We aim to adapt the existing techniques of reduced modelling and nonlinear approximation by exploiting the notion of sparsity, in order to build optimal methods (with respect to the computational cost and complexity) for controlling parameter-dependent PDEs in a robust manner. To this end, we rely on the use of greedy and weak-greedy algorithms to identify the most meaningful realizations of the parameters and ${L^1}$ minimisation to search for impulsional controls.

For doing this we view the parameter-dependent family of controls and controlled solutions as a manifold in the product control/state space.

These techniques are applied and tested in some classical benchmark control problems for parabolic and hyperbolic PDEs, but can also be adapted to more challenging problems as the optimal location of controllers and actuators.