Stability of a Variable Coefficient Star-Shaped Network with Distributed Delay

The paper deals with the exponential stability problem of a variable coefficient star-shaped network, whose strings are coupled at a common end in a star-shaped configuration and the common connection of all strings can be moved. Two kinds of media materials with a component of viscous and another simply elastic are distributed on each string. Under suitable hypothesis on the coefficient functions μ j ( x ) of damping terms and the kernels η j ( s ) of distributed delay terms, the well-posedness of the system is obtained by means of resolvent family theory. In addition, the allocation proportion of the two parts and the property of the material character functions are discussed when the star-shaped network is exponentially stable. Meanwhile, the sufficient condition of exponential stability is established. Numerical simulations are also included to verify the main results.