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We consider the problem of computing an approximate banded solution of thecontinuous-time Lyapunov equation$\underline{A}\underline{X}+\underline{X}\underline{A}^{T}=\underline{P}$,where the coefficient matrices $\underline{A}$ and $\underline{P}$ are large,symmetric banded matrices. The (sparsity) pattern of $\underline{A}$ describesthe interconnection structure of a large-scale interconnected system. Recently,it has been shown that the entries of the solution $\underline{X}$ arespatially localized or decaying away from a banded pattern. We show that thedecay of the entries of $\underline{X}$ is faster if the condition number of$\underline{A}$ is smaller. By exploiting the decay of entries of$\underline{X}$, we develop two computationally efficient methods forapproximating $\underline{X}$ by a banded matrix. For a well-conditioned andsparse banded $\underline{A}$, the computational and memory complexities of themethods scale linearly with the state dimension. We perform extensive numericalexperiments that confirm this, and that demonstrate the effectiveness of thedeveloped methods. The methods proposed in this paper can be generalized to(sparsity) patterns of $\underline{A}$ and $\underline{P}$ that are moregeneral than banded matrices. The results of this paper open the possibilityfor developing computationally efficient methods for approximating the solutionof the large-scale Riccati equation by a sparse matrix.

Sparse solution of the Lyapunov equation for large-scale interconnected systems