Minimal surfaces and optimal control

According to Pierre Ossian Bonnet’s theorem claiming that a surface is known up to a rigid body displacement from the data of its first and second fundamental forms, we give up to exhibit a minimal surface by finding directly its parametric equations. We prefer to discover the two fundamental forms in a first step, and to rebuild the surface in a second step. In this way we develop the following variational plan: “minimize the area regarded as a functional of the two fundamental forms taking into account the Gauss-Codazzi-Mainardi compatibility conditions by Lagrange multipliers”. We show that the introduced multipliers satisfy an adjoint partial differential equation, as in optimal control when one applies the Pontriagin principle. The well-known pioneering discovery of Jean-Baptiste Meusnier de la Place (1785) asserting that “the mean curvature should vanish” is revealed as a compatibility condition for this adjoint equation.