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First-order necessary conditions for quadratically optimal reduced-order modeling of linear time-invariant systems are derived in the form.of a pair of modified Lyapunov equations coupled by an oblique projection which determines the optimal reduced-order model. This form of the necessary conditions considerably simplifies previous results of Wilson 111 and clearly demonstrates the quadratic extremality and nonoptimality of the balancing method of Moore [Z]. The possible existence of mulsple solutions of the optimal projection equations is demonstrated and a relaxation-type algorithm is proposed for computing these local extrema. A component-cost analysis of the model-error criterion similar to the approach of Skelton [3] is utilized at each iteration to direct the algorithm to the global minimum. I. INTRODUCTIOKHE problem of approximating a high-order linear dynamical T s y s t e m with a relatively simpler system, i.e., the model-reduction problem, has received considerable attention in recent years. Among the myriad papers devoted to this problem are the notable contributions of Wilson [I], Moore [2], and Skelton [3] with which the present paper is concerned. In his 1970 paper, Wilson proposed an optimality-based approach to model reduction which involves minimizing the steady-state, quadratically weighted' output error when the original system and reduced-order model are subjected to white-noise inputs. For the resulting parameter-optimization problem, he obtained first-order paper is based on a prior submission

The real estate investment trust (REIT) industry has undergone three waves of initial public offerings (IPOs) since 1980. In this article we examine these waves within the context of the general IPO wave literature. We note that the unique nature of REITs may render them more transparent than other stocks, and that this may affect their IPO performance. Specifically we document the initial and long-run performance of equity REIT IPOs during and after each wave, and we then examine which of the general IPO clustering and pricing theories best fits that performance data. We find evidence of much smaller initial returns than is typically found for non-REIT IPOs. We also find no evidence of the longrun negative abnormal performance for REITs that is normally found for other stocks. Our findings support the idea that the Capital Demand Hypothesis best describes the REIT IPO market, although we also find some weak evidence in favor of the Information Asymmetry Hypothesis. We find evidence against the Investor Sentiment Hypothesis.

Abstruct-Relations between solutions of the algebraic Riccati equation and the associated quadratic matrix inequalities are discussed and explained.