Optimal Control Theory for a System of Partial Differential Equations Associated with Stratified Fluids

Abstract

In this paper, we investigate the existence of an optimal solution of a functional restricted to non-linear partial differential equations, which ruled the dynamics of viscous and incompressible stratified fluids in R3. Additionally, we use the first derivative of the considered functional to establish the necessary condition of the optimality for the optimal solution.

Study of a Nonlinear System of Partial Differential Equations Associated with Stratified Fluids in Three Dimensions

“…In this paper, we demonstrate the existence and unique solution of a nonlinear system of partial differential equations related to stratified fluids within a bounded domain in three dimensions. We open the door to future studies of more general nonlinear systems, considering stratification, heat transfer, and salinity, as well as exploring some optimization problems whose solutions are constraints to the systems discussed in our article, following the philosophy of [5]. We could also extend our results to environments that involve fractional derivatives like in [2], [3], [6] and [21].…”

“…On the other hand, let us denote the space of functions by D(Ω) such that φ : Ω −→ R of class C ∞ (Ω) with compact support and by D ′ (Ω) the space of distributions on Ω. Throughout this paper, we will use the standard notations for Lebesgue and Sobolev spaces as found in [5], in particular the norm in L 2 (Ω) and the scalar product in L 2 (Ω) will be represented by ∥ • ∥ and (•, •) respectively.…”

Simulating two-dimensional optimal control problem of fractional partial differential equations

“…Ref. [4] was investigated the optimal solution of a functional restricted to non-linear partial differential equation and established the necessary condition of the optimality for the optimal solution. Ref.…”