I am trying to solve the Kermak Mc-Kendrick SIR model using a non linear cost function, but I am stuck on how to possibly solve it.
I need to find an optimal control $u(t)$ in $[0,T]$ that minimize:
$min J = S(T) + \int\limits_{0}^{T} [u(t)S(t)]^2 dt$
with
$\frac{\partial S}{\partial t} = - \lambda S(t) [1 - S(t) - R(t)] - S(t)u(t)$,
$\frac{\partial R}{\partial t} = \gamma [1-S(t)-R(t)]$,
$S(t),R(t)\geq 0$, $ S(0) = s_0, R(0) = 0$.
It represents the evolution of an epidemic, where S(.) are the sane nodes, and R(.) the nodes that have recovered. The goal is to infect the most of the users by $T$, taking into account a quadratic cost for direct promotion (the $S(t)u(t)$ term) .
Classical optimal control does not work because of the non linear term in the cost function. Thanks in advance to anyone that will point me in the right direction.