Value function and optimal trajectories for a maximum running cost control problem with state constraints. Application to an abort landing problem
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 305-335.

The aim of this article is to study the Hamilton Jacobi Bellman (HJB) approach for state-constrained control problems with maximum cost. In particular, we are interested in the characterization of the value functions of such problems and the analysis of the associated optimal trajectories, without assuming any controllability assumption. The rigorous theoretical results lead to several trajectory reconstruction procedures for which convergence results are also investigated. An application to a five-state aircraft abort landing problem is then considered, for which several numerical simulations are performed to analyse the relevance of the theoretical approach.

DOI : 10.1051/m2an/2017064
Classification : 49L20, 49M30, 65M06
Mots-clés : Hamilton-Jacobi approach, state constraints, maximum running cost, trajectory reconstruction, aircraft landing in windshear
Assellaou, Mohamed 1 ; Bokanowski, Olivier 1 ; Desilles, Anya 1 ; Zidani, Hasnaa 1

1 
@article{M2AN_2018__52_1_305_0,
     author = {Assellaou, Mohamed and Bokanowski, Olivier and Desilles, Anya and Zidani, Hasnaa},
     title = {Value function and optimal trajectories for a maximum running cost control problem with state constraints. {Application} to an abort landing problem},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {305--335},
     publisher = {EDP-Sciences},
     volume = {52},
     number = {1},
     year = {2018},
     doi = {10.1051/m2an/2017064},
     mrnumber = {3808162},
     zbl = {1397.49038},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2017064/}
}
TY  - JOUR
AU  - Assellaou, Mohamed
AU  - Bokanowski, Olivier
AU  - Desilles, Anya
AU  - Zidani, Hasnaa
TI  - Value function and optimal trajectories for a maximum running cost control problem with state constraints. Application to an abort landing problem
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2018
SP  - 305
EP  - 335
VL  - 52
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2017064/
DO  - 10.1051/m2an/2017064
LA  - en
ID  - M2AN_2018__52_1_305_0
ER  - 
%0 Journal Article
%A Assellaou, Mohamed
%A Bokanowski, Olivier
%A Desilles, Anya
%A Zidani, Hasnaa
%T Value function and optimal trajectories for a maximum running cost control problem with state constraints. Application to an abort landing problem
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2018
%P 305-335
%V 52
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2017064/
%R 10.1051/m2an/2017064
%G en
%F M2AN_2018__52_1_305_0
Assellaou, Mohamed; Bokanowski, Olivier; Desilles, Anya; Zidani, Hasnaa. Value function and optimal trajectories for a maximum running cost control problem with state constraints. Application to an abort landing problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 305-335. doi : 10.1051/m2an/2017064. http://archive.numdam.org/articles/10.1051/m2an/2017064/

[1] A. Altarovici, O. Bokanowski and H. Zidani, A general Hamilton-Jacobi framework for non-linear state-constrained control problems. ESAIM: COCV 19 (2013) 337–357. | Numdam | MR | Zbl

[2] M. Assellaou, O. Bokanowski, A. Dsilles and H. Zidani, A Hamilton-Jacobi-Bellman approach for the optimal control of an abort landing problem, in IEEE 55th Conference on Decision and Control (CDC) (2016) 3630–3635.

[3] J.-P. Aubin and A. Cellina, Differential inclusions, in Set-Valued Maps and Viability Theory. Vol. 264 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo (1984). | DOI | MR | Zbl

[4] J.P. Aubin and H. Frankowska, The viability kernel algorithm for computing value functions of infinite horizon optimal control problems. J. Math. Anal. Appl. 201 (1996) 555–576. | DOI | MR | Zbl

[5] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Systems and Control: Foundations and Applications. Birkhäuser, Boston (1997). | MR | Zbl

[6] E.N. Barron, Viscosity solutions and analysis in L∞, in Nonlinear Analysis, Differential Equations and Control, Vol. 528 of Serie C: Mathematical and Physical Sciences. Springer Science, Business Media, Dordrecht (1999) 1–60. | MR | Zbl

[7] E.N. Barron and H. Ishii, The Bellman equation for minimizing the maximum cost. Nonlinear Anal.: Theory Methods Appl. 13 (1989) 1067–1090. | DOI | MR | Zbl

[8] O. Bokanowski, N. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption. SIAM J. Control Optim. 48 (2010) 4292–4316. | DOI | MR | Zbl

[9] O. Bokanowski, N. Forcadel and H. Zidani, Deterministic state-constrained optimal control problems without controllability assumptions. ESAIM: COCV 17 (2011) 995–1015. | Numdam | MR | Zbl

[10] N.D. Botkin and V.L. Turova, Dynamic programming approach to aircraft control in a windshear, in Advances in Dynamic Games. Vol. 13 of Ann. Int. Soc. Dyn. Games. Birkhäuser, Springer, Cham (2013) 53–69. | DOI | MR | Zbl

[11] R. Bulirsch, F. Montrone and H.J. Pesch, Abort landing in the presence of windshear as a minimax optimal control problem. I. Necessary conditions. J. Optim. Theory Appl. 70 (1991) 1–23. | DOI | MR | Zbl

[12] R. Bulirsch, F. Montrone and H.J. Pesch, Abort landing in the presence of windshear as a minimax optimal control problem. II. Multiple shooting and homotopy. J. Optim. Theory Appl. 70 (1991) 223–254. | DOI | MR | Zbl

[13] P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre, Optimal times for constrained nonlinear control problems without local controllability. Appl. Math. Optim. 36 (1997) 21–42. | DOI | MR | Zbl

[14] P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre, Numerical schemes for discontinuous value functions of optimal control. Set-Valued Anal. 8 (2000) 111–126. | DOI | MR | Zbl

[15] M.G. Crandall and P.L. Lions, Two approximations of solutions of Hamilton-Jacobi equations. Math. Comput. 43 (1984) 1–19. | DOI | MR | Zbl

[16] M. Falcone, Numerical solution of dynamic programming equations, in Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997).

[17] M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations. SIAM (2013). | DOI | MR

[18] H. Frankowska and R.B. Vinter, Existence of neighboring feasible trajectories: applications to dynamic programming for state-constrained optimal control problems. J. Optim. Theory Appl. 104 (2000) 20–40. | DOI | MR | Zbl

[19] C. Hermosilla and H. Zidani, Infinite horizon problems on stratifiable state-constraints sets. J. Differ. Equ. 258 (2015) 1420–1460. | DOI | MR | Zbl

[20] H. Ishii and S. Koike, A new formulation of state constraint problems for first-order pdes. SIAM J. Control Optim. 34 (1996) 554–571. | DOI | MR | Zbl

[21] A. Miele, T. Wang and W. Melvin, Quasi-steady flight to quasi-steady flight transition for abort landing in a windshear: trajectory optimization and guidance. J. Optim. Theory Appl. 58 (1988) 165–207. | DOI | MR | Zbl

[22] A. Miele,T. Wang, C.Y. Tzeng and W. Melvin, Optimal abort landing trajectories in the presence of windshear. J. Optim. Theory Appl. 55 (1987) 165–202. | DOI | Zbl

[23] S. Osher and C.-W. Shu, High essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28 (1991) 907–922. | DOI | MR | Zbl

[24] M. Quincampoix and O.S. Serea, A viability approach for optimal control with infimum cost. Ann. Univ. Al. I. Cuza Iasi 48 (2002) 113–132. | MR | Zbl

[25] J.D.L. Rowland and R.B. Vinter, Construction of optimal feedback controls. Syst. Control Lett. 16 (1991) 357–367. | DOI | MR | Zbl

[26] H. Soner, Optimal control with state-space constraint I. SIAM J. Control Optim. 24 (1986) 552–561. | DOI | MR | Zbl

Cité par Sources :