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

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.Keywords Optimal control · Numerical methods · Bang-bang control · Linear-quadratic optimal control problems · Time-discretization methods Mathematics Subject Classification

Gradient Methods on Strongly Convex Feasible Sets and Optimal Control of Affine Systems

“…In this section we turn back to the control-affine linear-quadratic problem (1)- (3) and prove that the gradient projection methods considered in the previous section are applicable to the (high order) discretization of the problem recently developed in [21,24]. (This also applies to the conditional gradient method, where the analysis is similar).…”

“…We also provide error estimates regarding both the errors due to discretization and those due to truncation of the gradient projection iterations. The first two subsections reproduce assumptions and results from [24] that are necessary for understanding the implementation of the GPM to the discretized version of problem (1)-(3). The next subsections prove the applicability of the abstract results obtained above, present details about the implementation of the gradient methods, and provide results of computational experiments.…”

“…Usually the discretization is performed by Runge-Kutta schemes (mainly the Euler scheme) and the accuracy is at most of first order due to the discontinuity of the optimal control. Discretization schemes of higher accuracy were recently proposed in [21,24] for systems with linear dynamics and Mayer or Bolza problems. In both cases the error analysis is based on the assumption that the optimal control is of purely bang-bang type.…”

“…Linear terms are not included in the integrand in (1), since they can be shifted in a standard way into the differential equation (2). For solving the above problem one can apply the high-order discretization scheme developed in [21,24]. It results in a discrete-time optimal control problem (a mathematical programming problem), where the gradient of the objective function can be calculated following a standard procedure involving the solution of the associated adjoint system, so that gradient-type methods are conveniently applicable.…”

“…3 we turn back to the optimal control problem (1)- (3). The first two subsections are preliminary, where we introduce notations, formulate assumptions and present the discrete approximation introduced in [21,24] and the error estimate proved in [24]. All this is needed for understanding of the implementation of the GPM and the CGM and of the proofs of the error estimations.…”