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This paper develops a new set of stability conditions for asymptotic stability of two-dimensional linear systems described by Roesser and Fornasini-Marchesini state-space models through extensive use of the Kalman-Yakubovich-Popov lemma. The resulting tests are formulated in terms of a convex optimization problem over linear matrix inequality constraints. Testing the resulting conditions only requires computations on matrices with constant entries with consequent computational load advantages when compared with alternatives, especially when direct extension to stabilizing control law design is required. Illustrative numerical examples are also given.