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A.L. Dontchev, I. Kolmanovsky, M. Krastanov, M. Nicotra, V.M. Veliov:
"Lipschitz Stability in Discretized Optimal Control with application to SQP";
SIAM Journal on Control and Optimization, 57 (2019), 1; 468 - 489.

We consider a control constrained nonlinear optimal control problem under perturbations represented by a parameter which is a function of time. 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 control 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 $\bar N$ such that if the number $N$ of the grid points is greater than $\bar N$, 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 involved do not depend on $N$. 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 convergence result.

optimal control, discrete approximation, uniform Lipschitz stability, Newton/SQP method

http://dx.doi.org/10.1137/18M1188483