Boundary control for the heat equation with nonlinear boundary condition (Q1822749)

This paper treats boundary control of the heat equation subject to a boundary condition which prescribes the normal derivative of the surface temperature as a function of the surface temperature and the ambient temperature, the latter being treated as the control. The existence and uniqueness of weak solutions is established; this permits one to prove the existence of optimal solutions to certain approximation problems. Finally a ``bang-bang'' principle is proved for a problem in 1 space dimension. This depends in a delicate way on unique backwards continuation as well as on results concerning strict total positivity and the heat equation due to the author [ibid. 62, 275--298 (1986; Zbl 0549.35052)].