Domain decomposition in exact controllability of second order hyperbolic systems on 1-d networks

This paper is concerned with domain decomposition in exact controllability of a class of linear second order hyperbolic systems on one-dimensional graphs in [R^3] that in particular serve as descriptive models of the dynamics of various multi-link structures consisting of one-dimensional elements, such as networks of Timoshenko beams in [R^3]. We first consider a standard unconstrained optimal control problem in which the cost functional penalizes the deviation of the final state of the global problem from a given target state. A convergent domain decomposition for the optimality system associated with this problem was recently given by G. Leugering. This decomposition depends on the penalty parameter. On each edge of the graph and at each iteration level the local problem is itself the optimality system associated with an unconstrained optimal control problem in which the cost functional penalizes the deviation of the final state of the particular edge from the target state for that edge. The main purpose of this paper is to show that at each iteration level and on each edge the local optimality system converges as the penalty parameter approaches its limit and that the limit system is a domain decomposition for the problem of norm minimum exact control to the target state.