  We introduce a vertical type relaxation for optimal control problems which
only have $L^1$-coercivity for their controls. Usually such problems feature
both concentration and oscillation effects at the same time. We propose
relaxing to an associated problem in space-time, where the controls can be
considered bounded in $L^\infty$, greatly simplifying any analysis. In this
relaxation, concentrations are transformed into vertical parts and oscillations
can be dealt with using Young-measures. This technique can be extended to
similar problems on infinite-dimensional spaces.

Optimization and Control;Analysis of PDEs