Modeling master’s students’ competence development based on the variational principle

<jats:p>&lt;p&gt;&lt;strong&gt;Context and relevance.&lt;/strong&gt; Despite the widespread adoption of the competency-based approach in higher education, a gap remains between the understanding of competence as a dynamic process and the tools available for its design and management. Dominant practices of learning outcomes assessment rely on static &amp;ldquo;snapshots,&amp;rdquo; which limits the possibilities for forecasting and purposeful development of competence. In this context, there is a growing need for formal modeling of competence development trajectories based on principles of optimality derived from the mathematical theories of the calculus of variations and optimal control. &lt;strong&gt;Objective.&lt;/strong&gt; To develop and empirically test a model of master&amp;rsquo;s students&amp;rsquo; competence development based on the variational principle, which conceptualizes the educational process as a problem of optimal control over the trajectory of key competence formation. &lt;strong&gt;Methods and materials.&lt;/strong&gt;The study involved 24 first-year master&amp;rsquo;s students enrolled in the program School Leadership and Educational Policy (83% women). Competence was assessed using seven indicators reflecting the ability to establish causal relationships between norms and practices (scale 0&amp;ndash;3; total score range 0&amp;ndash;21). Data were collected across three measurement points during the completion of an analytical learning task. The theoretical model was constructed using tools of the calculus of variations (Euler&amp;ndash;Lagrange equations, transversality conditions), and the empirical validation was conducted through multilevel regression analysis with random effects models. &lt;strong&gt;Results.&lt;/strong&gt; The analytical solution of the extremal problem demonstrated that the optimal trajectory of competence development is linear in nature and corresponds to the principle of minimizing the &amp;ldquo;length&amp;rdquo; of the educational path, interpreted as the minimization of cognitive and motivational overload. Empirical data confirmed a predominantly linear pattern of competence growth for most students. The multilevel regression model revealed a statistically significant increase in performance across measurement points, as well as a negative association between initial competence level and growth rate, indicating a compensatory pattern of development. The high value of the conditional coefficient of determination highlights the crucial role of individual trajectories in competence formation. &lt;strong&gt;Conclusions. &lt;/strong&gt;The findings suggest that competence development should be conceptualized as a non-ergodic process, which requires a shift from the analysis of interindividual differences to the modeling of intraindividual dynamics. The variational approach provides a foundation for the normative design of educational trajectories and enables the integration of mathematical models of optimality into educational analytics and learning support systems. The proposed model is recommended as a tool for forecasting and planning pedagogical interventions aimed at fostering sustainable and resource-efficient development of students&amp;rsquo; key competencies.&lt;/p&gt;</jats:p>