Uncertainty damping in kinetic traffic models by driver-assist controls

Figure 1.  The probability of accelerating $ P(\rho;\, z) $ given in (3) plotted for various $ z $

Figure 2.  The case $ z\in\{1, \, 3\} $ with $ \operatorname{Prob}(z = 1) = \alpha_1 $, $ \operatorname{Prob}(z = 3) = \alpha_2 $: fundamental diagram $ \rho\mapsto\rho\bar{V}_\infty(\rho) $ (solid line) and uncertainty lines $ \rho\mapsto\rho\bar{V}_\infty(\rho)\pm\rho\varsigma_\infty(\rho) $ (dash-dotted and dashed lines, respectively). The filled area is (12). The function $ P(\rho;\, z) $ is taken like in (3)

Figure 3.  The case $ z\sim \mathcal{U}([1, \, 3]) $: fundamental diagram $ \rho\mapsto\rho\bar{V}_\infty(\rho) $ (solid line) and uncertainty lines $ \rho\mapsto\rho\bar{V}_\infty(\rho)\pm\rho\varsigma_\infty(\rho) $ (dash-dotted and dashed lines, respectively). The filled area is (12). The function $ P(\rho;\, z) $ is taken like in (3)

Figure 4.  Representation of $ \bar{f}_\infty(v) $ (solid lines) and of its $ z $-standard deviation $ \pm\sqrt{ \operatorname{Var}_z(f_\infty(v;\, z))} $ (dashed lines) for $ z $ such that $ z-1 $ has a binomial distribution. In (a), $ \rho = 0.2 $. In (b), $ \rho = 0.4 $

Figure 5.  Representation of $ \bar{f}_\infty(v) $ (solid lines) and of its $ z $-standard deviation $ \pm\sqrt{ \operatorname{Var}_z(f_\infty(v;\, z))} $ (dashed lines) for different distributions of the uncertain parameter $ z $ and various traffic densities. In (a), $ z $ is uniformly distributed in $ [1, \, 3] $. In (b), $ z $ is such that $ z-2 $ has a gamma distribution, thus it is, in particular, unbounded

Figure 6.  Action of the uncertain control (19) on the fundamental diagram and its scattering for two possible distributions of the uncertain parameter $ z $ and two values of the effective penetration rate $ p^\ast $. We considered $ v_d(\rho) = 1-\rho $ as optimal speed. For uniformly distributed $ z $ (bottom row), a comparison is possible with Figure 3, which illustrates the fundamental diagram and its scattering in the uncontrolled case

Figure 7.  Representation of $ \bar{f}_\infty(v) $ (solid lines) and of its $ z $-standard deviation $ \pm\sqrt{ \operatorname{Var}_z(f_\infty(v;\, z))} $ (dashed lines) for different distributions of the uncertain parameter $ z $, different effective penetration rates $ p^\ast $ of the ADAS technology and various traffic densities. In (a)-(b), $ z $ is such that $ z-1 $ has a binomial distribution. In (c)-(d), $ z $ is uniformly distributed in $ [1, \, 3] $. We considered the optimal speed $ v_d(\rho) = 1-\rho $ and $ \lambda = 5\cdot 10^{-2} $ in (29)

Figure 8.  Comparison between the large time $ z $-averaged solution to the Boltzmann-type equation (32) with non-constant collision kernel (bulleted lines) and the average equilibrium solution to the Fokker-Planck equation (28) (solid line) for decreasing $ \epsilon $, mimicking the quasi-invariant limit $ \epsilon\to 0^+ $. The relevant parameters are $ \rho = 0.2, \, 0.4, \, 0.6 $, $ p^\ast = 1 $, $ z\sim \mathcal{U}([1, \, 3]) $

Figure 9.  Uncontrolled case. Convergence of the $ L^2 $-numerical error with respect to the exact solution (17) of the Fokker-Planck equation (15) for: (a) $ z\sim \mathcal{U}([1, \, 3]) $ (circular markers); (b) $ z $ such that $ z-1\sim \operatorname{B}\!\left(50, \, \frac{1}{50}\right) $ (triangular markers) and $ z $ with beta distribution in $ I_Z = [1, \, 3] $ with zero mean and variance equal to $ \frac{1}{3} $ (circular markers)

Figure 10.  Uncontrolled case, $ \boldsymbol{z\sim \mathcal{U}([1, \, 3])} $. Contours of $ \bar{f}(t, \, v) = \mathbb{E}_z(f(t, \, v;\, z)) $ (top row) and of $ \operatorname{Var}_z(f(t, \, v;\, z)) $ (bottom row), where $ f $ is the solution to (15) with $ \lambda = 5\cdot 10^{-2} $ issuing from the initial datum (44), for $ t\in [0, \, 20] $ and $ \rho = 0.2, \, 0.4, \, 0.6 $ in the case of uniformly distributed $ z $

Figure 11.  Uncontrolled case, $ \boldsymbol{z-1\sim \operatorname{B}\!\left(50, \, \frac{1}{50}\right)} $. Contours of $ \bar{f}(t, \, v)= \mathbb{E}_z(f(t, \, v;\, z)) $ (top row) and of $ \operatorname{Var}_z(f(t, \, v;\, z)) $ (bottom row), where $ f $ is the solution tok__ge (15), when $ z $ is such that $ z-1 $ has binomial distribution. All the parameters are like in Figure 10

Figure 12.  Controlled case, $ \boldsymbol{z\sim \mathcal{U}([1, \, 3])} $. Contours of $ \operatorname{Var}_z(f(t, \, v;\, z)) $, where $ f $ is the solution to (28) with $ \lambda = 5\cdot 10^{-2} $ issuing from the initial datum (44), for $ t\in [0, \, 20] $ and $ \rho = 0.2, \, 0.4, \, 0.6 $ in the case of uniformly distributed $ z $. Top row: $ p^\ast = 1 $; bottom row: $ p^\ast = 10 $

Figure 13.  Controlled case, $ \boldsymbol{z-1\sim \operatorname{B}\!\left(50, \, \frac{1}{50}\right)} $. Contours of $ \bar{f}(t, \, v)= \mathbb{E}_z(f(t, \, v;\, z)) $ (top row) and of $ \operatorname{Var}_z(f(t, \, v;\, z)) $ (bottom row), where $ f $ is the solution tok__ge (28), when $ z $ is such that $ z-1 $ has binomial distribution. All the parameters are like in Figure 12

Figure 14.  Controlled case. Asymptotic variance $ \operatorname{Var}_z(f_\infty(v;\, z)) $, where $ f_\infty(v;\, z) $ is like in (29), obtained with a uniform and a binomial distribution of the uncertainty for a decreasing penalisation of the control: from $ \kappa = 10^5 $, corresponding to a virtually uncontrolled setting, to $ \kappa = 10^{-1}, \, 10^{-2} $