First order necessary conditions of optimality for the two dimensional Tidal dynamics system

TL;DR

This paper investigates optimal control problems for two-dimensional tidal dynamics, establishing existence, necessary conditions, and uniqueness of optimal controls, with applications to energy minimization and data assimilation.

Contribution

It formulates and analyzes the first order necessary conditions for optimal control of tidal systems, including existence, characterization, and uniqueness results.

Findings

Existence of optimal control established.

First order necessary conditions derived.

Uniqueness of optimal control in small time intervals proved.

Abstract

In this work, we consider the two dimensional tidal dynamics equations in a bounded domain and address some optimal control problems like total energy minimization, minimization of dissipation of energy of the flow, etc. We also examine an another interesting control problem which is similar to that of the data assimilation problems in meteorology of obtaining unknown initial data, when the system under consideration is the tidal dynamics, using optimal control techniques. For these cases, different distributed optimal control problems are formulated as the minimization of suitable cost functionals subject to the controlled two dimensional tidal dynamics system. The existence of an optimal control as well as the first order necessary conditions of optimality for such systems is established and the optimal control is characterized via adjoint variable. We also establish the uniqueness of optimal control in small time interval.