What is Inventory penalty?

An inventory penalty replaces risk aversion in utility by a cost charged on the position while it is held: the market maker maximises $\E\bigl[X_T+q_TS_T-\alpha q_T^2-\phi\int_0^Tq_t^2\,dt\bigr]$, with a running penalty $\phi$ and a terminal one $\alpha$.