On descriptor systems and related linear quadratic problem

It is shown that a descriptor system under the condition of impulse controllability, Cobb (1984), may be converted, by means of linear transformations, to a system described in a state space and composed of state and output equations. The transfomations determine one to one correspondence between the solutions of both the systems. It is noted that the control in a feedback form may not determine a unique solution of the descriptor system what is often overlooked in many previous papers. It is also shown that the LCD problem formulated in a descriptor space for the impulse observable system, Cobb (1984); may be converted by means of linear transformations to the usual LQ problem formulated in the state space. It is stressed that the second problem may be regular even then, when the weighting matrix of the control, in the cost functional of the first problem, is singular. The proposed approach simplifies the calculations related to the LQ problem solution significantly.