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

We investigate the bang-bang property for fairly general classes of L∞−L1 constrained bilinear optimal control problems in case parabolic equations set one-dimensional torus. Such a study is motivated by several applications applied mathematics, most importantly reaction-diffusion models. The main equation writes ∂tum−∂xx2um=mum+f(t,x,um), where m=m(x) control, which must satisfy some L∞ bounds (0⩽m⩽1 a.e.) and an L1 constraint (∫m=m0 fixed), f non-linearity that only any solution this positive at given time. functionals we seek to optimise are rather general; they write J(m)=∬(0,T)×Tj1(t,x,um)+∫Tj2(x,um(T,⋅)). Roughly speaking prove article that, if j1 j2 increasing, then maximiser m⁎ J sense it m⁎=1E subset E It should be noted such result rewrites as existence shape optimisation problem. Our proof relies on second order optimality conditions, combined with fine two-scale asymptotic expansions. In conclusion article, offer possible generalisations our results more involved situations (for instance controls form mφ(um) or time-dependent controls), discuss limits methods explaining difficulties may arise other settings.