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This paper addresses a gradient tracking problem of a bilinear reaction–diffusion equation evolving in a spatial domain. Such an equation is excited with distributed and bounded controls. The problem is formulated by the minimisation of a functional constituted of the deviation between the desired gradient and the current one all over a time interval and the energy term. Then we prove the existence of an optimal control that we characterise by an optimality system. Moreover, we discuss two sets of particular controls: the set of time dependent controls and the space dependent ones. A computational approach and illustrative simulations are also given.

On the gradient tracking problem for a bilinear reaction–diffusion equation excited by distributed bounded controls