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Controllability of the heat equation with memory. (English) Zbl 0990.93008

The present paper is concerned with the controllability of the equation \[ \begin{cases} y_t(x,t)- \gamma\Delta y(x,t)- \int^t_0 a(t-s) \Delta y(x,s) ds= m(x) u(x,y)\text{ in }Q,\\ y(x,0)= y_0(x)\quad\text{in }\Omega,\quad y(x,t)= 0\quad\text{in }\Sigma,\end{cases}\tag{1.1} \] 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: \[ \begin{cases} y_t(x,t)- \gamma\Delta y(x,t)- \int^t_0 a(t-s)\Delta y(x,s) ds= 0\text{ in }Q,\\ y(x,0)= y_0(x)\quad\text{in }\Omega,\quad y(x,t)= u(x,t)\quad\text{in }\Sigma.\end{cases} \] In the last section the controllability of the one-dimensional linear viscoelasticity equation is studied: \[ \begin{cases} y_t(x,t)- \int^t_0 a(t-s) y_{xx}(x, s) ds= m(x) u(x,t),\;(x,t)\in Q,\\ y(0,t)= y(\ell,t)= 0,\quad t\in (0,T),\quad y(x,0)= y_0(x),\quad x\in (0,1),\end{cases} \] where \(Q= (0,\ell)\times (0,T)\), \(\omega= (a_1,a_2)\subset (0,\ell)\) and \(a(t)\in C[0,+\infty)\cap C^\infty (0,+\infty)\), \((-1)^j a^{(j)}(t)\geq 0\), \(t> 0\), \(j= 0,1,\dots\).

MSC:

93B05 Controllability
35B37 PDE in connection with control problems (MSC2000)
35K05 Heat equation
35R10 Partial functional-differential equations
93C20 Control/observation systems governed by partial differential equations
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