Abstract (EN)

We investigate the bang-bang property for fairly general classes of L ∞ − 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 onedimensional case writes ∂tum − ∆um = mum + f (t, x, um), where m = m(x) is the control, which must satisfy some L ∞ bounds (0 m 1 a.e.) and an L 1 constraint (m = m0 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 J (m) = (0,T)×T j1(t, x, um) + T j2(x, um(T, •)). Roughly speaking we prove in this article that, if j1 and j2 are increasing, then any maximiser m * of J is bang-bang in the sense that it writes m * = 1E 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. We prove an analogous result for time-discrete systems in any dimension. 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 mϕ(um)), and we discuss the limits of our methods by explaining which difficulties may arise in other contexts.