In this interesting article, the authors consider an optimal control problem where the dynamics is given by an hyperbolic equation and the cost is a classical tracking cost. The control acts both as distributed (on all components) and as a boundary control (on components corresponding to negative eigenvalues). The key novelty with respect to previous contributions is that the problem has initial and final boundary conditions on the time domain \([0,T]\).
The main result of the article is to prove that the time domain can be decomposed into \([T_k,T_{k+1}]\) and one can solve adapted optimal control problems on each interval in parallel. Indeed, in Section 3, one divides the time domain and introduce virtual boundary controls \(g_{k,k-1}\) at the interface; then, an iterative method (14-15-16) is provided to solve the problem.
The subsequent sections provide convergence of the iterative method to the actual unique solution, both when the controls are uncontrained (Section 5) and when they are constrained but the dynamics is linear (Section 6).