Robustness of controllability under some unbounded perturbations (Q1772310)

The authors consider the following linear control system \[ \left\{ \begin{aligned} &\dot x(t)=Ax(t)+Px(t)+B(t)u(t), \quad t\geq 0,\\ &x(0)=x_0, \end{aligned} \right. \] where the state \(x(\cdot)\) takes values in a Banach space \(X,\) the input function \(u\in L_{loc}^p(\mathbb R_{+},U),\) \(p\in [1,\infty),\) and \(U\) is a Banach space. The operator \((A, D(A))\) generates a \(C_0\)-semigroup on \(X\) and \(B(t)\in {\mathcal L}(U,X)\) for a.e. \(t\geq 0\) and for each \(u\in U,\) \(B(\cdot)u\in L_{loc}^{\infty}(\mathbb R_{+},X).\) Unbounded perturbations, the so-called Desch-Schappacher perturbations, are considered. The authors study the exact controllability of the linear system and show that this property is conserved when the operator \(P\) is close enough to \(0\) with respect to some metric. The results are illustrated by an application to controlled systems with dynamic and boundary perturbations.