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

IF 1.3 4区 数学 Q1 MATHEMATICS
Sakthivel Kumarasamy
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引用次数: 1

Abstract

In this paper, we consider the 3D Navier-Stokes-Voigt (NSV) equations with nonlinear damping \begin{document}$ |u|^{r-1}u, r\in[1, \infty) $\end{document} in bounded and space-periodic domains. We formulate an optimal control problem of minimizing the curl of the velocity field in the energy norm subject to the flow velocity satisfying the damped NSV equation with a distributed control force. The control also needs to obey box-type constraints. For any \begin{document}$ r\geq 1, $\end{document} the existence and uniqueness of a weak solution is discussed when the domain \begin{document}$ \Omega $\end{document} is periodic/bounded in \begin{document}$ \mathbb R^3 $\end{document} 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.

带控制约束的三维阻尼Navier-Stokes-Voigt方程的最优控制
In this paper, we consider the 3D Navier-Stokes-Voigt (NSV) equations with nonlinear damping \begin{document}$ |u|^{r-1}u, r\in[1, \infty) $\end{document} in bounded and space-periodic domains. We formulate an optimal control problem of minimizing the curl of the velocity field in the energy norm subject to the flow velocity satisfying the damped NSV equation with a distributed control force. The control also needs to obey box-type constraints. For any \begin{document}$ r\geq 1, $\end{document} the existence and uniqueness of a weak solution is discussed when the domain \begin{document}$ \Omega $\end{document} is periodic/bounded in \begin{document}$ \mathbb R^3 $\end{document} 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.
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来源期刊
Evolution Equations and Control Theory
Evolution Equations and Control Theory MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
3.10
自引率
6.70%
发文量
5
期刊介绍: EECT is primarily devoted to papers on analysis and control of infinite dimensional systems with emphasis on applications to PDE''s and FDEs. Topics include: * Modeling of physical systems as infinite-dimensional processes * Direct problems such as existence, regularity and well-posedness * Stability, long-time behavior and associated dynamical attractors * Indirect problems such as exact controllability, reachability theory and inverse problems * Optimization - including shape optimization - optimal control, game theory and calculus of variations * Well-posedness, stability and control of coupled systems with an interface. Free boundary problems and problems with moving interface(s) * Applications of the theory to physics, chemistry, engineering, economics, medicine and biology
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