最优控制Ginzburg-Landau方程的收敛速率
Convergence rates for an optimally controlled Ginzburg-Landau equation
AI总结：
针对热驱动Ginzburg-Landau方程的最优控制问题，证明了空间离散化后值函数二次收敛到原问题值函数，并采用辛欧拉方法实现时间离散化，在合理条件下达到一阶时间收敛。
摘要：
考虑一个与热驱动Ginzburg-Landau方程在稳定态之间跃迁概率相关的最优控制问题。证明了空间离散化后的最优控制问题的值函数二次收敛到原问题的值函数。这是通过利用值函数求解类似的Hamilton-Jacobi方程来实现的，其中原问题的方程定义在无限维Hilbert空间上。时间离散化采用辛欧拉方法。在施加合理条件的情况下，该方法在时间上达到一阶收敛，且常数与空间离散化无关。
英文摘要：
An optimal control problem related to the probability of transition between stable states for a thermally driven Ginzburg-Landau equation is considered. The value function for the optimal control problem with a spatial discretization is shown to converge quadratically to the value function for the original problem. This is done by using that the value functions solve similar Hamilton-Jacobi equations, the equation for the original problem being defined on an infinite dimensional Hilbert space. Time discretization is performed using the Symplectic Euler method. Imposing a reasonable condition this method is shown to be convergent of order one in time, with a constant independent of the spatial discretization.
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