Towards a multiresolution approach to linear control
Résumé
We develop a multiresolution approximation framework for linear control. We construct a multiresolution analysis of the set of input functions of a linear system. The approximation of an input $u$ at a scale $j$ is defined as the input $u_j$ of minimal energy such that the trajectories of the system associated with $u$ and $u_j$ coincide on a grid of step length $2^{-j}$. We propose a set of wavelet functions which generate this multiresolution analysis. These functions, called control theoretic wavelets, satisfy useful properties for the representation of control inputs of a linear system. As an example of application of our multiresolution approximation framework, we propose a method for efficient encoding of control inputs with regard to several criteria.