The problem under consideration is to plan sensor movements in prescribed feasible regions in such a way as to maximize the accuracy of parameter identification of a distributed system defined in a two-dimensional spatial domain. A general functional defined on the Fisher information matrix is used as the design criterion. Central to the approach is the formulation of the problem as an optimal control one, possibly with state inequality constraints. Its solution is obtained numerically with the use of a method of successive linearizations which is capable of handling various constraints imposed on the sensors' motions. Simple numerical examples are included which clearly demonstrate the ideas and trends presented in the paper.