A.L. Dontchev, I. Kolmanovsky, M. Krastanov, M. Nicotra, V.M. Veliov:
"Lipschitz Stability in Discretized Optimal Control";
Research Reports (Vienna University of Technology, Institute of Statistics and Mathematical Methods in Economics, Operations Research and Control Systems),
We consider a control constrained nonlinear optimal control problem under perturbations represented by changes of a vector parameter. Our main assumptions involve smoothness of the functions appearing in the integral functional and the state equations, an integral coercivity condition, and a condition that the reference optimal control is an isolated solution of the variational inequality for the control appearing in the maximum principle. We also consider a corresponding discrete-time optimal problem obtained from the continuous-time one by applying the Euler finite-difference scheme.
Based on an enhanced version of Robinson´s implicit function theorem, we establish that there exists a natural number N- such that if the number N of the grid points is greater than N, then the solution mapping of the discrete-time problem has a Lipschitz continuous single-valued localization with respect to the parameter whose Lipschitz constant and the sizes of the neighborhoods depend only on the ranges of values of the variables, the Lipschitz constants of the second derivatives of the functions involved, and the coercivity constant. As an application, we show that the Newton/SQP method converges uniformly with respect to the step-size of the discretization and small changes of the parameter. Numerical experiments with a satellite optimal control problem illustrate the results.
optimal control, discrete approximation, uniform Lipschitz stability, Newton/SQP method
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