On an optimal investment problem (Q1878733)

The paper considers the following optimal control problem describing a simplified model of producing a new technological product with account for investments to scientific research: \[ \begin{aligned} \dot x&=-\sigma x + u,\qquad u\in [0,p],\\ x(0) &=x_0,\qquad x(T)=x_1,\\ J(u) &= \int_0^Te^{-\gamma s}u^\alpha(s)\, ds \to \min, \end{aligned} \] where \(\alpha=1/\gamma>0\), \(p=q^\gamma>0\). The author first studies the attainability problem for the control system, and then he studies the structure of an optimal control in the problem using the Pontryagin maximum principle. In particular, he proves the differentiability of the optimal value of the functional considered as a function of \(T\).