Weak spreadability of reaction–diffusion processes via an optimal boundary control approach

This paper aims to address the weak spreadability problem for reaction–diffusion processes governed by semilinear parabolic partial differential equations (PDEs) with spatially-varying coefficients under Robin boundary control. We begin by recalling the notion of weak spreadability and reformulating the weak spreading control problem within an optimal boundary control framework. Subsequently, the associated optimal boundary control problem over a subinterval is analyzed via the Hilbert Uniqueness Method (HUM), and explicit necessary optimality conditions are derived. An optimal boundary control strategy is then proposed to achieve weak spreadability for the considered semilinear reaction–diffusion PDEs. Finally, simulation studies are presented to validate the key properties of our obtained theoretical results.