This article is cited in 7 papers

Applied mathematics

Boundary control of wave system in the space of generalized solutions on a graph

V. V. Provotorov, Yu. A. Gnilitskaya

Voroneczky State University, Voronecz 394036, Russian Federation

Abstract: A boundary value problem for the wave equation on an arbitrary geometrical graph is considered. To analyze this problem a spectral technique is used (Fourier analysis): the difficulties are generated by the geometry of the graph which can be relatively easely overcome especially in the case where the graph contains cycles. On the other hand, the possibility of expansion in generalized eigenfunctions of the corresponding boundary value problem is effectively used in the proofs of existence theorems by the methods represented in the known works of S. L. Sobolev, O. A. Ladyzhenskaya, V. I. Smirnov. On the example of a model problem on a star-graph the existence of boundary control actions is substantiated and the method for finding them is presented. To simplify the above formulas the length of edges is multiple $\pi$, the wave equation is used in its simplest form: $u_{tt}=u_{xx}$. The main results of the work are presented as formulas that determine unknown boundary control as a function of time. Bibliogr. 5.

Keywords: wave equation on a graph, boundary problem, unique solvability, boundary control.

UDC: 517.958

Received: March 21, 2013