The bang-bang property in some parabolic bilinear optimal control problems \emph{via} two-scale asymptotic expansions

Abstract

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 ${\partial }_t{u}_m-\Delta u_m=m{u}_m+f(t,x,u_m)$, where $m=m(x)$ is the control, which must satisfy some $L^{\infty }$ bounds ($0\le m\le 1$ 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 $m^{\ast }$ of $\mathcal{J}$ is bang-bang in the sense that it writes $m^{\ast }=1_E$ 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 $m\phi (u_m)$), and we discuss the limits of our methods by explaining which difficulties may arise in other contexts.