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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 $$\cases y_t(x,t)- \gamma\Delta y(x,t)- \int^t_0 a(t-s) \Delta y(x,s) ds= m(x) u(x,y)\text{ in }Q,\\ y(x,0)= y_0(x)\quad\text{in }\Omega,\quad y(x,t)= 0\quad\text{in }\Sigma,\endcases\tag 1.1$$ where $\gamma> 0$, $\Omega$ is an open bounded subset of $\bbfR^n$ 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: $$\cases y_t(x,t)- \gamma\Delta y(x,t)- \int^t_0 a(t-s)\Delta y(x,s) ds= 0\text{ in }Q,\\ y(x,0)= y_0(x)\quad\text{in }\Omega,\quad y(x,t)= u(x,t)\quad\text{in }\Sigma.\endcases$$ In the last section the controllability of the one-dimensional linear viscoelasticity equation is studied: $$\cases y_t(x,t)- \int^t_0 a(t-s) y_{xx}(x, s) ds= m(x) u(x,t),\ (x,t)\in Q,\\ y(0,t)= y(\ell,t)= 0,\quad t\in (0,T),\quad y(x,0)= y_0(x),\quad x\in (0,1),\endcases$$ where $Q= (0,\ell)\times (0,T)$, $\omega= (a_1,a_2)\subset (0,\ell)$ and $a(t)\in C[0,+\infty)\cap C^\infty (0,+\infty)$, $(-1)^j a^{(j)}(t)\ge 0$, $t> 0$, $j= 0,1,\dots$ .

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