ParaOpt for unstable systems
Résumé
We present a convergence analysis of the ParaOpt algorithm applied to linear optimal control problems where the linear operator involved in the constraint -- namely, a linear ordinary differential equation (ODE) -- has at least one positive eigenvalue, which usually corresponds to unstable systems. When the control problem is discretized using the explicit Euler scheme, the resulting formulation can be expressed as a stationary iterative method, in which a coarse Jacobian approximation acts as a preconditioner. An explicit bound on the spectral radius of the corresponding iteration matrix is proposed, depending on the discretization parameters as well as on the smallest negative and largest positive eigenvalues of the considered linear operator. Our theoretical results are completed by numerical experiments performed on both linear and nonlinear examples.
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