Finding stationary function of the functional with the condition that $y(x)$ is decreasing

Question

Given a functional $I[f] = \int_{x_1}^{x_2}F(x,y,y')\,dx$ and the boundary conditions $y(x_1)=y_1, y(x_2)=y_2$ with an additional condition that $y(x)$ is decreasing. How will this change Euler-Lagrange equation and what other conditions will be imposed? It could be done using lagrange multipliers and KKT conditions, but I cannot figure of will Lagrangian change.