arXiv subjects

Mikhail I. Gomoyunov

Publications and source records attributed to Mikhail I. Gomoyunov.

8 recordsLinked to original sources

arXiv2026

Path-dependent Hamilton--Jacobi equations: Uniqueness results for viscosity solutions defined via families of compact sets

We consider a path-dependent Hamilton--Jacobi equation with coinvariant derivatives over the space of continuous functions. We prove two uniqueness results for viscosity (generalized) solutions defined in terms of coinvariantly smooth test functionals and a dense family of compact subsets of the space of continuous functions. It is assumed that the Hamiltonian is continuous and satisfies a local Lipschitz condition in the functional variable with respect to the supremum norm. When the Lipschitz constant satisfies a sublinear growth condition in the gradient (impulse) variable, uniqueness is established in the class of continuous viscosity solutions. In the general case, without any such growth conditions, uniqueness is established in the class of continuous viscosity solutions that satisfy an additional local Lipschitz condition. The proofs are based on the standard method of doubling variables, but use a novel penalty functional for constructing coinvariantly smooth test functionals. The obtained results generalize previously known ones by relaxing the assumptions on the Hamiltonian and/or enlarging the class of functionals in which uniqueness is established.

math.AP↗

arXiv2024

Lipschitz Continuity Results for Minimax Solutions of Path-Dependent Hamilton--Jacobi Equations

We consider a Cauchy problem for a (first-order) path-dependent Hamilton--Jacobi equation with coinvariant derivatives and a right-end boundary condition. Such problems arise naturally in the study of properties of the value functional in (deterministic) optimal control problems and differential games for time-delay systems. We prove that, under certain assumptions on the Hamiltonian and the boundary functional, a minimax (generalized) solution of the Cauchy problem satisfies Lipschitz conditions both in time and functional variables. In the latter case, Lipschitz conditions are formulated with respect to the uniform norm and some special norm of the space of continuous functions.

math.OC↗

arXiv2024

Zero-Sum Games for Volterra Integral Equations and Viscosity Solutions of Path-Dependent Hamilton-Jacobi Equations

We consider a game, in which the dynamics is described by a non-linear Volterra integral equation of Hammerstein type with a weakly-singular kernel and the goals of the first and second players are, respectively, to minimize and maximize a given cost functional. We propose a way of how the dynamic programming principle can be formalized and the theory of generalized (viscosity) solutions of path-dependent Hamilton--Jacobi equations can be developed in order to prove the existence of the game value, obtain a characterization of the value functional, and construct players' optimal feedback strategies.

math.OC↗

arXiv2022

Value functional and optimal feedback control in linear-quadratic optimal control problem for fractional-order system

In this paper, a finite-horizon optimal control problem involving a dynamical system described by a linear Caputo fractional differential equation and a quadratic cost functional is considered. An explicit formula for the value functional is given, which includes a solution of a certain Fredholm integral equation. A step-by-step feedback control procedure for constructing $\varepsilon$-optimal controls with any accuracy $\varepsilon > 0$ is proposed. The basis for obtaining these results is the study of a solution of the associated Hamilton-Jacobi-Bellman equation with so-called fractional coinvariant derivatives.

math.OC↗

arXiv2021

On differentiability of solutions of fractional differential equations with respect to initial data

In this paper, we deal with a Cauchy problem for a nonlinear fractional differential equation with the Caputo derivative of order $α\in (0, 1)$. As initial data, we consider a pair consisting of an initial point, which does not necessarily coincide with the inferior limit of the fractional derivative, and a function that determines the values of a solution on the interval from this inferior limit to the initial point. We study differentiability properties of the functional associating initial data with the endpoint of the corresponding solution of the Cauchy problem. Stimulated by recent results on the dynamic programming principle and Hamilton--Jacobi--Bellman equations for fractional optimal control problems, we examine so-called fractional coinvariant derivatives of order $α$ of this functional. We prove that these derivatives exist and give formulas for their calculation.

math.OC↗

arXiv2021

On viscosity solutions of path-dependent Hamilton--Jacobi--Bellman--Isaacs equations for fractional-order systems

This paper deals with a two-person zero-sum differential game for a dynamical system described by a Caputo fractional differential equation of order $α\in (0, 1)$ and a Bolza cost functional. The differential game is associated to the Cauchy problem for the path-dependent Hamilton--Jacobi--Bellman--Isaacs equation with so-called fractional coinvariant derivatives of the order $α$ and the corresponding right-end boundary condition. A notion of a viscosity solution of the Cauchy problem is introduced, and the value functional of the differential game is characterized as a unique viscosity solution of this problem.

math.OC↗

arXiv2020

Path-Dependent Hamilton--Jacobi Equations: The Minimax Solutions Revised

Motivated by optimal control problems and differential games for functional differential equations of retarded type, the paper deals with a Cauchy problem for a path-dependent Hamilton--Jacobi equation with a right-end boundary condition. Minimax solutions of this problem are studied. The existence and uniqueness result is obtained under assumptions that are weaker than those considered earlier. In contrast to previous works, on the one hand, we do not require any properties concerning positive homogeneity of the Hamiltonian in the impulse variable, and on the other hand, we suppose that the Hamiltonian satisfies a Lipshitz continuity condition with respect to the path variable in the uniform (supremum) norm. The progress is related to the fact that a suitable Lyapunov--Krasovskii functional is built that allows to prove a comparison principle. This functional is in some sense equivalent to the square of the uniform norm of the path variable and, at the same time, it possesses appropriate smoothness properties. In addition, the paper provides non-local and infinitesimal criteria of minimax solutions, their stability with respect to perturbations of the Hamiltonian and the boundary functional, as well as consistency of the approach with the non-path-dependent case. Connection of the problem statement under consideration with some other possible statements (regarding the choice of path spaces and derivatives used) known in the theory of path-dependent Hamilton--Jacobi equations is discussed. Some remarks concerning viscosity solutions of the studied Cauchy problem are given.

math.OC↗

arXiv2019

Dynamic programming principle and Hamilton-Jacobi-Bellman equations for fractional-order systems

We consider a Bolza-type optimal control problem for a dynamical system described by a fractional differential equation with the Caputo derivative of an order $α\in (0, 1)$. The value of this problem is introduced as a functional in a suitable space of histories of motions. We prove that this functional satisfies the dynamic programming principle. Based on a new notion of coinvariant derivatives of the order $α$, we associate the considered optimal control problem with a Hamilton-Jacobi-Bellman equation. Under certain smoothness assumptions, we establish a connection between the value functional and a solution to this equation. Moreover, we propose a way of constructing optimal feedback controls. The paper concludes with an example.

math.OC↗