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

Abstract

We investigate the bang-bang property for fairly general classes of L∞−L1 constrained bilinear optimal control problems in the case of parabolic equations set in the one-dimensional torus. Such a study is motivated by several applications in applied mathematics, most importantly in the study of reaction-diffusion models. The main equation writes ∂tum−∂xx2um=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 L1 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 functionals we seek to optimise are rather general; they write J(m)=∬(0,T)×Tj1(t,x,um)+∫Tj2(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. Our proof relies 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) or for some time-dependent controls), and we discuss the limits of our methods by explaining which difficulties may arise in other settings.