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

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

“…A functional ϕ : [0, T ] × AC α → R is said to be ci-differentiable of the order α at a point (t, x(•)) ∈ [0, T ) × AC α (see, e.g., [18,7] and also [14]) if there exist…”

“…In this paper, we study a two-person zero-sum differential game (see, e.g., [15,19,30,2]) involving a dynamical system described by a Caputo fractional differential equation of order α ∈ (0, 1) (see, e.g., [25,17,5]) and a Bolza cost functional, which the first player tries to minimize while the second player tries to maximize. In accordance with [11,13], we associate the differential game to the Cauchy problem for the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation with socalled fractional coinvariant (ci-) derivatives of the order α (see, e.g., [18,7] and also [14]) and the corresponding right-end boundary condition. It should be noted that the path-dependent nature of the Caputo fractional derivative makes it necessary to consider the value of the differential game as a non-anticipative functional on a certain space of paths, and, respectively, the HJBI equation can be classified as pathdependent (in this connection, see, e.g., [23,16,29,6,1,3,26,14]).…”

“…In accordance with [11,13], we associate the differential game to the Cauchy problem for the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation with socalled fractional coinvariant (ci-) derivatives of the order α (see, e.g., [18,7] and also [14]) and the corresponding right-end boundary condition. It should be noted that the path-dependent nature of the Caputo fractional derivative makes it necessary to consider the value of the differential game as a non-anticipative functional on a certain space of paths, and, respectively, the HJBI equation can be classified as pathdependent (in this connection, see, e.g., [23,16,29,6,1,3,26,14]).…”

“…Following [28] and, in a path-dependent framework, [20] (see also [14] and the references therein), previous works in this direction [8,10,12] focused on the development of the minimax approach to a notion of a generalized solution. In particular, the questions of well-posedness of minimax solutions, consistency of minimax solutions with classical solutions, as well as and non-local and infinitesimal criteria of minimax solutions were addressed.…”

Minimax solutions of Hamilton–Jacobi equations with fractional coinvariant derivatives

“…Minimax solutions of Hamilton-Jacobi equations with first-order partial derivatives were proposed and comprehensively studied in [48] (see also [49]). Further, this technique was extended to Hamilton-Jacobi equations with first-order ci-derivatives, which arise in optimization problems for dynamical systems described by functional differential equations of a retarded type [37] (see also [17,[31][32][33][34]36], and [3] for an infinite dimensional case) and of a neutral type [38,39,42]. Note that the minimax approach was also applied to investigate generalized solutions of systems of equations arising in mean field games [1].…”