Quasi-Metric Spaces And Their Applications In Control

Quasi-metric, just as the name implies, is a kind of generalized measuring without guaranteeing symmetry. For it has been drawing the attentions of domestic and foreign researchers, its importance in the nonlinear system field has been seen recently. In the dissertation, some creative basic studies about quasi-metric space are carried out, and applications have been implemented in multiple objective optimizations, non-linear control and pattern recognition.In the dissertation, some basic concepts about quasi-metric space, such as upper limit, lower limit, upper completeness and others, are defined and its topological properties are investigated, and several important quasi-metrics are discussed as follows:1. Hausdorff semi-metric: In the research of fractal geometry and set-valued analysis, the complete metric space with symmetrical Hausdorff distance is utilized. In this paper, the upper completeness of Hausdorff semi-metric space is proved.2. Quasi-metric between measurable sets: The quasi-metric between measurable sets, say A and B, is defined, which is applied in measurable classifier space by taking the measure μ(A∩B~c) between set A and B as the quasi-metric from set A to B.3. Quasi-metric among random variables: It is established by conditional entropy, which is used for the relativity analysis between random variables.Considering the important role of Banach space in functional analysis, the quasi-metric Banach space is studied, in which the upper limit completeness is taken as its completeness. In the mean time, the non-symmetric norm of linear operator in non-symmetric norm space is constituted, the conclusion about the equivalence between bounded-increase and continuousness of linear operator is drawn. Also, both the G-differential and F- differential of non-linear operator in non-symmetric norm space are established. For the reason that the quasi-metric determines a partial order, the range of convex function is transferred into the non-asymmetric norm space, and some progress in the research is made.Besides, the application of quasi-metric in multiple objective optimization, nonlinear control and pattern recognition are investigated. The main achievements are as follows:1. Optimization of multiple objective functions: Currently, the study on the optimization of multiple objective functions is mainly based on orderly Banach space,...