Steady state controllability of a quasilinear parabolic system arising in modeling of a bacterial infection

We consider a radially symmetric model of a chronic wound featuring oxygen ($ w $), neutrophils ($ n $), bacteria ($ b $), and chemoattractant ($ c $). The dynamics couple three parabolic equations with a degenerate ODE and include a chemotactic drift for $ n $ driven by $ \nabla c $. We prove (ⅰ) local well-posedness and positivity of solutions for nonnegative data using maximal regularity and a contraction map, and (ⅱ) local null-controllability with minimally supported controls via an observability inequality obtained from global Carleman estimates. The nonlinear control follows from Kakutani's fixed point theorem applied to the control-to-state map. The result identifies a small set of localized controllers that steer the system to a steady state at a prescribed time.