带记忆的抛物方程的最优控制

IF 1.3 3区数学 Q4 AUTOMATION & CONTROL SYSTEMS Esaim-Control Optimisation and Calculus of Variations Pub Date : 2023-03-03 DOI:10.1051/cocv/2023013

E. Casas, J. Yong

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引用次数: 2

摘要

研究一类具有记忆的半线性抛物型偏微分方程的最优控制问题。利用Schauder不动点定理和隐式函数定理，建立了状态方程的适定性和一、二阶可微性。对于具有二次代价泛函的最优控制问题，证明了最优控制的存在性。给出了伴随方程的一阶和二阶必要条件，并研究了带测度为算子系数的线性抛物方程的伴随方程。最后，证明了局部最优控制的二阶最优条件的充分性。

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Optimal control of a parabolic equation with memory

An optimal control problem for a semilinear parabolic partial differential equation with memory is considered. The well-posedness as well as the first and the second order differentiability of the state equation is established by means of Schauder fixed point theorem and the implicity function theorem. For the corresponding optimal control problem with the quadratic cost functional, the existence of optimal control is proved. The first and the second order necessary conditions are presented, including the investigation of the adjoint equations which are linear parabolic equations with a measure as a coefficient of the operator. Finally, the sufficiency of the second order optimality condition for the local optimal control is proved.

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来源期刊

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

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7.10%

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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.