  We investigate the bang-bang property for fairly general classes of
$L^\infty-L^1$ constrained bilinear optimal control problems in two cases: that
of the one-dimensional torus, in which case we consider parabolic equations,
and that of general $d$ dimensional domains for time-discrete parabolic models.
Such a study is motivated by several applications in applied mathematics, most
importantly in the study of reaction-diffusion models. The main equation in the
one-dimensional case writes $\partial_t u_m-\Delta u_m=mu_m+f(t,x,u_m)$, where
$m=m(x)$ is the control, which must satisfy some $L^\infty$ bounds ($0\leq
m\leq 1$ a.e.) and an $L^1$ constraint ($\int m=m_0$ is fixed), and where $f$
is a non-linearity that must only satisfy that any solution of this equation is
positive at any given time. The time-discrete models are simply
time-discretisations of such equations. The functionals we seek to optimise are
rather general; in the case of the torus, they write $\mathcal
J(m)=\iint_{(0,T)\times \mathbb T} j_1(t,x,u_m)+\int_{\mathbb T}
j_2(x,u_m(T,\cdot))$. Roughly speaking we prove in this article that, if $j_1$
and $j_2$ are increasing, then any maximiser $m^*$ of $\mathcal J$ is bang-bang
in the sense that it writes $m^*=1_E$ for some subset $E$ of the torus. It
should be noted that such a result rewrites as an existence property for a
shape optimisation problem. Our proofs rely on second order optimality
conditions, combined with a fine study of two-scale asymptotic expansions. In
the conclusion of this article, we offer several possible generalisations of
our results to more involved situations (for instance for controls of the form
$m\varphi(u_m)$), and we discuss the limits of our methods by explaining which
difficulties may arise in other contexts.
