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

Èmanuilov, O. Yu.

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 \(\partial\Omega\).

Reviewer: Anatoly Martynyuk (Kyïv)

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