. We construct spaces of almost-periodic functions of the Sobolev space type, which induces a notion of weak almost-periodic solution. With these spaces, which are Hilbert spaces, we obtain existence theorems of almost-periodic solutions for almost-periodically forced equations.