A variational perspective on controllability
2010 - P. Pedregal
Inverse Problems 26, 015004 (2010)
Pedregal Tercero, Pablo.
We introduce and examine a variational approach to controllability problems based on the minimization of a suitable error functional that measures how far admissible functions are from being a solution of the underlying state law. This philosophy is based on imposing boundary, initial and desirable final conditions into the admissible class of functions, and focus on minimizing the departure of such admissible functions from being a solution of the state law. This strategy leads to considering a functional analytical framework which sometimes is a bit different from the standard one, but it permits to show that the unique continuation property is equivalent to exact controllability. Indeed, the unique continuation property can be interpreted as the fact that local minimizers of the error functional can only occur at zero error. The following two are the main ingredients in which this variational approach relies: the existence of minimizers for the error and the unique continuation property just stated. The main issue on the existence of minimizers is to overcome lack of coercivity in the classical sense. Two advantages of this approach are its natural extension to nonlinear situations, and its direct numerical implementation as a steepest descent procedure.