The maximum principle for lumped-distributed control systems

This paper concerns the optimal control of lumped-distributed systems, that is control systems comprising interacting infinite and finite dimensional subsystems. An examplar lumped- distributed system is an assembly of rotating components connected by flexible rods. The underlying mathematical model is a controlled semilinear evolution equation, in which nonlinear terms involve a projection of the full state onto a finite dimensional subspace. We derive necessary conditions of optimality in the form of a maximum principle, for a problem formulation which involves pathwise and end-point constraints on the lumped components of the state variable. A key feature of these necessary conditions is that they are expressed in terms of a costate variable taking values in a finite dimensional subspace (the subspace of the state space associated with the lumped variables). By contrast, costate trajectories in earlier-derived necessary conditions for optimal control of evolution equations evolve in the full (infinite dimensional) state space. The computational implications of the reduction techniques introduced in this paper to prove the maximum principle, which permit us to replace the original optimal control problem by one involving a reduced, finite dimensional, state space, will be explored in future work.