Inverse diffraction, duality and optimal control

The reduced wave equation can describe a non-dynamical system with infinite-dimensional state space and a control at the boundary. A generalised controllability property is defined and it is shown that it holds in a bounded domain. This property states that the linear subspace spanned by the trace of the system solution (i.e. the monochromatic scalar field) on a given boundary is dense in the corresponding Hilbert space as the boundary control is made to span an adequate Hilbert space. By duality, the 'controllability' property is shown to be equivalent to 'observability' of the adjoint system. The connection of this statement with the uniqueness proof for inverse diffraction problems is discussed. The adjoint system solution is a Green function. The inverse problem is ill posed. It calls for a regularisation procedure related to minimising a real-valued convex functional which depends on complex variables. At this point Lions's theory of optimal control is applied to state the existence and uniqueness of the minimising control. Lagrangian theory is used to derive the explicit form of the functional gradient. The latter allows one to design a minimisation algorithm where primal and adjoint systems must be solved sequentially