The effective control of the COVID-19 pandemic is one the most challenging issues of nowadays. The design of optimal control policies is perplexed from a variety of social, political, economical and epidemiological factors. Here, based on epidemiological data reported in recent studies for the Italian region of Lombardy, which experienced one of the largest and most devastating outbreaks in Europe during the first wave of the pandemic, we address a probabilistic model predictive control (PMPC) approach for the modelling and the systematic study of what if scenarios of the social distancing in a retrospective analysis for the first wave of the pandemic in Lombardy. The performance of the proposed PMPC scheme was assessed based on simulations of a compartmental model that was developed to quantify the uncertainty in the level of the asymptomatic cases in the population, and the synergistic effect of social distancing in various activities, and public awareness campaign prompting people to adopt cautious behaviors to reduce the risk of disease transmission. The PMPC scheme takes into account the social mixing effect, i.e. the effect of the various activities in the potential transmission of the disease. The proposed approach demonstrates the utility of a PMPC approach in addressing COVID-19 transmission and implementing public relaxation policies.