Reaction-diffusion equations appear frequently in natural phenomena such as:
- Population dynamics and invasion of species (see ).
- Neuroscience, where models for neuronal impulses exhibit traveling waves (see ).
- Chemical Reactions: modeling the evolution of concentrations of chemicals (see ).
- In evolutionary game theory (see [4, 5]).
- Magnetic systems in material science and their phase transitions (see ).
- Linguistics, we refer to  where the authors consider reaction-diffusion for analyzing language shift by means of a traveling wave.
In the majority of the systems mentioned above, the state $u$ of an equation of the type
represents a population, concentration or proportion. For this reason, any model intending to predict the behavior of such quantity must fulfill a maximum principle .
In this group of tutorials, our aim is to explain the phenomenology arising when considering a control problem in the contexts mentioned above.
In general, the controllability of parabolic equations has been widely studied (for instance [9,10,11]). However, in this literature, the requirement that the trajectory has to be positive or between prescribed bounds was not a concern.
In Figure 1 one can see a boundary control of the semilinear equation
with $a$ being the control and the target function $v\equiv 0.33$.
Control from $u(0)\equiv 0$ to $u(T)\equiv 0.33$.
This model models the evolution of a proportion, and we observe that the control does not preserve the positivity of the state neither the meaningful upper bound.
From the application point of view of several of the applications mentioned, any control action proposed must fulfill that the associated trajectory has meaning.
The ideas exposed in this tutorial group are the following:
- The presence of state-constraints can create intrinsic obstructions for achieving the controllability. This is due to the emergence on non-trivial solutions and the comparison principle. The topic is treated in this blog.
Control of reaction-diffusion under state constraints - Barriers
This tutorial is part of the control under state constraints. We will show how obstructions to the state constraint controllability can appear.
- In order to prove the existence of controls one can use the stair-case method. This method relies on the construction of paths of steady-states that the controlled trajectory can follow. One can find more information about these constructions in this blog.
Control of reaction-diffusion under state constraints - Application of the staircase method
In this tutorial, we will present how to generate admissible paths of steady states for the homogeneous reaction-diffusion equation
- In contrast to the unconstrained case, the presence of constraints induces a minimal controllability time. However, the construction done with the path of steady-states is only a way to control and it requires typically a large time, much larger than the minimal one. In this blog post, we explore different ways to control the system, a minimal controllability time control, a quasistatic control and we explore numerically the effect of the presence of barriers in an optimal control framework.
Control of reaction-diffusion under state constraints - Numerical exploration of controls
This tutorial is part of the control under state constraints. We will simulate different control strategies to the same target by minimizing different functionals.
- Many models in population dynamics take care about spatial heterogeneity, this can lead for example to new types obstructions. In the blog, we explain the main features and the influence of an heterogeneous drift in a bistable equation called gene-flow. The heterogeneity in the drift comes from an approximation of a system of a system of two equations with an heterogeneous environment.
Control of reaction-diffusion under state constraints - Heterogeneous setting: Gene-flow.
This tutorial is part of the control under state constraints. We will present the main features regarding the controllability of bistable reaction-diffusion equations with heterogeneous drifts.
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