Optimal Control in Linear Sobolev Type Mathematical Models

В статье представлен обзор работ челябинской математической школы по уравнениям соболевского типа при исследовании задачи оптимального управления для линейных моделей соболевского типа с начальным условием Коши (Шоуолтера - Сидорова)
или начально-конечным условием. Для выявления непустоты множества допустимых решений задачи управления используется уже хорошо зарекомендовавший себя при решении уравнений соболевского типа метод фазового пространства, заключающийся в редукции сингулярного уравнения к регулярному, определенному на некотором подпространстве исходного пространства и применении теории вырожденных (полу)групп операторов на случай относительно ограниченных, секториальных и радиальных операторов. В работе проводится редукция математических моделей к начальным (начально-конечным) задачам для абстрактного уравнения соболевского типа. Абстрактные результаты применены к исследованию задач управления для математической модели Баренблатта - Желтова - Кочиной, которая моделирует фильтрацию жидкости в трещинновато-пористой среде, модели Хоффа на графе, моделирующей динамику выпучивания двутавровых балок в конструкции, а также модели Буссинеска - Лява, описывающей продольные колебания в тонком упругом стержне с учетом инерции и при внешней нагрузке, либо распространения волн на мелкой воде.

Ключевые слова

уравнения соболевского типа; сильные решения; оптимальное управление; фазовое пространство; модель Баренблатта - Желтова - Кочиной модель Хоффа; модель Буссинеска - Лява; модель Девиса; модель Чена - Гетина.

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