  In the interaction between control and mathematics, mathematical tools are
fundamental for all the control methods, but it is unclear how control impacts
mathematics. This is the first part of our paper that attempts to give an
answer with focus on solving linear algebraic equations (LAEs) from the
perspective of systems and control, where it mainly introduces the
controllability-based design results. By proposing an iterative method that
integrates a learning control mechanism, a class of tracking problems for
iterative learning control (ILC) is explored for the problem solving of LAEs. A
trackability property of ILC is newly developed, by which analysis and
synthesis results are established to disclose the equivalence between the
solvability of LAEs and the controllability of discrete control systems. Hence,
LAEs can be solved by equivalently achieving the perfect tracking tasks of
resulting ILC systems via the classic state feedback-based design and analysis
methods. It is shown that the solutions for any solvable LAE can all be
calculated with different selections of the initial input. Moreover, the
presented ILC method is applicable to determining all the least squares
solutions of any unsolvable LAE. In particular, a deadbeat design is
incorporated to ILC such that the solving of LAEs can be completed within
finite iteration steps. The trackability property is also generalized to
conventional two-dimensional ILC systems, which creates feedback-based methods,
instead of the common used contraction mapping-based methods, for the design
and convergence analysis of ILC.
