Game-theoretical problems for fractional-order nonstationary systems

Nonstationary fractional-order systems represent a new class of dynamic systems characterized by time-varying parameters as well as memory effect and hereditary properties. Differential game described by system of linear nonstationary differential equations of fractional order is treated in the paper. The game involves two players, one of which tries to bring the system’s trajectory to a terminal set, whereas the other strives to prevent it. Using the technique of set-valued maps and their selections, sufficient conditions for reaching the terminal set in a finite time are derived. Theoretical results are supported by a model example.


Introduction
Although the concept of fractional integro-differentiation has been known since XVIII century, Fractional Calculus has become a very active field of research in the recent decades [1,12,16,26,27]. The reason for this is that a number of real-world phenomena and processes, such as viscoelasticity, anomalous diffusion in porous media, and Institute of Informatics, University of Gdańsk, Wita Stwosza 57, 80-308 Gdańsk, Poland supercapacitors to name a few, can be accurately described using fractional derivatives and integrals.
Fractional differential equations (FDEs) provide a powerful tool to describe memory effect and hereditary properties of various materials and processes. While linear systems of FDEs represent a fairly well investigated field of research, relatively few papers deal with nonstationary fractional-order systems described by linear FDEs with variable coefficients. Meanwhile, a number of real-life systems and processes can be described by linear FDEs with variable coefficients, e.g. linearized aircraft models, linearized models of population restricted growth, models related to the distribution of parameters in the charge transfer and the diffusion of the batteries etc.
Linear differential equations with variable coefficients arise in a natural way when modeling RLC-circuits with variable capacitance or inductance. With the advent of electronic components like super-capacitors (also called ultracapacitors) and fractances, one should employ fractional differential equations for circuit models. This provides motivation for research on FDEs with variable coefficients as well as related control and game problems.
Explicit solutions to linear systems of differential equations provide basis to perform stability analysis and to solve control and game-theoretical problems. Explicit solutions to linear systems of differential equations are usually expressed in terms of state transition matrix. In the case of FDEs with constant coefficients the state transition matrix can be represented using the matrix Mittag-Leffler function. Recently explicit solutions for the linear systems of FDEs with variable coefficients were obtained by the authors using state transition matrices expressed in terms of generalized Peano-Baker series.
Differential games with fractional derivatives were investigated for the first time in the papers [7,8] by Chikrii and Eidelman, who developed a generalization of the Method of Resolving Functions [6]. Papers [9,[28][29][30][31][32][33] represent a further development of this research. Differential games with fractional dynamics were examined in the papers [18,19] by Gomoyunov on the basis of Hamilton-Jacobi-Bellman equation. Pursuit-evasion games described by FDEs and involving multiple pursuers and one evader are treated in the papers [35,36]. This paper deals with differential games described by the systems of linear FDEs with variable coefficients involving Riemann-Liouville and Caputo derivatives. The game problem is treated from convex-analytical viewpoint using the method of resolving functions. On the basis of the resolving functions method sufficient conditions for the finite-time game termination from given initial states are derived. Theoretical results are supported by an illustrative example.

Fractional integrodifferentiation
Denote by R n the n-dimensional Euclidean space and by I some interval of the real line, I ⊂ R. In what follows we will assume that I = [t 0 , T ] for some T > t 0 and denoteI = (t 0 , T ). Let α, α > 0, and denote [α] and {α} integral and fractional part of α, respectively, i.e. α = [α] + {α}. Suppose the function f , f : I → R n is absolutely continuous along with its derivatives up to the order [α]. Let us recall that the Riemann-Liouville (left-sided) fractional integral and derivative of order α are defined as follows [39]: Hereafter, Γ (·) stands for the Gamma function defined by The Riemann-Liouville fractional derivative of a constant does not equal zero. Moreover, it becomes infinite as t approaches t 0 and due to this fact the FDEs with Riemann-Liouville derivative require initial conditions of special form lacking clear physical meaning. That is why the regularized Caputo derivative was introduced, which is free from these shortcomings.
The Caputo (regularized) derivative of fractional order α, m − 1 ≤ α < m, can be introduced by the following formula: The following properties of the fractional integrals and derivatives [22,37] will be used in the sequel.

Nonstationary linear systems with Riemann-Liouville fractional derivatives
In what follows we assume 0 < α < 1. Let us consider the following initial value problem: hereafter it is assumed that x(t) is a vector function taking values in R n and the matrix function A(·) ∈ L ∞ (I , R n×n ). By a solution to the problem (2.12), (2.13), we mean the function x(·) ∈ C(I ), satisfying the condition (2.13) and the equation (2.12) a.e. onI . According to [13,20], the problem (2.12), (2.13), has a unique solution x(t).
As shown in [13], Cauchy-type problem (2.12), (2.13), is equivalent to the following Volterra integral equation: (2.14) Applying Picard iterations to (2.14), we find that x(t) is a limit of the sequence: where I stands for an identity matrix. Repeating iterations in (2.15) infinitely, we find that where the state-transition matrix Φ(t, t 0 ) is defined as follows: We will refer to the series on the right-hand side of (2.17) as the generalized Peano-Baker series [3,15].
The following lemma was presented in [32].

Lemma 3
The state-transition matrix Φ(t, t 0 ) satisfies the following initial value problem On the other hand, the following lemma also holds true.

Lemma 4
Let the matrix function Φ(t, t 0 ) be a solution to the initial value problem (2.18). Then Φ(t, t 0 ) can be represented in the form of the generalized Peano-Baker series (2.17).
Proof As shown in [13], the initial value problem (2.18) is equivalent to the following Volterra integral equation: By means of a formal Picard iteration, this leads to the desired representation in the form of the generalized Peano-Baker series (2.17).
, it is bounded and continuous. This implies Φ(t, s) is measurable with respect to t and locally integrable with respect to s on [t 0 , t] for any t ∈ I .

Remark 2 If A(t)
is a constant matrix, i.e. A(t) ≡ A, then in view of (2.6) one gets is the matrix α-exponential function [22]. Equation (2.16) takes on the form which is consistent with the formulas, obtained for the systems of fractional differential equations with constant coefficients, [22,28].

Now consider the inhomogeneous linear initial value problem
We assume u : I → U ⊂ R n to be measurable on I , taking values from a nonempty compact set U ⊂ R n . Thus, u(·) ∈ L ∞ (I , U ). As before, a function x(·) ∈ C(I ), satisfying the condition (2.22) and the equation (2.21) a.e. on I , we call a solution to the problem (2.21), (2.22).

Theorem 1
The initial value problem (2.21), (2.22) has a unique solution, continuous onI , which can be written down as follows: is essentially bounded, the right-hand side of (2.21) fulfils a Lipschitz condition and, according to [13], there exists a unique solution to the initial value problem (2.21), (2.22), which is continuous onI . According to [4], solution to (2.21), (2.22) has the form where the matrix function Z (t, τ ) satisfies the initial value problem (2.18). In view of the uniqueness of solution to (2.18), we have Z (t, τ ) = Φ(t, τ ).

Remark 3 Since the Mittag-Leffler function becomes equal to an exponential when
is the matrix exponential defined as the sum of the following convergent series and expression (2.23) yields the well-known explicit formula for the solution of the integer-order Cauchy probleṁ

Nonstationary linear systems with Caputo fractional derivatives
We now examine homogeneous linear FDEs with variable coefficients involving Caputo derivatives. Let us consider the following initial value problem: where again the matrix function By a solution to the problem (2.26), (2.27), we mean the function x(·) ∈ C(I ), satisfying the condition (2.27) and the equation (2.26) a.e. on I . According to [13], the problem (2.26), (2.27), has a unique solution x(t).
Applying an iterative process similar to (2.15), we find that the state-transition matrix of the system (2.26) is defined as follows: Again, we will refer to the series on the right-hand side of (2.28) as the generalized Peano-Baker series [3,15].
In view of Lemma 1 and of (2.10), (2.11), the following lemma holds true.

Lemma 6
The state-transition matrix Ψ (t, t 0 ) satisfies the following initial value problem By a solution to the problem (2.26), (2.27), we mean the function x(·) ∈ J α t 0 (L 1 ), satisfying the condition (2.27) and the equation (2.26) a.e. on I .
Lemma 6 implies the following Theorem 2 Solution to the initial value problem (2.26), (2.27) is given by the following expression: Lemma 7 Let the matrix function Ψ (t, t 0 ) be a solution to the initial value problem (2.29). Then Ψ (t, t 0 ) can be represented in the form of the generalized Peano-Baker series (2.28).
The proof of this lemma is similar to that of Lemma 4.

Remark 4 If A(t) is a constant matrix, i.e. A(t) ≡
A, then in view of (2.6) one gets Equation (2.30) takes on the form which is consistent with the formulas, obtained for the systems of fractional differential equations with constant coefficients [22,28].

Now we consider the inhomogeneous linear initial value problem
As before, we assume u : I → U ⊂ R n to be measurable on I , with values from a nonempty compact set U ⊂ R n , hence, u(·) ∈ L ∞ (I , U ). By a solution to the problem (2.31), (2.32), we mean the function x(·) ∈ C(I ), satisfying the condition (2.32) and the equation (2.31) a.e. on I .

Theorem 3
There exists a unique solution to the initial value problem (2.31), (2.32), continuous on I , which has the form:

Remark 6
In view of Remark 4, since the Mittag-Leffler function becomes equal to an exponential when α = 1, i.e.
, for A(t) ≡ A = const and α = 1 we obtain Ψ (t, t 0 ) = e A(t−t 0 ) and expression (2.33) yields the same explicit formula (2.24) for the solution of the integer-order Cauchy problem (2.25).

Game problem statement
In this section a statement for the problem of approaching the terminal set will be given for the differential games, the dynamics of which is described by nonstationary equations involving fractional derivatives of either Riemann-Liouville or Caputo type. Now let I = [t 0 , ∞). Consider a system whose evolution is defined by the following nonstationary linear fractional differential equations: Hereafter D α stands for the operator of fractional differentiation in the sense of Riemann-Liouville or Caputo. It will be clear from the context which type of the fractional differentiation operator is meant. Here, as before, The control unit is specified by the vector-valued function ϕ(t, u, v), ϕ : I × U × V → R n satisfying the Carathéodory condition, i.e. it is measurable in t and jointly continuous with respect to (u, v) ∈ U × V , where u and v, u ∈ U , v ∈ V , are control parameters of the first and second players respectively, and the control sets U and V are from the set K (R k ) of all nonempty compact subsets of R k . Moreover, we assume that ϕ(·, ·, ·) is essentially bounded for all t When D α is the operator of fractional differentiation in the sense of Riemann-Liouville, i.e. D α ≡ t 0 D α t , the initial conditions for the process (3.1) are given in the form (2.22). When the derivative in (3.1) is understood in Caputo's sense, the initial conditions are of the form (2.32) and x(t 0 ) =x 0 . In what follows the initial conditions are assumed as fixed. We will denote by x 0 the initial time-state of the system (3.1), i.e.
Along with the process dynamics (3.1) and the initial conditions a terminal set of cylindrical form is given where M 0 is a linear subspace of R n , M ∈ K (L), and L = M ⊥ 0 is the orthogonal complement of the subspace M 0 in R n .
When the controls of the both players are chosen in the form of Lebesgue measurable functions u(t) and v(t) taking values from U and V , respectively, the Cauchy problem for the process (3.1) with corresponding initial values has a unique continuous solution in view of Theorems 1 and 3.
Consider the following dynamic game. The first player aims to bring a trajectory of the system (3.1) to the set (3.2), while the other player strives to delay the moment of hitting the terminal set as much as possible.
We assume that the second player's control is an arbitrary measurable function v(t) taking values from V , and the first player at each time instant t, t ≥ t 0 , forms her control on the basis of information about initial state x 0 and history v t (·) = {v(s) : v(s) ∈ V , s ∈ [t 0 , t]} of the control v(t): Therefore, u(t) represents a quasi-strategy [24]. By solving the problem stated above we employ the Method of Resolving Functions [6,34].

Method of Resolving Functions
Before we proceed let us formulate some results from the theory of set-valued maps, that will be used in the sequel.
Consider a set-valued map G(τ ), G : I → K (R n ), where K (R n ) is the set of all nonempty compacts (closed and bounded subsets of R n ). The following theorem can be found in [5] and is useful in integrating set-valued maps. In view of the properties of the Lebesgue integral, the interval I can be assumed either open or closed, without loss of generality. In what follows for any G ∈ K (R n ) we denote |G| = sup g∈G g .

Theorem 4 Let the set-valued map G(τ ) be measurable and satisfy the inequality |G(τ )| ≤ k(τ ), τ ∈ I , where k(τ ) is some scalar valued function integrable over I . Then the integral I G(τ )dτ is a convex compact set in R n .
The integral I G(τ )dτ is to be thought of in the sense of Aumann [21], i.e. as the set of integrals of all measurable selections of the set-valued map G(τ ).
Let us recall that The L×B-measurability of a function f (t, u), f : [11,Definition 6.33] means that the functions (and the sets) to which the property applies are measurable with respect to the σ -algebra generated by the product Let Π be the orthoprojector from R n onto L.
It should be noted that in view of Theorems 1, 3, the system (3.1) is equivalent to the integral equation where g(t) stands for the solution of homogeneous system D α x(t) = A(t)x(t), obtained from (3.1) by setting ϕ(t, u, v) ≡ 0. Thus, when D α is the Riemann-Liouville fractional differentiation operator (D α ≡ t 0 D α t ), in view of (2.16), we have For the Caputo regularized fractional derivative (D α ≡ t 0 D (α) t ), by (2.30) we have In the sequel we will follow the general scheme of the method of resolving functions [34].
Set ϕ(τ, U , v) = {ϕ(τ, u, v) : u ∈ U }, τ ∈ I , v ∈ V , and consider set-valued maps defined on the sets Δ × V and Δ, respectively. The condition is usually referred to as Pontryagin's condition. This condition reflects some kind of first player's advantage in resources over the second player.
Let us introduce the function where γ (t, ·) is a certain fixed selection, integrable on I . Consider the set-valued mapping Let us study its support function in the direction of +1 This function is called the resolving function [6].
It should be noted that for Assume Property 1 to hold true and denote (4.5)

Remark 7
If Property 1 is not true, the set T can be defined as where Ω V (I ) = {v(·) ∈ L 1 (I ) : v(t) ∈ V , t ∈ I } is the set of all functions integrable on I and taking values in V . The latter definition is valid, since the function ρ(t, τ, v) is superpositionally measurable as L × B-measurable with respect to τ , v [11].
If ξ(t) ∈ M, then ρ(t, τ, v) = +∞ for τ ∈ [t 0 , t] and in this case it is natural to set the value of the integral in (4.5) to be equal to +∞. Then the inequality in (4.5) is fulfilled by default. In the case when the inequality in braces in (4.5) fails for all t > t 0 , we set T = ∅. Let T ∈ T = ∅.
Let us recall that a set S ⊂ R n is referred to as star-shaped with respect to s 0 ∈ S if for any s ∈ S the line segment from s 0 to s lies in S.

Remark 8
In Property 2 the assumption of convex-valuedness of R(T , τ, v) can be replaced with the weaker one that R(T , τ, v) has values star-shaped with respect to the origin. Since ρ(T ) ≥ 1 due to (4.5) and Property 2 is fulfilled, the function ρ * (T , τ ), Consider the multivalued mapping In view of [10,Corollary 3], it is L × B-measurable; therefore, according to the measurable-selection theorem [2, p. 308], the multivalued mapping U (τ, v) contains at least one L × B-measurable selection u(τ, v), which, in turn, is a superpositionally measurable function. Set the first player's control to be u In the case when ξ(T ) ∈ M we construct the first player's control as follows. Let us set ρ * (T , τ ) ≡ 0 in (4.6) and denote by U 0 (τ, v) the set-valued mapping obtained in such a way from U (τ, v). Let us choose the first player's control in the Let us show that in each case treated above the trajectory of the process (3.1) hits the terminal set at the time instant T .
From (2.23), (2.33), we have Consider the case ξ(T ) / ∈ M. Let us add and subtract from the right-hand side of (4.7) the following vector Taking into account the control rule of the first player, we obtain from (4.7) the following inclusion Since M is a convex compact set and ρ * (T , τ ) is a non-negative function and [38, p. 342]. Again, the integral of the set valued map G(τ ) = ρ * (T , τ )M is to be thought of in the sense of Aumann [5,21], i.e. as the set of integrals of all measurable selections of the set-valued map G(τ ). Hence the inclusion Π x(T ) ∈ M. Now assume ξ(T ) ∈ M. Adding and subtracting from the right-hand side of (4.7) the vector (4.8) and taking into account the first player's control rule we immediately obtain the required inclusion Π x(T ) ∈ M.

Example
Here we consider an example illustrating the theoretical results. Let us study a differential game described by the following system of fractional differential equations: We also assume that |u| ≤ a, for some a > 1, and |v| ≤ 1.
The system (5.1) can be written down in matrix form as follows: under the initial condition Direct calculation yields that . Now assume the terminal set M * = {x ∈ R 2 : x 1 = 0}. Then M 0 = {x ∈ R 2 : x 1 = 0}, M = {x ∈ R 2 : x 1 = x 2 = 0}, L = {x ∈ R 2 : x 2 = 0}, and the orthoprojector from R 2 onto L is given by the matrix In view of (4.2), (4.3), we have and Pontryagin's condition (4.4) is fulfilled as a > 1. Let γ (t, τ ) ≡ 0. Then x 0 2 and assume that x 0 1 , x 0 2 are such that Now consider the resolving function which can be found as the greatest root of the following equation: Solving this equation yields that and the time T of game termination can be found from the equation: Denote ξ 2 (t) = x 0 1 + t α+1 Γ (α+2) x 0 2 and assume that x 0 1 , x 0 2 are such that Now consider the resolving function which can be found as the greatest root of the following equation: Solving this equation yields that which is equivalent to Dividing the both sides by T α+1 yields (a − 1)(α + 1)T α = Γ (2α + 2) 1 T α+1 x 0 1 + 1 Γ (α + 2) x 0 2 .
As T → 0, the left-hand side of this equation approaches zero, while the right-hand side tends to infinity. As T grows to infinity, the left-hand side also increases without bound, while the right-hand side approaches the finite limit of Γ (2α+2) Γ (α+2) |x 0 2 |. Thus, there exists a finite solution T < ∞ to the equation, which means the game can be terminated at the finite time T .
In view of (5.4), (5.5), (5.9), and (5.10), it follows from (4.6) that for both the Riemann-Liouville and Caputo cases the control of the first player that guarantees game termination at time T has the form u(τ ) ≡ a, τ ∈ [t 0 , T ].

Conclusions
A new class of differential games described by linear nonstationary fractional differential equations is studied in the paper. Sufficient conditions for the game termination in a finite time are derived in the framework of the method of resolving functions. The feasibility of the proposed approach is demonstrated by an example.
In further research the authors plan to adapt the developed method for the games with terminal payoff functional.

Conflict of interest
The authors declare that they have no conflict of interest.
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