On switching controls

In this talk, I will present recent results obtained on abstract control systems of the form $x' = A x + Bu$ when the control space $U$ is of the form $U_1 \times U_2$. In this case, it is natural to write $B(u_1, u_2) = B_1 u_1 + B_2 u_2$, and the question we address is the following: If we assume that the control system $x' = A x + Bu$ is null controllable, can we prove that null-controllability can be achieved with controls $u_1$, $u_2$ sucht that at all time, at most one control is active ? We will give sufficient conditions for this to be true, all of them in the context of analytic semigroups, which will allow us to strongly use analyticity properties. We shall also provide several examples of interest, in particular in the context of parabolic systems. This is a joint work with Felipe W. Chaves Silva (Joao Pessoa) and Diego A. de Souza (Recife).