Controller Design And Stability Anadlysis Of 1-D Wave Networks

In recent years, the control and stabilization of wave networks have beenstudied extensively and intensively. In this thesis, the feedback control strategyis designed for tree-shaped wave networks and bush-type wave networks, andthe stability of the corresponding closed loop systems is investigated.1. The exponential stability of tree-shaped wave networks with constantcoe?cients is considered. A"developing"viewpoint is proposed, from which atree-shaped wave network is regarded as developing from a single wave equa-tion. Then by analyzing the eigenvalue problem of the adjacent edges of thesystem in the developing order, an explicit recursive expression of the charac-teristic equation is obtained. In the exponential stability analysis, it su?ces toestimate the infimums of the recursive expressions in the inverse order, whichis relatively easy to carry out. As an application, the exponential stabilityof the tree-shaped wave network with constant coe?cients under boundaryvelocity feedback control is proved.2. The exponential stability of tree-shaped wave networks with variablecoe?cients is investigated. By a suitable variable transform, the system isrewritten into a tree-shaped wave network with constant coe?cients undera variable coe?cient perturbation. Using the"developing"viewpoint andasymptotic technique, an explicit recursive expression of the asymptotic char-acteristic equation of the system is derived. Based on the verification of Rieszbasis property of the system, it is asserted that the spectrum determinedgrowth assumption holds. Hence by the infimum estimate of those asymp-totic recursive expressions in the inverse order, an easy method is obtainedfor the exponential stability analysis. Then it is proved that the tree-shapedwave network with variable coe?cients is exponentially stable under boundaryvelocity feedback control.3. A new method, named"cutting-edge"method, for the feedback con- troller design for complex wave networks is introduced. It ensures the stabilityof the closed loop system. This method is based on the uniqueness theory ofthe ordinary di?erential equation and cutting-edge approach in the graph the-ory, but it is not an easy extension. As a realization, a bush-type wave networkis studied. The validity of"cutting-edge"method is proved rigorously by spec-tral analysis approach, and a detailed process of"cutting-edge"is presented.The result shows that if the controllers are imposed at the boundary verticesand at most 3 proper positions on the cycle, the closed loop system is thenasymptotically stable. The"cutting-edge"method can also be applied to thecontroller design for other complex wave networks.4. The stability of wave network on a generic tree is discussed when thefeedback gain constants fail to satisfy the conditions for Riesz basis genera-tion. By a detailed spectral analysis, the explicit expression of the spectra ispresented, which consist of simple eigenvalues locating on a vertical line in thecomplex left half-plane. But it is shown that the eigenvectors are not completein the state space. However, the state space decomposes into a topological di-rect sum of the spectral-subspace and another invariant subspace of infinitedimension. In the spectral-subspace, the solution can be expanded accordingto the eigenvectors, and hence it is exponentially stable; in the other subspace,the associated semigroup is super-stable, i.e., the solution is identical to zeroafter a finite time. In particular, the energy decay rate and the maximumexistence time of the super-stable part of the solution are given.