On the reachable space for the heat equation.

RESUMEN: The goal of this talk is to explain how perturbative arguments can be applied to derive a sharp description of the reachable space for heat equations having lower order terms. The main result I will present is the following one. Let us consider an abstract system $y’ = Ay + Bu$, where $A$ is an operator generating a $C^0$ semigroup $(exp(tA))_{t\geq 0}$ on a Hilbert space $X$, and $B$ is a control operator, for instance a linear operator from an Hilbert space $U$ to $X$, and let us assume that this system is null-controllable in $X$ in any positive time. Then, setting $R$ the reachable set of the system (that is all the states that can be achieved by $y$ solution of $y’ = Ay + Bu$, $y(0) = 0$), the restriction of $(exp(tA))_{t \geq 0}$ to $R$ forms a $C^0$ semigroup on $R$. Accordingly, the system $y’ = Ay + Bu$ is exactly controllable on $R$, and one can then perform classical perturbative arguments to handle lower order terms, as I will explain on a few examples.

Posted on Oct 7, 2024 in Seminario IPCT, Seminars