Stabilization Of The 1-d Wave Equation On Complex Networks

The thesis is devoted to study the stabilization of the 1-d wave equationson the complex network by using the graph theory, the semigroup theory andthe Riesz basis approach, that is, to study the Riesz basis property of theclosed loop system, which is generated by the network being controlled, andto analyze the stabilization of the controlled network after the controllers arelocated on different vertexes of the network.First, the connections of the network is formulated by the graph theory,and the definitions of a geometrically continuous type wave equations on thenetwork are presented, the mathematical model of the dynamic behavior ofthe one-dimensional wave equations on the complex continuous type networkis derived. Based on the energy analysis of the system, the feedback controllersof the system are designed so that the system energy decays, and a variety ofsetting modes of controller are discussed.Second, the well-posedness of the closed loop system is proven by thesemigroup theory. By the spectral analysis, it is shown that the spectra of thesystem operator lie in a strip parallel to the imaginary axis in the left half com-plex plane. Thus, the completeness of (generalized) eigenvectors and the Rieszbasis property of the closed loop system are proven, and it is concluded thatthe closed loop system satisfies the spectrum determined growth condition.In additional, the characteristic determinant of system is compactly expressedwith the edge-edge joint matrix of digraph. Then, the stability of the net-work system under various control schemes is analyzed by using the infimumestimates and the spectral analysis in the imaginary axis, such as completecontrol, non-complete control and boundary control and so on. At the sametime, some necessary and suffcient conditions are provided for the asymptoticstability and the exponential stability of the controlled network. The problemhow to stabilize the network with parallel edges and circuits asymptotically is solved when it is not exact controllable.Third, based on the above theoretical analysis, the controller design andthe stabilization of some concrete elastic strings networks are discussed, suchas the star-shaped network, the tree-shaped network, the network with one ormore circuits and the honeycomb-shaped network. According to the"irrationaldependence", a simple path algorithm is presented to analyze the asymptoticstability of the completely controlled network.In the end, the Riesz basis property and stabilization of two specifictypes of non-geometrical continuous type network are analyzed, one is a forcecontinuous and serial connection string network, the other is a hybrid typetriangular circuit network.