Crank-Nicolson finite difference schemes for parabolic optimal Dirichlet boundary control problem

In this paper, we adopt the optimize-then-discretize approach to solve parabolic optimal Dirichlet boundary control problem. First, we derive the first-order necessary optimality system, which includes the state, co-state equations and the optimality condition. Then, we propose Crank-Nicolson finite difference schemes to discretize the optimality system in 1D and 2D cases, respectively. In order to build the second order spatial approximation, we use the ghost points on the boundary in the schemes. We prove that the proposed schemes are unconditionally stable, compatible and second-order convergent in both time and space. To avoid solving the large coupled schemes directly, we use the iterative method. Finally, we present a numerical example to validate our theoretical analysis.

The problem of boundary control of string vibrationsby displacement of the left end when the right end is fixedwith the given values of the deflection functionat intermediate times

We consider the boundary control problem for the homogeneous string vibrationequation with given the classical boundary (initial and final) conditions and with given valuesof the deflection function at intermediate times. The control is performed by displacementof the left end of the string when the right end is fixed. The problem is reduced to thecontrol problem with zero boundary conditions. We propose the constructive method forconstructing the boundary control of the process of string vibrations with given values ofthe deflection function at intermediate times.We present the results of numerical experimentsand the corresponding graphs confirm the validity of the results.

BOUNDARY CONTROL PROBLEM FOR ONE DEGENERATE HIBERBOLIC EQUATION

В работе изучается задача граничного управления для вырождающегося гиперболического уравнения второго порядка. Установлены необходимые и достаточные условия управляемости данными Коши за минимальный промежуток времени. Граничные управления предъявлены в явном аналитическом виде. The paper studies the boundary control problem for a degenerate second-order hyperbolic equation. Necessary and sufficient conditions are established for minimal time controllability over Cauchy data. Boundary controls are presented in an explicit analytical form.