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A fast second-order parareal solver for fractional optimal control problems. (English) Zbl 1400.93082

Summary: The gradient projection technique has recently been used to solve the optimal control problems governed by a fractional diffusion equation. It lies in repeatedly solving the state and co-state equations derived from the optimality conditions, and the Crank-Nicolson (CN) scheme, which gives a second-order numerical solution, is a widely used method to solve these two equations. The goal of this paper is to implement the CN scheme in a parallel-in-time manner in the framework of the parareal algorithm. Because of the stiffness of the approximation matrix of the fractional operator, direct use of the CN scheme results in a convergence factor \(\rho\) satisfying \(\rho\rightarrow 1\) as \(\Delta x\rightarrow 0\) for the parareal algorithm, where \(\Delta x\) denotes the space step-size. Here, we provide a new idea to let the parareal algorithm use the CN scheme as the basic component possessing a constant convergence factor \(\rho\approx 1/5\), which is independent of \(\Delta x\). Numerical results are provided to show the efficiency of the proposed algorithm.

MSC:

93B40 Computational methods in systems theory (MSC2010)
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
34A08 Fractional ordinary differential equations
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