CAA Talk: Control under constraints of reaction-diffusion equations (Ruiz-Balet, Madrid)

Mathematical control in deep learning.
Domènec Ruiz-Balet, (Universidad Autónoma de Madrid & Chair of Computational Mathematics, DeustoTech)

Abstract: Reaction-diffusion models typically intend to predict the behavior of quantities such as concentrations, densities, or proportions. These quantities are, by nature, nonnegative or taking values between [0, 1]. For this reason, in practice, any control strategy that one might propose should satisfy that the controlled trajectory is between the proper bounds. The talk will be centered in the boundary control of scalar reaction-diffusion equations of the form:

u_t – \triangle u = f(u) , (x,t)\in \Omega \times (0,T),
u = a, (x,t) \in \partial\Omega \times (0,T),
u(x, 0) = u_0 .

The first issue to notice is that one should notice is that, nontrivial solutions of the problem

-\triangle u = f(u), x\in\Omega,
u = 0 , x \in \partial\Omega,
1>u> 0, x in\Omega,

Whenever controllability is possible, the primary strategy is to use the staircase method. This method guarantees the controllability in large time among admissible connected steady-states. Furthermore, the results for spatially heterogeneous drifts will be presented.