  In this letter, we investigate sufficient conditions for the exponential
stability of LTI systems driven by controllers derived from parametric
optimization problems. Our primary focus is on parametric projection
controllers, namely parametric programs whose objective function is the squared
distance to a nominal controller. Leveraging the virtual system method of
analysis and a novel contractivity result for Lur'e systems, we establish a
sufficient LMI condition for the exponential stability of an LTI system with a
parametric projection-based controller. Separately, we prove additional results
for single-integrator systems. Finally, we apply our results to state-dependent
saturated control systems and control barrier function-based control and
provide numerical simulations.
