Editha C. Jose, Ph D., et. al.

Partial differential equations (PDEs) are known to exhibit interesting properties of materials. For instance, the thermal conductivity of a material is described using the heat equation (a parabolic PDE). However, in general, solutions to differential equations are very difficult to get and, in this case, a proper or suitable mathematical formulation of the concerned system is needed. Homogenization and control theory offer alternative techniques of representing solutions of PDES or characterizing them, showing the interplay of theoretical and applied mathematics (in this case in physical sciences and engineering).

A control system is a dynamical system on which one can act by using suitable controls. A dynamical model can be modeled by partial differential equations. Controllability seeks to find a control that will lead the given system in a desirable situation. Roughly speaking, given two states, is it possible to steer or drive the control system from the first one to the second one? There are several types of controllability problem, namely, exact, null and approximate controllability.

In this paper, the authors studied the approximate controllability of a semilinear second order boundary value problem in a periodically perforated domain. The internal control was distributed in an open subset of the domain, where the holes could intersect the control region. They expressed the control in terms of the solution of an adjoint problem. They also homogenized the state and adjoint equations to prove that the limit of the Ɛ- problems converged to that of the homogenized problems.

See full paper here.

Complete Citation: C. Conca, E.C. Jose, I. Mishra. (2024) Approximate controllability of a semilinear elliptic problem in periodically perforated domains, Journal of Mathematical Analysis and Applications (530) 1: 127676