On the null-controllability of the heat exchange process

A mathematical model of the heat exchange process, where the temperature inside some domain is controlled by $m$ convectors acting on the boundary, is considered. The control parameter is a vector-function, whose components are equal to the magnitude of the output of hot or cold air produced by each convector. The necessary and sufficient conditions, which initial temperature must satisfy for achieving the zero value by the projection of the temperature into some $m$-dimensional subspace, are studied.