scholarly journals Optimal control for quasilinear retarded parabolic systems

2001 ◽  
Vol 42 (4) ◽  
pp. 532-551 ◽  
Author(s):  
Liping Pan ◽  
Jiongmin Yong
Keyword(s):  
Optimal Control ◽  
Parabolic Equation ◽  
Maximum Principle ◽  
State Constraints ◽  
Quasilinear Parabolic Equation ◽  
Parabolic Systems ◽  
Terminal State ◽  
Spatial Derivative ◽  
The Cost ◽  
Quasilinear Parabolic

AbstractWe study an optimal control problem for a quasilinear parabolic equation which has delays in the highest order spatial derivative terms. The cost functional is Lagrange type and some terminal state constraints are presented. A Pontryagin-type maximum principle is derived.

scholarly journals On boundary optimal control of a coefficient in a nonlinear parabolic equation

2021 ◽  
pp. 263-280
Author(s):  
Н.Л. Гольдман
Keyword(s):  
Optimal Control ◽  
Parabolic Equation ◽  
Quasilinear Parabolic Equation ◽  
Time Derivative ◽  
Parabolic Systems ◽  
Conjugate Problem ◽  
Unknown Coefficient ◽  
Smooth Functions ◽  
Modeling And Control ◽  
Nonlinear Parabolic

Работа связана с изучением нелинейных параболических систем, возникающих при моделировании и управлении физико-химическими процессами, в которых происходят изменения внутренних свойств материалов. Исследовано оптимальное управление одной из таких систем, которая включает в себя краевую задачу третьего рода для квазилинейного параболического уравнения с неизвестным коэффициентом при производной по времени, а также уравнение изменения по времени этого коэффициента. Обоснована постановка оптимальной задачи с финальным наблюдением искомого коэффициента, в которой управлением является граничный режим на одной из границ области. Получено явное представление дифференциала минимизируемого функционала через решение сопряженной задачи. Доказаны условия ее однозначной разрешимости в классе гладких функций. Полученные результаты имеют практическое значение для приложений в различных технических областях, медицине, геологии и т.п. Приведены некоторые примеры таких приложений. The work is connected with investigation of nonlinear parabolic systems arising in the mathematical modeling and control of physical-chemical processes in which inner properties of materials are subjected to changes. We consider optimal control in one of such systems that involves a boundary value problem of the third kind for a quasilinear parabolic equation with an unknown coefficient at the time derivative and, moreover, an additional equation for a time dependence of this coefficient. The optimal problem with a boundary control regime is justified for the given final observation of the sought coefficient. The exact representation for the differential of the minimization functional in terms of the solutions of the conjugate problem is obtained. The form of this conjugate problem and conditions of unique solvability in a class of smooth functions are shown. The obtained results are important for applications in various technical fields, medicine, geology, etc. Some examples of such applications are discussed.

scholarly journals A Maximum Principle for Controlled Time-Symmetric Forward-Backward Doubly Stochastic Differential Equation with Initial-Terminal Sate Constraints

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Shaolin Ji ◽  
Qingmeng Wei ◽  
Xiumin Zhang
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Maximum Principle ◽  
Stochastic Differential Equation ◽  
State Constraints ◽  
Terminal State ◽  
Necessary Condition ◽  
Variation Principle ◽  
Linear Quadratic ◽  
Doubly Stochastic

We study the optimal control problem of a controlled time-symmetric forward-backward doubly stochastic differential equation with initial-terminal state constraints. Applying the terminal perturbation method and Ekeland’s variation principle, a necessary condition of the stochastic optimal control, that is, stochastic maximum principle, is derived. Applications to backward doubly stochastic linear-quadratic control models are investigated.

A Problem with Nonlocal Boundary Conditions for a Quasilinear Parabolic Equation

1999 ◽  
Vol 6 (5) ◽  
pp. 421-428
Author(s):  
T. D. Dzhuraev ◽  
J. O. Takhirov
Keyword(s):  
Parabolic Equation ◽  
Maximum Principle ◽  
Quasilinear Parabolic Equation ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Nonlocal Boundary Value Problem ◽  
Solution Uniqueness ◽  
The Maximum Principle ◽  
Class Of Functions ◽  
Quasilinear Parabolic

Abstract The solvability of the nonlocal boundary value problem 𝑢𝑡 = 𝑎(𝑡, 𝑥, 𝑢, 𝑢𝑥)𝑢𝑥𝑥 + 𝑏(𝑡, 𝑥, 𝑢, 𝑢𝑥), 0 ≤ 𝑡 ≤ 𝑇, |𝑥| ≤ 𝑙, 𝑢(0, 𝑥) = 0, 𝑢(𝑡, –𝑙) = 𝑢(𝑡, 𝑙), 𝑢𝑥(𝑡, –𝑙) = 𝑢𝑥(𝑡, 𝑙) in a class of functions is investigated for a quasilinear parabolic equation. The solution uniqueness follows from the maximum principle.

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