Modeling student engagement using optimal control theory

Figure 1.  Left: graphs of $ u_c(\mu, \cdot ) $ for sample values of $ \mu $; the lowest curve is the graph of $ u_c(1, \cdot ) $. Right: contour plot of $ u_{\rm opt} $, with contour values $ \frac j {10}, j = 0, \ldots, 10 $. Black: u = 0; white: u = 1

Figure 2.  Left: $ K_{\frac \pi {10}}(r) $ for $ r $ near $ \tan \frac \pi {10} $, right: $ \upsilon_\circ(r) $ associated to $ \xi_\circ(u) = \sqrt{1 - u^2} $

Figure 3.  Graphs of $ X_{h_0}^\pm $ for $ \mathcal{S} = [0, 1] $, $ \gamma(m) = (1 - m)^2 $, $ \mu(m) = {1 \over 2} \left ( 1 - \frac m 2 \right )^2 $, constant functions $ \psi_{\rm pe} = {1 \over 2} $ and $ \phi = \frac \pi {12} $, and representative values of $ h_0 $ for which $ \breve C_{h_0} $ has a strict global minimum $ {c_{h_0}} $. Solid curves: $ X_{h_0}^+ $; dashed curves: $ X_{h_0}^- $

Figure 4.  Representative vertically aligned graphs of $ K_\phi(r) $ and $ \breve C_{h_0}(m) $ for constant $ \phi $ and three values of $ h_0 $. Solid: projections of the trajectories with initial data $ (m_0, r_0) $; dotted: points on the graphs outside those projections. Upper row: $ r_0 < r_\phi $ and $ m_0 > m_c $; initially, $ m $ decreases and $ r $ increases. Lower row: $ r_0 > r_\phi $ and $ m_0 < m_c $; initially, $ m $ increases and $ r $ decreases. Left: $ m' $ changes sign as $ r $ passes through $ r_\phi $; center: $ m $ and $ r $ asymptotically approach the equilibrium $ (m_c, r_\phi) $; right: $ r' $ changes sign as $ m $ passes through $ m_c $