Advertisement

Vestnik St. Petersburg University, Mathematics

, Volume 51, Issue 4, pp 397–406 | Cite as

A Numerical Method for Finding the Optimal Solution of a Differential Inclusion

  • A. V. FominyhEmail author
Mathematics
  • 1 Downloads

Abstract

In the paper, we study a differential inclusion with a given continuous convex multivalued mapping. For a prescribed finite time interval, it is required to construct a solution to the differential inclusion, which satisfies the prescribed initial and final conditions and minimizes the integral functional. By means of support functions, the original problem is reduced to minimizing some functional in the space of partially continuous functions. When the support function of the multivalued mapping is continuously differentiable with respect to the phase variables, this functional is Gateaux differentiable. In the study, the Gateaux gradient is determined and the necessary conditions for the minimum of the functional are obtained. Based on these conditions, the method of steepest descent is applied to the original problem. The numerical examples illustrate the implementation of the constructed algorithm.

Keywords

differential inclusion support function steepest descent method 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. S. Polovinkin Multivalued Analysis and Differential Inclusions (Fizmatlit, Moscow, 2014) [in Russian].Google Scholar
  2. 2.
    J.-P. Aubin and H. Frankowska Set-Valued Analysis (Birkhauser, Boston, 1990).zbMATHGoogle Scholar
  3. 3.
    V. I. Blagodatskih and A. F. Filippov “Differential inclusions and optimal control,” Proc. Steklov. Inst. Math. 169, 199–259 (1985).MathSciNetGoogle Scholar
  4. 4.
    A. Cernea and C. Georgescu “Necessary optimality conditions for differential–difference inclusions with state constraints,” J. Math. Anal. Appl. 344, 43–53 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    B. Sh. Mordukhovich, “Discrete approximations and refined Euler–Lagrange conditions for nonconvex differential inclusions,” SIAM J. Control Optim. 33, 887–915 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    G. S. Pappas “Optimal solutions to differential inclusions in presence of state constraints,” J. Optim. Theory Appl. 44, 657–679 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Q. J. Zhu “Necessary optimality conditions for nonconvex differential inclusions with endpoint constraints,” J. Differ. Equations 124, 186–204 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    A. V. Arutyunov S. M. Aseev and V. I. Blagodatskikh “First-order necessary conditions in the problem of optimal control of a differential inclusion with phase constraints,” Russ. Acad. Sci. Sb. Math. 79, 117–139 (1994).Google Scholar
  9. 9.
    S. M. Aseev “A method of smooth approximation in the theory of necessary optimality conditions for differential inclusions,” Izv. Math. 61, 235–258 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    A. V. Fominyh V. V. Karelin and L. N. Polyakova “Exact penalties and differential inclusions,” Electron. J. Differ. Equations 2015, 1–13 (2015).zbMATHGoogle Scholar
  11. 11.
    M. S. Nikol’skii “On approximation of the attainability domain of control process,” Mat. Zametki 41, 71–76 (1987).MathSciNetGoogle Scholar
  12. 12.
    M. S. Nikol’skii “On a method for approximation of attainable set for a differential inclusion,” J. Vychisl. Mat. Mat. Fiz. 28, 1252–1254 (1988).Google Scholar
  13. 13.
    R. Baier M. Gerdts and I. Xausa “Approximation of reachable sets using optimal control algorithms,” Numer. Algebra Control Optim. 3, 519–548 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    A. I. Panasyuk “Equations of attainable set dynamics, Part 1: Integral funnel equations,” J. Optim. Theory Appl. 64, 349–366 (1990).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    A. Puri V. Borkar and P. Varaiya “ε-Approximation of differential inclusions,” in Proc. Int. Hybrid Systems Workshop Hybrid Systems III, New Brunswick, NJ, Oct. 22–25,1995 (Springer-Verlag, Berlin, 1995), in Ser.: Lecture Notes in Computer Science, Vol. 1066, pp. 362–376.Google Scholar
  16. 16.
    V. I. Blagodatskih Introduction to Optimal Control (Vysshaya Shkola, Moscow, 2001) [in Russian].Google Scholar
  17. 17.
    F. P. Vasil’ev Optimization Methods (Factorial, Moscow, 2002) [in Russian]Google Scholar
  18. 18.
    J. F. Bonnans and A. Shapiro Perturbation Analysis of Optimization Problems (Springer-Verlag, New York, 2000).CrossRefzbMATHGoogle Scholar
  19. 19.
    V. F. Demyanov Extremum Conditions and Variation Calculus (Vysshaya Shkola, Moscow, 2005) [in Russian].Google Scholar
  20. 20.
    L. V. Kantorovich and G. P. Akilov Functional Analysis (Nauka, Moscow, 1977) [in Russian].zbMATHGoogle Scholar
  21. 21.
    J. P. Penot “On the convergence of descent algorithms,” Comput. Optim. Appl. 23, 279–284 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    S. P. Iglin Mathematical Calculations on the Basis of MATLAB (BKhV-Peterburg, St. Petersburg, 2005) [in Russian].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

Personalised recommendations