How Much Spatial Control Is Enough? Subdomain Optimal Control of Reaction-Diffusion Systems in Synthetic Developmental Biology

Reaction-diffusion systems can produce spatial patterns such as stripes and spots through diffusion-driven instability.

Steering these patterns from one configuration to another can be formulated as an optimal control problem.

When the control acts on the entire spatial domain, existence and optimality conditions are well understood.

Yet in practice, the control can only act on a part of the domain.

Taking the Nodal--Lefty reaction--diffusion system as a case study, we consider the setting where the control is restricted to a subdomain.

We derive an explicit upper bound on the optimality loss, defined as the difference between the subdomain optimal cost and the full-domain optimal cost.

From this bound, we obtain an explicit formula for the minimum size of the control region needed to reach a target pattern with prescribed accuracy.

We also consider the case where the control is distributed over several disjoint regions instead of a single one, with the same total area, and prove that the distributed configuration gives a tighter bound under natural conditions on the spatial structure of the target.

Numerical illustrations confirm the theoretical results and show that a control region covering roughly forty percent of the domain is sufficient to drive the system from stripes to spots with high accuracy.