In the following, I am referring to the general "Hamiltonian control theory" using the conventions defined here.
I am working on a very simple S($x_1$)I($x_2$)R($x_3$) model for infectious disease spreading with the additional control function $u$. The model is explicated by the ODE system $\dot{\bf x}={\bf f}({\bf x},{\bf u}) \bf x$:
$$\begin{aligned} \dot{x}_1 &= -c(1-u) x_1 x_2 \\ \dot{x}_2 &= c(1-u) x_1 x_2 - k x_2 \\ \dot{x}_3 &= k x_2 \end{aligned}$$
plus suitable initial ($t=t_0$) and end ($t=t_1$) values for the functions. I am now particularly interested in constraining the control function $u(t)$ to a fixed integral value, rather then restraining it to a minimal contribution. So that my problem boils down like this:
in the non-explicitely-time-dependent problem the objective function $J$ shall be minimized to
$$ J^* = \min_{u(t)\in\mathcal{U}}J = \min_{u(t)\in\mathcal{U}} \int_{t_0}^{t_1}I({\bf x}(t), u(t))\mathrm{d}t = \int_{t_0}^{t_1}I({\bf x}(t), u^*(t))\mathrm{d}t \tag{1a} $$
thus to find the "best" $u^* \in \mathcal{U}$ with the performance index $I$ given as
$$ I({\bf x}(t), u(t)) = h({\bf x}(t)) \tag{1b}$$
but under the constraint of
$$\int_{t_0}^{t_1} u(t) \mathrm{d}t = C > 0 \tag{1c}$$
with $u \ge 0$ and a fixed $C$.
A formal application of the theory would suggest to use something like
$$ I({\bf x}(t), u(t)) = h({\bf x}(t)) + g(u(t))$$ such that for example the objective function becomes
$$ J = \int_{t_0}^{t_1}h({\bf x}(t)) \mathrm{d}t + A \int_{t_0}^{t_1} u(t) -\frac{B}{t_1 - t_0} \mathrm{d}t $$
and approach large values of $A$ in order to achieve constraining a fixed integral value. But as far as I understand it is problematic to use linear or even other than quadratic functions of the control function $u$ in the integrand. Problems arise since the existence of solutions cannot be warranted in general and also the solution of the equations systems can get difficult. I suppose this is connected with the convexity condition (also in $u$) for the Hamiltionian ($\mathcal{H} = I + \lambda^T \bf f$).
So typically with conventional control theory such problems as mine are usually tackled by using
$$ J = \int_{t_0}^{t_1}f({\bf x}(t)) \mathrm{d}t + B \int_{t_0}^{t_1} u^2(t) \mathrm{d}t \tag{2}$$
This is working smoothly, but addresses not directly my problem.
As a workaround I have thought about optimizing $B$ in $(2)$ such that $(1c)$ holds. But I am not really sure if the resulting $u^*$ function can be expected to be equivalent to the one from $(1a-c)$.
So the questions would be how to best (with the possibility of numerical inplementation) solve problem $(1a-c)$, and if solutions of (2) can be expected to be equivalent to those from $(2)$ with suitable $B$?
