Regularity of Optimal Cost Functional Applied to Study of Environmental Pollution

Abstract

We study environmental pollution problems by using the optimal control theory applied to partial differential equations. We consider the problem to find the optimal way to eliminate pollution in the time, such that the concentration is close to a standard level which does not affect the ecological equilibrium when the source is pointwise. We consider a quadratic cost functional and we prove the existence and uniqueness of optimal control. We find a characterization which makes possible the computing of optimal control. Additionally, we consider the problem moving the pointwise source. So we define a functional j(b) that associates to any point b in a region of the space the optimal cost functional applied to the optimal control. We show that j(b) is differentiable, provided that the controls are taken in a convenient subset of admissible functions satisfying the cone properties. We also find the point in the region, for which the cost functional is minimum.