Please help me how to deal with maximization of functional like this:
$$F\{a(s)\} = \int\limits_0^t \left( g(a(s)) - \alpha\, v(s)^2 \right) ds, \ a(s) \in \left[0, \infty\right)$$
where $g(x) = x e^{-x}$ , $\alpha=\mathrm{const}$, and $\displaystyle v(t) = \int\limits_0^t a(s) ds$
Thank you for reading.
Reformulate the problem in terms of $v$. That is, you seek the maximum of $$F\{v\} = \int\limits_0^t \left( g(v'(s)) - \alpha\, v(s)^2 \right) ds, \ a(s) \in \left[0, \infty\right) \tag1$$ over increasing differentiable functions $v$ with $v(0)=0$. The Euler-Lagrange equation for (2) is easy to state: $$ -\frac{d}{ds}(g'(v'(s))) -2\alpha v(s)\equiv 0 \tag2 $$ Since $g''(x)=(x-2)e^{-x}$, the equation (2) becomes $$ (2-v'(s))e^{-v'(s)}v''(s) -2\alpha v(s)\equiv 0 \tag3 $$ I wouldn't expect an explicit solution of (3), but a numeric solution should not be hard to obtain.