Optimal periodic control for scalar dynamics under integral constraint on the input

Figure 1.  Functions $ \gamma = \psi\circ \ell^{-1} $ and $ \hat\gamma $ defined above

Figure 2.  $ T $-periodic solutions $ x(\cdot,u^-,\bar x) $ and $ x(\cdot,u^+,\bar x) $

Figure 3.  The solution $ \tilde x $ in thick line, $ x $ in thin line

Figure 4.  Optimal criterion $ J_{T}(\hat u_{T}) $ (left) and $ x_m $, $ x_M $ (right) as functions of the period $ T $ for the logistic growth

Figure 5.  Graphs of the functions $ h $ (left) and $ \psi $ (right) for $ r = 0.3 $, $ K = 5 $, $ \alpha = 2.5 $, $ E_{max} = 0.5893 $, $ E^\star = 0.6235 $

Figure 6.  Optimal criterion $ J_{T}(\hat u_{T}) $ (left) and $ x_m $, $ x_M $ (right) as functions of the period $ T $ for the depensation model (case 1)

Figure 7.  Plot of the function $ F $ defined by (22) (left), and $ x_m $, $ x_M $, $ x_T^- $, $ x_T^+ $ (right) as functions of the period $ T $ $ (T<6) $ for the depensation model (case 2)

Figure 8.  Optimal criterion $ J_{T}(\hat u_{T}) $ for the depensation model (case 2)