Algebraic solvability tests for linear matrix inequalities

Discusses algebraic tests for the solvability of the indefinite linear matrix inequality (LMI) (A*P+PA+Q/B*P+S* PB+S/R)⩾0 which arises in the general LQ problem and in H_∞-control. The author presents a new geometric algorithm which allows one to directly reduce the LMI to a certain algebraic Riccati inequality (ARI). Under a mild regularity assumption the author describes how to further reduce the Riccati inequality to an indefinite Lyapunov inequality with a matrix whose eigenvalues are located at the imaginary axis. Finally, the author derives new general necessary conditions for the solvability of such Lyapunov inequalities and discusses cases under which these conditions are also sufficient