  We introduce a theoretical framework for resource-efficient characterization
and control of non-Markovian open quantum systems, which naturally allows for
the integration of given, experimentally motivated, control capabilities and
constraints. This is achieved by developing a transfer filter-function
formalism based on the general notion of a frame and by appropriately tying the
choice of frame to the available control. While recovering the standard
frequency-based filter-function formalism as a special instance, this
control-adapted generalization affords intrinsic flexibility and, crucially, it
permits an efficient representation of the relevant control matrix elements and
dynamical integrals if an appropriate finite-size frame condition is obeyed.
Our frame-based formulation overcomes important limitations of existing
approaches. In particular, we show how to implement quantum noise spectroscopy
in the presence of nonstationary noise sources, and how to effectively achieve
control-driven model reduction for noise-optimized prediction and quantum gate
design.
