Calculus of Variations: What if the functional is an integral with boundaries at infinity?

Question

I am trying to grasp the basics of Calculus of Variations. The problem seems to be concentrated on functionals of the form : $$ F[y] = \int_{a}^{b} G(y,y(x),y'(x))dx$$ where $y(x)$ is assumed to be fixed at boundaries of integration. What if the boundaries $a$ and $b$ lie at infinity, $a=-\infty$ and $b=\infty$? This is the case most of the time in Variational Machine Learning problems, where $y$ is a probability distribution over whole $\mathbb{R^n}$. The book I am following (Christopher Bishop's Pattern Recognition and Machine Learning) says that these boundaries may be at infinity; but all other resources on Calculus of Variations treat $F[y]$ as a functional with finite boundaries. How can this be extended to bounds at infinity?