Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T03:44:41.491Z Has data issue: false hasContentIssue false
  • English
  • Français

Feedback in state constrained optimal control

Published online by Cambridge University Press:  15 September 2002

Francis H. Clarke ,
Ludovic Rifford  and
R. J. Stern
Francis H. Clarke
Affiliation:
Institut Desargues, bâtiment 101, Université Claude Bernard Lyon I, 69622 Villeurbanne, France; clarke@desargues.univ-lyon1.fr.
R. J. Stern
Affiliation:
Department of Mathematics and Statistics, Concordia University, Montréal, Quebec H4B 1R6, Canada.
Get access

Abstract

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.

Keywords

Optimal controlstate constraintnear-optimal control feedbacknonsmooth analysis.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 7 , 2002 , pp. 97 - 133
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ancona, F. and Bressan, A., Patchy vector fields and asymptotic stabilization. ESAIM: COCV 4 (1999) 445-471. CrossRef
Barabanova, N.N. and Subbotin, A.I., On continuous evasion strategies in game theoretic problems on the encounter of motions. Prikl. Mat. Mekh. 34 (1970) 796-803.
Barabanova, N.N. and Subbotin, A.I., On classes of strategies in differential games of evasion. Prikl. Mat. Mekh. 35 (1971) 385-392.
M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997).
Berkovitz, L.D., Optimal feedback controls. SIAM J. Control Optim. 27 (1989) 991-1006. CrossRef
Cannarsa, P. and Frankowska, H., Some characterizations of optimal trajectories in control theory. SIAM J. Control Optim. 29 (1991) 1322-1347. CrossRef
Capuzzo-Dolcetta, I. and Lions, P.L., Hamilton-Jacobi equations with state constraints. Trans. Amer. Math. Soc. 318 (1990) 643-683. CrossRef
F.H. Clarke, Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983). Republished as Vol. 5 of Classics in Appl. Math. SIAM, Philadelphia (1990).
F.H. Clarke, Methods of Dynamic and Nonsmooth Optimization, Vol. 57 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1989).
Clarke, F.H., Ledyaev, Yu.S., Rifford, L. and Stern, R.J., Feedback stabilization and Lyapunov functions. SIAM J. Control Optim. 39 (2000) 25-48. CrossRef
Clarke, F.H., Ledyaev, Yu.S., Sontag, E.D. and Subbotin, A.I., Asymptotic controllability implies control feedback stabilization. IEEE Trans. Automat. Control 42 (1997) 1394. CrossRef
Clarke, F.H., Ledyaev, Yu.S. and Stern, R.J., Proximal analysis and control feedback construction. Proc. Steklov Inst. Math. 226 (2000) 1-20.
Clarke, F.H., Ledyaev, Yu.S., Stern, R.J. and Wolenski, P.R., Qualitative properties of trajectories of control systems: A survey. J. Dynam. Control Systems 1 (1995) 1-48. CrossRef
Clarke, F.H., Ledyaev, Yu.S. and Subbotin, A.I., Universal feedback strategies for differential games of pursuit. SIAM J. Control Optim. 35 (1997) 552-561. CrossRef
Clarke, F.H., Ledyaev, Yu.S. and Subbotin, A.I., Universal positional control. Proc. Steklov Inst. Math. 224 (1999) 165-186. Preliminary version: Preprint CRM-2386. Univ. de Montréal (1994).
Clarke, F.H., Ledyaev, Yu.S. and Stern, R.J., Complements, approximations, smoothings and invariance properties. J. Convex Anal. 4 (1997) 189-219.
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).
Clarke, F.H., Stern, R.J. and Wolenski, P.R., Proximal smoothness and the lower-C 2 property. J. Convex Anal. 2 (1995) 117-145.
Forcellini, F. and Rampazzo, F., 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.
Frankowska, H. and Rampazzo, F., Filippov's and Filippov-Wazewski's theorems on closed domains. J. Differential Equations 161 (2000) 449-478. CrossRef
Garnysheva, G.G. and Subbotin, A.I., Suboptimal universal strategies in a game-theoretic time-optimality problem. Prikl. Mat. Mekh. 59 (1995) 707-713.
Hiriart-Urruty, J.-B., New concepts in nondifferentiable programming. Bull. Soc. Math. France 60 (1979) 57-85.
Ishii, H. and Koike, S., On ε-optimal controls for state constraint problems. Ann. Inst. H. Poincaré Anal. Linéaire 17 (2000) 473-502. CrossRef
Krasovskii, N.N., Differential games. Approximate and formal models. Mat. Sb. (N.S.) 107 (1978) 541-571.
Krasovskii, N.N., Extremal aiming and extremal displacement in a game-theoretical control. Problems Control Inform. Theory 13 (1984) 287-302.
N.N. Krasovskii, Control of dynamical systems. Nauka, Moscow (1985).
N.N. Krasovskii and A.I. Subbotin, Positional Differential Games. Nauka, Moscow (1974). French translation: Jeux Différentielles. Mir, Moscou (1979).
N.N. Krasovskii and A.I. Subbotin, Game-Theoretical Control Problems. Springer-Verlag, New York (1988).
P. Loewen, Optimal Control via Nonsmooth Analysis. CRM Proc. Lecture Notes Amer. Math. Soc. 2 (1993).
Nobakhtian, S. and Stern, R.J., Universal near-optimal control feedbacks. J. Optim. Theory Appl. 107 (2000) 89-123. CrossRef
L. Rifford, Problèmes de Stabilisation en Théorie du Contrôle, Doctoral Thesis. Univ. Claude Bernard Lyon 1 (2000).
Rifford, L., Stabilisation des systèmes globalement asymptotiquement commandables. C. R. Acad. Sci. Paris 330 (2000) 211-216. CrossRef
L. Rifford, Existence of Lipschitz and semiconcave control-Lyapunov functions. SIAM J. Control Optim. (to appear).
Rockafellar, R.T., Clarke's tangent cones and boundaries of closed sets in ${\mathbb R}^n$ . Nonlinear Anal. 3 (1979) 145-154. CrossRef
R.T. Rockafellar, Favorable classes of Lipschitz continuous functions in subgradient optimization, in Nondifferentiable Optimization, edited by E. Nurminski. Permagon Press, New York (1982).
Rowland, J.D.L. and Vinter, R.B., Construction of optimal control feedback controls. Systems Control Lett. 16 (1991) 357-357. CrossRef
Soner, M., Optimal control problems with state-space constraints I. SIAM J. Control Optim. 24 (1986) 551-561.
E.D. Sontag, Mathematical Control Theory, 2nd Ed.. Springer-Verlag, New York, Texts in Appl. Math. 6 (1998).
Sontag, E.D., Clock and insensitivity to small measurement errors. ESAIM: COCV 4 (1999) 537-557. CrossRef
A.I. Subbotin, Generalized Solutions of First Order PDE's. Birkhäuser, Boston (1995).
Subbotina, N.N., Universal optimal strategies in positional differential games. Differential Equations 19 (1983) 1377-1382.
Subbotina, N.N., The maximum principle and the superdifferential of the value function. Problems Control Inform. Theory 18 (1989) 151-160.
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).
R.B. Vinter, Optimal Control. Birkhäuser, Boston (2000).