крайові задачі, краевые задачи ; матричні рівняння, матричные уравнения ; простори Гільберта ; рівняння Ляпунова, уравнение Ляпунова
У статті розглянуто двоточкову крайову задачу в критичному випадку, яка виникає в теорії оптимального керування для матричних диференціальних рівнянь Ріккаті та рівнянь типу Ляпунова. Досліджено задачу в припущенні, що оператор, який описує однорідну лінійну крайову задачу, є нетеровим. Запропоновано підхід до знаходження її розв'язку за допомогою теорії псевдообернених матриць. Знайдено умову розв'язності таких задач. В статье рассмотрено двухточечную краевую задачу в критическом случае, которая возникает в теории оптимального управления для матричных дифференциальных уравнений Риккати и уравнений типа Ляпунова. Исследовано задачу в предположении, что оператор, который описывает линейную краевую задачу, является нетеровым. Предложен подход к нахождению её решения с помощью теории псевдообратных матриц. Найдено условие разрешимости таких задач.
The paper is devoted to investigation of the two-point boundary-value problem in the critical case. This problem has many applications in the optimal controltheory. Under assumption that the corresponding generating operator is Fredholm the given boundary-value problem is studied. The set of solutions are constructed with using the theory of pseudoinvertible operators. Conditions of solvability of such problems are found. It should be noted that such equations very often use in the games theory and the variational calculations. There are exist many papers where matrix Riccati equations and operator differential Riccati equations was investigated. As a rule such equations was investigated in the regular case where the given problem has a unique solution. In the nonregular case such equation was investigated (in the periodic case) in the work [1,5] of Boichuk O.A., Krivosheya S.A. It should be noted that this equation was investigated as in the operator and matrix case as in the operator-differential and differential case. Riccati equation plays important role in the theory of optimal control, calculus of variations, physics and many others applications. It should be noted here that in general many papers are devoted to obtaining the conditions of solvability in the regular case. We can be noted such papers as [1-3, 5-7] where this equation was investigated in finite-dimensional case and papers where such equation was investigated in the infinite-dimensional case. Examples of countable system of such equations is presented. We find the necessary and sufficient conditions of the existence of solutions of boundary value problem for Lyapunov equation in the Hilbert space. The controllability of such operator-differential equation is investigated. We find the control function, which satisfy the next condition: the solutions of boundary value problem have to go through the given operator at the given moment of time.
