Alternative algebraic approach to stabilization for linear parabolic boundary control systems

In this paper, we develop an alternative algebraic approach to feedback stabilization problems of linear parabolic boundary control systems. The control system is general in the sense that no Riesz basis associated with the elliptic operator is expected. The control system contains a finite dimensional dynamic compensator, the dimension of which has been so far determined only by the actuators on the boundary. One of the purposes of the paper is to show a new alternative scheme that the dimension of the compensator could be determined by the sensors on the boundary: it leads to a realization of stabilizing compensators of lower dimension by comparison. This is achieved by introducing a new operator equation connecting two states of the controlled plant and the compensator. The other purpose is a generalization of the condition posed on the sensors. When the elliptic operator admits generalized eigenfunctions, the sharpest criterion is proposed on the minimum choice of the necessary number of the sensors. In fact, only one sensor is enough for the stabilization in the best case (the algebraic multiplicities of the corresponding eigenvalues may, of course, be greater than 1).