Abstract
This paper considers a linear-quadratic optimal control problem where the control function appears linearly and takes values in a hypercube. It is assumed that the optimal controls are of purely bang–bang type and that the switching function, associated with the problem, exhibits a suitable growth around its zeros. The authors introduce a scheme for the discretization of the problem that doubles the rate of convergence of the Euler’s scheme. The proof of the accuracy estimate employs some recently obtained results concerning the stability of the optimal solutions with respect to disturbances.
Highlights
Discretization schemes for optimal control problems have been largely investigated in the last 60 years
Error estimates for the accuracy of the Euler discretization scheme applied to various classes of affine optimal control problems were obtained in [1,2,13,18,26,27]
The first paper that addresses the issue of accuracy of discrete approximations for affine problems is [30], where a higher order Runge–Kutta scheme is applied to a linear system, but the error estimate is of first order or less
Summary
Discretization schemes for optimal control problems have been largely investigated in the last 60 years (see, e.g., [6,7,8,9,17,21], and the more recent paper [5] and the references therein). Results on the stability of solutions with respect to disturbances were recently obtained, see [4,12,14,25] and the bibliography therein Based on these results, error estimates for the accuracy of the Euler discretization scheme applied to various classes of affine optimal control problems were obtained in [1,2,13,18,26,27]. We assume that the optimal controls are of strictly bang–bang type with a finite number of switches, and the components of the switching function have a certain growth rate at their zeros, characterized by a number κ ≥ 1 This number appears in the error estimate obtained in this paper for the proposed discretization scheme.
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