Singular perturbations in infinite-dimensional control systems

SIAM Journal on Control and Optimization

Volumn 42, Issue 5, 2003, Pages 1795-1812

  Donchev, T D ^a   Dontchev, A L ^b  

^a UNIVERSITY OF ARCHITECTURE   (Bulgaria)

^b Mathematical Reviews   (United States)

Author keywords

Control systems; Convergence of solutions; Differential inclusions; Order reduction; Singular perturbations; Time scales

Indexed keywords

DIFFERENTIAL INCLUSION; ORDER REDUCTION; SINGULAR PERTURBATIONS; TIME SCALES;

CONVERGENCE OF NUMERICAL METHODS; DIFFERENTIAL EQUATIONS; FUNCTIONS; PERTURBATION TECHNIQUES; THEOREM PROVING;

CONTROL SYSTEMS;

EID: 4944239107     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012902416080     Document Type: Article

References (29)

1. Z. ARTSTEIN, Invariant measures of differential inclusions applied to singular perturbations, J. Differential Equations, 152 (1999), pp. 289-307.
2. Z. ARTSTEIN, Invariant measures of set-valued maps, J. Math. Anal. Appl., 252 (2000), pp. 696-706.
3. K. DEIMLING, Multivalued Differential Equations, de Gruyter Ser. Nonlinear Anal. Appl. 1, Walter de Gruyter & Co., Berlin, 1992.
4. T. DONCHEV, Functional differential inclusions with monotone right hand side, Nonlinear Anal., 16 (1991), pp. 543-552.
5. T. DONCHEV, Semicontinuous differential inclusions, Rend. Sem. Mat. Univ. Padova, 101 (1999), pp. 147-160.
6. T. DONCHEV AND E. FARKHI, Stability and Euler approximation of one-sided Lipschitz differential inclusions, SIAM J. Control Optim., 36 (1998), pp. 780-796.
7. T. DONCHEV AND E. FARKHI, Approximation of one-sided Lipschitz differential inclusions with discontinuous right-hand sides, in Calculus of Variations and Differential Equations, Chapman & Hall/CRC Res. Notes Math. 410, A. Ioffe, S. Reich, and I. Shafrir, eds., Chapman & Hall/CRC, Boca Raton, FL, 1999, pp. 101-118.
8. T. DONCHEV AND R. IVANOV, On the existence of solutions of differential inclusions in uniformly convex Banach spaces, Math. Balkanica, 6 (1992), pp. 13-24.
9. T. DONCHEV AND I. SLAVOV, Singularly perturbed functional differential inclusions, Set-Valued Anal., 3 (1995), pp. 113-128.
10. T. DONCHEV AND I. SLAVOV, Averaging method for one-sided Lipschitz differential inclusions with generalized solutions, SIAM J. Control Optim., 37 (1999), pp. 1600-1613.
11. A. DONTCHEV, T. DONCHEV, AND I. SLAVOV, A Tikhonov-type theorem for singularly perturbed differential inclusions, Nonlinear Anal., 26 (1996), pp. 1547-1554.
12. A. DONTCHEV AND V. VELIOV, Singular perturbation in Mayer's problem for linear systems, SIAM J. Control Optim., 21 (1983), pp. 566-581.
13. O. FILATOV AND M. HAPAEV, Averaging of differential inclusions with fast and slow variables, Math. Notes, 47 (1990), pp. 102-109.
14. O. FILATOV AND M. HAPAEV, Averaging of Systems of Differential Inclusions, Moskow University Press, Moskow, 1998 (in Russian).
15. V. GAITSGORY, Use of averaging method in control problems, Differ. Equ., 22 (1986), pp. 1290-1299.
16. V. GAITSGORY, Suboptimization of singularly perturbed control systems, SIAM J. Control Optim., 30 (1992), pp. 1228-1249.
17. G. GRAMMEL, Averaging of singularly perturbed systems, Nonlinear Anal., 28 (1997), pp. 1851-1865.
18. G. GRAMMEL, Singularly perturbed differential inclusions: An averaging approach, Set-Valued Anal., 4 (1996), pp. 361-374.
19. F. HOPPENSTEADT, Stability in systems with parameter, J. Math. Anal. Appl., 18 (1967), pp. 129-134.
20. F. HOPPENSTEADT, Properties of solutions of ordinary differential equations with small parameters, Comm. Pure Appl. Math., 24 (1971), pp. 807-840.
21. S. HU AND N. PAPAGEORGIOU, Handbook of Multivalued Analysis, Vol. II, Kluwer, Dodrecht, The Netherlands, 2000.
22. V. LAKSHMIKANTHAM AND S. LEELA, Nonlinear Differential Equations in Abstract Spaces, Pergamon, Oxford, UK, 1981.
23. F. LEMPIO AND V. VELIOV, Discrete approximations of differential inclusions, Bayreuth. Math. Sehr., 54 (1998), pp. 149-232.
24. N. LEVINSON, Perturbations of discontinuous solutions of nonlinear systems of differential equations, Proc. Natl. Acad. Sci. USA, 33 (1947), pp. 214-218.
25. A. TIKHONOV, On the dependence of the solutions of differential equations on a small parameter, Mat. Sb. (N.S.), 22 (1948), pp. 193-204.
26. A. TIKHONOV, On systems of differential equations containing parameters, Mat. Sb. (N.S.), 27 (1950), pp. 147-156.
27. A. TIKHONOV, Systems of differential equations containing small parameters in the derivatives, Mat. Sb. (N.S.), 31 (1952), pp. 575-586.
28. V. VELIOV, Differential inclusions with stable subinclusions, Nonlinear Anal., 23 (1994), pp. 1027-1038.
29. V. VELIOV, A generalization of Tikhonov theorem for singularly perturbed differential inclusions, J. Dynam. Control Systems, 3 (1997), pp. 1-28.