Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions

Abstract

We analyze the Euler discretization to a class of linear-quadratic optimal control problems. First we show convergence of order h for the optimal values of the objective function, where h is the mesh size. Under the additional assumption that the optimal control has bang-bang structure we show that the discrete and the continuous controls coincide except on a set of measure O( √ h). Under a slightly stronger assumption on the smoothness of the coefficients of the system equation we obtain an error estimate of order O(h).

Convergence results for the discrete regularization of linear-quadratic control problems with bang–bang solutions

“…1 we give some preliminaries and known auxiliary results for the linear-quadratic control problem (OS). Moreover we generalize the second-order condition, which has been an important tool of the analysis in [6,Lemma 4.1,Theorem 4.2], under weaker assumptions on the structure of the switching function to order k + 1. In Sect.…”

“…[18]). In [33] the error estimates from [4] have been improved to order O(h) by combining proof techniques from [6,7] and [4].…”

“…First results on the error analysis for bang-bang controls can be found in [36]. In [6,7] Euler discretizations for a class of linear-quadratic control problems with bang-bang solutions have been investigated. These results have been extended to a stable implicit discretization scheme in [5].…”

Higher-order numerical scheme for linear quadratic problems with bang–bang controls

“…1 Due to the compactness and the convexity of U , the set F is compact with respect to the L 2 -weak topology for u and the uniform norm for x. Thus a minimizer (x,û) does exist in the space W…”

“…We recall that the Euler method has already been profoundly investigated in the case where bang-bang controls appear (e.g. [1,2,13,18,26]). As mentioned in the introduction, in doing this we use an idea that originates from [15,28] and was implemented in [23] in the case of Mayer's problems.…”