Euler discretization for a class of nonlinear optimal control problems with control appearing linearly

Alt, Walter; Felgenhauer, Ursula; Seydenschwanz, Martin

doi:10.1007/s10589-017-9969-7

Cited by 23 publications

(15 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Error estimates and numerical analysis for inverse problems involving BV-functions are studied in [5,6]. Related discussion of ODE-constrained control problems involving discontinuous functions and their numerical analysis can be found in, e.g., [1,2,10,21,25,26,37,38].The main difficulty in deriving error estimates for the above problem is given by the fact that it lacks certain coercivity properties that are usually employed to obtain error estimates for the controls, for instance by suitably testing the first order necessary optimality conditions. Hence, only error estimates for the state and the adjoint state can be proven in a rather direct manner; these are, however, suboptimal.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

Hafemeyer¹,

Mannel²,

Neitzel³

et al. 2019

Mathematical Control &Amp; Related Fields

View full text Add to dashboard Cite

Here, u ∈ V := H 1 0 (Ω) is the state and q ∈ Q := BV (Ω) is the control, where BV (Ω) denotes the space of functions of bounded variation (BV) on the interval Ω := (0, 1). The operator A is elliptic and α is a positive real number. The two finite element schemes that will be analyzed are identical in regard to the discretization of state and adjoint state, but they differ in the treatment of the control. In the variational discretization the control is not discretized, while in the second scheme the control is discretized by piecewise constant functions.The significance of the above control problem is given by the use of the BVseminorm q ′ M(Ω) in the objective. This favors piecewise constant controls with only a limited number of jumps, which makes this problem type interesting in many practical applications. The precise functional analytic setting will be provided in the next section.Optimal control problems with BV-controls defined in one space dimension are strongly related to control problems with measures as controls. Both BV optimal control problems and optimal control problems with measures have attracted significant research interest in the recent past, see, e.g. [8,13,14,17,22,23] for the former and [11,12,15,16,29,30] for the latter.Error estimates for PDE-constrained optimal control problems involving measures have been presented in [11,30,31,34,35]. For error estimates of further sparsity promoting optimal control problems with PDEs see for example [19,30]. The literature on error estimates for optimal control problems with controls in BV is rather limited. We are only aware of [14,18]. Error estimates and numerical analysis for inverse problems involving BV-functions are studied in [5,6]. Related discussion of ODE-constrained control problems involving discontinuous functions and their numerical analysis can be found in, e.g., [1,2,10,21,25,26,37,38].The main difficulty in deriving error estimates for the above problem is given by the fact that it lacks certain coercivity properties that are usually employed to obtain error estimates for the controls, for instance by suitably testing the first order necessary optimality conditions. Hence, only error estimates for the state and the adjoint state can be proven in a rather direct manner; these are, however, suboptimal. To obtain an error estimate for the control and also to improve the error estimates for state and adjoint state, we make use of a structural assumption on the Lagrange multiplierΦ arising from the convex subdifferential of the term q ′ M(Ω) . Specifically, we assume thatΦ, which is a C 2 function inΩ, has only finitely many global extreme points and that it exhibits quadratic growth near those points (i.e.,Φ ′′ = 0 near those points; see Assumption 4.4 and Assumption 4.5). Since the jump set of the optimal control is contained in the set of global extreme points ofΦ, see Corollary 1, this assumption implies that the optimal control admits only finitely many jumps, which is a rather typical situation in practice. In addition, it ensures ...

show abstract

mentioning

confidence: 99%

mentioning

confidence: 99%

Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

Hafemeyer¹,

Mannel²,

Neitzel³

et al. 2019

Mathematical Control &Amp; Related Fields

View full text Add to dashboard Cite

show abstract

“…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).…”

Section: The Affine Optimal Control Problemmentioning

confidence: 85%

“…Considerable progress was made in the past decade in the analysis of discretization schemes in combination with various methods of solving the resulting discrete-time optimization problems. The papers [1,2,25,27] apply to problems with linear dynamics, while [3,11] address nonlinear affine (in the control) dynamics. 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.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Veliov

Vuong

2018

Appl Math Optim

View full text Add to dashboard Cite

The paper presents new results about convergence of the gradient projection and the conditional gradient methods for abstract minimization problems on strongly convex sets. In particular, linear convergence is proved, although the objective functional does not need to be convex. Such problems arise, in particular, when a recently developed discretization technique is applied to optimal control problems which are affine with respect to the control. This discretization technique has the advantage to provide higher accuracy of discretization (compared with the known discretization schemes) and involves strongly convex constraints and possibly non-convex objective functional. The applicability of the abstract results is proved in the case of linear-quadratic affine optimal control problems. A numerical example is given, confirming the theoretical findings. Keywords Optimal control • Mathematical programming • Numerical methods • Gradient methods • Affine control systems • Bang-bang control Mathematics Subject Classification 49M25 • 90C25 • 90C48 • 49M37

show abstract

On the regularity and stability of the Mayer‐type convex optimal control problems

Djendel

Zhang

Zhou

2022

Asian Journal of Control

View full text Add to dashboard Cite

This paper is devoted to sufficient condition for Strong Metric sub‐Regularity (SMsR for short) of the set‐valued mapping corresponding to the local description of Pontryagin maximum principle for the Mayer‐type optimal control problems with convexity condition of the Hamiltonian and functional. In particular, stability property of optimal control for the Mayer‐type problem has been established for the occasion of a polyhedral control set and entirely bang‐bang solution structure. Moreover, based on the sufficiency of SMsR and stability property of optimal control, we give the approximate errors of Euler discretization methods utilized to such problems.

show abstract

Euler discretization for a class of nonlinear optimal control problems with control appearing linearly

Cited by 23 publications

References 28 publications

Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

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

On the regularity and stability of the Mayer‐type convex optimal control problems

Contact Info

Product

Resources

About