Geodesic
Some estimates for the jump of the derivative of the Lagrange multiplier function in optimal control problems with second-order state constraints
The Bulletin of Irkutsk State University. Series Mathematics, Tome 49 (2024), pp. 3-15 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The optimal control problem for a nonlinear dynamic system of a cascade type with endpoint and irregular pointwise state constraints (the so-called state constraints of depth 2) is studied. This problem admits a refined formulation of Pontryagin’s maximum principle in terms of a (non-standard) Hamilton-Pontryagin function of the second order. The question of estimating the jump of the derivative of the Lagrange multiplier corresponding to the state constraint is studied. Some sufficient conditions have been obtained under which the maximum principle implies uniform in time estimates for the jump of the specified function. In particular, sufficient conditions have been given for the absence of a jump (i.e., continuous differentiability) of the multiplier. These results are based on the concepts of 2-regularity of the state constraint and the so-called regularity zone of the problem. The obtained estimates are of interest for the theory of Pontryagin’s maximum principle and can be used in practice, including the implementation of the known shooting method within the framework of one of the standard approaches to the numerical interpretation of the necessary optimality condition.
Keywords: optimal control, state constraints, Pontryagin's maximum principle, regularity assumptions, numerical methods. 
@article{IIGUM_2024_49_a0,
     author = {D. Yu. Karamzin},
     title = {Some estimates for the jump of the derivative of the {Lagrange} multiplier function in optimal control problems with second-order state constraints},
     journal = {The Bulletin of Irkutsk State University. Series Mathematics},
     pages = {3--15},
     year = {2024},
     volume = {49},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IIGUM_2024_49_a0/}
}
TY  - JOUR
AU  - D. Yu. Karamzin
TI  - Some estimates for the jump of the derivative of the Lagrange multiplier function in optimal control problems with second-order state constraints
JO  - The Bulletin of Irkutsk State University. Series Mathematics
PY  - 2024
SP  - 3
EP  - 15
VL  - 49
UR  - http://geodesic.mathdoc.fr/item/IIGUM_2024_49_a0/
LA  - ru
ID  - IIGUM_2024_49_a0
ER  - 
%0 Journal Article
%A D. Yu. Karamzin
%T Some estimates for the jump of the derivative of the Lagrange multiplier function in optimal control problems with second-order state constraints
%J The Bulletin of Irkutsk State University. Series Mathematics
%D 2024
%P 3-15
%V 49
%U http://geodesic.mathdoc.fr/item/IIGUM_2024_49_a0/
%G ru
%F IIGUM_2024_49_a0
D. Yu. Karamzin. Some estimates for the jump of the derivative of the Lagrange multiplier function in optimal control problems with second-order state constraints. The Bulletin of Irkutsk State University. Series Mathematics, Tome 49 (2024), pp. 3-15. http://geodesic.mathdoc.fr/item/IIGUM_2024_49_a0/

[1] Arutyunov A., Optimality Conditions: Abnormal and Degenerate Problems, Springer, Dordrecht, 2000 | MR

[2] Arutyunov A. V., Karamzin D. Yu., “A Survey on Regularity Conditions for State-Constrained Optimal Control Problems and the Non-degenerate Maximum Principle”, Journal of Optimization Theory and Applications, 184:3 (2020), 697–723 | DOI | MR | Zbl

[3] Arutyunov A. V., Karamzin D. Yu., Pereira F. L., “Investigation of Controllability and Regularity Conditions for State Constrained Problems”, IFAC-PapersOnLine, 50:1 (2017), 6295–6302 | DOI

[4] R. Chertovskih, D. Karamzin, N. T. Khalil, F. L. Pereira, “An indirect numerical method for a time-optimal state-constrained control problem in a steady two-dimensional fluid flow”, 2018 IEEE/OES Autonomous Underwater Vehicle Workshop, 2018, 1–6

[5] An Indirect Method for Regular State-Constrained Optimal Control Problems in Flow Fields / R. Chertovskih, D. Karamzin, N. T. Khalil, F. L. Pereira, IEEE Transactions on Automatic Control, 66:2 (2021), 787–793 | DOI | MR

[6] Karamzin D. Yu., Pereira F. L., “On Higher-Order State Constraints”, SIAM Journal on Control and Optimization, 61:4 (2023), 1913–1933 | DOI | MR | Zbl

[7] Mordukhovich B. Sh., “Maximum principle in the problem of time optimal response with nonsmooth constraints”, Journal of Applied Mathematics and Mechanics, 40:6 (1976), 960–969 | DOI | MR | Zbl

[8] Neustadt L. W., “An Abstract Variational Theory with Applications to a Broad Class of Optimization Problems. II. Applications”, SIAM Journal on Control, 5:1 (1967), 90–137 | DOI | MR | Zbl

[9] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience, New York, 1962 | MR | Zbl

[10] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, New York, 2007 | MR | Zbl