In this paper it is aimed to make an explicit expression of controllability condition of the system which is described by a linear partial differential equation. At first is shown the input-state relation of the system, and next is discussed the controllability condition by virtue of authors' already presented theorem.
The “genuine solution” of differential equation for square integrable input function can not be known in general, but the “weak solution” which is equal to the “genuine solution” almost everywhere in its domain, can be obtained by the method proposed in this paper. In order to use this “weak solution” as the input-state relation, it is necessary to introduce the topology under which every functions equal each other almost everywhere are identified as the same. From this point of view, the L₂ topology is introduced into the state space.
In the partial differential equation systems, there are two types of controls, one is spatially distributed, the other is concentrated on the boundary. In this paper are derived input-state relations for these two kinds of control and reduced to congruent equations in l₂ space. To these relations in l₂ space, is applied the theorem obtained by authors in ref. (2) treating the necessary and sufficient condition for a system to be controllable. And then this condition is expressed explicitly in terms of both the elliptic differential opereator of the differential equation and the operators which describe the control mechanisms. The operators which describe the boundary control mechnisms are bounded operators defined on the boundary, but they become unbounded ones in the input-state relation in l₂ space. Therefore, in the cases of boundary controls can not be applied the same reasoning as in distributed controls. However, it is also possible in this case to obtain the explicit expression of controllability condition by the appropriate modification of the theorem.