In this lecture, a new method is proposed to solve elliptic problems.
It is based on rewriting the governing PDE system in form of an optimal
control problem in such a way that the optimal states of the control 
problem coincide with the original solution. Using duality techniques and 
Pontryagin's maximum principle, one can in many cases derive better 
regularity and approximation results than using standard methods like 
Dirichlet's principle or Korn's inequality.
We demonstrate the validity and applicability of this approach to a number
of mechanical systems by discussing *Kirchhoff-Love arches and isotropic 
linear elasticity. *