Dynamic Programming Principle and Hamilton--Jacobi--Bellman Equations for Fractional-Order Systems

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

“…In this paper, we consider the space [0, T ] × AC α , endowed with the standard product metric, and require that all mappings defined on this space are non-anticipative. Note that this framework differs from previous studies [7,8,9,10,11,12,13], which deal with a certain metric space consisting of all pairs (t, w(•)) where t ∈ [0, T ] and w(•) is a restriction of a function x(•) ∈ AC α to [0, t]. Nevertheless, as shown in [14, Subsection 5.1], these two approaches are closely related, which allows us to use the previously obtained results in this paper, after their corresponding reformulation.…”

“…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]).…”

Minimax solutions of Hamilton–Jacobi equations with fractional coinvariant derivatives

“…In [12], the dynamic programming principle was extended to a Bolza-type optimal control problem for a dynamical system described by a fractional differential equation with the Caputo derivative of an order α ∈ (0, 1). In particular, it was shown that the value of this problem should be introduced as a functional in a suitable space of paths.…”

Causal State Feedback Representation for Linear Quadratic Optimal Control Problems of Singular Volterra Integral Equations

“…In the recent years, optimal control problems have been studied for fractional differential equations by a number of authors. We mention the works of Agrawal [1,2], Agrawal-Defterli-Baleanu [3], Bourdin [12], Frederico-Torres [24], Hasan-Tangpong-Agrawal [27] and Kamocki [29,30], Gomoyunov [25], Koenig [32].…”