Controllability of the heat equation with memory (Q5933758)

The present paper is concerned with the controllability of the equation NEWLINE\[NEWLINE\begin{cases} y_t(x,t)- \gamma\Delta y(x,t)- \int^t_0 a(t-s) \Delta y(x,s) ds= m(x) u(x,y)\text{ in }Q,\\ y(x,0)= y_0(x)\quad\text{in }\Omega,\quad y(x,t)= 0\quad\text{in }\Sigma,\end{cases}\tag{1.1}NEWLINE\]NEWLINE where \(\gamma> 0\), \(\Omega\) is an open bounded subset of \(\mathbb{R}^n\) with the boundary \(\partial\Omega\) of class \(C^2\), \(Q=\Omega\times (0,T)\), \(\Sigma= \partial\Omega\times (0, T)\), \(m(\cdot)\) is the characteristic function of an open subset \(\omega\) of \(\Omega\), and \(a\in C^\infty(0,+\infty)\) is a locally integrable completely monotone kernel. The main result tells us that under some assumptions related to the kernel \(a\), the problem (1.1) is approximately controllable. In particular, the approximate boundary controllability of the problem follows: NEWLINE\[NEWLINE\begin{cases} y_t(x,t)- \gamma\Delta y(x,t)- \int^t_0 a(t-s)\Delta y(x,s) ds= 0\text{ in }Q,\\ y(x,0)= y_0(x)\quad\text{in }\Omega,\quad y(x,t)= u(x,t)\quad\text{in }\Sigma.\end{cases}NEWLINE\]NEWLINE In the last section the controllability of the one-dimensional linear viscoelasticity equation is studied: NEWLINE\[NEWLINE\begin{cases} y_t(x,t)- \int^t_0 a(t-s) y_{xx}(x, s) ds= m(x) u(x,t),\;(x,t)\in Q,\\ y(0,t)= y(\ell,t)= 0,\quad t\in (0,T),\quad y(x,0)= y_0(x),\quad x\in (0,1),\end{cases}NEWLINE\]NEWLINE where \(Q= (0,\ell)\times (0,T)\), \(\omega= (a_1,a_2)\subset (0,\ell)\) and \(a(t)\in C[0,+\infty)\cap C^\infty (0,+\infty)\), \((-1)^j a^{(j)}(t)\geq 0\), \(t> 0\), \(j= 0,1,\dots\)\ .