Dynamic phenomena in social and biological sciences can often be modeled by reaction-diffusion equations. When addressing the control from a mathematical viewpoint, one of the main challenges is that, because of the intrinsic nature of the models under consideration, the solution, typically a proportion or a density function, needs to preserve given lower and upper bounds (taking values in $[0,1])$). Controlling the system to the desired final configuration then becomes complex, and sometimes even impossible. In the present work, we analyze the controllability to constant steady states of spatially homogeneous monostable and bistable semilinear heat equations, with constraints in the state, and using boundary controls. We prove that controlling the system to a constant steady state may become impossible when the diffusivity is too small due to the existence of barrier functions. We build sophisticated control strategies combining the dissipativity of the system, the existence of traveling waves, and some connectivity of the set of steady states to ensure controllability whenever it is possible. This connectivity allows building paths that the controlled trajectories can follow, in a long time, with small oscillations, preserving the natural constraints of the system. This kind of strategy was successfully implemented in one-space dimension, where phase plane analysis techniques allowed to decode the nature of the set of steady states. These techniques fail in the present multi-dimensional setting. We employ a fictitious domain technique, extending the system to a larger ball, and building paths of radially symmetric solution that can then be restricted to the original domain.

社会和生物科学中的动态现象通常可以通过反应扩散方程来建模。从数学角度解决控制问题时，主要挑战之一是，由于所考虑模型的固有性质，解决方案（通常为比例或密度函数）需要保留给定的上下限（取值）。在$[0，\mathrm{1个}]）$）。将系统控制到所需的最终配置变得很复杂，有时甚至是不可能的。在目前的工作中，我们分析了空间均质单稳态和双稳态半线性热方程的恒定稳态的可控制性，该方程具有状态约束，并使用边界控制。我们证明，由于势垒功能的存在，当扩散率太小时，无法将系统控制到恒定的稳态。我们建立了复杂的控制策略，结合了系统的耗散性，行波的存在以及一组稳态集的连通性，以确保在可能的情况下可控。这种连通性允许构建路径，使受控轨迹可以在很长的时间内产生很小的振荡，保留系统的自然约束。这种策略已在一个空间维度上成功实施，其中相平面分析技术允许解码稳态集的性质。这些技术在当前的多维设置中失败。我们采用虚拟域技术，将系统扩展到更大的球，并建立径向对称解的路径，然后可以将其限制为原始域。