On the Solution of Bilevel Optimal Control Problems  to Increase the Fairness in Air Races

Fisch, F.; Lenz, Jakob; Holzapfel, Florian; Sachs, Gottfried

doi:10.2514/1.54407

Cited by 22 publications

(9 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Now, we are in a position to apply Lemmas 5.5, 5.6, and 5.7 to (12) which allows us to infer that ( x, ȳ, ū) is a W-or even (under the additional compactness of B or C) C-stationary point of (12).…”

Section: Remark 52mentioning

confidence: 99%

“…the monographs [4,7,10,32] and [19,25,33,34] for detailed introductions to bilevel programming and optimal control (of ordinary as well as partial differential equations), respectively. Some more applications of bilevel optimal control can be found in [12,17,22,24] while necessary optimality conditions are the subject in e.g. [5,26,27,35,36].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Solving inverse optimal control problems via value functions to global optimality

et al. 2019

View full text Add to dashboard Cite

In this paper, we show how a special class of inverse optimal control problems of elliptic partial differential equations can be solved globally. Using the optimal value function of the underlying parametric optimal control problem, we transfer the overall hierarchical optimization problem into a nonconvex single-level one. Unfortunately, standard regularity conditions like Robinson's CQ are violated at all the feasible points of this surrogate problem. It is, however, shown that locally optimal solutions of the problem solve a Clarke-stationarity-type system. Moreover, we relax the feasible set of the surrogate problem iteratively by approximating the lower level optimal value function from above by piecewise affine functions. This allows us to compute globally optimal solutions of the original inverse optimal control problem. The global convergence of the resulting algorithm is shown theoretically and illustrated by means of a numerical example. KeywordsBilevel optimal control • Global optimization • Inverse optimal control • Optimality conditions • Solution algorithm This work is supported by the DFG Grant Analysis and Solution Methods for Bilevel Optimal Control Problems (Grant Nos. DE 650/10-1 and WA3636/4-1) within the Priority Program SPP 1962 (Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization).

show abstract

Section: Remark 52mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Solving inverse optimal control problems via value functions to global optimality

et al. 2019

View full text Add to dashboard Cite

show abstract

“…The paper [47] deals with the scheduling of multiple agents which are controlled at the lower level stage. In [20], the authors discuss a bilevel optimal control problem where airplanes are controlled at multiple lower levels in order to increase the fairness in air racing. Finally, it is possible that leader and follower have to solve an optimal control problem.…”

Section: What Is Bilevel Optimal Control?mentioning

confidence: 99%

“…x ∈ X ad ∩ B ε ( x) (y, u) ∈ Ψ(x). (20) Combining Theorem 4.9 as well as Lemmas 4.11 and 4.12, ( x, ȳ, ū) is a C-stationary point of (20). Noting that the derivative of the functional R n x → 1 2 |x − x| 2 2 ∈ R vanishes at x while N X ad ∩B ε ( x) ( x) = N X ad ( x) holds since x is an interior point of B ε ( x), the C-stationarity conditions of ( 20) and (IOC) coincide at ( x, ȳ, ū).…”

Section: Derivation Of Stationarity Conditionsmentioning

confidence: 99%

Bilevel optimal control: existence results and stationarity conditions

Mehlitz¹,

Wachsmuth²

2019

Preprint

View full text Add to dashboard Cite

The mathematical modeling of numerous real-world applications results in hierarchical optimization problems with two decision makers where at least one of them has to solve an optimal control problem of ordinary or partial differential equations. Such models are referred to as bilevel optimal control problems. Here, we first review some different features of bilevel optimal control including important applications, existence results, solution approaches, and optimality conditions. Afterwards, we focus on a specific problem class where parameters appearing in the objective functional of an optimal control problem of partial differential equations have to be reconstructed. After verifying the existence of solutions, necessary optimality conditions are derived by exploiting the optimal value function of the underlying parametric optimal control problem in the context of a relaxation approach.

show abstract

“…An overview of existing theoretical literature on bilevel optimal control problems may be found in [5,6] and [23]. On the other hand, studies related to applications are presented in [2,12] and [18]. In these contributions the authors consider an optimal control problem in each level, some of them including mixed control-state constraints.…”

Section: Introductionmentioning

confidence: 99%

Bilevel Optimal Control Problems with Pure State Constraints and Finite-dimensional Lower Level

Benita¹,

Dempe²,

Mehlitz³

2016

SIAM J. Optim.

View full text Add to dashboard Cite

This paper focuses on the development of optimality conditions for a bilevel optimal control problem with pure state constrains in the upper level and a finite-dimensional parametric optimization problem in the lower level. After transforming the problem into an equivalent single-level problem we concentrate on the derivation of a necessary optimality condition of Pontryagin-type. We point out some major difficulties arising from the bilevel structure of the original problem and its pure state constraints in the upper level leading to a degenerated maximum principle in the absence of constraint qualifications. Hence, we use a partial penalization approach and a well-known regularity condition for optimal control problems with pure state constraints to ensure the non-degeneracy of the derived maximum principle. Finally, we illustrate the applicability of the derived theory by means of a small example.

show abstract

On the Solution of Bilevel Optimal Control Problems to Increase the Fairness in Air Races

Cited by 22 publications

References 15 publications

Solving inverse optimal control problems via value functions to global optimality

Solving inverse optimal control problems via value functions to global optimality

Bilevel optimal control: existence results and stationarity conditions

Bilevel Optimal Control Problems with Pure State Constraints and Finite-dimensional Lower Level

Contact Info

Product

Resources

About