On the construction of nearly time optimal continuous feedback laws around switching manifolds
ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 4.

In this paper, we address the question of the construction of a nearly time optimal feedback law for a minimum time optimal control problem, which is robust with respect to internal and external perturbations. For this purpose we take as starting point an optimal synthesis, which is a suitable collection of optimal trajectories. The construction we exhibit depends exclusively on the initial data obtained from the optimal feedback which is assumed to be known.

Reçu le :
Accepté le :
Première publication :
Publié le :
DOI : 10.1051/cocv/2019002
Classification : 49J15, 49J30, 93B52
Mots-clés : Feedback controls, nearly time optimal control, minimum time problems 
@article{COCV_2020__26_1_A4_0,
     author = {Ancona, Fabio and Hermosilla, Cristopher},
     title = {On the construction of nearly time optimal continuous feedback laws around switching manifolds},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {26},
     year = {2020},
     doi = {10.1051/cocv/2019002},
     mrnumber = {4055456},
     zbl = {1440.49041},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2019002/}
}
TY  - JOUR
AU  - Ancona, Fabio
AU  - Hermosilla, Cristopher
TI  - On the construction of nearly time optimal continuous feedback laws around switching manifolds
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2020
VL  - 26
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2019002/
DO  - 10.1051/cocv/2019002
LA  - en
ID  - COCV_2020__26_1_A4_0
ER  - 
%0 Journal Article
%A Ancona, Fabio
%A Hermosilla, Cristopher
%T On the construction of nearly time optimal continuous feedback laws around switching manifolds
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2020
%V 26
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2019002/
%R 10.1051/cocv/2019002
%G en
%F COCV_2020__26_1_A4_0
Ancona, Fabio; Hermosilla, Cristopher. On the construction of nearly time optimal continuous feedback laws around switching manifolds. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 4. doi : 10.1051/cocv/2019002. http://archive.numdam.org/articles/10.1051/cocv/2019002/

[1] F. Ancona and A. Bressan, Patchy vector fields and asymptotic stabilization. ESAIM: COCV 4 (1999) 445–471. | Numdam | MR | Zbl

[2] F. Ancona and A. Bressan, Flow stability of patchy vector fields and robust feedback stabilization. SIAM J. Cont. Optim. 41 (2002) 1455–1476. | DOI | MR | Zbl

[3] F. Ancona and A. Bressan, Nearly time optimal stabilizing patchy feedbacks. Ann. Inst. Henri Poincaré (C) Nonlinear Anal. 24 (2007) 279–310. | DOI | Numdam | MR | Zbl

[4] D. Anisi, J. Hamberg and X. Xiaoming, Nearly time optimal paths for a ground vehicle. J. Cont. Theory Appl. 1 (2003) 2–8. | DOI | MR

[5] J.-P. Aubin and A. Cellina, Differential inclusions: Set-valued maps and viability theory, in Vol. 264 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, New York, Inc. (1984). | DOI | MR | Zbl

[6] L. Berkovitz, Optimal feedback controls. SIAM J. Cont. Optim. 27 (1989) 991–1006. | DOI | MR | Zbl

[7] V. Boltyanskii, Sufficient conditions for optimality and the justification of the dynamic programming principle. SIAM J. Cont. Optim. 4 (1966) 326–361. | DOI | MR | Zbl

[8] U. Boscain and B. Piccoli, Optimal syntheses for control systems on 2-D manifolds, in Vol. 43 of Mathématiques & Applications. Springer (2004). | MR | Zbl

[9] R. Brockett, Asymptotic stability and feedback stabilization, in Differential Geometric Control Theory. Birkhäuser Boston (1983) 181–191. | MR | Zbl

[10] P. Brunovskỳ, Every normal linear system has a regular time optimal synthesis. Math. Slov. 28 (1978) 81–100. | MR | Zbl

[11] P. Brunovskỳ, Existence of regular synthesis for general control problems. J. Differ. Equ. 38 (1980) 317–343. | DOI | MR | Zbl

[12] I. Capuzzo-Dolcetta and P.-L. Lions, Hamilton-Jacobi equations with state constraints. Trans. Am. Math. Soc. 318 (1990) 643–683. | 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] F. Clarke, Discontinuous feedback and nonlinear systems, in 8th IFAC Symposium on Nonlinear Control Systems (2010).

[15] F. Clarke, Functional analysis, calculus of variations and optimal control, in Vol. 264 of Graduate Text in Mathematics. Springer (2013). | MR | Zbl

[16] F.H. Clarke, L. Rifford and R. Stern, Feedback in state constrained optimal control. ESAIM: COCV 7 (2002) 97–133. | Numdam | MR | Zbl

[17] T. Faulwasser, M. Korda, C. Jones and D. Bonvin, On turnpike and dissipativity properties of continuous-time optimal control problems. Automatica 81 (2017) 297–304. | DOI | MR | Zbl

[18] O. Hájek, Terminal manifolds and switching locus. Math. Syst. Theory 6 (1972) 289–301. | DOI | MR | Zbl

[19] C. Hermosilla, Stratified discontinuous differential equations and sufficient conditions for robustness. Discr. Continu. Dyn. Syst. A 35 (2015) 4415–4437. | DOI | MR | Zbl

[20] C. Hermosilla and H. Zidani, Infinite horizon problems on stratifiable state constraints sets. J. Differ. Equ. 258 (2015) 1430–1460. | DOI | MR | Zbl

[21] C. Hermosilla, P. Wolenski and H. Zidani, The Mayer and Minimum time problems with stratified state constraints. Set-Valued Variat. Anal. 26 (2018) 643–662. | DOI | MR | Zbl

[22] H. Ishii and S. Koike, On ε-optimal controls for state constraint problems. Ann. Inst. Henri Poincaré (C) Non Linear Anal. 17 (2000) 473–502. | DOI | Numdam | MR | Zbl

[23] J.M. Lee, Introduction to smooth manifolds, in Vol. 218 of Graduate Text in Mathematics. Springer (2013). | DOI | MR | Zbl

[24] K. Mall and M.J. Grant, Epsilon-trig regularization method for bang-bang optimal control problems. J. Optim. Theory Appl. 174 (2017) 500–517. | DOI | MR | Zbl

[25] A. Marigo and B. Piccoli, Regular syntheses and solutions to discontinuous ODEs. ESAIM: COCV 7 (2002) 291–307. | Numdam | MR | Zbl

[26] A. Marigo and B. Piccoli, Safety controls and applications to the Dubins’ car. Nonlin. Differ. Equ. Appl. 11 (2004) 73–94. | DOI | MR | Zbl

[27] L.D. Meeker, Local time optimal feedback control of strictly normal two-input linear systems. SIAM J. Cont. Optim. 27 (1989) 53–82. | DOI | MR | Zbl

[28] B. Piccoli, Optimal syntheses for state constrained problems with application to optimization of Cancer therapies. Math. Control Related Fields 2 (2012) 383–398. | DOI | MR | Zbl

[29] B. Piccoli and H.J. Sussmann, Regular synthesis and sufficiency conditions for optimality. SIAM J. Cont. Optim. 39 (2000) 359–410. | DOI | MR | Zbl

[30] F.S. Priuli, State constrained patchy feedback stabilization. Math. Control Related Fields 5 (2015) 141–163. | DOI | MR | Zbl

[31] L. Rifford, Semiconcave control-Lyapunov functions and stabilizing feedbacks. SIAM J. Cont. Optim. 41 (2002) 659–681. | DOI | MR | Zbl

[32] L. Rifford, Stratified semiconcave control-Lyapunov functions and the stabilization problem. Ann. Inst. Henri Poincaré (C) Non Linear Anal. 22 (2005) 343–384. | DOI | Numdam | MR | Zbl

[33] L. Rifford, On the existence of local smooth repulsive stabilizing feedbacks in dimension three. J. Differ. Equ. 226 (2006) 429–500. | DOI | MR | Zbl

[34] R. Rockafellar, Convex analysis, in Vol. 28 of Princeton Mathematical Series. Princeton University Press (1970). | MR | Zbl

[35] J. Rowland and R. Vinter, Construction of optimal feedback controls. Syst. Cont. Lett. 16 (1991) 357–367. | DOI | MR | Zbl

[36] C. Silva and E. Trélat, Smooth regularization of bang-bang optimal control problems. IEEE Trans. Autom. Cont. 55 (2010) 2488–2499. | DOI | MR | Zbl

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

[38] P. Spinelli and G.S. Rakotonirainy, Minimum Time Problem Synthesis. Syst. Cont. Lett. 10 (1988) 281–290. | DOI | MR | Zbl

[39] H. Sussmann, Regular synthesis for time optimal control of single-input real analytic systems in the plane. SIAM J. Cont. Optim. 25 (1987) 1145–1162. | DOI | MR | Zbl

Cité par Sources :

This work was supported by the European Union under the 7th Framework Programme FP7-PEOPLE-2010-ITN Grant agreement number 264735-SADCO.

C. Hermosilla was supported by CONICYT-Chile through FONDECYT grant number 3170485 and Proyecto REDES ETAPA INICIAL, Convocatoria 2017 REDI170200.