Social robots (SRs) should autonomously interact with humans, while exhibiting proper social behaviors associated to their role. By contributing to health-care, education, and companionship, SRs will enhance life quality. However, personalization and sustaining user engagement remain a challenge for SRs, due to their limited understanding of human mental states. Accordingly, we leverage a recently introduced mathematical dynamic model of human perception, cognition, and decision-making for SRs. Identifying the parameters of this model and deploying it in behavioral steering system of SRs allows to effectively personalize the responses of SRs to evolving mental states of their users, enhancing long-term engagement and personalization. Our approach uniquely enables autonomous adaptability of SRs by modeling the dynamics of invisible mental states, significantly contributing to the transparency and awareness of SRs. We validated our model-based control system in experiments with 10 participants who interacted with a Nao robot over three chess puzzle sessions, 45 - 90 minutes each. The identified model achieved a mean squared error (MSE) of 0.067 (i.e., 1.675% of the maximum possible MSE) in tracking beliefs, goals, and emotions of participants. Compared to a model-free controller that did not track mental states of participants, our approach increased engagement by 16% on average. Post-interaction feedback of participants (provided via dedicated questionnaires) further confirmed the perceived engagement and awareness of the model-driven robot. These results highlight the unique potential of model-based approaches and control theory in advancing human-SR interactions.