A COMPLETE ALGEBRAIC SOLVABILITY TEST FOR THE NONSTRICT LYAPUNOV INEQUALITY

For arbitrary equally sized square complex matrices A and Q (Q Hermiti an), the paper provides a complete algebraic test for verifying: the e xistence of a Hermitian solution X of the nonstrict Lyapunov inequalit y AX + XA + Q greater than or equal to 0. If existing, we exhibit how to construct a solution. Our approach involves the validation problem for the linear matrix inequality Sigma(j=1)(k) (A(j)X(j)B(j) + B-j*X (j)A(j)) + Q > 0 in X(j), for which we provide an algebraic solvabili ty test and a procedure to construct solutions if the kernels of A(j) or, dually, those of B-j form an isotonic sequence.