Global boundary null-controllability of one-dimensional semilinear heat equations

This paper addresses the boundary null-controllability of the semi linear heat equation partial derivative(ty) - partial derivative(xxy) + f(y) = 0, (x, t) is an element of (0, 1) x (0, T). Assuming that the function f is an element of C-1(R) satisfies lim sup(|r|->+infinity) |f(r)|/(|r| ln(3/2) |r | ) <= beta for some beta > 0 small enough and that the initial datum belongs to L-infinity(0, 1), we prove the global null-controllability using the Schauder fixed point theorem and a linearization for which the term f(y) is seen as a right side of the equation. Then, assuming that f satisfies lim sup(|r|->infinity) |f '(r)|/ ln(3/2) |r | <= beta for some beta small enough, we show that the fixed point application is contracting yielding a constructive method to approximate boundary controls for the semilinear equation. The crucial technical point is a regularity property of a state-control pair for a linear heat equation with L-2 right hand side obtained by using a global Carleman estimate with boundary observation. Numerical experiments illustrate the results. The arguments developed can notably be extended to the multi-dimensional case.