Home Mathematics 9. Non-degenerate forms of the generalized Euler–Lagrange condition for state-constrained optimal control problems
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9. Non-degenerate forms of the generalized Euler–Lagrange condition for state-constrained optimal control problems

  • Piernicola Bettiol and Nathalie Khalil
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Variational Methods
This chapter is in the book Variational Methods

Abstract

We establish nondegeneracy of the generalized Euler-Lagrange conditions for state-constrained optimal control problems, in which the dynamic is represented in terms of a differential inclusion depending on both time and state variables (allowing the casewhere the velocity set is justmeasurablew.r.t. the time variable), the state and the end-point constraints are closed sets.We propose here a new constraint qualification involving just tangent vectors to the state constraint (no inward pointing condition of the velocity set is required) and distance properties of trajectories to the end point constraints; this condition can be applied also when the minimizer has left end-point in a region where the state constraint set is nonsmooth. We finally provide an illustrative example.

Abstract

We establish nondegeneracy of the generalized Euler-Lagrange conditions for state-constrained optimal control problems, in which the dynamic is represented in terms of a differential inclusion depending on both time and state variables (allowing the casewhere the velocity set is justmeasurablew.r.t. the time variable), the state and the end-point constraints are closed sets.We propose here a new constraint qualification involving just tangent vectors to the state constraint (no inward pointing condition of the velocity set is required) and distance properties of trajectories to the end point constraints; this condition can be applied also when the minimizer has left end-point in a region where the state constraint set is nonsmooth. We finally provide an illustrative example.

Chapters in this book

  1. Frontmatter I
  2. Contents V
  3. Part I
  4. 1. Second-order decomposition model for image processing: numerical experimentation 5
  5. 2. Optimizing spatial and tonal data for PDE-based inpainting 35
  6. 3. Image registration using phase–amplitude separation 84
  7. 4. Rotation invariance in exemplar-based image inpainting 108
  8. 5. Convective regularization for optical flow 184
  9. 6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients 202
  10. 7. On optical flow models for variational motion estimation 225
  11. 8. Bilevel approaches for learning of variational imaging models 252
  12. Part II
  13. 9. Non-degenerate forms of the generalized Euler–Lagrange condition for state-constrained optimal control problems 295
  14. 10 The Purcell three-link swimmer: some geometric and numerical aspects related to periodic optimal controls 314
  15. 11. Controllability of Keplerian motion with low-thrust control systems 344
  16. 12. Higher variational equation techniques for the integrability of homogeneous potentials 365
  17. 13. Introduction to KAM theory with a view to celestial mechanics 387
  18. 14. Invariants of contact sub-pseudo-Riemannian structures and Einstein–Weyl geometry 434
  19. 15. Time-optimal control for a perturbed Brockett integrator 454
  20. 16. Twist maps and Arnold diffusion for diffeomorphisms 473
  21. 17. A Hamiltonian approach to sufficiency in optimal control with minimal regularity conditions: Part I 496
  22. Index 517
https://doi.org/10.1515/9783110430394-009

Chapters in this book

  1. Frontmatter I
  2. Contents V
  3. Part I
  4. 1. Second-order decomposition model for image processing: numerical experimentation 5
  5. 2. Optimizing spatial and tonal data for PDE-based inpainting 35
  6. 3. Image registration using phase–amplitude separation 84
  7. 4. Rotation invariance in exemplar-based image inpainting 108
  8. 5. Convective regularization for optical flow 184
  9. 6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients 202
  10. 7. On optical flow models for variational motion estimation 225
  11. 8. Bilevel approaches for learning of variational imaging models 252
  12. Part II
  13. 9. Non-degenerate forms of the generalized Euler–Lagrange condition for state-constrained optimal control problems 295
  14. 10 The Purcell three-link swimmer: some geometric and numerical aspects related to periodic optimal controls 314
  15. 11. Controllability of Keplerian motion with low-thrust control systems 344
  16. 12. Higher variational equation techniques for the integrability of homogeneous potentials 365
  17. 13. Introduction to KAM theory with a view to celestial mechanics 387
  18. 14. Invariants of contact sub-pseudo-Riemannian structures and Einstein–Weyl geometry 434
  19. 15. Time-optimal control for a perturbed Brockett integrator 454
  20. 16. Twist maps and Arnold diffusion for diffeomorphisms 473
  21. 17. A Hamiltonian approach to sufficiency in optimal control with minimal regularity conditions: Part I 496
  22. Index 517
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