Linearization and Boundary Trajectories of Nonsmooth Control Systems

Frankowska, H. ; Kaśkosz, B.

Canadian journal of mathematics, Tome 40 (1988) no. 3, pp. 589-609

Résumé

This paper deals with boundary trajectories of non-smooth control systems and differential inclusions.Consider a control system (1.1) and denote by R(t) its reachable set at time t. Let (z, u *) be a trajectory-control pair. If for every t from the time interval [0, 1], z(t) lies on the boundary of R(t) then z is called a boundary trajectory. It is known that for systems with Lipschitzian in x right-hand side, z is a boundary trajectory if and only if z(1) belongs to the boundary of the set R(1). If z is not a boundary trajectory, that is, z(1) ∊ Int R(1) then the system is said to be locally controllable around z at time 1.A first-order necessary condition for boundary trajectories of smooth systems comes from the Pontriagin maximum principle, (see e.g. [12]).

Détail

DOI : 10.4153/CJM-1988-025-7

Bibliographie

[1] 1. Aubin, J. P. and Cellina, A., Differential inclusions (Springer-Verlag, 1984). Google Scholar | DOI

[2] 2. Clarke, F. H., Nonsmooth analysis and optimization (Wiley Interscience, 1983). Google Scholar

[3] 3. Clarke, F. H., The maximum principle under minimal hypotheses, SIAM J. Control Optim. 14 (1976), 1078–1091. Google Scholar

[4] 4. Clarke, F. H., Necessary conditions for a general control problem, Proc. Int. Symp. on the Calculus of Variations and Control Theory (Academic Press, New York, 1976). Google Scholar

[5] 5. Ekeland, I., Nonconvex minimization problems, Bull. Am. Math. Soc. 1 (1979), 443–474. Google Scholar

[6] 6. Frankowska, H., The maximum principle for the differential inclusions with end point constraints, SIAM J. of Control 25 (1987), (to appear). Google Scholar

[7] 7. Frankowska, H., Local controllability and infinitesimal generators of semigroups of set-valued maps, SIAM J. of Control, 24 (1986), (to appear). Google Scholar

[8] 8. Frankowska, H., Local controllability of control systems with feedback, submitted. Google Scholar | DOI

[9] 9. Gamkrelidze, R. V., On some extremal problems in the theory of differential equations with applications to optimal control, SIAM J. of Control 3 (1965), 106–128. Google Scholar

[10] 10. Kaskosz, B. and Łojasiewicz, S., A maximum principle for generalized control systems, Nonlinear Analysis, Theory, Meth. & Appl. 9 (1985), 109–130. Google Scholar

[11] 11. LeDonne, A. and Marchi, V., Representations of Lipschitzian compact-convex valued mappings, Lincei-Rend. Sc.fis.mat. e nat. 68 (1980), 278–280. Google Scholar

[12] 12. Lee, E. B. and Markus, L., Foundation of optimal control theory, (1969), Wiley. Google Scholar

[13] 13. Łojasiewicz, S., Lipschitz selections of orientor fields (to appear). Google Scholar

[14] 14. McMullen, P. and Schneider, R., Valuations of convex bodies, in Convexity and its applications (Birkhàuser, Basel, 1983). Google Scholar

[15] 15. Olech, C., Existence theory in optimal control, Control Theory and Topics in Functional Analysis, 1, Internat. At. Energy Agency, Vienna (1976), 291–328. Google Scholar

[16] 16. Vitale, R., The Steiner point in infinite dimensions, Israel J. of Math. 52 (1985), 245–250. Google Scholar

[17] 17. Warga, J., Optimal control of differential and functional equations (Academic Press, New York, 1972). Google Scholar

[18] 18. Warga, J., Derivative containers, inverse functions and controllability, Proc. Int. Symp. on the Calculus of Variations and Control Theory (Academic Press, New York, 1976). Google Scholar

[19] 19. Warga, J., Optimization and controllability without differentiability assumptions, SIAM J. on Control 21 (1983), 837–855. Google Scholar

[20] 20. Warga, J., Controllability, extremality and abnormality in nonsmooth optimal control, J. Opt. Theory and Appl. 41 (1983), 239–259. Google Scholar