This article concerns the question, Which subsets of R m can be represented with linear matrix inequalities (LMIs)? This gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory. Also, before having much hope of representing engineering problems as LMIs by automatic methods, one needs a good idea of which problems can and cannot be represented by LMIs. Little is currently known about such problems. In this article we give a necessary condition that we call "rigid convexity," which must hold for a set C ⊆ R m in order for C to have an LMI representation. Rigid convexity is proved to be necessary and sufficient when m = 2. This settles a question formally stated by Pablo Parrilo and Berndt Sturmfels in [15]. As shown by Lewis, Parillo, and Ramana [11], our main result also establishes (in the case of three variables) a 1958 conjecture by Peter Lax on hyperbolic polynomials.

No abstract

The basic question of nonlinear H" control theory is to decide, for a given two port system, when does feedback exist which makes the full system dissipative and internally stable. This problem can also be viewed as an interesting question about circuits. Also, after translation, the problem has a game theoretic statement. This paper presents several necessary conditions for solutions to exist and gives sufficient conditions for a certain construction to lead to a solution.

Abstract. We show that the quadratic matrix equation V W + η(W )W = I, for given V with positive real part and given analytic mapping η with some positivity preserving properties, has exactly one solution W with positive real part. Also we provide and compare numerical algorithms based on the iteration underlying our proofs.This work bears on operator-valued free probability theory, in particular on the determination of the asymptotic eigenvalue distribution of band or block random matrices.

2016-12-26T15:11:00