I have an objective function:
$$O(x_1, x_2, x_3, x_4) = \int_{T_1}^{T_2} \sqrt{F_1(t)^2 + F_2(t)^2} dt$$ , where
$$F_1(t) = x_1a_1(t) + x_2a_2(t) $$ and $$F_2(t) = x_3a_1(t) + x_4a_2(t). $$
$a_1(t)$ and $a_2(t)$ are known functions.
I'd like to obtain $\mathbf{x} = [x_1, x_2, x_3, x_4]^T$ that maximizes $O$ with constraints $$ x_1^2 + x_2^2 \leq k $$ $$ x_3^2 + x_4^2 \leq k $$
How can I approach this problem? Any hint would be appreciated.
I think I can convince myself that you want $x_1=x_3, x_2=x_4$. Then you have $O=\sqrt 2\int_{T_1}^{T_2}|x_1a_1(t)+x_2a_2(t)|dt$ Now you can take partials to find $\frac {\partial O}{\partial x_1}=\sqrt 2\int_{T_1}^{T_2}a_1(t)\operatorname{sgn}(x_1a_1(t)+x_2a_2(t))dt$ and the other, set them both to zero and solve the simultaneous equations. To check the boundary, you need to set $x_1=\pm\sqrt{k-x_2^2}$ and take the derivative with respect to $x_2$