A polynomial stability criterion for 2-D systems is taken as a starting point for introducing a frequency dependent Lyapunov equation. The Fourier analysis of its matrix solution leads to an infinite dimensional quadratic form which provides a Lyapunov function for the global state of the system. The Fourier coefficients are explicitly obtained as the sum of series involving the system matrices. The convergence of these series constitutes a necessary and sufficient stability condition, which generalizes the analogous condition for 1-D systems.
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Fornasini et al. (1980) studied this question.
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