Pancyclic Out-Arcs Of Vertex In Digraphs And Stability Of A Class Of Beam Systems

In this paper we deal with two aspects of applied mathematics, one is digraph theory and the other is infinite-dimensional linear system theory. Both of them play important roles in mathematics. This paper consists of two chapters: Chapter I is about digraph theory and Chapter II is about infinite-dimensional linear system theory.Graph theory is a very popular area of discrete mathematics with not only numerous theoretical developments, but also countless applications to practical problems. The theory of graphs can be roughly partitioned into two branches: the areas of undirected graphs and directed graphs (digraphs). The theory of directed graphs has developed enormously within the last three decades. In chapter I, we mainly study the pancyclicity of arcs in local tournaments. In 1980, Thomassen proved that every strong tournament contains a vertex v such that each out-arc of v is contained in a Hamiltonian cycle, (see [6]). In 2000, Yao etc. extended the result of Thomassen [6] and proved that every strong tournament contains an out-arc-pancyclic vertex, (see [8]). Here, we extend the result of Yao etc. and obtain a meaningful result.Infinite-dimensional linear systems is now an established area of research with a long list of journal articles, conference proceedings, and several textbooks to its credit. All of them reflect the overwhelming popularity of. the infinite-dimensional linear systems. In Chapter II, by using the method of regular systems we investigate the stability and exact observability of the non-dissipative flexible beam systems with boundary feedback described byNow we outline the contents of this chapter and state some of the results. In section 2.1, we introduce some of the recent research results about the relationship between the internal stability and external stability. In section 2.2, we state some necessary background, definitions and basic facts about infinite-dimensional linear systems. In section 2.3, we are concerned with the admissibility of the control operator and the observation operator of system (1.1)- (1.4). In section 2.4 and 2.5, we show that the system (1.1)-(1.4) is externally and internally stable, respectively. In section 2.6, we show the open-loop system (1.1)-(1.3) is exactly observable. This provides us with an effective method to study the stability of non-dissipitiwe flexible beam systems with boundary feedback.