Kontrol Optimal Dinamika Miskonsepsi dengan Intervensi Konseptual Bergantung Waktu

Abstract

This study aimed to formulate and solve an optimal control problem for a nonlinear differential equation model to minimize the cumulative burden of misconceptions and the quadratic cost of intervention. The state variables represent the proportions of learners with insufficient understanding, misconceptions, and conceptual understanding. The time-dependent control regulated the transition from misconceptions to conceptual understanding, with an upper intensity bound of 0.30 over a horizon of 100 time units. The necessary conditions for optimality were derived using Pontryagin’s minimum principle. The numerical solution was obtained by reducing the model from three to two dimensions and applying the forward-backward sweep method based on fourth-order Runge–Kutta integration with damping. For the assumed parameter values α = 0.20, β = 0.75, γ = 0.15, δ = 0.10, and baseline weights A = B = 1, the optimal control yielded an objective value of 2.7064, lower than those obtained under the no-control scenario and constant controls of 0.10 and 0.20. The cumulative burden of misconceptions was 1.5492, with a control energy of 2.3143. Compared with the constant control of 0.20, the optimal strategy reduced the cumulative burden of misconceptions by 26.1% and used 42.1% less energy. The control intensity remained near its upper bound until time 9.50, then gradually decreased to zero at the terminal time. Numerical verification yielded a maximum mass conservation error of 2.55 × 10⁻¹⁵, a state difference of 2.00 × 10⁻¹⁵ between the two- and three-dimensional models, and a projection residual of 8.66 × 10⁻⁹. Sensitivity analysis at B = 0.5, 1, 2, and 5 showed a consistent trade-off between intervention cost and the persistence of misconceptions. This study provides a mathematical framework for examining time-dependent conceptual interventions. Because all parameters and cost weights are assumed, the findings are limited to the model’s mathematical implications and cannot be treated as empirical recommendations.