A Numerical Method for Finding the Optimal Solution of a Differential Inclusion
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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.
Keywordsdifferential inclusion support function steepest descent method
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- 1.E. S. Polovinkin Multivalued Analysis and Differential Inclusions (Fizmatlit, Moscow, 2014) [in Russian].Google Scholar
- 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
- 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
- 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.V. I. Blagodatskih Introduction to Optimal Control (Vysshaya Shkola, Moscow, 2001) [in Russian].Google Scholar
- 17.F. P. Vasil’ev Optimization Methods (Factorial, Moscow, 2002) [in Russian]Google Scholar
- 19.V. F. Demyanov Extremum Conditions and Variation Calculus (Vysshaya Shkola, Moscow, 2005) [in Russian].Google Scholar
- 22.S. P. Iglin Mathematical Calculations on the Basis of MATLAB (BKhV-Peterburg, St. Petersburg, 2005) [in Russian].Google Scholar