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Sensitivity and Controllability of Systems Governed by Integral Equations Via Proximal Analysis

Published online by Cambridge University Press:  20 November 2018

A. Yezza
A. Yezza*
Affiliation:
Departement de mathematiques et de statistique Universite de Montreal C.P. 6128-A Montreal, Quebec H3C 3J7
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Abstract

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In this paper, we are concerned with the basic problem defined in [9]. Formulas for δV(0)and δ∞V(0),respectively the generalized and asymptotic gradient of the value function at zero, corresponding to an L2 -additive perturbation of dynamics are given. Under the normality condition, δV(0)turns out to be a compact subset of L2, formed entirely of arcs, and V is locally finite and Lipschitz at 0. Moreover, estimations of the generalized directional derivative and Dini's derivative of V at 0 are derived. Supplementary conditions imply that Dini's derivative of V at 0 exists, and V is actually strictly differentiate at this point.

Keywords

49E2593B0593B3593C22Optimal controlvalue functionsensitivitycontrollabilityproximal normal analysisgeneralized gradientgeneralized asymptotic gradientVolterra integral equation
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 5 , 01 October 1993 , pp. 1104 - 1120
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Borwein, J.M. and Giles, J.R., The proximal normal formula in Banach space, Trans. Amer. Math. Soc. 302(1987), 371–381.Google Scholar
2. Borwein, J.M. and Strojwas, H.M., Proximal analysis and boundaries of closed sets in Banach space, Part I: Theory, Can. J. Math. (2) XXXVIII(1986), 431–452.Google Scholar
3. Borwein, J.M. and Strojwas, H.M., Proximal analysis and boundaries of closed sets in Banach space, Part II: Applications, Can. J. Math. (2) XXXIX(1987), 428–472.Google Scholar
4. Clarke, F.H., Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983. second edition 1990, Classics in Applied Math., SIAM, Philadelphia.Google Scholar
5. Clarke, F.H., Perturbed optimal control problems, IEEE Transactions on Automatic Control (6) AC31(1986), 535–542.Google Scholar
6. Loewen, P.D., The proximal normal formula in Hubert space, Nonlinear Analysis, Theory and Applications (9) 11(1987), 979–995.Google Scholar
7. Raissi, N., Analyse Proximate en Optimisation, Ph.D. thesis, Université de Montréal, 1987.Google Scholar
8. Yezza, A., Optimisation des Systèmes Gouvernés par des Equations Intégrales, Ph.D. thesis, Université de Montréal, 1991.Google Scholar
9. Yezza, A., Relaxed optimal control problems governed by integral equations,]. Math. Anal and Appl., (1993), to appear.Google Scholar
10. Warga, J., Optimal Control of Differential and Functional Equations, Academic Press, New York/London, 1972.Google Scholar