Control-theoretic methods for solving linear algebraic equations: A continuous-time perspective

In the applications of control theory, how to improve the solution methods of mathematical problems with some control methods is an attractive problem. This paper aims at exploiting a class of continuous-time solvers for linear algebraic equations (LAEs) via the incorporation of the idea of “control design”. The equivalence is established between the solving problem of LAEs and two basic control problems of continuous-time systems, namely, the output tracking problem and the state observation problem, in a unified manner. Then, the equivalence between the properties of LAEs and the controllability and observability properties of continuous-time systems are disclosed from the perspective of the output tracking and the state observation, respectively. Moreover, a continuous-time solver is developed based on the feedback control design such that all (least squares) solutions to any (un)solvable LAE can be determined by the selection of different initial conditions. Particularly, the design conditions to derive the best approximate solutions to LAEs are presented for the continuous-time solver.