Convergence rates for an optimally controlled Ginzburg-Landau equation
Provides rigorous convergence rates for optimal control of stochastic PDEs, relevant to researchers in control theory and applied mathematics.
The paper proves that the value function for an optimally controlled Ginzburg-Landau equation converges quadratically with spatial discretization and linearly in time using the Symplectic Euler method, with constants independent of spatial discretization.
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.