M. I. Gomoyunov, N.Yu. Lukoyanov, A.R. Plaksin. Approximation of minimax solutions to Hamilton–Jacobi functional equations for delay systems ... P. 53-62

A minimax solution of the Cauchy problem for a functional Hamilton–Jacobi equation with coinvariant derivatives and a condition at the right end is considered. Hamilton–Jacobi equations of this type arise in dynamical optimization problems for time-delay systems. Their approximation is associated with additional questions of the correct transition from the infinite-dimensional functional argument of the desired solution to the finite-dimensional one. Earlier, the schemes based on the piecewise linear approximation of the functional argument and the correctness properties of minimax solutions were studied. In this paper, a scheme for the approximation of Hamilton–Jacobi functional equations with coinvariant derivatives by ordinary Hamilton–Jacobi equations with partial derivatives is proposed and justified. The scheme is based on the approximation of the characteristic functional–differential inclusions used in the definition of the desired minimax solution by ordinary differential inclusions.

Keywords: Hamilton–Jacobi equations, generalized solutions, coinvariant derivatives, finite-dimensional approximations, time-delay systems.

The paper was received by the Editorial Office on October 1, 2017.

Funding Agency: Ministry of Education and Science of the Russian Federation (project no. МК-3047.2017.1).

Mikhail Igorevich Gomoyunov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and
Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia; Ural
Federal University, Yekaterinburg, 620002 Russia, e-mail: m.i.gomoyunov@gmail.com.

Nikolai Yur’evich Lukoyanov, Dr. Phys.-Math. Sci., Corresponding Member of RAS, Prof., Krasovskii
Institute ofMathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg,
620990 Russia; Ural Federal University, Yekaterinburg, 620002 Russia,
e-mail: nyul@imm.uran.ru .

Anton Romanovich Plaksin, Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the
Russian Academy of Sciences, Yekaterinburg, 620990 Russia; Ural Federal University, Yekaterinburg,
620002 Russia, e-mail: a.r.plaksin@gmail.com.


1. Krasovskii N.N., Subbotin A.I. Game-theoretical control problems. New York, Springer. 1988. 517 p. ISBN: 978-1-4612-8318-8.Original Russian text published in Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial’nye igry. Moscow, Nauka Publ., 1974, 456 p.

2. Krasovskii N.N. On the problem of unifying differential games. Sov. Math., Dokl., 1976, vol. 17, no. 1, pp. 269–273.

3. Osipov Yu.S. Differential games of systems with aftereffect. Sov. Math., Dokl., 1971, vol. 12, pp. 262–266.

4. Subbotin A.I., Chentsov A.G. Optimizatsiya garantii v zadachakh upravleniya [Optimization of guarantee in control problems]. Moscow, Nauka Publ., 1981, 286 p.

5. Chentsov A.G. On a game problem of converging at a given instant of time. Math. USSR-Sb., 1976, vol. 28, no. 3, pp. 353–376. doi: 10.1070/SM1976v028n03ABEH001657 .

6. Subbotin A.I. Minimaksnye neravenstva i uravneniya Gamil’tona-Yakobi. [Minimax inequalities and Hamilton-Jacobi equations]. Moscow, Nauka Publ., 1991, 216 p.

7. Crandall M.G., Lions P.-L. Viscosity solutions of Hamilton–Jacobi equations. Trans. Amer. Math. Soc., 1983, vol. 277, no. 1, pp. 1–42. doi: 10.1090/S0002-9947-1983-0690039-8 .

8. Lukoyanov N.Yu. Functional equations of Hamilton–Jacobi type and differential games with hereditary information. Dokl. Math., 2000, vol. 61, no. 2, pp. 301–304.

9. Krasovskii N.N., Lukoyanov N.Yu. Equations of Hamilton–Jacobi type in hereditary systems: minimax solutions. Proc. Steklov Inst. Math. (Suppl.), 2000, suppl. 1, pp. S136–S153.

10. Lukoyanov N.Yu. Funktsional’nye uravneniya Gamil’tona-Yakobi i zadachi upravleniya s nasledstvennoi informatsiei [Functional Hamilton-Jacobi equations and control problems with hereditary information]. Ekaterinburg, Ural Federal University Publ., 2011, 243 p.

11. Kim A.V. Functional differential equations. Application of i-smooth calculus. Dordrecht, Kluwer, 1999, 165 p. doi: 10.1007/978-94-017-1630-7 .

12. Taras’ev A.M., Uspenskij A.A., Ushakov V.N. Approximation schemes and finite-difference operators for constructing generalized solutions of Hamilton–Jacobi equations. J. Comput. Syst. Sci. Int., 1995, vol. 33, no. 6, pp. 127–139.

13. Falcone M., Ferretti R. Discrete time high order schemes for viscosity solutions of Hamilton–Jacobi–Bellman equations. Numer. Math., 1994, vol. 67, no. 3, pp. 315–344. doi: 10.1007/s002110050031 .

14. Subbotin A.I., Chentsov A.G. An iteration procedure for constructing minimax and viscous solutions to Hamilton–Jacobi equations. Dokl. Math., 1996, vol. 53, no. 3, pp. 416–419.

15. Lukoyanov N.Yu. Approximation of the Hamilton–Jacobi functional equations in systems with hereditary information. Proc. Internat. Seminar Control Theory and Theory of Generalized Solutions of Hamilton–Jacobi eguations, dedicated to the 60th birthday of Academician A.I. Subbotin, June 22-26, 2005, Ekaterinburg, Russia. Vol. 1. Ekaterinburg, Ural State University Publ., 2006, pp. 108–115 (in Russian). ISBN: 5-7996-0318-4 .

16. Krasovskii N.N. The approximation of a problem of analytic design of controls in a system with time-lag. PMM, J. Appl. Math. Mech., 1964, vol. 28, no. 4, pp. 876–885. doi: 10.1016/0021-8928(64)90073-5 .

17. Repin Yu.M. On the approximate replacement of systems with lag by ordinary dynamical systems. PMM, J. Appl. Math. Mech., 1965, vol. 29, no. 2, pp. 254–264. doi: 10.1016/0021-8928(65)90029-8 .

18. Kurzhanskii A.B. On the approximation of linear differential equations with lag. Differ. Uravn., 1967, vol. 3, no. 12, pp. 2094–2107 (in Russian).

19. Lukoyanov N.Yu., Plaksin A.R. Finite-dimensional modeling guides in systems with delay. Tr. Inst. Mat. Mekh. UrO RAN, 2013, vol. 19, no. 1, pp. 182–195 (in Russian).

20. Bellman R., Cooke K.L. Differential–Difference Equations. N Y, Acad. Press, 1963, 462 p. ISBN: 9780080955148 . Translated to Russian under the title Differentsial’no-raznostnye uravneniya. Moscow, Mir Publ., 1967, 548 p.