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The strong [[gamma].sub.[kappa]]-[[gamma].sub.cl][H.sub.[infinity]] stabilization problem is addressed in this paper. Based on the bounded real lemma, an improved sufficient condition on strong [H.sub.[infinity]] stabilization as well as its dual form is derived, which extends the ARE constraint in the former results into an inequality constraint. Two path-following algorithms are proposed to solve the associated optimization problems with BMI constraints, which lead to full order controllers. A numerical example is presented to demonstrate the effectiveness of the result. Index Terms--Bilinear matrix inequality (BMI), [H.sub.[infinity]] control, linear matrix inequality (LMI), strong stabilization.

This note demonstrated that a certain class of Petri nets called timed-event multigraphs can be represented as a set of linear equations by using a novel dioid. With this new algebra, some properties are presented in the text. An example is discussed in the end, and an interesting result is computed. Index Terms--Discrete-event systems, max-plus algebra, timed Petri nets, timed-event graph.

In the above paper, new methods to obtain upper matrix bounds for a discrete algebraic Riccati equation (DARE) (1) have been established, but the author assumes that the system matrix A is nonsingular and DARE (1) has a positive-definite solution. In this note, some similar results can be obtained when a solution is positive-semidefinite and there are no extra assumption that A is nonsingular. Index Terms--Discrete algebraic Riccati equation, matrix bound.