On representation formulas for solutions of linear differential equations with Caputo fractional derivatives

In the paper, a linear differential equation with variable coefficients and a Caputo fractional derivative is considered. For this equation, a Cauchy problem is studied, when an initial condition is given at an intermediate point that does not necessarily coincide with the initial point of the fractional differential operator. A detailed analysis of basic properties of the fundamental solution matrix is carried out. In particular, the Hölder continuity of this matrix with respect to both variables is proved, and its dual definition is given. Based on this, two representation formulas for the solution of the Cauchy problem are proposed and justified. © 2020 Diogenes Co., Sofia.This work was supported by RSF, Project No 19-11-00105

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∈ (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. © 2022, Diogenes Co.Ltd.Russian Science Foundation, RSF: 19-11-00105This work was supported by RSF, Project no. 19-11-00105

Экстремальный сдвиг на сопутствующие точки в позиционной дифференциальной игре для системы дробного порядка

A two-person zero-sum differential game is considered. The motion of the dynamical system is described by an ordinary differential equation with a Caputo fractional derivative of order α ∈ (0, 1). The performance index consists of two terms: the first depends on the motion of the system realized by the terminal time and the second includes an integral estimate of the realizations of the players’ controls. The positional approach is applied to formalize the game in the “strategies — counter-strategies” and “counter-strategies — strategies” classes as well in the “strategies — strategies” class under the additional saddle point condition in the small game. In each case, the existence of the value and of the saddle point of the game is proved. The proofs are based on an appropriate modification of the method of extremal shift to accompanying points, which takes into account the specific properties of fractional-order systems. © 2019 Krasovskii Institute of Mathematics and Mechanics. All rights reserved

Extremal Shift to Accompanying Points in a Positional Differential Game for a Fractional-Order System

A two-person zero-sum differential game is considered. The motion of the dynamic system is described by an ordinary differential equation with a Caputo fractional derivative of order α ∈ (0, 1). The quality index consists of two terms: the first depends on the motion of the system realized by the terminal time and the second includes an integral estimate of the realizations of the players’ controls. The positional approach is applied to formalize the game in the “strategy–counterstrategy” and “counterstrategy–strategy” classes as well as in the “strategy–strategy” classes under the additional saddle point condition in the small game. In each case, the existence of the value and of the saddle point of the game is proved. The proofs are based on an appropriate modification of the method of extremal shift to accompanying points, which takes into account the specific properties of fractional-order systems. © 2020, Pleiades Publishing, Ltd

Minimax Solutions of Homogeneous Hamilton–Jacobi Equations with Fractional-Order Coinvariant Derivatives

The Cauchy problem is considered for a homogeneous Hamilton–Jacobi equation with fractional-order coinvariant derivatives,which arises in problems of dynamic 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, dependscontinuously on the parameters of the problem, and is consistent with the classical solution. An infinitesimal criterion of the minimaxsolution is obtained in the form of a pair of differential inequalities for suitable directional derivatives. An illustrative example is given. © 2021, Pleiades Publishing, Ltd.Russian Science Foundation, RSF: 19-71-00073This work was supported by the Russian Science Foundation (project no. 19-71-00073)

Minimax solutions of Hamilton- Jacobi equations with fractional coinvariant derivatives

We consider a Cauchy problem for a Hamilton- Jacobi equation with coinvariant derivatives of an order α e (0, 1). Such problems arise naturally in optimal control problems for dynamical systems which evolution is described by differential equations with the Caputo fractional derivatives of the order α. We propose a notion of a generalized in the minimax sense solution of the considered problem. We prove that a minimax solution exists, is unique, and is consistent with a classical solution of this problem. In particular, we give a special attention to the proof of a comparison principle, which requires construction of a suitable Lyapunov- Krasovskii functional. ©Russian Science Foundation, RSF: 19-71-00073∗This work was supported by RSF (project no. 19-71-00073)

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∈ (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. © 2022, Diogenes Co.Ltd.Russian Science Foundation, RSF: 19-11-00105This work was supported by RSF, Project no. 19-11-00105

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 \alpha \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 \alpha, 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. © 2020 Society for Industrial and Applied Mathematics.This work was supported by the RSF, project 19-71-00073

Fractional derivatives of convex Lyapunov functions and control problems in fractional order systems

The paper is devoted to the development of control procedures with a guide for fractional order dynamical systems controlled under conditions of disturbances, uncertainties or counteractions. We consider a dynamical system which motion is described by ordinary fractional differential equations with the Caputo derivative of an order α ∈ (0, 1). For the case when the guide is, in a certain sense, a copy of the system, we propose a mutual aiming procedure between the original system and guide. The proof of proximity between motions of the systems is based on the estimate of the fractional derivative of the superposition of a convex Lyapunov function and a function represented by the fractional integral of an essentially bounded measurable function. This estimate can be considered as a generalization of the known estimates of such type. We give an example that illustrates the workability of the proposed control procedures with a guide. © 2018 Diogenes Co., Sofia

Минимаксные решения однородных уравнений Гамильтона — Якоби с коинвариантными производными дробного порядка

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. © 2020 Krasovskii Institute of Mathematics and Mechanics. All rights reserved.This work was supported by RSF (project no. 19-71-00073)