Optimal control problems governed by two dimensional convective Brinkman-Forchheimer equations

Abstract

The convective Brinkman-Forchheimer (CBF) equations describe the motion of incompressible viscous fluids through a rigid, homogeneous, isotropic, porous medium and is given byIn this work, we consider some distributed optimal control problems like total energy minimization, minimization of enstrophy, etc governed by the two dimensional CBF equations with the absorption exponent r = 1, 2 and 3. We show the existence of an optimal solution and the first order necessary conditions of optimality for such optimal control problems in terms of the Euler-Lagrange system. Furthermore, for the case r = 3, we show the second order necessary and sufficient conditions of optimality. We also investigate an another 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 2D CBF equations, using optimal control techniques.

Optimal control of the 3D damped Navier-Stokes-Voigt equations with control constraints

“…By Lemma 2.1 of [21] (also refer to [29]), there exists a constant C(r) > 0 such that the following lower bound holds:…”

“…Recently, a control problem for the regularized 3D NS equations (NSV equation) in a bounded domain with distributed control and tracking type cost functional has been studied in [4], and the time-optimal control of this model has been considered in [5]. Optimal control of 2D CBF equations is examined for r = 1, 2 and 3 in [29].…”

“…Let us further analyze these results closely associated with the existing literature on the bounded domain. For the optimal control problem of 2D CBF equation (µ = 0 in (1)) in the bounded domain, a first-order necessary condition was derived in [29] for r = 2, 3. The results obtained in this paper for the regularized (OCP) (( 1)-( 3)) generalizes [29] to the 3D bounded domain for any r ∈ [2,5].…”

“…For the optimal control problem of 2D CBF equation (µ = 0 in (1)) in the bounded domain, a first-order necessary condition was derived in [29] for r = 2, 3. The results obtained in this paper for the regularized (OCP) (( 1)-( 3)) generalizes [29] to the 3D bounded domain for any r ∈ [2,5]. The regularized model further helps to prove the crucial Fréchet differentiability, Lipschitz continuity of the control-to-state operator, and that of the control-to-costate operator in better function spaces.…”