Optimal Borel measure-valued controls to the viscous Cahn--Hilliard--Oberbeck--Boussinesq phase-field system on two-dimensional bounded domains

IF 1.3 3区数学 Q4 AUTOMATION & CONTROL SYSTEMS

Esaim-Control Optimisation and Calculus of Variations Pub Date : 2023-04-04 DOI:10.1051/cocv/2023025

Gilbert Peralta

引用次数: 0

Abstract

We consider an optimal control problem for the two-dimensional viscous Cahn--Hilliard--Oberbeck--Boussinesq system with controls that take values in the space of regular Borel measures. The state equation models the interaction between two incompressible non-isothermal viscous fluids. Local distributed controls with constraints are applied in either of the equation governing the dynamics for the concentration, mean velocity, and temperature. Necessary and sufficient conditions characterizing local optimality in terms of the Lagrangian will be demonstrated. These conditions will be obtained through regularity results for the associated adjoint system, a priori estimates for the solutions of the linearized system in a weaker norm compared to that of the state space, and the Lebesgue decomposition of Borel measures.

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二维有界域上粘性Cahn—Hilliard—Oberbeck—Boussinesq相场系统的最优Borel测量值控制

我们考虑二维粘性Cahn—Hilliard—Oberbeck—Boussinesq系统的最优控制问题，该系统的控制在正则Borel测度空间中取值。状态方程模拟了两种不可压缩非等温粘性流体之间的相互作用。在控制浓度、平均速度和温度的动力学方程中，应用带有约束的局部分布控制。用拉格朗日定理证明了局部最优性的充分必要条件。这些条件将通过相关伴随系统的正则性结果、相对于状态空间的弱范数的线性化系统解的先验估计以及Borel测度的Lebesgue分解得到。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Esaim-Control Optimisation and Calculus of Variations Mathematics-Computational Mathematics

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7.10%

发文量

期刊介绍： ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations. Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines. Targeted topics include: in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory; in optimisation: mathematical programming, large scale systems, stochastic optimisation, combinatorial optimisation, shape optimisation, convex or nonsmooth optimisation, inverse problems, interior point methods, duality methods, numerical methods, convergence and complexity, global optimisation, optimisation and dynamical systems, optimal transport, machine learning, image or signal analysis; in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.