Research

Homogenization

Homogenization is a mathematical procedure to understand heterogeneous materials (or media) with highly oscillating heterogeneities (at the microscopic scale) via a homogeneous material. In a composite, the heterogeneities are small compared to its global dimension. Two scales characterize the material, namely, microscopic scale and macroscopic scale. The microscopic scale describes the distribution of heterogeneities, and the macroscopic scale describes the global behavior of the composite. The main aim of homogenization theory is to establish the macroscopic behavior of a microscopically heterogeneous system. This theory characterizes the heterogeneities present in the material (or media). The homogenization deals with the asymptotic analysis of such solutions and obtaining the equation satisfied by the limit. This limit equation will characterize the overall behavior of the material.

• Mathematically, it  is a Limiting Analysis.

Optimal Control Problems (OCPs)

In the calculus of variation, one minimizes certain associated functional over a class of trajectories, the optimal control problems deal with a wider class of minimization problems where the trajectories are defined via certain dynamic constraints. The dynamic constraints may be ordinary differential equations (ODE) or partial differential equations (PDE) giving rise to trajectories, but the crucial point is that these trajectories can be varied by suitable action on the constraint system by applying what is known as controls. Thus the application of optimal control problems is much wider especially in engineering sciences as problems are modeled via differential equations with controls.