The state equation is
where is a linear elliptic differential operator in a domain with boundary The (controlled) boundary condition is
where is a directional derivative on associated with The problem includes constraints on the control as well as on the state and the cost functional to be minimized in a time interval is
The author derives a maximum principle of Pontryagin’s type for controls minimizing the cost functional under all constraints. Besides the new result, the paper is a very good survey of the different approaches to this sort of problems (for instance, spike vs. patch or diffuse perturbations) and of the difficulties associated with the adjoint variational equation, whose inhomogeneous term is a measure rather than a function.