On descriptor systems and related linear quadratic problem.

*(English)*Zbl 1007.49022Summary: It is shown that, under the condition of impulse controllability [D. Cobb, IEEE Trans. Autom. Control 29, No. 12, 1076-1082 (1984; MR 86g:93008)], a descriptor system may be converted, by means of linear transformations, to a system described in a state space and composed of state and output equations. The transformations determine a one-to-one correspondence between the solutions of both systems. It is noted that the control in a feedback form may not determine a unique solution of the descriptor system, which was often overlooked in many previous papers. It is also shown that the LQ problem formulated in a descriptor space for the impulse observable system [op. cit.] 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 when the weighting matrix of the control, in the cost functional of the first problem, is singular. The proposed approach simplifies significantly the calculations related to the LQ problem solution.