The Cauchy problem is considered for a homogeneous Hamilton–Jacobi equation with fractional-order coinvariant derivatives, which arises in problems of dynamical optimization of systems described by differential equations with Caputo fractional derivatives. A generalized solution of the problem in the minimax sense is defined. It is proved that such a solution exists, is unique, depends continuously on the parameters of the problem, and is consistent with the classical solution. An infinitesimal criterion of the minimax solution is obtained in the form of a pair of differential inequalities for suitable directional derivatives. An illustrative example is given.
Keywords: Hamilton–Jacobi equations, generalized solutions, coinvariant derivatives, fractional-order derivatives
Received August 17, 2020
Revised October 15, 2020
Accepted October 26, 2020
Funding Agency: This work was supported by RSF (project no. 19-71-00073).
Mikhail Igorevich Gomoyunov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: m.i.gomoyunov@gmail.com
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Cite this article as: M.I. Gomoyunov. Minimax solutions of homogeneous Hamilton–Jacobi equations with fractional-order coinvariant derivative, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2020, vol. 26, no. 4, pp. 106–125; Proceedings of the Steklov Institute of Mathematics, 2021, Vol. 315, Suppl. 1, pp. S97–S116.
