We discuss a variational approach to abstract doubly nonlinear evolution systems defined on the time half line . This relies on the minimization of weighted energy-dissipation (WED) functionals, namely a family of -dependent functionals defined over entire trajectories. We prove WED functionals admit minimizers and that the corresponding Euler-Lagrange system, which is indeed an elliptic-in-time regularization of the original problem, is strongly solvable. Such WED minimizers converge, up to subsequences, to a solution of the doubly nonlinear system as . The analysis relies on a specific estimate on WED minimizers, which is specifically tailored to the unbounded time interval case. In particular, previous results on the bounded time interval are extended and generalized. Applications of the theory to classes of nonlinear PDEs are also presented.