Domain decomposition, optimal control of systems governed by partial differential equations, and synthesis of feedback laws

We present an iterative domain decomposition method for the optimal control of systems governed by linear partial differential equations. The equation s can be of elliptic, parabolic, or hyperbolic type. The space region suppo rting the partial differential equations is decomposed and the original glo bal optimal control problem is reduced to a sequence of similar local optim al control problems set on the subdomains. The local problems communicate t hrough transmission conditions, which take the form of carefully chosen bou ndary conditions on the interfaces between the subdomains. This domain deco mposition method can be combined with any suitable numerical procedure to s olve the local optimal control problems. We remark that it offers a good po tential for using feedback laws (synthesis) in the case of time-dependent p artial differential equations. A test problem for the wave equation is solv ed using this combination of synthesis and domain decomposition methods. Nu merical results are presented and discussed. Details on discretization and implementation can be found in Ref. 1.