In applications, models are not completely known, since the relevant parameters (deterministic or stochastic) are subject to uncertainty. It is therefore essential to develop robust analytical and computational methods that not only allow a given realisation of the model to be controlled, but also to deal with parameter-dependent families of systems in a stable and computationally efficient way. This issue requires the development of specific new tools, and constitutes one of our major objectives. We focus on the deterministic setting, although similar ones arise in the stochastic and random setting as well.
Nevertheless, averaged control does not guarantee the efficiency in the control of specific realisations if the variance of the states is not controlled and, accordingly, the development of new concepts and techniques is required. The innovative approach we propose in DYCON consists in adapting the existing theory 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 shall rely on the recent works by A. Cohen, W. Dahmen, R. DeVore, Ch. Schwab et al., based on the use of greedy and weak-greedy algorithms to identify the most meaningful realizations of the parameters.
More precisely, we shall consider the parameter-dependent family of controls and controlled solutions as a manifold in the product control/state space, thereby allowing us to quarry for specific values of the parameters defining an optimal approximation space in the sense of the Kolmogorov width and entropy number.
These methods will also be employed to revisit the work by the team on the optimal location of controllers and actuators.
As main output of the research conducted in this working package, a rather complete theory will be developed for the controllability of parametric PDE and the corresponding numerical algorithms so to be able to address applications in a robust manner.