Bang-bang property of time optimal controls of semilinear parabolic equation

The bang-bang property of time optimal controls for a semilinear parabolic equation, with homogeneous Dirichlet boundary condition and distributed controls acting on an open subset of the domain is established. This relies on an observability estimate from a measurable set in time for a linear parabolic equation, with potential depending on both space and time variables. The proof of the bang-bang property relies on a Kakutani fixed point argument.