We develop the method of discrete approximations to study optimal control problems for differential inclusions by using advanced tools of variational analysis and generalized differentiation. The first part describes the method, appropriate machinery of variational analysis and then presents the main result on necessary optimality conditions in maximum principle form for Lipschitzian differential inclusions.

Boris S. Mordukhovich, Ph.D. (Applied Mathematics), Distinguished University Prof., Wayne State University, Detroit, Michigan, 48202, USA, e-mail: boris@math.wayne.edu

Mordukhovich B.S. Optimal Control of Differential Inclusions, I: Lipschitzian case. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 29, pp. 45-58. https://doi.org/10.26516/1997-7670.2019.30.45

1. Clarke F.H. Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
2. Donchev T., Farkhi E., Mordukhovich B.S., Discrete approximations, relaxation, and optimization of one-sided Lipschitzian differential inclusions in Hilbert spaces. J. Diff. Eqs, 2007, vol. 243, pp. 301-328.
3. Euler L. Methodus Inveniendi Curvas Lineas Maximi Minimive Proprietate Gaudentes Sive Solution Problematis Viso Isoperimetrici Latissimo Sensu Accepti, Lausanne, 1774; reprinted in Opera Omnia, 1952, Ser. 1, vol. 24.
4. Gabasov R., Kirillova F.M. The Qualitative Theory of Optimal Processes. Marcel Dekker, New York, 1976.
5. Mordukhovich B.S. Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mech., 1976, vol. 40, pp. 960-969.
6. Mordukhovich B.S. Approximation Methods in Problems of Optimization and Control. Moscow, Nauka Publ.,1988.
7. Mordukhovich B.S. Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Amer. Math. Soc., 1993, vol. 340, pp. 1-35.
8. Mordukhovich B.S. Discrete approximations and refined Euler-Lagrange conditions for differential inclusions. SIAM J. Control Optim., 1995, vol. 33, pp. 882-915.
9. Mordukhovich B.S. Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin, 2006.
10. Mordukhovich B.S. Variational Analysis and Generalized Differentiation, II: Applications. Springer, Berlin, 2006.
11. Mordukhovich B.S. Variational Analysis and Applications. Springer, Switzerland, 2018.
12. Mordukhovich B.S. Optimal control of Lipschitzian and discontinuous differential inclusions with various applications. Proc. Inst. Math. Mech., 2019, vol. 45, pp. 52-74.
13. Mordukhovich B.S. Optimal control of differential inclusions, II: Sweeping. The Bulletin of Irkutsk State University. Series Mathematics, 2020, vol. 31.
14. Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R. V., Mishchenko E.F. The Mathematical Theory of Optimal Processes. Wiley, New York, 1962.
15. Rockafellar R.T., Wets R.J-B. Variational Analysis. Springer, Berlin, 1998.
16. Tolstonogov A.A. Differential Inclusions in a Banach Spaces. Kluwer, Dordrecht, the Netherlands, 2000.
17. Vasiliev O.V. Optimization Methods. World Federation Publishers, Atlanta, GA, 1996.
18. Vinter R.B. Optimal Control. Birkhauser, Boston, 2000.