CoV28

Compute the first variation $J(y) = \int_0^1 3y^2 + x dx + y^2(0), y_0(x) = x, h(x)=x+1\,$

$\delta J(y_0,h) = \frac{d}{d\epsilon} \left[ \int_0^1 3(y_0(x) + \epsilon h(x))^2 + xdx + (y_0(0) + \epsilon h(0))^2\right] \Bigg|_{\epsilon=0} \,$

$=\frac{d}{d\epsilon} \left[ \int_0^1 3(x+\epsilon(x+1))^2 + xdx + (0+\epsilon)^2\right]\Bigg|_{\epsilon=0} \,$

$=\frac{d}{d\epsilon} \left[ \int_0^1 3x^2 + 6\epsilon x (x+1) + 3\epsilon^2(x+1)^2 + xdx + \epsilon^2\right]\Bigg|_{\epsilon=0} \,$

$=\frac{d}{d\epsilon} \left[ 7\epsilon^2 + 5\epsilon + \epsilon/2 + \epsilon^2 \right]\Bigg|_{\epsilon=0} \,$

$=14\epsilon + 5 + 2\epsilon \Big|_{\epsilon=0} \,$

$=5\,$