Feedback in state constrained optimal control

Clarke, Francis H.; Rifford, Ludovic; Stern, R. J.

doi:10.1051/cocv:2002005

Clarke, Francis H. ; Rifford, Ludovic ; Stern, R. J.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 97-133.

Résumé

An optimal control problem is studied, in which the state is required to remain in a compact set $S$ . A control feedback law is constructed which, for given $ε > 0$ , produces $ε$ -optimal trajectories that satisfy the state constraint universally with respect to all initial conditions in $S$ . The construction relies upon a constraint removal technique which utilizes geometric properties of inner approximations of $S$ and a related trajectory tracking result. The control feedback is shown to possess a robustness property with respect to state measurement error.

MR Zbl | 1 citation dans Numdam

DOI : 10.1051/cocv:2002005

Classification : 49J24, 49J52, 49N55, 90D25
Mots clés : optimal control, state constraint, near-optimal control feedback, nonsmooth analysis

@article{COCV_2002__7__97_0,
     author = {Clarke, Francis H. and Rifford, Ludovic and Stern, R. J.},
     title = {Feedback in state constrained optimal control},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {97--133},
     publisher = {EDP-Sciences},
     volume = {7},
     year = {2002},
     doi = {10.1051/cocv:2002005},
     mrnumber = {1925023},
     zbl = {1033.49004},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2002005/}
}

TY  - JOUR
AU  - Clarke, Francis H.
AU  - Rifford, Ludovic
AU  - Stern, R. J.
TI  - Feedback in state constrained optimal control
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2002
SP  - 97
EP  - 133
VL  - 7
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2002005/
DO  - 10.1051/cocv:2002005
LA  - en
ID  - COCV_2002__7__97_0
ER  -

%0 Journal Article
%A Clarke, Francis H.
%A Rifford, Ludovic
%A Stern, R. J.
%T Feedback in state constrained optimal control
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2002
%P 97-133
%V 7
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2002005/
%R 10.1051/cocv:2002005
%G en
%F COCV_2002__7__97_0

Clarke, Francis H.; Rifford, Ludovic; Stern, R. J. Feedback in state constrained optimal control. ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 97-133. doi : 10.1051/cocv:2002005. http://www.numdam.org/articles/10.1051/cocv:2002005/

Bibliographie
Cité par

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

[2] N.N. Barabanova and A.I. Subbotin, On continuous evasion strategies in game theoretic problems on the encounter of motions. Prikl. Mat. Mekh. 34 (1970) 796-803. | MR | Zbl

[3] N.N. Barabanova and A.I. Subbotin, On classes of strategies in differential games of evasion. Prikl. Mat. Mekh. 35 (1971) 385-392. | MR | Zbl

[4] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997). | Zbl

[5] L.D. Berkovitz, Optimal feedback controls. SIAM J. Control Optim. 27 (1989) 991-1006. | MR | Zbl

[6] P. Cannarsa and H. Frankowska, Some characterizations of optimal trajectories in control theory. SIAM J. Control Optim. 29 (1991) 1322-1347. | MR | Zbl

[7] I. Capuzzo-Dolcetta and P.L. Lions, Hamilton-Jacobi equations with state constraints. Trans. Amer. Math. Soc. 318 (1990) 643-683. | Zbl

[8] F.H. Clarke, Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983). Republished as Vol. 5 of Classics in Appl. Math. SIAM, Philadelphia (1990). | MR | Zbl

[9] F.H. Clarke, Methods of Dynamic and Nonsmooth Optimization, Vol. 57 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1989). | MR | Zbl

[10] F.H. Clarke, Yu.S. Ledyaev, L. Rifford and R.J. Stern, Feedback stabilization and Lyapunov functions. SIAM J. Control Optim. 39 (2000) 25-48. | MR | Zbl

[11] F.H. Clarke, Yu.S. Ledyaev, E.D. Sontag and A.I. Subbotin, Asymptotic controllability implies control feedback stabilization. IEEE Trans. Automat. Control 42 (1997) 1394. | MR | Zbl

[12] F.H. Clarke, Yu.S. Ledyaev and R.J. Stern, Proximal analysis and control feedback construction. Proc. Steklov Inst. Math. 226 (2000) 1-20. | MR

[13] F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Qualitative properties of trajectories of control systems: A survey. J. Dynam. Control Systems 1 (1995) 1-48. | MR | Zbl

[14] F.H. Clarke, Yu.S. Ledyaev and A.I. Subbotin, Universal feedback strategies for differential games of pursuit. SIAM J. Control Optim. 35 (1997) 552-561. | MR | Zbl

[15] F.H. Clarke, Yu.S. Ledyaev and A.I. Subbotin, Universal positional control. Proc. Steklov Inst. Math. 224 (1999) 165-186. Preliminary version: Preprint CRM-2386. Univ. de Montréal (1994). | MR | Zbl

[16] F.H. Clarke, Yu.S. Ledyaev and R.J. Stern, Complements, approximations, smoothings and invariance properties. J. Convex Anal. 4 (1997) 189-219. | MR | Zbl

[17] F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory. Springer-Verlag, New York, Grad. Texts in Math. 178 (1998). | MR | Zbl

[18] F.H. Clarke, R.J. Stern and P.R. Wolenski, Proximal smoothness and the lower- $C^{2}$ property. J. Convex Anal. 2 (1995) 117-145. | MR | Zbl

[19] F. Forcellini and F. Rampazzo, On nonconvex differential inclusions whose state is constrained in the closure of an open set. Applications to dynamic programming. Differential and Integral Equations 12 (1999) 471-497. | MR | Zbl

[20] H. Frankowska and F. Rampazzo, Filippov's and Filippov-Wazewski's theorems on closed domains. J. Differential Equations 161 (2000) 449-478. | Zbl

[21] G.G. Garnysheva and A.I. Subbotin, Suboptimal universal strategies in a game-theoretic time-optimality problem. Prikl. Mat. Mekh. 59 (1995) 707-713. | MR | Zbl

[22] J.-B. Hiriart-Urruty, New concepts in nondifferentiable programming. Bull. Soc. Math. France 60 (1979) 57-85. | Numdam | Zbl

[23] H. Ishii and S. Koike, On $ε$ -optimal controls for state constraint problems. Ann. Inst. H. Poincaré Anal. Linéaire 17 (2000) 473-502. | Numdam | MR | Zbl

[24] N.N. Krasovskiĭ, Differential games. Approximate and formal models. Mat. Sb. (N.S.) 107 (1978) 541-571. | MR | Zbl

[25] N.N. Krasovskiĭ, Extremal aiming and extremal displacement in a game-theoretical control. Problems Control Inform. Theory 13 (1984) 287-302. | MR | Zbl

[26] N.N. Krasovskiĭ, Control of dynamical systems. Nauka, Moscow (1985).

[27] N.N. Krasovskiĭ and A.I. Subbotin, Positional Differential Games. Nauka, Moscow (1974). French translation: Jeux Différentielles. Mir, Moscou (1979). | MR | Zbl

[28] N.N. Krasovskiĭ and A.I. Subbotin, Game-Theoretical Control Problems. Springer-Verlag, New York (1988). | MR | Zbl

[29] P. Loewen, Optimal Control via Nonsmooth Analysis. CRM Proc. Lecture Notes Amer. Math. Soc. 2 (1993). | MR | Zbl

[30] S. Nobakhtian and R.J. Stern, Universal near-optimal control feedbacks. J. Optim. Theory Appl. 107 (2000) 89-123. | MR | Zbl

[31] L. Rifford, Problèmes de Stabilisation en Théorie du Contrôle, Doctoral Thesis. Univ. Claude Bernard Lyon 1 (2000).

[32] L. Rifford, Stabilisation des systèmes globalement asymptotiquement commandables. C. R. Acad. Sci. Paris 330 (2000) 211-216. | MR | Zbl

[33] L. Rifford, Existence of Lipschitz and semiconcave control-Lyapunov functions. SIAM J. Control Optim. (to appear). | MR | Zbl

[34] R.T. Rockafellar, Clarke’s tangent cones and boundaries of closed sets in $ℝ^{n}$ . Nonlinear Anal. 3 (1979) 145-154. | Zbl

[35] R.T. Rockafellar, Favorable classes of Lipschitz continuous functions in subgradient optimization, in Nondifferentiable Optimization, edited by E. Nurminski. Permagon Press, New York (1982). | MR

[36] J.D.L. Rowland and R.B. Vinter, Construction of optimal control feedback controls. Systems Control Lett. 16 (1991) 357-357. | MR | Zbl

[37] M. Soner, Optimal control problems with state-space constraints I. SIAM J. Control Optim. 24 (1986) 551-561. | Zbl

[38] E.D. Sontag, Mathematical Control Theory, 2nd Ed.. Springer-Verlag, New York, Texts in Appl. Math. 6 (1998). | MR | Zbl

[39] E.D. Sontag, Clock and insensitivity to small measurement errors. ESAIM: COCV 4 (1999) 537-557. | Numdam | MR | Zbl

[40] A.I. Subbotin, Generalized Solutions of First Order PDE's. Birkhäuser, Boston (1995). | Zbl

[41] N.N. Subbotina, Universal optimal strategies in positional differential games. Differential Equations 19 (1983) 1377-1382. | MR | Zbl

[42] N.N. Subbotina, The maximum principle and the superdifferential of the value function. Problems Control Inform. Theory 18 (1989) 151-160. | MR | Zbl

[43] N.N. Subbotina, On structure of optimal feedbacks to control problems, Preprints of the eleventh IFAC International Workshop, Control Applications of Optimization, edited by V. Zakharov (2000).

[44] R.B. Vinter, Optimal Control. Birkhäuser, Boston (2000). | MR | Zbl

Cité par Sources :