## Controllability of parabolic equations.(English. Russian original)Zbl 0845.35040

Sb. Math. 186, No. 6, 879-900 (1995); translation from Mat. Sb. 186, No. 6, 109-132 (1995).
This paper studies two problems of exact controllability of semilinear parabolic equations. Consider the semilinear parabolic equation $Ly+ f(t, x, y)= g+ u\quad\text{in} \quad Q_T= [0, T]\times \Omega,\tag{1}$
$y|_{[0, T]\times \partial\Omega}= 0,\;y(0, x)= v_0(x),\;y(T, x)= v_1(x),$ where $Ly= \partial_t y- \sum^n_{i, j= 1} \partial x_i(a_{ij} \partial x_j y)+ \sum^n_{i= 1} b_i \partial x_i y+ cy$ is a linear parabolic operator, $$v_0$$, $$v_1$$, $$g$$ are given function $$u$$ with support in an arbitrary fixed subdomain $$\omega\in \Omega$$. In addition to (1) this paper studies the problem of exact boundary controllability of $Ly+ f(t, x, y)= g\quad\text{in }Q_T,$
$y|_{[0, T]\times (\partial\Omega\backslash\Gamma_0)}= 0,\;y|_{[0, T]\times \Gamma_0}= v,\;y(0, x)= v_0(x),\;y(t, x)= v_1(x),$ where $$v_0$$, $$v_1$$, $$g$$ are given and the control $$v(t, x)$$ is concentrated on an open subset $$\Gamma_0$$ of the boundary $$\partial\Omega$$.

### MSC:

 93C20 Control/observation systems governed by partial differential equations 35K20 Initial-boundary value problems for second-order parabolic equations

### Keywords:

exact controllability; exact boundary controllability
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