Exact null-controllability of interconnected abstract evolution equations with unbounded input operators (Q2062716)

In 2014, the author analyzed the exact null-controllability problem in the class of smooth controls with applications to interconnected systems for the case of bounded input operators appearing in systems with distributed controls. The present manuscript is an extension of this study for the case of unbounded input operators. \par In this paper, the author considers the control evolution equation with scalar control \begin{align*} \dot{x}_1(t) & = A_1x_1(t)+b_1v(t), \ x_1(0)=x_1^{0}, \tag{1} \\ v(t) & = (c,x_2(t)), \ 0 \leq t < +\infty, \ c \in X_2, \tag{2}\end{align*} where \(x_2(t)\) is a mild solution of the another control equation of the form \[ \dot{x}_2(t) = A_2x_2(t)+b_2u(t), \ 0 \leq t < +\infty,\ x_2(0)=x_2^{0}. \tag{3}\] In these equations, \(x_1(t), x_1^{0} \in X_1\), where \(X_1\) is the Hilbert space of equation (1), \(v(t) \in \mathbb{R}\), \(x_2(t), x_2^{0} \in X_2\), where \(X_2\) is the Hilbert space of equation (3), and linear operators \(A_1\) and \(A_2\) generate strongly continuous \(C_0\)-semigroups \(S_1(t)\) in \(X_1\) and \(S_2(t)\) in \(X_2\) respectively. \par The exact null-controllability conditions for two interconnected abstract control equations (1)--(3) governed by an unbounded control \(u(t)\) of equation (3) are obtained. These results make it possible to investigate the controllability of two or more interconnected boundary control objects of different structure, governed by the control of the latter.