A State Feedback Controller Used to Solve an Ill-posed Linear System by a GL(n, R) Iterative Algorithm

Starting from a quadratic invariant manifold in terms of the residual vector ${\textbf r}={\textbf B}{\textbf x}-{\textbf b}$ for an $n$-dimensional ill-posed linear algebraic equations system ${\textbf B}{\textbf x}={\textbf b}$, we derive an ODEs system for ${\textbf x}$ which is equipped with a state feedback controller to enforce the orbit of the state vector ${\textbf x}$ on a specified manifold, whose residual-norm is exponentially decayed. To realize the above idea we develop a very powerful implicit scheme based on the novel $GL(n,{\mathbb R})$ Lie-group method to integrate the resultant differential algebraic equation (DAE). Through numerical tests of inverse problems we find that the present Lie-group DAE algorithm can significantly accelerate the convergence speed, and is robust enough against the random noise.

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