Numerical boundary control for semilinear hyperbolic systems

Figure 1.  Network with $ n $ arcs and coupling conditions $ B:\mathbb{R}^{2n}\rightarrow \mathbb{R}^{2n} $

Figure 2.  Continuous setting for the computation of stabilizing boundary controls $ \mathcal{B}_t $

Figure 3.  Semi-discrete setting for the computation of stabilizing boundary controls $ \mathcal{B}_t $

Figure 4.  Deviations of the density to the steady state. The upper panels show a simulation with desired decay rate $ \mu = 0.1 $ where the parameters $ {{\hat{\mu}^s}_{{\mathcal{M}}}(t)} $ (left) and $ {{\hat{\mu}^s}_{\mathcal{Q}}(t)} $ (right) are used. The lower panels show simulations for the decay rate $ \mu = 1 $, respectively

Figure 5.  The $ L^2 $-norm $ \big\lVert {\mathcal{R}}(t, \cdot) \big\rVert^2_{L^2} $ and the scaled Lyapunov function (22) obtained by simulations with the parameter $ {\hat{\mu}^s}_{{\mathcal{M}} }(t) $ are plotted in blue. Simulations that are based on the parameter $ {\hat{\mu}^s}_{\mathcal{Q}}(t) $ are shown in red, respectively. The desired exponential decay is black dashed

Figure 6.  Left $ y $-axis shows the obtained boundary control $ \kappa^*(t) $ for the parameters $ {\hat{\mu}^s}_{\mathcal{Q}}(t) $ and $ {\hat{\mu}^s}_{\mathcal{M} }(t) $ which are plotted against the right axis as dashed line

Figure 7.  Deviations of the density to the steady state with desired decay rate $ \mu = 1 $, where the parameter $ {{\hat{\mu}^s}_{{\mathcal{M}}}(t)} $ (left) and $ {{\hat{\mu}^s}_{\mathcal{Q}}(t)} $ (right) are used for the boundary control

Figure 8.  The left panel shows the scaled Lyapunov function (22) and the $ L^2 $-norm obtained by simulations with the parameter $ {\hat{\mu}^s}_{{\mathcal{M}} }(t) $ in blue, while those corresponding to the parameter $ {\hat{\mu}^s}_{\mathcal{Q}}(t) $ are shown in red. The desired decay is black dashed. The right panel states the control $ \kappa^*(t) $ at the left $ y $-axis and the corresponding parameters $ {\hat{\mu}^s}_{{\mathcal{Q}} }(t) $ and $ {\hat{\mu}^s}_{{\mathcal{M}} }(t) $ at the right axis

Figure 9.  Deviations of the density from steady state for the nonlinear systems with the control based on the parameter $ {\hat{\mu}^s}_{{\mathcal{M}} }(t) $

Figure 10.  Deviations from steady state are stated in terms of the scaled Lyapunov function (22) as dashed line, which is an upper bound of the $ L^2 $-norm. Here, simulations for both parameters differ in a magnitude less than $ \sim10^{-5} $

Figure 11.  Discontinuous initial data $ {\mathcal{R}}^\pm(0, x) = 2\, \text{sign}(x) $ and feedback control based on the weighted Rayleigh quotient. Simulations for the conservation law without source term are shown in blue. The scaled Lyapunov function (22) yields an upper bound on the $ L^2 $-norm that decays at least with the desired rate $ \mu = 0.1 $ (left panel) and $ \mu = 1 $ (right panel). Simulations for the semilinear system are shown in red. The corresponding deviations decrease non-monotonically