Optimal Control of After-School Tutoring: A Cost–Benefit Dynamic System Model for Enhancing College Students’ Professional Competence

After-school tutoring has become a prominent form of personalized learning in higher education worldwide. It not only makes up for the insufficiency of classroom learning but also provides an important channel for the cultivation of college students' academic literacy and the improvement of their vocational skills. Given the scarcity of high-quality after-school tutoring and the resulting tutoring costs, this paper starts from the perspective of the cost-benefit trade-off of rational college students accepting after-school tutoring, rationalizes after-school tutoring as a control factor in the growth system of college students, and proposes an optimal control model to analyze the impact of after-school tutoring on college students' professional literacy under the comprehensive consideration of costs and benefits. Our model simulation shows that college students' acceptance of after-school tutoring can significantly improve their professional quality, and the earlier they accept it, the better. In addition, the benefit of after-school tutoring under cost-benefit trade-off is only slightly lower than that of full-time complete after-school tutoring, with extremely high cost-effectiveness. The conclusions of this paper not only are consistent with existing research but also deepen and expand related viewpoints.