DISTRIBUTED CONTROL OF AN INCOMPRESSIBLE NAVIER-STOKES EQUATIONS UNDER CONJUGATION CONDITIONS

Abstract

This paper is concerned with the study of first-order necessary optimality conditions for the non-stationary incompressible uid ows described by the Navier-Stokes equa-tions subject to conjugation conditions within a bounded domain ϱ⊂ℝⁿ (2≤n≤3). First,we establish the existence and uniqueness of the weak solution to this system by using the Lax-Milgram lemma then, we prove the existence and uniqueness of distributed control for aquadratic cost functional, subject to the non-stationary incompressible fluid ows governedby Navier-Stokes equations with conjugation and Dirichlet conditions.