Control of reaction-diffusion models in biology and social sciences

Figure 1.  Left (resp. Right): A bistable (resp. monostable) nonlinearity

Figure 2.  Profile $ U(x) $ for the traveling wave for the cubic nonlinearity $ f(u) = u(1-u)(u-\theta) $ restricted in the interval $ [-10,10] $. This profile $ U $ is independent of the value of $ \theta $

Figure 3.  Original domain $ \Omega = (0,L) $, extended domain $ \Omega_E = (-1,L+1) $ and the control region $ \omega $

Figure 4.  Left: Control function $ a(0,t) $ steering the semilinear equation with cubic bistable nonlinearity $ f(s) = s(1-s)(s-\theta) $ to the steady-state $ w\equiv\theta $ in time $ T = 5 $. Right: Snapshot at time $ t = 2.2688 $ of the controlled trajectory $ v_a(\cdot,t) $ violating the constraints $ 0\le v \le 1 $

Figure 5.  Qualitative representation of the staircase strategy

Figure 6.  A barrier function for the semilinear heat equation with cubic nonlinearity $ f(u) = u(1-u)(u-1/3) $ for $ L = 20 $

Figure 7.  Separatrix of the system and phase portrait inside the region it determines, $ f(s) = s(1-s) $

Figure 8.  Qualitative bifurcation diagram for the stationary solutions. The black curve represents a nontrivial solution. At the left, the diagram for convave monostable nonlinearities is plotted and, at the right, for a general monostable nonlinearity [67]

Figure 9.  Function $ v_\delta $

Figure 10.  Bounds on $ L^* $ and $ L^*_\theta $ for different values of $ \theta $ in the nonlinearity $ f(s) = s(1-s)(1-\theta) $. The red dots represent upper bounds and the blue dots lower bounds. Left: Bounds on $ L^* $. Right: bounds on $ L_\theta^* $ (Definition 5.3)

Figure 11.  Phase portraits for $ f(s) = s(1-s)(s-\theta) $, when $ \theta<1/2 $ ($ F(1)>0 $). Left/The phase plane: in black lines the stable and unstable manifolds of the points $ (0,0) $ and $ (1,0) $, and, in blue lines, the trajectories of the ODE system. Right: The separatrix, stable and unstable manifolds of the points $ (0,0) $ and $ (1,0) $ depicted outside of the admisible region $ 0\leq u\leq 1 $ (in blue)

Figure 12.  Phase portraits for $ f(s) = s(1-s)(s-1/2) $ ($ \theta = 1/2, \,F(1) = 0 $). Left/The phase plane: in black lines the stable and unstable manifolds of the points $ (0,0) $ and $ (1,0) $, and, in blue lines, the trajectories of the ODE system. Right: The separatrix, stable and unstable manifolds of the points $ (0,0) $ and $ (1,0) $ depicted outside of the admisible region $ 0\leq u\leq 1 $ (in blue)

Figure 13.  Left: Qualitative bifurcation diagram for the stationary solutions of a bistable nonlinearity that is convex in $ (0,\theta) $ and concave in $ (\theta,1) $. Richt: bifurcation dyagram for a general non-convex nonlinearity in $ (0,\theta) $

Figure 14.  Functional $ J^{e_1,e_3}_\lambda(\alpha,\beta,0) $ for different values of $ \lambda $

Figure 15.  Graph of the functional $ J^{e_1,e_3}_\lambda(\alpha,\beta,\theta = 0.33) $ for different pairs $ (\alpha,\beta) $ and values of $ \lambda $

Figure 16.  Energy functional $ J^{e_1,e_3}_\lambda(\alpha,\beta,\theta) $. The values of the functional above $ 3 $ are represented as $ 3 $ in the picture. The white crosses indicate the critical points of the functional

Figure 17.  The blue and red lines corresponds to nontrivial solutions with Dirichlet conditions $ 0 $ and $ \theta $ respectively. In green, a section of the ground state solution defined in the whole space $ \mathbb{R} $

Figure 18.  Invariant region $ \Gamma $ (in blue) for the nonlinearity $ f(s) = s(1-s)(s-1/3) $

Figure 19.  $ L = 8>L_\theta $, $ \theta = 0.33 $. Left: A path of steady-states of system (34) connecting the steady-states $ (0,0) $ and $ (\theta,0) $. In black the steady-states of (49) that are part of a continuous path connecting to the constant stationary solution $ \theta $ are represented. The red crosses are the corresponding boundary controls. Right: The stationary path plotted in the space domain. In red the curve of maximum value in the invariant region $ \Gamma $, and in green the initial condition and in blue the constant steady-state $ \theta $

Figure 20.  $ L = 1 $, $ \theta = 0.33 $. Representation of the "even strategy" in the phase portrait of system (34). Left: in black, the steady-states of (49) that constitute the connected path of stationary solutions, connecting to $ \theta $. In red, the values of the Dirichlet controls. Right: the stationary path in the space domain. The red curve represents the nontrivial elliptic solution in $ \mathbb{R} $, that corresponds to the homoclinic orbit in the phase plane. In green the initial steady-state from which the path of steady-states departs

Figure 21.  $ L = 8 $, $ \theta = 0.33 $. Representation of the "even strategy" in the phase portrait of system (34). Left: in black, the representation of the steady-states of (49) connecting to $ \theta $. In red, the values of the Dirichlet controls. Right: the plot of the stationary states along the space domain. The red curve represents the nontrivial elliptic solution in $ \mathbb{R} $, i. e. the homoclinic orbit in the phase plane. In green the initial condition

Figure 22.  $ L = 8 $, $ \theta = 0.33 $. Left: Trace $ a(s) $ of the connected path of the steady states in equation (50) as a function of the shooting parameter $ 0\le s \le 1 $. Middle: The minimum of the steady-states $ u_{a} $ depending on the boundary value $ a $ along the path. Note that for a given boundary value $ a $, there might be two solutions associated with $ a $ inside the path. Right: The maximum of the steady-states $ u_a $ depending on the boundary value $ a $ along the path.

Figure 23.  $ L = 20 $, $ \theta = 0.33 $. The "even strategy" is represented in the phase portrait of system (34). Left: in black, the plot of the steady-states of (49), constituting a connected path leading to the stationary solution $ \theta $. In red, the values of the corresponding boundary controls. Right: the stationary path plotted in the space domain. The red curve is the nontrivial elliptic solution in $ \mathbb{R} $, corresponding to the homoclinic orbit in the phase plane. In green the steady-state $ v_\epsilon $ from which the path departs

Figure 24.  $ L = 20 $, $ \theta = 0.33 $. Left: trace of the connected path of the steady states $ a(s) $ as a function of the shooting parameter $ 0\le s \le 1 $ in equation (50). Middle: The minimum of the steady-states $ u_a $ depending on the boundary value $ a $ along the path. Note that for a given boundary value $ a $, there might be two solutions associated with $ a $ inside the path. Right: The maximum of the steady-states $ u_a $ depending on the boundary value $ a $ along the path

Figure 25.  $ f(s) = s(1-s)(s-\theta) $, $ \theta = 0.33 $, $ L = 12 $. Left: Connected path of steady-states from $ 0 $ to the minimal nontrivial solution $ \underline{u}_{L} $ in the phase plane. This path has elements outside the invariant region $ \Gamma $. Right: The connected path in the physical space

Figure 26.  $ L = 20 $, $ \theta = 0.5 $. The even path of steady-states from an initial condition to the stationary solution $ \theta $ of system (34). Left: In black, the steady-states of (49), constituting a connected path of stationary solutions leading to $ \theta $. In red, the corresponding values of the controls. Right: the stationary path plotted in the space domain, the green curve being the initial steady-state

Figure 27.  Graph associated with the path shown in Figure 26. Left: Connected path of steady-states, in the horizontal axis the parameter $ s\in [0,1] $ in the vertical axis the boundary value. Center: The minimum value of $ u_{a} $, as a function of the boundary value $ a $. Right: The maximum value of $ u_{a} $, as a function of the boundary value $ a $

Figure 28.  $ L = 20 $, $ \theta = 0.5 $. Phase portrait representation of the path of even steady-states of system (34) connecting the initial condition with the constant steady-state $ 0 $. Left: In black, the path of steady-states of (49) connecting to the stationary solution $ 0 $. In red, the values of the corresponding controls. Right: the stationary path plotted in the space domain, the green curve being the initial steady-state

Figure 29.  Graph associated with the path shown in Figure 28. Left: Boundary values of the connected path of steady-states as a function of the parameter $ s\in [0,1] $. Center: The minimum value of $ u_{a} $, with respect to $ a $, for the connected path of steady-states. Right: The maximum value of $ u_{a} $, with respect to $ a $

Figure 30.  $ L = 20 $, $ \theta = 0.5 $. Phase portrait representation of the path of even steady-states of system (34) connecting the initial condition with the constant steady-state $ 1 $. Left: In black, the path of steady-states of (49) connecting to the stationary solution $ 1 $. In red, the values of the corresponding controls. Right: the stationary path plotted in the space domain, the green curve being the initial steady-state

Figure 31.  Graph associated with the path shown in Figure 30. Left: Boundary values of the connected path of steady-states as a function of the parameter $ s\in [0,1] $. Center: The minimum value of $ u_{a} $, with respect to $ a $, for the connected path of steady-states. Right: The maximum value of $ u_{a} $, with respect to $ a $

Figure 32.  Left: Connectivity map for $ F(1)>0 $. In red, an admissible continuous path of steady-states connecting $ 0 $ and $ \theta $, existing for all $ L>0 $. In green, an admissible and continuous path of steady-states connecting $ \theta $ and $ 1 $, whose existence is guaranteed for $ L<L^* $. In black, traveling waves for the Cauchy problem. The traveling wave from $ 0 $ to $ 1 $ is unique, while there are infinitely many traveling waves from $ \theta $ to $ 1 $ or to $ 0 $, since the nonlinearity, when restricted to $ [\theta,1] $ or in $ [0,\theta] $ is monostable and Theorem 3.9 applies. Right: Connectivity map for $ F(1) = 0 $. In red, admissible continuous paths of steady-states connecting $ 0 $ and $ \theta $ and $ \theta $ to $ 1 $, respectively, existing for any $ L>0 $. The traveling wave from $ 0 $ to $ 1 $ is unique and stationary, giving a continuous path of admissible steady-states connecting $ 0 $ and $ 1 $. In black, infinitely many non-stationary traveling waves connecting $ \theta $ to $ 1 $ and to $ 0 $, respectively

Figure 33.  Non admissible continuous path from $ 0 $ to $ 1 $

Figure 34.  $ f_d(s) = s\left(d+(12-3d)s+(3d-12)s^2\right) $ for different values of $ d $. The integral between $ (0,1) $ of the three functions is the same, i.e. $ \int_0^1f_d(s)ds = 1 $. The red dashed line indicates the slope at the origin

Figure 35.  Experiment 1: Control towards $ \theta $. Controlled state for $ L = 8 $, $ T = 30 $ and initial datum $ u_0(x) = 1 $

Figure 36.  Experiment 2: Control in the presence of a barrier. Controlled state for $ L = 20 $, $ T = 30 $ and initial datum $ u_0 \equiv 1 $

Figure 37.  Experiment 2: Control in the presence of a barrier. Controlled state for $ L = 20 $, $ T = 60 $ and initial datum $ u_0\equiv 0 $

Figure 38.  Experiment 3. Positivity of the minimal control time. Controlled state for $ L = 20 $, $ T = 15 $ and initial datum $ u_0 \equiv 0 $

Figure 39.  Experiment 3. Positivity of the minimal control time. Controlled state in minimal time for $ L = 20 $ and initial datum $ u_0 \equiv 1 $

Figure 40.  Experiment 4. Quasistatic control strategy. Controlled state for $ L = 20 $, $ T = 400 $ and initial datum $ u_0 \equiv 0 $

Figure 41.  Experiment 4. Quasistatic control strategy. Left: Optimal constrained control associated with Figure 40. Right: Phase-plane plot of the steady-states (in black) and snapshots of the parabolic state (in red)

Figure 42.  Numerical simulation of the semilinear heat equation with nonlinearity $ f(s) = s(1-s)(1-\theta) $ with boundary value $ a(x,t) = 0 $ leading to a barrier steady-state

Figure 43.  Ball containing the original domain $ \Omega $

Figure 44.  $ \theta = 1/3 $, $ R = 30 $ and $ d = 2 $. In blue the phase space description of the invariant region. In black the radial trajectories forming the continuous path of steady-states. The red stars indicate the values taken over the boundary

Figure 45.  In blue the curve $ N(x) $, in orange the quotient $ -N_x(x)/N(x) $ responsible of the drift effect. Left $ N(x) = e^{-\frac{x^2}{\sigma}} $, right $ N(x) = e^{\frac{x^2}{\sigma}} $

Figure 46.  Minimal controllability time from $ 0 $ to $ \theta = 1/3 $ as a function $ 1/\sigma $. Left: minimal controllability time for the Gaussian $ N(x) = e^{-\frac{x^2}{\sigma}} $. Right: $ N(x) = e^{\frac{x^2}{\sigma}} $

Figure 47.  The blue dotted line represents the continuous path of steady-states for the homogeneous equation. In red, the perturbed steady-states, linked to the unperturbed steady-states (black) that belong to a continuous path for the homogeneous equation

Figure 48.  Left: Upper barrier, solution of (62) for $ \sigma = 40 $. Right: Sketch of the phase-plane analysis for the trajectory leading to a solution of (62)

Figure 49.  Space-time representation of the optimal control to $ w\equiv\theta $ with initial data $ w\equiv 0 $. The spatial domain is the unit interval. The nonlinearity is $ f(s) = s(s-\theta)(1-s) $ with $ \theta = 0.33 $ (therefore $ \int_0^1f(s)>0 $) and the controls have been limited to take values in $ [0,1] $. The optimal control problem consists on minimizing the $ L^2 $ distance of the final state to $ w\equiv\theta $. At the left, with diffusivity $ \mu(t) = 0.125\, \mathrm{exp}(-4t) $, we observe a lack of controllability from the initial state $ w\equiv 0 $ to the target $ w\equiv \theta $. At the right, with diffusivity $ \mu(t) = 0.125\, \mathrm{exp}(-2.5t) $, we observe controllability to the steaty-state $ w\equiv \theta $