Spectral Analysis And Expansion Of Solution To A Class Of Transport Network

The paper studied a model of transport network, at present, the existence, stability ofthe solution are the main aspects in the study of transport networks, but there is a lack ofanalysis for solution structure and operation strategy of the system. In order to study thedynamic behavior of the solution and inner structure well, we employ operator semigrouptheory to make a spectral analysis and to expand solution of the system.The paper established a mathematical model according with the change of passengersin buses for a class of transport network. Under some special assumptions, we derive amodel of diferential equations. Then the diferential equations are transformed into someabstract Cauchy problem on some suitable state space by method of functional analysis.And then we prove the well-posedness of the network system and give the existence ofpositive solution, we make a detailed spectral analysis of the system operator and give theexpression for calculating eigenvalues of system operator we also discuss the geometric andalgebraic multiplicity of eigenvalue of and also give the essential condition that eigenvaluesare simple. we derive the dual operator and the corresponding eigenvectors of the originaloperator by dual operator spectral theory. we prove the incompleteness of the eigenvectorsunder the assumption that the eigenvalues are simple. Since discussing the solution of thesystem in Banach space is difcult, we put the system in a suitable space-Hilbert space andmainly study its solution structure including the completeness of the eigenvectors and thesolution expansion, we study convergence property of the expansion term and prove thatthe system solution can be expanded according to its eigenvectors when t is larger than acertain moment. At last, we give a simple operation strategy. When spectrum are located indiferent palaces, transport company will take diferent strategies to balance the system.