Optimal Control of the 3D Damped Navier-Stokes-Voigt Equations with Control Constraints

TL;DR

This paper studies the optimal control of 3D damped Navier-Stokes-Voigt equations with control constraints, establishing existence, uniqueness, and optimality conditions for solutions in various boundary conditions.

Contribution

It introduces a comprehensive analysis of optimal control for damped NSV equations, including existence, uniqueness, and first- and second-order optimality conditions.

Findings

Existence and uniqueness of weak and strong solutions under different boundary conditions.

Derivation of first-order necessary optimality conditions via adjoint problem.

Establishment of second-order sufficient optimality conditions for local optimality.

Abstract

In this paper, we consider the 3D Navier-Stokes-Voigt (NSV) equations with nonlinear damping $|u|^{r-1}u, r\in[1,\infty)$ in bounded and space-periodic domains. We formulate an optimal control problem of minimizing the curl of the velocity field in the energy norm subject to the flow velocity satisfying the damped NSV equation with a distributed control force. The control also needs to obey box-type constraints. For any $r\geq 1,$ the existence and uniqueness of a weak solution is discussed when the domain $\Omega$ is periodic/bounded in $\mathbb R^3$ while a unique strong solution is obtained in the case of space-periodic boundary conditions. We prove the existence of an optimal pair for the control problem. Using the classical adjoint problem approach, we show that the optimal control satisfies a first-order necessary optimality condition given by a variational inequality. Since the optimal control problem is non-convex, we obtain a second-order sufficient optimality condition showing that an admissible control is locally optimal. Further, we derive optimality conditions in terms of adjoint state defined with respect to the growth of the damping term for a global optimal control.