Limits of singularly perturbed control problems with statistical dynamics of fast motions

SIAM Journal on Control and Optimization

Volumn 35, Issue 1, 1997, Pages 1-28

  Vigodner, Alexander ^a,b  

^a WEIZMANN INSTITUTE OF SCIENCE   (Israel)

^b INTEL CORPORATION   (Israel)

Author keywords

Invariant measure; Relatively slow control; Relaxed control; Singular perturbations; Statistical convergence

Indexed keywords

APPROXIMATION THEORY; CONTROL THEORY; DIFFERENTIAL EQUATIONS; INTEGRAL EQUATIONS; OPTIMIZATION; STATISTICAL METHODS; SYSTEM STABILITY;

INVARIANT MEASURE; SINGULAR PERTURBATIONS; STATISTICAL DYNAMICS;

OPTIMAL CONTROL SYSTEMS;

EID: 0030826053     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012994264207     Document Type: Article

References (33)

1. Z. ARTSTEIN, A variational convergence that yields chattering systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, supp., 6 (1989), pp. 49-71.
2. Z. ARTSTEIN, Chattering linear systems: A model of rapidly oscillating coefficients, Math. Control Signals Systems, 2 (1989), pp. 49-71.
3. Z. ARTSTEIN, Chattering variational limits of control systems, Forum Math., 5 (1993), pp. 369-403.
4. Z. ARTSTEIN, Rapid oscillations, chattering systems, and relaxed controls, SIAM J. Control Optim., 27 (1989), pp. 940-948.
5. Z. ARTSTEIN AND A. VIGODNER, Singularly perturbed ordinary differential equations with dynamic limits, Proc. Roy. Soc. Edinburgh Sect. A, (1996), pp. 541-569.
6. A. BENSOUSSAN, Singular perturbations for deterministic control problems, in Singular Perturbations and Asymptotic Analysis in Control Systems, Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin, 1987, pp. 9-170
7. A. BENSOUSSAN, Perturbation Methods in Optimal Control Problems, John Wiley, New York, 1989.
8. A. BENSOUSSAN AND G. BLANKENSHIP, Singular perturbations in stochastic control, in Singular Perturbations and Asymptotic Analysis in Control Systems, Lecture Notes in Control and Inform. Sci., Springer-Verlag, Berlin, 1987, pp. 171-287.
9. N. P. BHATIA AND G. P. SZEGÖ, Stability Theory of Dynamical Systems, Springer-Verlag, Berlin, 1970.
10. P. BILLINGSLEY, Convergence of Probability Measures, John Wiley, New York, 1968.
11. S. L. CAMPBELL, Singular Systems of Differential Equations, Pitman, San Francisco, 1980.
12. A. L. DONTCHEV AND V. M. VELIOV, Singular perturbations in Mayer's problem for linear systems, SIAM J. Control Optim., 21 (1983), pp. 566-581.
13. A. L. DONTCHEV AND V. M. VELIOV, Singular perturbations in linear control systems with weakly coupled stable and unstable fast subsystems, J. Math. Anal. Appl., 110 (1985), pp. 1-30.
14. V. GAITSGORY, Systems with Divided Motions: Control and Suboptimization, Nauka, Moscow, 1991 (in Russian).
15. V. GAITSGORY, Suboptimization of singularly perturbed control systems, SIAM J. Control Optim., 30 (1992), pp. 1228-1249.
16. V. GAITSGORY, Suboptimal control of singularly perturbed systems and periodic optimization, IEEE Trans. Automat. Control, 38 (1993), pp. 888-903.
17. G. GRAMMEL, Averaging of singularly perturbed systems, Nonlinear Anal., to appear.
18. P. R. HALMOS, Measure Theory, Van Nostrand Reinhold, New York, 1950.
19. YU. KABANOV AND S. PERGAMENSHCHIKOV, Optimal control of singularly perturbed stochastic linear systems, Stochastics Stochastics Rep., 36 (1991), pp. 109-135.
20. P. V. KOKOTOVIC, Applications of singular perturbations techniques to control problems, SIAM Rev., 26 (1984), pp. 501-550.
21. P. V. KOKOTOVIC, R. E. O'MALLEY, AND P. SANNUTI, Singular perturbations and order reduction in control theory, Automatica, 12 (1976), pp. 123-132.
22. V. V. NEMYTSKII AND V. V. STEPANOV, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, NJ, 1960.
23. R. E. O'MALLEY, Boundary layer methods for certain non-linear singularly perturbed optimal control problems, J. Math. Anal. Appl., 45 (1974), pp. 468-484.
24. R. E. O'MALLEY, Introduction to Singular Perturbations, Academic Press, New York, 1974.
25. G. R. SELL, Topological Dynamics and Ordinary Differential Equations, Van Nostrand Reinhold, London, 1971.
26. A. N. TICHONOV, Systems of differential equations containing small parameters near derivatives, Mat. Sb., 31 (1952), pp. 575-586.
27. H. D. TUAN, On reachable set of singularly perturbed differential inclusions and optimal control problems, Optimization, 26 (1992), pp. 325-338.
28. A. B. VASIL'EVA, Asymptotic behavior of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivative, Russian Math. Surveys, 18 (1963), pp. 15-86.
29. A. B. VASIL'EVA AND V. F. BUTUZOV, Asymptotic expansions of solutions to singularly perturbed equations, Nauka, Moscow, 1973 (in Russian).
30. A. VIGODNER, Limits of singularly perturbed control problems: Dynamical systems approach, Ph.D. thesis, Weizmann Institute of Sciences, Rehovot, Israel, 1995.
31. V. M. VOLOSOV, Averaging in systems of differential and functional equations, Uspekhi Mat. Nauk, 17 (1962), pp. 3-126.
32. J. WARGA, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.
33. W. WASOW, Asymptotic Expansions for Ordinary Differential Equations, Wiley-Interscience, New York, 1965.