---
_id: '30733'
abstract:
- lang: eng
  text: Hamilton-Jacobi reachability methods for safety-critical control have been
    well studied, but the safety guarantees derived rely on the accuracy of the numerical
    computation. Thus, it is crucial to understand and account for any inaccuracies
    that occur due to uncertainty in the underlying dynamics and environment as well
    as the induced numerical errors. To this end, we propose a framework for modeling
    the error of the value function inherent in Hamilton-Jacobi reachability using
    a Gaussian process. The derived safety controller can be used in conjuncture with
    arbitrary controllers to provide a safe hybrid control law. The marginal likelihood
    of the Gaussian process then provides a confidence metric used to determine switches
    between a least restrictive controller and a safety controller. We test both the
    prediction as well as the correction capabilities of the presented method in a
    classical pursuit-evasion example.
author:
- first_name: Nikolaus
  full_name: Vertovec, Nikolaus
  id: '93930'
  last_name: Vertovec
- first_name: Sina
  full_name: Ober-Blöbaum, Sina
  id: '16494'
  last_name: Ober-Blöbaum
- first_name: Kostas
  full_name: Margellos, Kostas
  last_name: Margellos
citation:
  ama: 'Vertovec N, Ober-Blöbaum S, Margellos K. Verification of safety critical control
    policies using kernel methods. In: ; 2022:1870-1875.'
  apa: Vertovec, N., Ober-Blöbaum, S., &#38; Margellos, K. (2022). <i>Verification
    of safety critical control policies using kernel methods</i>. 1870–1875.
  bibtex: '@inproceedings{Vertovec_Ober-Blöbaum_Margellos_2022, title={Verification
    of safety critical control policies using kernel methods}, author={Vertovec, Nikolaus
    and Ober-Blöbaum, Sina and Margellos, Kostas}, year={2022}, pages={1870–1875}
    }'
  chicago: Vertovec, Nikolaus, Sina Ober-Blöbaum, and Kostas Margellos. “Verification
    of Safety Critical Control Policies Using Kernel Methods,” 1870–75, 2022.
  ieee: N. Vertovec, S. Ober-Blöbaum, and K. Margellos, “Verification of safety critical
    control policies using kernel methods,” London, 2022, pp. 1870–1875.
  mla: Vertovec, Nikolaus, et al. <i>Verification of Safety Critical Control Policies
    Using Kernel Methods</i>. 2022, pp. 1870–75.
  short: 'N. Vertovec, S. Ober-Blöbaum, K. Margellos, in: 2022, pp. 1870–1875.'
conference:
  end_date: 2022-07-15
  location: London
  name: 2022 European Control Conference (ECC)
  start_date: 2022-07-12
date_created: 2022-03-31T11:14:13Z
date_updated: 2023-11-29T10:00:18Z
ddc:
- '510'
department:
- _id: '636'
has_accepted_license: '1'
language:
- iso: eng
page: 1870-1875
status: public
title: Verification of safety critical control policies using kernel methods
type: conference
user_id: '15694'
year: '2022'
...
---
_id: '21592'
abstract:
- lang: eng
  text: We propose a reachability approach for infinite and finite horizon multi-objective
    optimization problems for low-thrust spacecraft trajectory design. The main advantage
    of the proposed method is that the Pareto front can be efficiently constructed
    from the zero level set of the solution to a Hamilton-Jacobi-Bellman equation.
    We demonstrate the proposed method by applying it to a low-thrust spacecraft trajectory
    design problem. By deriving the analytic expression for the Hamiltonian and the
    optimal control policy, we are able to efficiently compute the backward reachable
    set and reconstruct the optimal trajectories. Furthermore, we show that any reconstructed
    trajectory will be guaranteed to be weakly Pareto optimal. The proposed method
    can be used as a benchmark for future research of applying reachability analysis
    to low-thrust spacecraft trajectory design.
author:
- first_name: Nikolaus
  full_name: Vertovec, Nikolaus
  id: '87056'
  last_name: Vertovec
- first_name: Sina
  full_name: Ober-Blöbaum, Sina
  id: '16494'
  last_name: Ober-Blöbaum
- first_name: Kostas
  full_name: Margellos, Kostas
  last_name: Margellos
citation:
  ama: 'Vertovec N, Ober-Blöbaum S, Margellos K. Multi-objective minimum time optimal
    control for low-thrust trajectory design. In: ; :1975-1980.'
  apa: Vertovec, N., Ober-Blöbaum, S., &#38; Margellos, K. (n.d.). <i>Multi-objective
    minimum time optimal control for low-thrust trajectory design</i>. 1975–1980.
  bibtex: '@inproceedings{Vertovec_Ober-Blöbaum_Margellos, title={Multi-objective
    minimum time optimal control for low-thrust trajectory design}, author={Vertovec,
    Nikolaus and Ober-Blöbaum, Sina and Margellos, Kostas}, pages={1975–1980} }'
  chicago: Vertovec, Nikolaus, Sina Ober-Blöbaum, and Kostas Margellos. “Multi-Objective
    Minimum Time Optimal Control for Low-Thrust Trajectory Design,” 1975–80, n.d.
  ieee: N. Vertovec, S. Ober-Blöbaum, and K. Margellos, “Multi-objective minimum time
    optimal control for low-thrust trajectory design,” Rotterdam, the Netherlands,
    pp. 1975–1980.
  mla: Vertovec, Nikolaus, et al. <i>Multi-Objective Minimum Time Optimal Control
    for Low-Thrust Trajectory Design</i>. pp. 1975–80.
  short: 'N. Vertovec, S. Ober-Blöbaum, K. Margellos, in: n.d., pp. 1975–1980.'
conference:
  end_date: 2021-07-02
  location: Rotterdam, the Netherlands
  name: 2021 European Control Conference (ECC)
  start_date: 2021-06-29
date_created: 2021-04-03T03:00:35Z
date_updated: 2023-11-29T10:26:49Z
department:
- _id: '636'
external_id:
  arxiv:
  - '2103.08813'
language:
- iso: eng
page: 1975-1980
publication_status: accepted
status: public
title: Multi-objective minimum time optimal control for low-thrust trajectory design
type: conference
user_id: '15694'
year: '2021'
...
---
_id: '10594'
abstract:
- lang: eng
  text: "Multiobjective optimization plays an increasingly important role in modern
    applications, where several criteria are often of equal importance. The task in
    multiobjective optimization and multiobjective optimal control is therefore to
    compute\r\nthe set of optimal compromises (the Pareto set) between the conflicting
    objectives.\r\n\r\nSince – in contrast to the solution of a single objective optimization
    problem – the\r\nPareto set generally consists of an infinite number of solutions,
    the computational\r\neffort can quickly become challenging. This is even more
    the case when many problems have to be solved, when the number of objectives is
    high, or when the objectives\r\nare costly to evaluate. Consequently, this thesis
    is devoted to the identification and\r\nexploitation of structure both in the
    Pareto set and the dynamics of the underlying\r\nmodel as well as to the development
    of efficient algorithms for solving problems with\r\nadditional parameters, with
    a high number of objectives or with PDE-constraints.\r\nThese three challenges
    are addressed in three respective parts.\r\n\r\nIn the first part, predictor-corrector
    methods are extended to entire Pareto sets.\r\nWhen certain smoothness assumptions
    are satisfied, then the set of parameter dependent Pareto sets possesses additional
    structure, i.e. it is a manifold. The tangent\r\nspace can be approximated numerically
    which yields a direction for the predictor\r\nstep. In the corrector step, the
    predicted set converges to the Pareto set at a new\r\nparameter value. The resulting
    algorithm is applied to an example from autonomous\r\ndriving.\r\n\r\nIn the second
    part, the hierarchical structure of Pareto sets is investigated. When\r\nconsidering
    a subset of the objectives, the resulting solution is a subset of the Pareto\r\nset
    of the original problem. Under additional smoothness assumptions, the respective
    subsets are located on the boundary of the Pareto set of the full problem. This\r\nway,
    the “skeleton” of a Pareto set can be computed and due to the exponential\r\nincrease
    in computing time with the number of objectives, the computations of\r\nthese
    subsets are significantly faster which is demonstrated using an example from\r\nindustrial
    laundries.\r\n\r\nIn the third part, PDE-constrained multiobjective optimal control
    problems are\r\naddressed by reduced order modeling methods. Reduced order models
    exploit the\r\nstructure in the system dynamics, for example by describing the
    dynamics of only the\r\nmost energetic modes. The model reduction introduces an
    error in both the function values and their gradients, which has to be taken into
    account in the development of\r\nalgorithms. Both scalarization and set-oriented
    approaches are coupled with reduced\r\norder modeling. Convergence results are
    presented and the numerical benefit is\r\ninvestigated. The algorithms are applied
    to semi-linear heat flow problems as well\r\nas to the Navier-Stokes equations.\r\n"
author:
- first_name: Sebastian
  full_name: Peitz, Sebastian
  id: '47427'
  last_name: Peitz
  orcid: https://orcid.org/0000-0002-3389-793X
citation:
  ama: Peitz S. <i>  Exploiting Structure in Multiobjective Optimization and Optimal
    Control</i>.; 2017. doi:<a href="https://doi.org/10.17619/UNIPB/1-176">10.17619/UNIPB/1-176</a>
  apa: Peitz, S. (2017). <i>  Exploiting structure in multiobjective optimization
    and optimal control</i>. <a href="https://doi.org/10.17619/UNIPB/1-176">https://doi.org/10.17619/UNIPB/1-176</a>
  bibtex: '@book{Peitz_2017, title={  Exploiting structure in multiobjective optimization
    and optimal control}, DOI={<a href="https://doi.org/10.17619/UNIPB/1-176">10.17619/UNIPB/1-176</a>},
    author={Peitz, Sebastian}, year={2017} }'
  chicago: Peitz, Sebastian. <i>  Exploiting Structure in Multiobjective Optimization
    and Optimal Control</i>, 2017. <a href="https://doi.org/10.17619/UNIPB/1-176">https://doi.org/10.17619/UNIPB/1-176</a>.
  ieee: S. Peitz, <i>  Exploiting structure in multiobjective optimization and optimal
    control</i>. 2017.
  mla: Peitz, Sebastian. <i>  Exploiting Structure in Multiobjective Optimization
    and Optimal Control</i>. 2017, doi:<a href="https://doi.org/10.17619/UNIPB/1-176">10.17619/UNIPB/1-176</a>.
  short: S. Peitz,   Exploiting Structure in Multiobjective Optimization and Optimal
    Control, 2017.
date_created: 2019-07-10T08:12:22Z
date_updated: 2022-01-06T06:50:46Z
ddc:
- '510'
department:
- _id: '101'
doi: 10.17619/UNIPB/1-176
file:
- access_level: closed
  content_type: application/pdf
  creator: speitz
  date_created: 2020-03-13T12:52:50Z
  date_updated: 2020-03-13T12:52:50Z
  file_id: '16298'
  file_name: Dissertation_Peitz.pdf
  file_size: 16636801
  relation: main_file
  success: 1
file_date_updated: 2020-03-13T12:52:50Z
has_accepted_license: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://d-nb.info/1139356542/34
oa: '1'
project:
- _id: '52'
  name: Computing Resources Provided by the Paderborn Center for Parallel Computing
publication_status: published
status: public
title: " \tExploiting structure in multiobjective optimization and optimal control"
type: dissertation
user_id: '47427'
year: '2017'
...