Saddle-point solution to zero-sum games subject to noncausal systems

Abstract

A singular system, assumed to possess both regularity and freedom from impulses, is categorized as a causal system. Noncausal systems (NSs) are a class of singular systems anticipated to exhibit regularity. This study focuses on investigating zero-sum games (ZSGs) in the context of NSs. We introduce recurrence equations grounded in Bellman’s optimality principle. The saddle-point solution for multistage two-player ZSGs can be obtained by solving these recurrence equations. This methodology has demonstrated its effectiveness in addressing two-player ZSGs involving NSs. Analytical expressions that characterize saddle-point solutions for two types of two-player ZSGs featuring NSs, encompassing both linear and quadratic control scenarios, are derived in this paper. To enhance clarity, we provide an illustrative example that effectively highlights the utility of our results. Finally, we apply our methodology to analyze a ZSG in the realm of environmental management, showcasing the versatility of our findings.

Keywords

Zero-sum game noncausal system saddle-point solution recurrence equations

1 Introduction

A zero-sum game (ZSG) is a non-cooperative game [1] where the sum of gains and losses among the participating players equals zero. Zero-sum games (ZSGs) represent a significant research domain within modern game theory [2]. Von Neumann and Morgenstern [3] developed the theory of ZSG, and introduced axiomatic utility theory to analyze economic issues. The theory of ZSGs has garnered considerable attention among scholars due to their practical implications. The contributions of Von Neumann, Nash, and numerous other researchers have been instrumental in advancing the ZSG theory. In contemporary times, ZSGs have emerged as a focal point of extensive research, yielding substantial theoretical insights [4, 5] and demonstrating broad application prospects [6, 7].

Scholars have examined ZSGs within the context of various system types. References [8, 9] demonstrated that the saddle-point solutions of ZSGs subject to ordinary differential equations may be obtained by solving Hamilton-Jacobi equations and provided algorithms for finding the solution of the Hamilton-Jacobi equations. Using backward stochastic differential equations, El-Karoui and Hamadene [10] studied the existence of a risk-sensitive ZSG subject to a stochastic differential equation with the Isaacs’ condition. Utilizing uncertain differential equations as a foundation, Yang and Gao [11] established the max-min theorem for a linear-quadratic ZSG. They successfully applied this theorem to address an economic concern related to resource extraction. Building upon on optimal control technique, Li et al. [12] conducted further investigations on a linear-quadratic ZSG. They employed a parametric approximate optimization method to simplify the forms of the saddle-point solution, thus advancing the understanding of this game. Most of these studies primarily revolve around continuous-time ZSGs, as their systems are described using ordinary or partial differential equations [8 –12]. Discrete-time systems have widespread applications in practical problems such as battery life prediction [13 –15], profit allocation [16], and portfolio analysis [17]. As a result, Sun et al. [18] extended their exploration to discrete-time ZSGs by utilizing difference equations. They demonstrated that the saddle-point solutions of ZSGs under uncertain difference equations can be obtained by solving recurrence equations. Moreover, Sun et al. [19] introduced a hybrid intelligent algorithm for resolving recurrence equations. The majority of the aforementioned studies are centered on ZSGs associated with normal systems. These pivotal findings serve as a foundation for future research aimed at investigating ZSGs within the framework of singular systems.

The study of singular systems originated in the late 1970s when Rosenbrock [20] introduced a model of a singular system while discussing complex electrical network systems. Shortly thereafter, Luenberger and Arbel [21] identified that certain models used in economic theory belonged to the category of singular systems. Since then, research on singular systems has continued to advance, steadily gaining momentum and flourishing. Cobb [22] introduced the concepts of controllability and observability for singular systems, while the admissibility of singular switching systems was defined by Darouach and Chadli [23]. The progress in the study of singular systems has been remarkable in recent years [24 –26]. Linear matrix inequalities have been utilized to analyze stability in singular systems [27]. A singular system that satisfies both regularity and impulse-free conditions is referred to as a causal system, while a singular system that only satisfies regularity conditions is categorized as a noncausal system [28]. Based on causal systems, references [29, 30] created appropriate equations for addressing optimal control issues using Bellman’s optimality principle. Based on noncausal systems (NSs), recurrence equations have been suggested to deal with optimal control issues [31, 32]. ZSGs are essentially multi-player multi-objective optimal control issues. The investigation of two-player ZSGs based on singular systems has given several successful outcomes [29 , 33].

The majority of current research on ZSGs subject to singular systems was based on the premise that the singular systems are regular and impulse-free [5, 30]. Singular systems adhering to both regularity and impulse-free conditions demonstrate adherence to the causality rule. This study delves deeper by exploring a more advanced system, referred to as a noncausal system (NS), which arises from singular systems with regular assumption along. Noncausal singular systems characterize a way of diverse connections between entities in the real world and have practical backgrounds. For instance, in economic theory, both the Leontief dynamic input-output model [21] and the Von Neumann model [34] under certain conditions can be regarded as typical examples of noncausal discrete-time systems. Moreover, in circuit networks, noncausal continuous-time systems can be realized through appropriate design of differential operators [35]. These examples shown that noncausal systems play a significant role in our life. Studying NSs is both important and challenging. Despite the difficulties encountered during the investigation, a few solutions for two-player ZSGs with NSs have been developed. This study expands the realm of ZSGs by exploring two-player ZSGs subject to discrete-time NSs. The key contributions and advantages are outlined as follows:

We employ an algebraic transformation approach to convert NSs into subsystems in Section 2. This transformation effectively emphasizes noncausal characteristics, making the subsystems more amenable to analysis. It is noteworthy that the singular systems discussed in [36, 37] can be regarded as particular instances of NSs.

ZSGs for NSs are transformed into equivalent ZSGs. Then we develop recurrence equations specifically tailored for ZSGs involving NSs. In comparison to [5 , 31], Section 3 presents recurrence equations that have the potential to address a wider range of problems related to singular systems.

We employ recurrence equations to derive analytical expressions for the saddle-point equilibria of ZSGs. This achievement, influenced by the research in [19 , 31], is elaborated on in Sections 4 and 6. It is worth highlighting that these saddle-point solutions for two-player ZSGs are precisely articulated, facilitating their implementation.

The following sections of this paper are organized as follows: In Section 2, we present a concise introduction to a noncausal system (NS) and explore its corresponding equivalent subsystems. Section 3 demonstrates the derivation of saddle-point solutions for ZSGs under NS conditions using recurrence equations. We extend this approach to ZSGs involving NSs with linear control in Section 4, deriving analytical solutions of equilibrium results. Furthermore, we provide a step-by-step procedure to solve an illustrative example in Section 5 to demonstrate the effectiveness of our approach. Finally, we employ zero-sum game analysis to address environmental management issues in Section 6, proposing fund allocation strategies by utilizing recurrence equations.

In this paper, we adopt the following notation conventions: $ℝ^{m_{y}}$ and $ℝ^{m_{y} \times n_{z}}$ represent the m_y-dimensional real Euclidean space and the set of all m_y × n_z matrices, respectively. For a vector α = (α₁, α₂, ⋯ , α_p) ^T, we have α^T = (α₁, α₂, ⋯ , α_p), $∣ ∣ α ∣ ∣_{1} = \sum_{l = 1}^{γ} α_{l}$ , and ( α) _l = α_l. I _ω symbolizes the unit matrix with dimensions ω × ω.

2 Singular system

To begin with, a singular system is illutrated below:

$\begin{matrix} H y (ξ + 1) = A y (ξ) + B w (ξ) + D v (ξ), \\ ξ = 0, 1, 2, \dots, Ξ - 1, \end{matrix}$ (1) where $y (ξ) \in ℝ^{m}$ , $w (ξ) \in ℝ^{n_{1}}$ , and $v (ξ) \in ℝ^{n_{2}}$ . $H, A \in ℝ^{m_{y} \times m_{y}}, B \in ℝ^{m_{y} \times n_{1}}, D \in ℝ^{m_{y} \times n_{2}}$ are deterministic matrices with rank H = q < m.

Definition 2.1. (Dai [28]) The system (1) is termed regular if det(x H - Λ ) 0, and it is considered impulse-free if deg(det(x H - Λ )) = rank H , where x is a complex number, det represents the determinant of a matrix, and deg refers to the degree of a function. The system (1) qualifies as a causal system when it satisfies both regularity and the impulse-free condition. Conversely, the system (1) qualifies as a noncausal system when it maintains regularity but does not adhere to the impulse-free criterion.

Example 2.1. (Causal System) We investigate a singular system $\begin{matrix} {\begin{matrix} H y (ξ + 1) = A y (ξ) + B w (ξ) + D v (ξ), \\ ξ = 0, 1, 2, \dots, Ξ - 1, \end{matrix} \end{matrix}$ (2) with $\begin{matrix} H = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], A = [\begin{matrix} 1 & 1 & 0 & 2 \\ 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}] . \end{matrix}$ (3) We have $\begin{matrix} det (x H - A) = det [\begin{matrix} x - 1 & - 1 & 0 & - 2 \\ - 1 & - 1 & - 1 & - 1 \\ - 1 & 0 & x & 0 \\ 0 & - 1 & 0 & x \end{matrix}] \\ = - x^{3} - x^{2} - 2 x - 20, \end{matrix}$ (4) and deg(det(x H - A )) = rank ( H ) =3. Consequently, the system (5) falls under the category of causal systems. That is a causal system.

Example 2.2. (Noncausal System) For a singular system $\begin{matrix} {\begin{matrix} H y (ξ + 1) = A y (ξ) + B w (ξ) + D v (ξ), \\ ξ = 0, 1, 2, \dots, Ξ - 1, \end{matrix} \end{matrix}$ (5) with $\begin{matrix} H = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & - 1 \\ 0 & 0 & 0 & 0 \end{matrix}], A = [\begin{matrix} 1 & 1 & 0 & - 2 \\ 0 & - 1 & 1 & 0 \\ 0 & 0 & 2 & - 1 \\ 0 & 0 & 0 & - 1 \end{matrix}] . \end{matrix}$ (6) We have $\begin{matrix} det (x H - A) = det [\begin{matrix} x - 1 & - 1 & 0 & 2 \\ 0 & 1 & x - 1 & 0 \\ 0 & 0 & x - 2 & - x + 1 \\ 0 & 0 & 0 & 1 \end{matrix}] \\ = x^{2} - 3 x + 20, \end{matrix}$ (7) and deg(det(x H - A )) =2 < rank ( H ) =3. Based on Definition 2.1, it can be inferred that the system (5) satisfies regularity conditions but lacks impulse-free conditions. Consequently, the system (5) is classified as a NS.

For a NS (1), Dai [28] suggested that there exist nonsingular matrices $F_{nc}, M_{nc} \in ℝ^{m_{y} \times m_{y}}$ such that

$F_{nc} H M_{nc} = [\begin{matrix} I_{q} & 0 \\ 0 & H_{2} \end{matrix}]$ (8) where $H_{2} \in ℝ^{(m_{y} - q) \times (m_{y} - q)}$ is a nilpotent matrix, and $l_{min} = min {l ∣ H_{2}^{l} = 0, l \in Z^{+}}$ is the nilpotent. Then the system (1) is equivalent to the system (9):

${\begin{matrix} {\tilde{y}}_{1} (ξ + 1) = A_{1} {\tilde{y}}_{1} (ξ) + B_{1} w (ξ) + D_{1} v (ξ), \\ {\tilde{y}}_{2} (ξ) = H_{2} {\tilde{y}}_{2} (ξ + 1) - B_{2} w (ξ) - D_{2} v (ξ), \\ ξ = 0, 1, 2, \dots, Ξ - 1, \end{matrix}$ (9) where

$\begin{matrix} M_{nc}^{- 1} y (ξ) = [\begin{matrix} {\tilde{y}}_{1} (ξ) \\ {\tilde{y}}_{2} (ξ) \end{matrix}], F_{nc} A M_{nc} = [\begin{matrix} A_{1} & 0 \\ 0 & I_{m_{y} - q} \end{matrix}], \\ F_{nc} B = [\begin{matrix} B_{1} \\ B_{2} \end{matrix}], F_{nc} D = [\begin{matrix} D_{1} \\ D_{2} \end{matrix}], \end{matrix}$ (10) and ${\tilde{y}}_{1} (ξ) \in ℝ^{q}, {\tilde{y}}_{2} (ξ) \in ℝ^{m_{y} - q}, A_{1} \in ℝ^{q \times q}, B_{1} \in ℝ^{q \times n_{1}}, D_{1} \in ℝ^{q \times n_{2}}, B_{2} \in ℝ^{(m_{y} - q) \times n_{1}}, D_{2} \in ℝ^{(m_{y} - q) \times n_{2}}$ .

Remark 2.1. The first Equation in (9) represents a forward difference equation, i.e., $\begin{matrix} {\tilde{y}}_{1} (Ξ) = A_{1}^{Ξ - ξ} {\tilde{y}}_{1} (ξ) + \sum_{i = 0}^{Ξ - ξ - 1} A_{1}^{i} B_{1} w (Ξ - 1 - i) \\ + \sum_{i = 0}^{Ξ - ξ - 1} A_{1}^{i} D_{1} v (Ξ - 1 - i), \\ ξ = 0, 1, 2, \dots, Ξ - 1 . \end{matrix}$ (11) In contrast to the first Equation in (9), the second Equation in (9) represents a backward difference equation, i.e., $\begin{matrix} {\tilde{y}}_{2} (ξ) = H_{2}^{Ξ - ξ} {\tilde{y}}_{2} (Ξ) - \sum_{i = 0}^{Ξ - ξ - 1} H_{2}^{i} B_{2} w (ξ + i) \\ - \sum_{i = 0}^{Ξ - ξ - 1} H_{2}^{i} D_{2} v (ξ + i), ξ = 0, 1, 2, \dots, Ξ - 1 . \end{matrix}$ (12)

It’s worth noting that this violates the rule of causality [28], as the update of the substate ${\tilde{y}}_{2} (ξ)$ involves the use of the terminal substate ${\tilde{y}}_{2} (Ξ)$ and the controls w (ξ) , w (ξ+ 1) , ⋯, w (Ξ - 1) , v (ξ) , v (ξ+ 1) , ⋯, v (Ξ - 1).

Remark 2.2. References [36, 37] explored singular systems adhering to regularity and impulse-free conditions, representing causal systems in essence. When the noncausal system (1) satisfies the impulse-free condition, it transforms into a causal system, and its equivalent formulation is expressed as

${\begin{matrix} {\bar{y}}_{1} (ξ + 1) = {\bar{A}}_{1} {\bar{y}}_{1} (ξ) + {\bar{B}}_{1} w (ξ) + {\bar{D}}_{1} v (ξ), \\ {\bar{y}}_{2} (ξ) = - {\bar{B}}_{2} w (ξ) - {\bar{D}}_{2} v (ξ), \\ ξ = 0, 1, 2, \dots, Ξ - 1, \end{matrix}$ (13) owing to deg(det(x H - A )) = rank H , H ₂ = 0 , and l_min = 1. Hence, noncausal systems encompass a broader spectrum of systems compared to causal systems, underscoring the wider applicability of the main conclusions drawn in this paper. Additionally, these conclusions are not confined solely to noncausal systems; they can be derived by suitable degeneration for the corresponding causal systems as well.

From the Definition 2.1, and Examples 2.1–2.2, it can be seen that causal and noncausal systems are singular systems that satisfy different assumptions. The NS (1) is transformed into the system (9). In next sections, we will investigate ZSGs subject to NSs.

3 Zero-sum game

A two-player ZSG subject to the NS is presented:

$\begin{matrix} {\begin{matrix} J (y (0), y (Ξ), 0) = max_{\overset{w (ξ) \in W (ξ)}{0 \leq ξ \leq Ξ - 1}} min_{\overset{v (ξ) \in V (ξ)}{0 \leq ξ \leq Ξ - 1}} U (y (0), y (Ξ), w, v, 0) \\ = min_{\overset{v (ξ) \in V (ξ)}{0 \leq ξ \leq Ξ - 1}} max_{\overset{w (ξ) \in W (ξ)}{0 \leq ξ \leq Ξ - 1}} U (y (0), y (Ξ), w, v, 0), \\ s . t . \\ H y (ξ + 1) = A y (ξ) + B w (ξ) + D v (ξ), \\ ξ = 0, 1, 2, \dots, Ξ - 1, \end{matrix} \end{matrix}$ (14) with the objective function

$\begin{matrix} U (y (0), y (Ξ), w, v, 0) \\ = \sum_{ξ = 0}^{Ξ - 1} φ_{ξ} (y (ξ), w (ξ), v (ξ), ξ) + φ_{Ξ} (y (Ξ), Ξ), \end{matrix}$ (15) where $φ_{ξ} : ℝ^{m_{y}} \times ℝ^{n_{1}} \times ℝ^{n_{2}} \times ℝ \to ℝ$ and $φ_{Ξ} : ℝ^{m_{y}} \times ℝ \to ℝ$ are real-valued functions. The control sequence $w = {w (ξ) \in W (ξ) \subset ℝ^{n_{1}}, 0 \leq ξ \leq Ξ - 1}$ seeks to maximize the objective function U ( y (0) , y (Ξ) , w , v , 0), whereas the control sequence $v = {v (ξ) \in V (ξ) \subset ℝ^{n_{2}}, 0 \leq ξ \leq Ξ - 1}$ is dedicated to the minimization of the objective function. The two-player ZSG (14) is identical to the following game:

$\begin{matrix} {\begin{matrix} J ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), 0) = max_{\overset{w (ξ) \in W (ξ)}{0 \leq ξ \leq Ξ - 1}} min_{\overset{v (ξ) \in V (ξ)}{0 \leq ξ \leq Ξ - 1}} \tilde{U} ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), w, v, 0) \\ = min_{\overset{v (ξ) \in V (ξ)}{0 \leq ξ \leq Ξ - 1}} max_{\overset{w (ξ) \in W (ξ)}{0 \leq ξ \leq Ξ - 1}} \tilde{U} ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), w, v, 0), \\ s . t . \\ {\tilde{y}}_{1} (ξ + 1) = A_{1} {\tilde{y}}_{1} (ξ) + B_{1} w (ξ) + D_{1} v (ξ), \\ {\tilde{y}}_{2} (ξ) = H_{2} {\tilde{y}}_{2} (ξ + 1) - B_{2} w (ξ) - D_{2} v (ξ), \\ ξ = 0, 1, 2, \dots, Ξ - 1, \end{matrix} \end{matrix}$ (16) with $\begin{matrix} \tilde{U} ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), w, v, 0) \\ = \sum_{ξ = 0}^{Ξ - 1} φ_{ξ} (M_{nc, 1} {\tilde{y}}_{1} (ξ) + M_{nc, 2} {\tilde{y}}_{2} (ξ), w (ξ), v (ξ), ξ) \\ + φ_{Ξ} (M_{nc, 1} y_{1} (Ξ) + M_{nc, 2} y_{2} (Ξ), Ξ) \\ = \sum_{ξ = 0}^{Ξ - 1} {\tilde{φ}}_{ξ} ({\tilde{y}}_{1} (ξ), {\tilde{y}}_{2} (Ξ), w (ξ), w (ξ + 1), \dots, \\ w (Ξ - 1), v (ξ), v (ξ + 1), \dots, v (Ξ - 1), ξ) \\ + {\tilde{φ}}_{Ξ} ({\tilde{y}}_{1} (Ξ), {\tilde{y}}_{2} (Ξ), Ξ) \end{matrix}$ (17) where

$\begin{matrix} {\tilde{φ}}_{ξ} ({\tilde{y}}_{1} (ξ), {\tilde{y}}_{2} (Ξ), w (ξ), w (ξ + 1), \dots, w (Ξ - 1), v (ξ), v (ξ + 1), \dots, v (Ξ - 1), ξ) \\ = φ_{ξ} (M_{nc, 1} {\tilde{y}}_{1} (ξ) + M_{nc, 2} (H_{2}^{Ξ - ξ} {\tilde{y}}_{2} (Ξ) - \sum_{i = 0}^{Ξ - ξ - 1} H_{2}^{i} [B_{2} w (ξ + i) + D_{2} v (ξ + i)]), w (ξ), v (ξ), ξ), \end{matrix}$ (18)

$\begin{matrix} {\tilde{φ}}_{Ξ} ({\tilde{y}}_{1} (Ξ), {\tilde{y}}_{2} (Ξ), Ξ) \\ = φ_{Ξ} (M_{nc, 1} {\tilde{y}}_{1} (Ξ) + M_{nc, 2} {\tilde{y}}_{2} (Ξ), Ξ) . \end{matrix}$ (19) and $M_{nc} = [M_{nc, 1} M_{nc, 2}], M_{nc, 1} \in ℝ^{m_{y} \times q}, M_{nc, 1} \in ℝ^{m_{y} \times (m_{y} - q)}$ . Denote $\begin{matrix} J ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), τ) \\ = max_{\overset{w (ξ) \in W (ξ)}{τ \leq ξ \leq Ξ - 1}} min_{\overset{v (ξ) \in V (ξ)}{τ \leq ξ \leq Ξ - 1}} \tilde{U} ({\tilde{y}}_{1} (τ) \tilde{,} y_{2} (Ξ), w, v, τ) \end{matrix}$ (20) $= min_{\overset{v (ξ) \in V (ξ)}{τ \leq ξ \leq Ξ - 1}} max_{\overset{w (ξ) \in W (ξ)}{τ \leq ξ \leq Ξ - 1}} \tilde{U} ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), w, v, τ),$ (21) with

$\begin{matrix} \tilde{U} ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), w, v, τ) \\ = \sum_{ξ = τ}^{Ξ - 1} {\tilde{φ}}_{ξ} ({\tilde{y}}_{1} (ξ), {\tilde{y}}_{2} (Ξ), w (ξ), w (ξ + 1), \dots, \\ w (Ξ - 1), v (ξ), v (ξ + 1), \dots, v (Ξ - 1), ξ) \\ + {\tilde{φ}}_{Ξ} ({\tilde{y}}_{1} (Ξ), {\tilde{y}}_{2} (Ξ), Ξ) . \end{matrix}$ (22) Notice that the terminal condition is $J ({\tilde{y}}_{1} (Ξ), {\tilde{y}}_{2} (Ξ), Ξ) = {\tilde{φ}}_{Ξ} ({\tilde{y}}_{1} (Ξ), {\tilde{y}}_{2} (Ξ), Ξ)$ .

Theorem 3.1. For each τ = Ξ - 1, Ξ - 2, ⋯ , 1, 0, the optimal value $J ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), τ)$ is governed by the following recurrence eqiations:

$\begin{array}{l} J ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), τ) = \\ \max_{w (τ) \in W (τ)} \min_{v (τ) \in V (τ)} {{\tilde{ϕ}}_{ξ} ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), w (τ), w^{*} (τ + 1), \\ \dots, w^{*} (Ξ - 1), v (τ), v^{*} (τ + 1), \dots, v^{*} (Ξ - 1), τ) \\ + J ({\tilde{y}}_{1} (τ + 1), {\tilde{y}}_{2} (Ξ), τ + 1)} \end{array}$ (23) $\begin{array}{l} \max_{w (τ) \in W (τ)} \min_{v (τ) \in V (τ)} {{\tilde{ϕ}}_{ξ} ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), w (τ), w^{*} (τ + 1), \\ \dots, w^{*} (Ξ - 1), v (τ), v^{*} (τ + 1), \dots, v^{*} (Ξ - 1), τ) \\ + J ({\tilde{y}}_{1} (τ + 1), {\tilde{y}}_{2} (Ξ), τ + 1)} \end{array}$ (24) $\begin{matrix} = {\tilde{φ}}_{ξ} ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), w^{*} (τ), w^{*} (τ + 1), \dots, w^{*} \\ (Ξ - 1), v^{*} (τ), v^{*} (τ + 1), \dots, v^{*} (Ξ - 1), τ) \\ + J ({\tilde{y}}_{1} (τ + 1), {\tilde{y}}_{2} (Ξ), τ + 1) \end{matrix}$ (25) where w ^* (τ) , w ^* (τ + 1) , ⋯ , w ^* (Ξ - 1) , v ^* (τ) , v ^* (τ + 1) , ⋯ , v ^* (Ξ - 1) are the equilibrium solution, and the terminal condition is $J ({\tilde{y}}_{1} (Ξ), {\tilde{y}}_{2} (Ξ), Ξ) = {\tilde{φ}}_{Ξ} ({\tilde{y}}_{1} (Ξ), {\tilde{y}}_{2} (Ξ), Ξ)$ .

Proof. Let ( w ^*, v ^*) be the saddle-point solution of game (16) where w ^* = ( w ^* (1) , w ^* (2) , ⋯ , w ^* (Ξ - 1)), and v ^* = ( v ^* (1) , v ^* (2), ⋯, v ^* (Ξ - 1)). Then w ^* and v ^* are the solutions of the optimal control issues (26) and (28), respectively.

$\begin{matrix} {\begin{matrix} J ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), 0) = max_{\overset{w (ξ) \in W (ξ)}{0 \leq ξ \leq Ξ - 1}} \tilde{U} ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), w, v^{*}, 0) \\ s . t . \\ {\tilde{y}}_{1} (ξ + 1) = A_{1} {\tilde{y}}_{1} (ξ) + B_{1} w (ξ) + D_{1} v (ξ), \\ {\tilde{y}}_{2} (ξ) = H_{2} {\tilde{y}}_{2} (ξ + 1) - B_{2} w (ξ) - D_{2} v (ξ), \\ ξ = 0, 1, 2, \dots, Ξ - 1, \end{matrix} \end{matrix}$ (26)

with $\begin{matrix} \tilde{U} ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), w, v^{*}, 0) \\ = \sum_{ξ = 0}^{Ξ - 1} {\tilde{φ}}_{ξ} ({\tilde{y}}_{1} (ξ), {\tilde{y}}_{2} (Ξ), w (ξ), w (ξ + 1), \dots, w \\ (Ξ - 1), v^{*} (ξ), v^{*} (ξ + 1), \dots, v^{*} (Ξ - 1), ξ) \\ + {\tilde{φ}}_{Ξ} ({\tilde{y}}_{1} (Ξ), {\tilde{y}}_{2} (Ξ), Ξ), \end{matrix}$ (27)

$\begin{matrix} {\begin{matrix} J ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), 0) = min_{\overset{v (ξ) \in V (τ)}{0 \leq ξ \leq Ξ - 1}} \tilde{U} ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), w^{*}, v, 0) \\ s . t . \\ {\tilde{y}}_{1} (ξ + 1) = A_{1} {\tilde{y}}_{1} (ξ) + B_{1} w (ξ) + D_{1} v (ξ), \\ {\tilde{y}}_{2} (ξ) = H_{2} {\tilde{y}}_{2} (ξ + 1) - B_{2} w (ξ) - D_{2} v (ξ), \\ ξ = 0, 1, 2, \dots, Ξ - 1, \end{matrix} \end{matrix}$ (28)

with $\begin{matrix} \tilde{U} ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), w^{*}, v, 0) \\ = \sum_{ξ = 0}^{Ξ - 1} {\tilde{φ}}_{ξ} ({\tilde{y}}_{1} (ξ), {\tilde{y}}_{2} (Ξ), w^{*} (ξ), w^{*} (ξ + 1), \dots, w^{*} \\ (Ξ - 1), v (ξ), v (ξ + 1), \dots, v (Ξ - 1), ξ) \\ + {\tilde{φ}}_{Ξ} ({\tilde{y}}_{1} (Ξ), {\tilde{y}}_{2} (Ξ), Ξ) . \end{matrix}$ (29) Applying the dynamic programming method to the optimal control issues (26) and (28) yields $\begin{matrix} J ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), τ) \\ = max_{w (τ) \in W (τ)} {{\tilde{φ}}_{ξ} ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), w (τ), w^{*} (τ + 1), \dots, \\ w^{*} (Ξ - 1), v^{*} (τ), v^{*} (τ + 1), \dots, v^{*} (Ξ - 1), τ) \\ + J ({\tilde{y}}_{1} (τ + 1), {\tilde{y}}_{2} (Ξ), τ + 1)} \end{matrix}$ (30) $\begin{matrix} = min_{v (τ) \in V (τ)} {{\tilde{φ}}_{ξ} ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), w^{*} (τ), w^{*} (τ + 1), \dots, \\ w^{*} (Ξ - 1), v (τ), v^{*} (τ + 1), \dots, v^{*} (Ξ - 1), τ) \\ + J ({\tilde{y}}_{1} (τ + 1), {\tilde{y}}_{2} (Ξ), τ + 1)} . \end{matrix}$ (31) It follows from the Equation (30) that

$\begin{matrix} J ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), τ) \\ \geq max_{w (τ) \in W (τ)} min_{v (τ) \in V (τ)} {{\tilde{φ}}_{ξ} ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), w (τ), w^{*} (τ + 1), \\ \dots, w^{*} (Ξ - 1), v (τ), v^{*} (τ + 1), \dots, v^{*} (Ξ - 1), τ) \\ + J ({\tilde{y}}_{1} (τ + 1), {\tilde{y}}_{2} (Ξ), τ + 1)} . \end{matrix}$ (32) According to the Equation (31), we obtain

$\begin{matrix} J ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), τ) \\ \leq min_{v (τ) \in V (τ)} max_{w (τ) \in W (τ)} {{\tilde{φ}}_{ξ} ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), w (τ), w^{*} (τ + 1), \\ \dots, w^{*} (Ξ - 1), v (τ), v^{*} (τ + 1), \dots, v^{*} (Ξ - 1), τ) \\ + J ({\tilde{y}}_{1} (τ + 1), {\tilde{y}}_{2} (Ξ), τ + 1)} . \end{matrix}$ (33) We have

$\begin{matrix} max_{w (τ) \in W (τ)} min_{v (τ) \in V (τ)} {{\tilde{φ}}_{ξ} ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), w (τ), w^{*} (τ + 1), \\ \dots, w^{*} (Ξ - 1), v (τ), v^{*} (τ + 1), \dots, v^{*} (Ξ - 1), τ) \\ + J ({\tilde{y}}_{1} (τ + 1), {\tilde{y}}_{2} (Ξ), τ + 1)} \\ = min_{v (τ) \in V (τ)} max_{w (τ) \in W (τ)} {{\tilde{φ}}_{ξ} ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), w (τ), w^{*} (τ + 1), \\ \dots, w^{*} (Ξ - 1), v (τ), v^{*} (τ + 1), \dots, v^{*} (Ξ - 1), τ) \\ + J ({\tilde{y}}_{1} (τ + 1), {\tilde{y}}_{2} (Ξ), τ + 1)} \end{matrix}$ (34) due to

$\begin{matrix} max_{w (τ) \in W (τ)} min_{v (τ) \in V (τ)} {{\tilde{φ}}_{ξ} ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), w (τ), w^{*} (τ + 1), \\ \dots, w^{*} (Ξ - 1), v (τ), v^{*} (τ + 1), \dots, v^{*} (Ξ - 1), τ) \\ + J ({\tilde{y}}_{1} (τ + 1), {\tilde{y}}_{2} (Ξ), τ + 1)} \\ \geq min_{v (τ) \in V (τ)} {{\tilde{φ}}_{ξ} ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), w (τ), w^{*} (τ + 1), \dots, \\ w^{*} (Ξ - 1), v (τ), v^{*} (τ + 1), \dots, v^{*} (Ξ - 1), τ) \\ + J ({\tilde{y}}_{1} (τ + 1), {\tilde{y}}_{2} (Ξ), τ + 1)}, \\ \forall w (τ) \in W (τ) . \end{matrix}$ (35) Then the Equations (23)–(25) are proved based on the Equations (32)–(34).

Note that the Equations (23)–(25) may be reformulated as $\begin{matrix} J ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), τ) \\ = max_{w (τ) \in W (τ)} min_{v (τ) \in V (τ)} {{\tilde{φ}}_{ξ} ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), w (τ), w^{*} (τ + 1), \dots, w^{*} (Ξ - 1), v (τ), v^{*} (τ + 1), \dots, v^{*} (Ξ - 1), τ) \\ + J (A_{1} {\tilde{y}}_{1} (τ) + B_{1} w (τ) + D_{1} v (τ), {\tilde{y}}_{2} (Ξ), τ + 1)} \\ = min_{v (τ) \in V (τ)} max_{w (τ) \in W (τ)} {{\tilde{φ}}_{ξ} ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), w (τ), w^{*} (τ + 1), \dots, w^{*} (Ξ - 1), v (τ), v^{*} (τ + 1), \dots, v^{*} (Ξ - 1), τ) \\ + J (A_{1} {\tilde{y}}_{1} (τ) + B_{1} w (τ) + D_{1} v (τ), {\tilde{y}}_{2} (Ξ), τ + 1)} \\ = {\tilde{φ}}_{ξ} ({\tilde{y}}_{1} (τ), {\tilde{y}}_{2} (Ξ), w^{*} (τ), w^{*} (τ + 1), \dots, w^{*} (Ξ - 1), v^{*} (τ), v^{*} (τ + 1), \dots, v^{*} (Ξ - 1), τ) \\ + J (A_{1} {\tilde{y}}_{1} (τ) + B_{1} w (τ) + D_{1} v (τ), {\tilde{y}}_{2} (Ξ), τ + 1) ∥ \end{matrix}$ (36) for τ = Ξ - 1, Ξ - 2, ⋯ , 1, 0 .

Remark 3.1. References [37] and [5] address optimal control problems and ZSGs within the context of singular causal systems. Meanwhile, reference [31] focuses on optimal control problems linked to singular noncausal systems. Our study delves into ZSG (14) within the framework of singular noncausal systems, leveraging optimal control techniques. As game problems inherently involve multi-agent multi-objective optimal control scenarios, the recurrence equations from Theorem 3.1 apply not only to ZSG (14) but also extend to solving optimal control problems [31]. This research’s implications in the control theory field are significant. In future research, we could enhance this study by integrating recurrence equations with hybrid intelligent algorithms [19] for optimized solutions. Alternatively, by incorporating existing parameterized models from reference [12], we could address parameter optimization challenges in general noncausal system ZSGs.

4 Zero-sum game with noncausal system considering linear control

Based on recurrence equations, we consider a ZSG under the influence of a NS considering linear control:

$\begin{matrix} {\begin{matrix} J (y (0), y (Ξ), 0) = max_{\overset{w (ξ) \in [- 1, 1]^{n_{1}}}{0 \leq ξ \leq Ξ - 1}} min_{\overset{v (ξ) \in [- 1, 1]^{n_{2}}}{0 \leq ξ \leq Ξ - 1}} U (y (0), y (Ξ), w, v, 0) \\ = min_{\overset{v (ξ) \in [- 1, 1]^{n_{2}}}{0 \leq ξ \leq Ξ - 1}} max_{\overset{w (ξ) \in [- 1, 1]^{n_{1}}}{0 \leq ξ \leq Ξ - 1}} U (y (0), y (Ξ), w, v, 0), \\ s . t . \\ H y (ξ + 1) = A y (ξ) + B w (ξ) + D v (ξ), \\ ξ = 0, 1, 2, \dots, Ξ - 1, \end{matrix} \end{matrix}$ (37)

with the objective function

$\begin{matrix} U (y (0), y (Ξ), w, v, 0) \\ = \sum_{ξ = 0}^{Ξ - 1} φ_{ξ}^{T} y (ξ) + φ_{w, ξ}^{T} w (ξ) + φ_{v, ξ}^{T} v (ξ) + φ_{Ξ}^{T} y (Ξ), \end{matrix}$ (38)

where $φ_{1, ξ}, φ_{Ξ} \in ℝ^{m_{y}}, φ_{2, ξ} \in ℝ^{n_{1}}, φ_{3, ξ} \in ℝ^{n_{2}}$ are deterministic vectors. The game (14) is identical to

$\begin{matrix} {\begin{matrix} J (y_{1} (0), y_{2} (Ξ), 0) = max_{\overset{w (ξ) \in [- 1, 1]^{n_{1}}}{0 \leq ξ \leq Ξ - 1}} min_{\overset{v (ξ) \in [- 1, 1]^{n_{2}}}{0 \leq ξ \leq Ξ - 1}} \tilde{U} ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), w, v, 0) \\ = min_{\overset{v (ξ) \in [- 1, 1]^{n_{2}}}{0 \leq ξ \leq Ξ - 1}} max_{\overset{w (ξ) \in [- 1, 1]^{n_{1}}}{0 \leq ξ \leq Ξ - 1}} \tilde{U} ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), w, v, 0) \\ s . t . \\ {\tilde{y}}_{1} (ξ + 1) = A_{1} {\tilde{y}}_{1} (ξ) + B_{1} w (ξ) + D_{1} v (ξ), \\ {\tilde{y}}_{2} (ξ) = H_{2} {\tilde{y}}_{2} (ξ + 1) - B_{2} w (ξ) - D_{2} v (ξ), \\ ξ = 0, 1, 2, \dots, Ξ - 1, \end{matrix} \end{matrix}$ (39) with

$\begin{matrix} \tilde{U} ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), w, v, 0) \\ = \sum_{ξ = 0}^{Ξ - 1} φ_{ξ}^{T} [M_{nc, 1} {\tilde{y}}_{1} (ξ) + M_{nc, 2} {\tilde{y}}_{2} (ξ)] + φ_{w, ξ}^{T} w (ξ) + φ_{v, ξ}^{T} v (ξ) + φ_{Ξ}^{T} [M_{nc, 1} {\tilde{y}}_{1} (Ξ) + M_{nc, 2} {\tilde{y}}_{2} (Ξ)] . \end{matrix}$ (40)Theorem 4.1.The saddle-point solution ( w ^*, v ^*) for the ZSGP (39) is supplied by $w_{i_{1}}^{*} (ξ) = {\begin{matrix} sign {l_{i_{1}}^{w, ξ}}, & if l_{i_{1}}^{w, ξ} \neq 0, \\ sign {u_{i_{1}}^{w, ξ, 1}}, & if l_{i_{1}}^{w, ξ} = 0, and u_{i_{1}}^{w, ξ, 1} \neq 0, \\ sign {u_{i_{1}}^{w, ξ, 1}}, & if l_{i_{1}}^{w, ξ} = 0, u_{i_{1}}^{w, ξ, 1} = 0, and u_{i_{1}}^{w, ξ, 2} \neq 0 \\ \dots \\ sign {u_{i_{1}}^{w, ξ, ξ}}, & if l_{i_{1}}^{w, ξ} = 0, u_{i_{1}}^{w, ξ, 1} = 0, \dots, u_{i_{1}}^{w, ξ, ξ - 1} = 0, and u_{i_{1}}^{w, ξ, ξ} \neq 0 \\ undetermined, & if l_{i_{1}}^{w, ξ} = 0, u_{i_{1}}^{w, ξ, 1} = 0, \dots, u_{i_{1}}^{w, ξ, ξ - 1} = 0, and u_{i_{1}}^{w, ξ, ξ} = 0, \end{matrix}$ (41) and

$v_{i_{2}}^{*} (ξ) = {\begin{matrix} - sign {l_{i_{2}}^{v, ξ}}, & if l_{i_{2}}^{v, ξ} \neq 0, \\ - sign {u_{i_{2}}^{v, ξ, 1}}, & if l_{i_{2}}^{v, ξ} = 0, and u_{i_{2}}^{v, ξ, 1} \neq 0, \\ - sign {u_{i_{2}}^{v, ξ, 1}}, & if l_{i_{2}}^{v, ξ} = 0, u_{i_{2}}^{v, ξ, 1} = 0, and u_{i_{2}}^{v, ξ, 2} \neq 0 \\ \dots \\ - sign {u_{i_{2}}^{v, ξ, ξ}}, & if l_{i_{2}}^{v, ξ} = 0, u_{i_{2}}^{v, ξ, 1} = 0, \dots, u_{i_{2}}^{v, ξ, ξ - 1} = 0, and u_{i_{2}}^{v, ξ, ξ} \neq 0 \\ undetermined, & if l_{i_{2}}^{v, ξ} = 0, u_{i_{2}}^{v, ξ, 1} = 0, \dots, u_{i_{2}}^{v, ξ, ξ - 1} = 0, and u_{i_{2}}^{v, ξ, ξ} = 0, \end{matrix}$ (42) where $l^{w, ξ} = r_{ξ + 1}^{T} B_{1} - φ_{ξ}^{T} M_{nc, 2} B_{2}, u^{w, ξ, \tilde{l}} = - \sum_{τ = 1}^{\tilde{l}} φ_{ξ - τ}^{T} M_{nc, 2} H_{2}^{τ} B_{2},$ (43) $l^{v, ξ} = r_{ξ + 1}^{T} D_{1} - φ_{ξ}^{T} M_{nc, 2} D_{2}, u^{w, ξ, \tilde{l}} = - \sum_{τ = 1}^{\tilde{l}} φ_{ξ - τ}^{T} M_{nc, 2} H_{2}^{τ} D_{2},$ (44) and

$r_{ξ}^{T} = φ_{ξ}^{T} M_{1} + r_{ξ + 1}^{T} A_{1},$ (45)

for $i_{1} = 1, 2, \dots, n_{1}, i_{2} = 1, 2, \dots, n_{2}, \tilde{l} = 1, 2, \dots, ξ, ξ = 1, 2, \dots, Ξ - 1$ . The optimal values are $J ({\tilde{y}}_{1} (Ξ), {\tilde{y}}_{2} (Ξ), Ξ) = r_{Ξ}^{T} {\tilde{y}}_{1} (Ξ) + s_{Ξ}^{T} {\tilde{y}}_{2} (Ξ),$ (46) $\begin{matrix} J ({\tilde{y}}_{1} (ξ), {\tilde{y}}_{2} (Ξ), ξ) = r_{ξ}^{T} {\tilde{y}}_{1} (ξ) + s_{ξ}^{T} {\tilde{y}}_{2} (Ξ) \\ + \sum_{τ = ξ + 1}^{Ξ - 1} [u^{w, τ, τ - ξ} w^{*} (τ) + u^{v, τ, τ - ξ} v^{*} (τ)] \\ + \sum_{τ = ξ}^{Ξ - 1} (∣ ∣ l^{w, τ} ∣ ∣_{1} + ∣ ∣ l^{v, τ} ∣ ∣_{1}), \end{matrix}$ (47) where

$s_{ξ}^{T} = φ_{ξ}^{T} M_{2} H_{2}^{Ξ - ξ} + s_{ξ + 1}^{T},$ (48)τ = ξ, ξ + 1, ⋯ , Ξ - 1, ξ = 0, 1, 2, ⋯ , Ξ - 1, with $r_{Ξ}^{T} = φ_{Ξ}^{T} M_{nc, 1}$ , $s_{Ξ}^{T} = φ_{Ξ}^{T} M_{nc, 2}$ .

$\begin{matrix} J ({\tilde{y}}_{1} (Ξ - 1), {\tilde{y}}_{2} (Ξ - 1), Ξ - 1) \\ = max_{w (Ξ - 1) \in [- 1, 1]^{n_{1}}} min_{v (Ξ - 1) \in [- 1, 1]^{n_{2}}} {φ_{Ξ - 1}^{T} [M_{nc, 1} {\tilde{y}}_{1} (Ξ - 1) \\ + M_{nc, 2} {\tilde{y}}_{2} (Ξ - 1)] + φ_{w, Ξ - 1}^{T} w (Ξ - 1) \\ + φ_{v, Ξ - 1}^{T} v (Ξ - 1) + r_{Ξ}^{T} {\tilde{y}}_{1} (Ξ) + s_{Ξ}^{T} {\tilde{y}}_{2} (Ξ)} \\ = max_{w (Ξ - 1) \in [- 1, 1]^{n_{1}}} min_{v (Ξ - 1) \in [- 1, 1]^{n_{2}}} {φ_{Ξ - 1}^{T} [M_{nc, 1} {\tilde{y}}_{1} (Ξ - 1) \\ + M_{nc, 2} (H_{2} {\tilde{y}}_{2} (Ξ) - B_{2} w (Ξ - 1) - D_{2} v (Ξ - 1))] \\ + r_{Ξ}^{T} (A_{1} {\tilde{y}}_{1} (Ξ - 1) + B_{1} w (Ξ - 1) \\ + D_{1} v (Ξ - 1)) + s_{Ξ}^{T} {\tilde{y}}_{2} (Ξ)} \\ = (φ_{Ξ - 1}^{T} M_{nc, 1} + r_{Ξ}^{T} A_{1}) {\tilde{y}}_{1} (Ξ - 1) \\ + (φ_{Ξ - 1}^{T} M_{nc, 2} H_{2} + s_{Ξ}^{T}) {\tilde{y}}_{2} (Ξ) \\ + max_{w (Ξ - 1) \in [- 1, 1]^{n_{1}}} [(r_{Ξ}^{T} B_{1} - φ_{Ξ - 1}^{T} M_{nc, 2} B_{2}) w (Ξ - 1)] \\ + min_{v (Ξ - 1) \in [- 1, 1]^{n_{2}}} [(r_{Ξ}^{T} D_{1} - φ_{Ξ - 1}^{T} M_{nc, 2} D_{2}) v (Ξ - 1)] . \end{matrix}$ (49) Denote

$\begin{matrix} l^{w, Ξ - 1} = r_{Ξ}^{T} B_{1} - φ_{Ξ - 1}^{T} M_{nc, 2} B_{2}, \\ l^{v, Ξ - 1} = r_{Ξ}^{T} D_{1} - φ_{Ξ - 1}^{T} M_{nc, 2} D_{2} . \end{matrix}$ (50) For each i₁ = 1, 2, ⋯ , n₁ (i₂ = 1, 2, ⋯ , n₂), we have

$\begin{matrix} w_{i_{1}}^{*} (Ξ - 1) = {\begin{matrix} sign {l_{i_{1}}^{w, Ξ - 1}}, & if l_{i_{1}}^{w, Ξ - 1} \neq 0, \\ undetermined, & if l_{i_{1}}^{w, Ξ - 1} = 0, \end{matrix} \\ and v_{i_{2}}^{*} (Ξ - 1) = {\begin{matrix} - sign {l_{v, i_{2}}^{Ξ - 1}}, & if l_{i_{2}}^{v, Ξ - 1} \neq 0, \\ undetermined, & if l_{i_{2}}^{v, Ξ - 1} = 0 . \end{matrix} \end{matrix}$ (51) and

$\begin{matrix} J ({\tilde{y}}_{1} (Ξ - 1), {\tilde{y}}_{2} (Ξ - 1), Ξ - 1) \\ = r_{Ξ - 1}^{T} {\tilde{y}}_{1} (Ξ - 1) + s_{Ξ - 1}^{T} {\tilde{y}}_{2} (Ξ) \\ + ∣ ∣ l^{w, Ξ - 1} ∣ ∣_{1} + ∣ ∣ l^{v, Ξ - 1} ∣ ∣_{1}, \end{matrix}$ (52) where

$\begin{matrix} r_{Ξ - 1}^{T} = φ_{Ξ - 1}^{T} M_{nc, 1} + r_{Ξ}^{T} A_{1}, \\ s_{Ξ - 1}^{T} = φ_{Ξ - 1}^{T} M_{nc, 2} H_{2} + s_{Ξ}^{T}, \end{matrix}$ (53) and ∣∣ l ^w,Ξ-1 ∣∣ ₁ and ∣∣ l ^v,Ξ-1 ∣∣ ₁ are the 1-norm of the vectors l ^w,Ξ-1 and l ^v,Ξ-1, respectively.

For ξ = Ξ - 2, we have the following result based on the Theorem 3.1: $\begin{matrix} J ({\tilde{y}}_{1} (Ξ - 2), {\tilde{y}}_{2} (Ξ), Ξ - 2) \\ = max_{w (Ξ - 2) \in [- 1, 1]^{n_{1}}} min_{v (Ξ - 2) \in [- 1, 1]^{n_{2}}} {φ_{Ξ - 2}^{T} [M_{nc, 1} {\tilde{y}}_{1} (Ξ - 2) \\ + M_{nc, 2} {\tilde{y}}_{2} (Ξ - 2)] + φ_{w, Ξ - 2}^{T} \\ w (Ξ - 2) + φ_{v, Ξ - 2}^{T} v (Ξ - 2) + r_{Ξ - 1}^{T} {\tilde{y}}_{1} (Ξ - 1) \\ + s_{Ξ - 1}^{T} {\tilde{y}}_{2} (Ξ) + ∣ ∣ l^{w, Ξ - 1} ∣ ∣_{1} + ∣ ∣ l^{v, Ξ - 1} ∣ ∣_{1}} \\ = max_{w (Ξ - 2) \in [- 1, 1]^{n_{1}}} min_{v (Ξ - 2) \in [- 1, 1]^{n_{2}}} {φ_{Ξ - 2}^{T} [M_{nc, 1} {\tilde{y}}_{1} (Ξ - 2) \\ + M_{nc, 2} (H_{2}^{2} {\tilde{y}}_{2} (Ξ) - B_{2} w (Ξ - 2) - D_{2} v (Ξ - 2) \\ - H_{2} B_{2} w^{*} (Ξ - 1) - H_{2} D_{2} v^{*} (Ξ - 1))] \\ + r_{Ξ - 1}^{T} (A_{1} {\tilde{y}}_{1} (Ξ - 2) + B_{1} w (Ξ - 2) \\ + D_{1} v (Ξ - 2)) + s_{Ξ - 1}^{T} {\tilde{y}}_{2} (Ξ) + ∣ ∣ l^{w, Ξ - 1} ∣ ∣_{1} \\ + ∣ ∣ l^{v, Ξ - 1} ∣ ∣_{1}} \\ = (φ_{Ξ - 2}^{T} M_{nc, 1} + r_{Ξ - 1}^{T} A_{1}) {\tilde{y}}_{1} (Ξ - 2) \\ + (φ_{Ξ - 2}^{T} M_{nc, 2} H_{2}^{2} + s_{Ξ - 1}^{T}) {\tilde{y}}_{2} (Ξ) \\ - φ_{Ξ - 2}^{T} M_{nc, 2} H_{2} B_{2} w^{*} (Ξ - 1) \\ - φ_{Ξ - 2}^{T} M_{nc, 2} H_{2} D_{2} v^{*} (Ξ - 1) \\ + max_{w (Ξ - 2) \in [- 1, 1]^{n_{1}}} [(r_{Ξ - 1}^{T} B_{1} - φ_{Ξ - 2}^{T} M_{nc, 2} B_{2}) w (Ξ - 2)] \\ + min_{v (Ξ - 2) \in [- 1, 1]^{n_{2}}} [(r_{Ξ - 1}^{T} D_{1} - φ_{Ξ - 2}^{T} M_{nc, 2} D_{2}) v \\ (Ξ - 2)] + ∣ ∣ l^{w, Ξ - 1} ∣ ∣_{1} + ∣ ∣ l^{v, Ξ - 1} ∣ ∣_{1} . \end{matrix}$ (54) Denote

$\begin{matrix} l^{w, Ξ - 2} = r_{Ξ - 1}^{T} B_{1} - φ_{Ξ - 2}^{T} M_{nc, 2} B_{2}, \\ l^{v, Ξ - 2} = r_{Ξ - 1}^{T} D_{1} - φ_{Ξ - 2}^{T} M_{nc, 2} D_{2} . \end{matrix}$ (55) Then we have

$\begin{matrix} w_{i_{1}}^{*} (Ξ - 2) = {\begin{matrix} sign {l_{i_{1}}^{w, Ξ - 2}}, & if l_{i_{1}}^{w, Ξ - 2} \neq 0, \\ undetermined, & if l_{i_{1}}^{w, Ξ - 2} = 0, \end{matrix} \\ i_{1} = 1, 2, \dots, n_{1}, \end{matrix}$ (56) and

$\begin{matrix} v_{i_{2}}^{*} (Ξ - 2) = {\begin{matrix} - sign {l_{i_{2}}^{v, Ξ - 2}}, & if l_{i_{2}}^{v, Ξ - 2} \neq 0, \\ undetermined, & if l_{i_{2}}^{v, Ξ - 2} = 0, \end{matrix} \\ i_{2} = 1, 2, \dots, n_{2} . \end{matrix}$ (57) In order to maximize the expression $- φ_{Ξ - 2}^{T} M_{2} H_{2} B_{2} w^{*} (Ξ - 1)$ , and minimize the expression $- φ_{Ξ - 2}^{T} M_{2} H_{2} D_{2} v^{*} (Ξ - 1)$ . Hence those elements of the controls w ^* (Ξ - 1) and v ^* (Ξ - 1) that cannot be calculated in the previous step should be updated. For convenience, we acquire the new controls w ^* (Ξ - 1) and v ^* (Ξ - 1) listed below:

$w_{i_{1}}^{*} (Ξ - 1) = {\begin{matrix} sign {l_{i_{1}}^{w, Ξ - 1}}, & if l_{i_{1}}^{w, Ξ - 1} \neq 0, \\ sign {u_{i_{1}}^{w, Ξ - 1, 1}}, & if l_{i_{1}}^{w, Ξ - 1} = 0, and u_{i_{1}}^{w, Ξ - 1, 1} \neq 0, \\ undetermined, & if l_{i_{1}}^{w, Ξ - 1} = 0, and u_{i_{1}}^{w, Ξ - 1, 1} = 0, \end{matrix} i_{1} = 1, 2, \dots, n_{1},$ (58) and

$v_{i_{2}}^{*} (Ξ - 1) = {\begin{matrix} - sign {l_{i_{2}}^{v, Ξ - 1}}, & if l_{i_{2}}^{v, Ξ - 1} \neq 0, \\ - sign {u_{i_{2}}^{v, Ξ - 1, 1}}, & if l_{i_{2}}^{v, Ξ - 1} = 0, and u_{i_{2}}^{v, Ξ - 1, 1} \neq 0, \\ undetermined, & if l_{i_{2}}^{v, Ξ - 1} = 0, and u_{i_{2}}^{v, Ξ - 1, 1} = 0, \end{matrix} i_{2} = 1, 2, \dots, n_{2},$ (59) where

$\begin{matrix} u^{w, Ξ - 1, 1} = - φ_{Ξ - 2}^{T} M_{nc, 2} H_{2} B_{2}, \\ u^{v, Ξ - 1, 1} = - φ_{Ξ - 2}^{T} M_{nc, 2} H_{2} D_{2} . \end{matrix}$ (60) The corresponding optimal value is $\begin{matrix} J ({\tilde{y}}_{1} (Ξ - 2), {\tilde{y}}_{2} (Ξ), Ξ - 2) \\ = r_{Ξ - 2}^{T} {\tilde{y}}_{1} (Ξ - 2) + s_{Ξ - 2}^{T} {\tilde{y}}_{2} (Ξ) \\ + u^{w, Ξ - 1, 1} w^{*} (Ξ - 1) + u^{v, Ξ - 1, 1} v^{*} (Ξ - 1) \\ + \sum_{τ = Ξ - 2}^{Ξ - 1} (∣ ∣ l^{w, τ} ∣ ∣_{1} + ∣ ∣ l^{v, τ} ∣ ∣_{1}), \end{matrix}$ (61) where

$\begin{matrix} r_{Ξ - 2}^{T} = φ_{Ξ - 2}^{T} M_{nc, 1} + r_{Ξ - 1}^{T} A_{1}, \\ s_{Ξ - 2}^{T} = φ_{Ξ - 2}^{T} M_{nc, 2} H_{2}^{2} + s_{Ξ - 1}^{T} . \end{matrix}$ (62)

For ξ = Ξ - 3, we have the following result based on the Theorem 3.1:

$\begin{matrix} J ({\tilde{y}}_{1} (Ξ - 3), {\tilde{y}}_{2} (Ξ), Ξ - 3) \\ = max_{w (Ξ - 3) \in [- 1, 1]^{n_{1}}} min_{v (Ξ - 3) \in [- 1, 1]^{n_{2}}} {φ_{Ξ - 3}^{T} [M_{nc, 1} {\tilde{y}}_{1} (Ξ - 3) \\ + M_{nc, 2} {\tilde{y}}_{2} (Ξ - 3)] + φ_{w, Ξ - 3}^{T} w (Ξ - 3) \\ + φ_{v, Ξ - 3}^{T} v (Ξ - 3) + r_{Ξ - 2}^{T} {\tilde{y}}_{1} (Ξ - 2) + s_{Ξ - 2}^{T} {\tilde{y}}_{2} (Ξ) \\ + u^{w, Ξ - 1, 1} w^{*} (Ξ - 1) + u^{v, Ξ - 1, 1} v^{*} (Ξ - 1) \\ + \sum_{τ = Ξ - 2}^{Ξ - 1} (∣ ∣ l^{w, τ} ∣ ∣_{1} + ∣ ∣ l^{v, τ} ∣ ∣_{1})} \end{matrix}$

$\begin{matrix} = (φ_{Ξ - 3}^{T} M_{nc, 1} + r_{Ξ - 2}^{T} A_{1}) {\tilde{y}}_{1} (Ξ - 3) \\ + (φ_{Ξ - 3}^{T} M_{nc, 2} H_{2}^{3} + s_{Ξ - 2}^{T}) {\tilde{y}}_{2} (Ξ) \\ - φ_{Ξ - 3}^{T} M_{nc, 2} H_{2} B_{2} w^{*} (Ξ - 2) \\ + (u^{w, Ξ - 1, 1} - φ_{w, Ξ - 3}^{T} M_{nc, 2} H_{2}^{2} B_{2}) w^{*} (Ξ - 1) \\ - φ_{Ξ - 3}^{T} M_{nc, 2} H_{2} D_{2} v^{*} (Ξ - 2) \\ + (u^{v, Ξ - 1, 1} - φ_{v, Ξ - 3}^{T} M_{nc, 2} H_{2}^{2} D_{2}) v^{*} (Ξ - 1) \\ + max_{w (Ξ - 3) \in [- 1, 1]^{n_{1}}} [(r_{Ξ - 2}^{T} B_{1} - φ_{Ξ - 3}^{T} M_{nc, 2} B_{2}) w (Ξ - 3)] \\ + min_{v (Ξ - 3) \in [- 1, 1]^{n_{2}}} [(r_{Ξ - 2}^{T} D_{1} - φ_{Ξ - 3}^{T} M_{nc, 2} D_{2}) v (Ξ - 3)] \\ + \sum_{τ = Ξ - 2}^{Ξ - 1} (∣ ∣ l^{w, τ} ∣ ∣_{1} + ∣ ∣ l^{v, τ} ∣ ∣_{1}) . \end{matrix}$ (63)

Denote

$\begin{matrix} l^{w, Ξ - 3} = r_{Ξ - 2}^{T} B_{1} - φ_{Ξ - 3}^{T} M_{nc, 2} B_{2}, \\ l^{v, Ξ - 3} = r_{Ξ - 2}^{T} D_{1} - φ_{Ξ - 3}^{T} M_{nc, 2} D_{2} . \end{matrix}$ (64) Then we have

$\begin{matrix} w_{i_{1}}^{*} (Ξ - 3) = {\begin{matrix} sign {l_{i_{1}}^{w, Ξ - 3}}, & if l_{i_{1}}^{w, Ξ - 3} \neq 0, \\ undetermined, & if l_{i_{1}}^{w, Ξ - 3} = 0, \end{matrix} \\ i_{1} = 1, 2, \dots, n_{1}, \end{matrix}$ (65) and

$\begin{matrix} v_{i_{2}}^{*} (Ξ - 3) = {\begin{matrix} - sign {l_{i_{1}}^{v, Ξ - 3}}, & if l_{i_{2}}^{v, Ξ - 3} \neq 0, \\ undetermined, & if l_{i_{2}}^{v, Ξ - 3} = 0, \end{matrix} \\ i_{2} = 1, 2, \dots, n_{2} . \end{matrix}$ (66) We define $\begin{matrix} u^{w, Ξ - 1, 2} = - \sum_{τ = 1}^{2} φ_{Ξ - 1 - τ}^{T} M_{nc, 2} H_{2}^{τ} B_{2}, \\ u^{w, Ξ - 2, 1} = - φ_{Ξ - 3}^{T} M_{nc, 2} H_{2}^{2} B_{2}, \end{matrix}$ (67) $\begin{matrix} u^{v, Ξ - 1, 2} = - \sum_{τ = 1}^{2} φ_{Ξ - 1 - τ}^{T} M_{nc, 2} H_{2}^{τ} D_{2}, \\ u^{v, Ξ - 2, 1} = - φ_{Ξ - 3}^{T} M_{nc, 2} H_{2}^{2} D_{2} . \end{matrix}$ (68) Similarly, we update the elements of the controls w ^* (Ξ - 1) , w ^* (Ξ - 2) and v ^* (Ξ - 1) , v ^* (Ξ - 2). The new controls w ^* (Ξ - 1) , w ^* (Ξ - 2) and v ^* (Ξ - 1) , v ^* (Ξ - 2) are still symbolized by the same symbols in the following:

$w_{i_{1}}^{*} (Ξ - 1) = {\begin{matrix} sign {l_{i_{1}}^{w, Ξ - 1}}, & if l_{i_{1}}^{w, Ξ - 1} \neq 0, \\ sign {u_{i_{1}}^{w, Ξ - 1, 1}}, & if l_{i_{1}}^{w, Ξ - 1} = 0, and u_{i_{1}}^{w, Ξ - 1, 1} \neq 0, \\ sign {u_{i}^{Ξ - 1, 2}}, & if l_{i_{1}}^{w, Ξ - 1} = 0, u_{i_{1}}^{w, Ξ - 1, 1} = 0, and u_{i_{1}}^{w, Ξ - 1, 2} \neq 0 \\ undetermined, & if l_{i_{1}}^{w, Ξ - 1} = 0, u_{i_{1}}^{w, Ξ - 1, 1} = 0, and u_{i_{1}}^{w, Ξ - 1, 2} = 0, \end{matrix}$ (69)

$w_{i_{1}}^{*} (Ξ - 2) = {\begin{matrix} sign {l_{i_{1}}^{w, Ξ - 2}}, & if l_{i_{1}}^{w, Ξ - 2} \neq 0, \\ sign {u_{i_{1}}^{w, Ξ - 2, 1}}, & if l_{i_{1}}^{w, Ξ - 2} = 0, and u_{i_{1}}^{w, Ξ - 2, 1} \neq 0, \\ undetermined, & if l_{i_{1}}^{w, Ξ - 2} = 0, and u_{i_{1}}^{w, Ξ - 2, 1} = 0, \end{matrix}$ (70)i₁ = 1, 2, ⋯ , n₁, and

$v_{i_{2}}^{*} (Ξ - 1) = {\begin{matrix} - sign {l_{i_{2}}^{v, Ξ - 1}}, & if l_{i_{2}}^{w, Ξ - 1} \neq 0, \\ - sign {u_{i_{2}}^{v, Ξ - 1, 1}}, & if l_{i_{2}}^{v, Ξ - 1} = 0, and u_{i_{2}}^{v, Ξ - 1, 1} \neq 0, \\ - sign {u_{i_{2}}^{v, Ξ - 1, 2}}, & if l_{i_{2}}^{v, Ξ - 1} = 0, u_{i_{1}}^{v, Ξ - 1, 2} = 0, and u_{i_{2}}^{v, Ξ - 1, 2} \neq 0 \\ undetermined, & if l_{i_{2}}^{v, Ξ - 1} = 0, u_{i_{2}}^{v, Ξ - 1, 1} = 0, and u_{i_{2}}^{v, Ξ - 1, 2} = 0, \end{matrix}$ (71)

$v_{i_{2}}^{*} (Ξ - 2) = {\begin{matrix} - sign {l_{i_{2}}^{v, Ξ - 2}}, & if l_{i_{2}}^{v, Ξ - 2} \neq 0, \\ - sign {u_{i_{2}}^{v, Ξ - 2, 1}}, & if l_{i_{2}}^{v, Ξ - 2} = 0, and u_{i_{2}}^{v, Ξ - 2, 1} \neq 0, \\ undetermined, & if l_{i_{2}}^{v, Ξ - 2} = 0, and u_{i_{2}}^{v, Ξ - 2, 1} = 0, \end{matrix}$ (72)

The corresponding optimal value is

$\begin{matrix} J ({\tilde{y}}_{1} (Ξ - 3), {\tilde{y}}_{2} (Ξ), Ξ - 3) & = r_{Ξ - 3}^{T} {\tilde{y}}_{1} (Ξ - 3) + s_{Ξ - 2}^{T} {\tilde{y}}_{2} (Ξ) + \sum_{τ = Ξ - 2}^{Ξ - 1} [u^{w, τ, τ - Ξ + 3} w^{*} (τ) + u^{v, τ, τ - Ξ + 3} v^{*} (τ)] \\ + \sum_{τ = Ξ - 3}^{Ξ - 1} (∣ ∣ l^{v, τ} ∣ ∣_{1} + ∣ ∣ l^{w, τ} ∣ ∣_{1}) \end{matrix}$ (73)

where

$r_{Ξ - 3}^{T} = φ_{Ξ - 3}^{T} M_{nc, 1} + r_{Ξ - 2}^{T} A_{1}, s_{Ξ - 3}^{T} = φ_{Ξ - 3}^{T} M_{nc, 2} H_{2}^{3} + s_{Ξ - 2}^{T} .$ (74)

By induction, the theorem is proved.

In Theorem 4.1, we have derived analytical expressions for both the saddle-point solution and the optimal values of the game. Expanding on the relationships across different stages, we introduce the subsequent algorithm that enables the calculation of equilibrium outcomes for ZSG (37).

Algorithm 1 Equilibrium outcomes for ZSG (37) with noncausal system considering linear control

Input:System matrices H , A , B , D , and weight vectors φ _ξ, φ _w,ξ, φ _v,ξ, φ _Ξ

Output: Saddle-point solution in equations (41)–(42), and optimal values in equations (46)-(47)

1: Check the regularity of ( H , A ). If the conditions do not hold, then Exit.

2: Find nonsingular matrices F _nc and M _nc, and transform ZSG (37) into an equivalent game (39).

3: Calculate $l^{w, ξ}, u^{w, ξ, \tilde{l}}, l^{v, ξ}, u^{v, ξ, \tilde{l}}, r_{ξ}, s_{ξ}, ξ = 0, 1, 2, \dots, Ξ - 1$ , by equations (43)–(45), (48), with $r_{Ξ}^{T} = φ_{Ξ}^{T} M_{nc, 1}$ and $s_{Ξ}^{T} = φ_{Ξ}^{T} M_{nc, 2}$ where M _nc = [ M _nc,1 M _nc,2].

4: Calculate saddle-point solution w ^* (ξ) , v ^* (ξ) , ξ = 0, 1, 2, ⋯ , Ξ - 1, by equations (41)-(42).

5: Update the substates according to $\begin{matrix} {\begin{matrix} {\tilde{y}}_{1} (ξ + 1) = A_{1} {\tilde{y}}_{1} (ξ) + B_{1} w^{*} (ξ) + D_{1} v^{*} (ξ), \\ {\tilde{y}}_{2} (ξ) = H_{2} {\tilde{y}}_{2} (ξ + 1) - B_{2} w^{*} (ξ) - D_{2} v^{*} (ξ), \\ ξ = 0, 1, 2, \dots, Ξ - 1, \end{matrix} \end{matrix}$ with initial condition ${\tilde{y}}_{1} (0)$ and terminal condition ${\tilde{y}}_{2} (Ξ)$ .

6: Calculate optimal values by equations (46)-(47).

5 Numerical example

We consider a ZSG subject to the NS:

$\begin{matrix} {\begin{matrix} J (y (0), y (7), 0) = max_{\overset{w (ξ) \in [- 1, 1]^{3}}{0 \leq ξ \leq 6}} min_{\overset{v (ξ) \in [- 1, 1]^{4}}{0 \leq ξ \leq 6}} [\sum_{ξ = 0}^{6} φ_{ξ}^{T} y (ξ) + φ_{7}^{T} \tilde{y} (7)] \\ = min_{\overset{v (ξ) \in [- 1, 1]^{4}}{0 \leq ξ \leq 6}} max_{\overset{w (ξ) \in [- 1, 1]^{3}}{0 \leq ξ \leq 6}} [\sum_{ξ = 0}^{6} φ_{ξ}^{T} y (ξ) + φ_{7}^{T} y (7)] \\ s . t . : \\ H y (ξ + 1) = A \tilde{y} (ξ) + B w (ξ) + D v (ξ), \\ ξ = 0, 1, 2, \dots, 6, \end{matrix} \end{matrix}$ (75)

where

$\begin{matrix} H = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & - 1 \\ 0 & 0 & 0 & 0 \end{matrix}], A = [\begin{matrix} 1 & 1 & 1 & - 3 \\ 0 & - 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix}], \\ B = [\begin{matrix} 1 & - 2 & 3 \\ 2 & 1 & - 1 \\ - 2 & 1 & 4 \\ 2 & - 1 & 0 \end{matrix}], D = [\begin{matrix} 2 & - 4 & 1 & - 5 \\ - 1 & 2 & 0 & 3 \\ 0 & 0 & 1 & 1 \\ 0 & - 1 & 0 & - 1 \end{matrix}] . \end{matrix}$ (76) The state weighting vectors are

$\begin{matrix} [\begin{matrix} φ_{0}^{T} \\ φ_{1}^{T} \\ φ_{2}^{T} \\ φ_{3}^{T} \\ φ_{4}^{T} \\ φ_{5}^{T} \\ φ_{6}^{T} \\ φ_{7}^{T} \end{matrix}] = [\begin{matrix} - 3 & 1 & 1 & - 9 \\ 6 & - 3 & 2 & 5 \\ 4 & 3 & - 1 & 13 \\ 5 & - 1 & 5 & - 1 \\ 2 & - 7 & 1 & 10 \\ 7 & - 3 & 9 & 1 \\ 5 & 3 & 8 & 1 \\ 7 & 4 & 6 & 3 \end{matrix}] . \end{matrix}$ (77)

First, we have

$\begin{matrix} det (x H - A) = det [\begin{matrix} x - 1 & - 1 & - 1 & 3 \\ 0 & 1 & x & - 1 \\ 0 & 0 & x - 1 & - x \\ 0 & 0 & 0 & 1 \end{matrix}] \\ = x^{2} - 2 x + 10 \end{matrix}$ (78) and deg det(x H - A ) =2 < rank ( H ) =3, i.e., the system is a NS. There exist nonsingular matrices

$\begin{matrix} F_{nc} = [\begin{matrix} - 1 & - 1 & 0 & 2 \\ 0 & 0 & - 0.5 & - 0.5 \\ 0 & - 1 & 1 & - 2 \\ 0 & 0 & 0 & - 1 \end{matrix}], \\ M_{nc} = [\begin{matrix} - 1 & 2 & 0 & - 1 \\ 0 & 2 & 1 & - 2 \\ 0 & - 2 & 0 & 1 \\ 0 & 0 & 0 & 1 \end{matrix}], \end{matrix}$ (79) such that $\begin{matrix} F_{nc} H M_{nc} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & 0 \end{matrix}], \\ F_{nc} A M_{nc} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], \end{matrix}$ (80) $\begin{matrix} F_{nc} B = [\begin{matrix} 1 & - 1 & - 1 \\ 0 & 0 & - 2 \\ - 8 & 2 & 5 \\ - 2 & 1 & 0 \end{matrix}], \\ F_{nc} D = [\begin{matrix} - 1 & 0 & - 1 & 0 \\ 0 & 0.5 & - 0.5 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{matrix}] . \end{matrix}$ (81) We obtain $\begin{matrix} H_{2} = [\begin{matrix} 0 & - 1 \\ 0 & 0 \end{matrix}], A_{1} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}], \\ B_{1} = [\begin{matrix} 1 & - 1 & - 1 \\ 0 & 0 & - 2 \end{matrix}], B_{2} = [\begin{matrix} - 8 & 2 & 5 \\ - 2 & 1 & 0 \end{matrix}], \end{matrix}$ (82) $D_{1} = [\begin{matrix} - 1 & 0 & - 1 & 0 \\ 0 & 0.5 & - 0.5 & 0 \end{matrix}], D_{2} = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]$ (83) and

M_{nc, 1} = [\begin{matrix} - 1 & 2 \\ 0 & 2 \\ 0 & - 2 \\ 0 & 0 \end{matrix}], M_{nc, 2} = [\begin{matrix} 0 & - 1 \\ 1 & - 2 \\ 0 & 1 \\ 0 & 1 \end{matrix}]

(84)

Table 1

The equilibrium results of ZSG (85)

Stage	( w ^* (ξ)) ^T	( v ^* (ξ)) ^T	$({\tilde{y}}_{1} (ξ))^{T}$	$({\tilde{y}}_{2} (ξ))^{T}$	$J ({\tilde{y}}_{1} (ξ), {\tilde{y}}_{2} (7), ξ)$
0	(-1, 1, - 1)	(-1, - 1, - 1, - 1)	(1, 1)	(-3, - 1)	497
1	(-1, 1, - 1)	(-1, - 1, 1, 1)	(2, 3)	(-5, - 3)	448
2	(1, - 1, - 1)	(-1, 1, - 1, 1)	(1, 4)	(17, 1)	480
3	(-1, 1, 1)	(-1, 1, - 1, 1)	(6, 7)	(-13, - 5)	312
4	(-1, - 1, 1)	(-1, 1, - 1, 1)	(5, 6)	(-9, - 3)	164
5	(-1, 1, 1)	(-1, 1, - 1, 1)	(6, 5)	(-13, - 5)	53
6	(1, 1, - 1)	(-1, - 1, 1, - 1)	(5, 4)	(9, 3)	38
7	(6, 5)	(2, 2)	4

The ZSG (75) correspondings to

$\begin{matrix} {\begin{matrix} J ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (7), 0) = max_{\overset{w (ξ) \in [- 1, 1]^{3}}{0 \leq ξ \leq 6}} min_{\overset{v (ξ) \in [- 1, 1]^{4}}{0 \leq ξ \leq 6}} [\sum_{ξ = 0}^{6} φ_{ξ}^{T} + φ_{7}^{T} [M_{1} {\tilde{y}}_{1} (7) + M_{2} {\tilde{y}}_{2} (7)]] \\ = min_{\overset{v (ξ) \in [- 1, 1]^{4}}{0 \leq ξ \leq 6}} max_{\overset{w (ξ) \in [- 1, 1]^{3}}{0 \leq ξ \leq 6}} [\sum_{ξ = 0}^{6} φ_{ξ}^{T} [M_{1} {\tilde{y}}_{1} (ξ) + M_{2} {\tilde{y}}_{2} (ξ)] + φ_{7}^{T} [M_{1} {\tilde{y}}_{1} (7) + M_{2} {\tilde{y}}_{2} (7)]] \\ s . t . \\ {\tilde{y}}_{1} (ξ + 1) = A_{1} {\tilde{y}}_{1} (ξ) + B_{1} w (ξ) + D_{1} v (ξ), \\ {\tilde{y}}_{2} (ξ) = H_{2} {\tilde{y}}_{2} (ξ + 1) - B_{2} w (ξ) + D_{2} v (ξ), \\ ξ = 0, 1, 2, \dots, 6 . \end{matrix} \end{matrix}$ (85) Set ${\tilde{y}}_{1} (0)^{T} = (1, 1), {\tilde{y}}_{2} (7)^{T} = (2, 2)$ , the equilibrium results of ZSG (85) are acquired by Algorithm 1 and listed in Table 1. The trajectories associated with the controls $w^{*} (ξ) = (w_{1}^{*} (ξ), w_{2}^{*} (ξ), w_{3}^{*} (ξ))^{T}$ and $v^{*} (ξ) = (v_{1}^{*} (ξ), v_{2}^{*} (ξ), v_{3}^{*} (ξ))^{T}$ are depicted in Figs. 1 and 2, respectively.

Fig. 1

Trajectories Concerning $w^{*} (ξ) = (w_{1}^{*} (ξ), w_{2}^{*} (ξ), w_{3}^{*} (ξ))^{T}$ .

Fig. 2

Trajectories Concerning $v^{*} (ξ) = (v_{1}^{*} (ξ), v_{2}^{*} (ξ), v_{3}^{*} (ξ), v_{4}^{*} (ξ))^{T}$ .

Form the columns 4 and 5 of Table 1, the trajectories concerning ${\tilde{y}}_{1} (ξ)$ and ${\tilde{y}}_{2} (ξ)$ are displayed in Fig. 3. Notice that the substate ${\tilde{y}}_{2, 1} (ξ)$ changes dramatically due to that it is determined by the controls w ^* (ξ) and v ^* (ξ), i.e., ${\bar{y}}_{2, 1} (ξ) = - 8 w_{1}^{*} (ξ) + 2 w_{2}^{*} (ξ) + 5 w_{3}^{*} (ξ) + v_{1}^{*} (ξ) + v_{3}^{*} (ξ)$ .

Based on the transformation $M_{nc}^{- 1} y (ξ) = [\begin{matrix} {\tilde{y}}_{1} (ξ) \\ {\tilde{y}}_{2} (ξ) \end{matrix}]$ , we have $\begin{matrix} y (0) = (2, 1, - 3, 1)^{T}, y (1) = (7, 7, - 9, - 3)^{T}, \\ y (2) = (6, 23, - 7, 1)^{T}, y (3) = (13, 11, - 19, - 5)^{T}, \end{matrix}$ (86)

$\begin{matrix} y (4) = (10, 9, - 15, - 3)^{T}, y (5) = (9, 7, - 15, - 5)^{T}, \\ y (6) = (0, 11, - 5, 3)^{T}, y (7) = (2, 8, - 8, 2)^{T} . \end{matrix}$ (87)

Fig. 3

Trajectories Concerning ${\tilde{y}}_{1} (ξ) = ({\tilde{y}}_{1, 1} (ξ), {\tilde{y}}_{1, 2} (ξ))^{T}, {\tilde{y}}_{2} (ξ) = ({\tilde{y}}_{2, 1} (ξ), {\tilde{y}}_{2, 2} (ξ))^{T}$ .

The trajectories concerning the concerning the states $\tilde{y} (ξ) = (y_{1} (ξ), y_{2} (ξ), y_{3} (ξ), y_{4} (ξ))^{T}$ of game (75) are displayed in Fig. 4. The solutions acquired through this investigation are not only more precise but also more practical for real-world implementation. However, a drawback of this approach lies in the increased complexity of computing nonsingular matrices F _nc and M _nc as the order of matrices H and A escalates.

Fig. 4

Trajectories Concerning y (ξ) = (y₁ (ξ) , y₂ (ξ) , y₃ (ξ) , y₄ (ξ)) ^T.

6 Application to environmental management

Amidst rapid economic development and a burgeoning population, the depletion of natural resources and escalating environmental pollution have reached critical levels. The concerns surrounding resource and environmental issues are gaining substantial traction in both academia and society [38, 39]. Addressing these challenges, researchers such as [38, 39] have focused on optimal control problems within the domain of environmental management. Building on their insights, we now extend our examination to a game-theoretic scenario within the context of environmental management. The specific problem is formulated as follows.

In an oil refinery, the oil output in the j-th month is denoted as y ₁ (j) (in kilograms) with initial state y ₁ (0). The coefficients for raw material input, fixed-asset investment, and consumption are represented by A ₁, B ₁, and C ₁, respectively. It’s evident that A ₁ y ₁ (j) (in kilograms) corresponds to the input, B ₁ [ y ₁ (j + 1) - y ₁ (j)] (in kilograms) reflects the increase in fixed assets from the j-th month to the (j + 1)-th month, and C ₁ y ₁ (j) (in kilograms) represents consumption. The promotional expenses invested in the j-th month, denoted as w (j) ∈ [γ_w, η_w] (in million dollars), result in a product consumption of ${\tilde{h}}_{0} (w (j)) = {\tilde{b}}_{0}^{w} w (j) + {\tilde{c}}_{0}^{w} w (j)^{2}$ . To control wastewater production during the manufacturing process, the refinery allocates a pollution control fund v (j) ∈ [γ_v, η_v] (in million dollars) for managing industrial wastewater. The resulting product consumption ${\tilde{h}}_{1} (v (j))$ (in kilograms) due to the pollution control fund is modeled using a quadratic function: ${\tilde{h}}_{1} (v (j)) = {\tilde{b}}_{1}^{v} v (j) + {\tilde{c}}_{1}^{v} v (j)^{2}$ . Drawing upon the Leontief dynamic input-output model, we obtain the first equilibrium equation:

$\begin{matrix} y_{1} (j) = A_{1} y_{1} (j) + B_{1} [y_{1} (j + 1) - y_{1} (j)] \\ + C_{1} y_{1} (j) + {\tilde{h}}_{0} (w (j)) + {\tilde{h}}_{1} (v (j)) . \end{matrix}$ (88) Similarly, we can establish the second equilibrium equation as follows:

$\begin{matrix} y_{2} (j + 1) = y_{2} (j) + A_{2} y_{2} (j) \\ + B_{2} [y_{2} (j + 1) - y_{2} (j)] + C_{2} y_{2} (j) - {\tilde{h}}_{2} (v (j)), \end{matrix}$ (89) where y ₂ (j) (in kilograms) represents the output of industrial wastewater in the j-th month. The terms A ₂ y ₂ (j) , B ₂ [ y ₂ (j + 1) - y ₂ (j)] , and C ₂ y ₂ (j) (in kilograms) correspond to the contribution of industrial wastewater due to input, changes in fixed assets, and consumption, respectively. The reduction in industrial wastewater after investing in pollution control fund, denoted as ${\tilde{h}}_{2} (v (j))$ (in kilograms), is modeled by the expression ${\tilde{h}}_{2} (v (j)) = {\tilde{b}}_{2}^{v} v (j) + {\tilde{c}}_{2}^{v} v (j)^{2}$ . Due to strict industrial wastewater emission standards, here we assume that the wastewater discharge amount at the terminal time is denoted as y ₂ (Ξ).

The refinery’s aim is to optimize production output while minimizing industrial wastewater generation. In light of this context, we will delve into the allocation of promotional investment costs and industrial wastewater treatment funds for an oil refinery over a span of Ξ months. This brings us to the exploration of an environmental management game involving the allocation of resources in the context of oil production.

$\begin{matrix} {\begin{matrix} J (y_{1} (0), y (Ξ), 0) = max_{\overset{w (j) \in [γ_{w}, η_{w}]}{0 \leq j \leq Ξ - 1}} min_{\overset{v (j) \in [γ_{v}, η_{v}]}{0 \leq j \leq Ξ - 1}} [\sum_{j = 0}^{Ξ} a^{T} y_{1} (j) - a^{T} y_{2} (j)], \\ = min_{\overset{v (j) \in [γ_{v}, η_{v}]}{0 \leq j \leq Ξ - 1}} max_{\overset{w (j) \in [γ_{w}, η_{w}]}{0 \leq j \leq Ξ - 1}} [\sum_{j = 0}^{Ξ} a^{T} y_{1} (j) - a^{T} y_{2} (j)], \\ s . t . \\ y_{1} (j) = A_{1} y_{1} (j) + B_{1} [y_{1} (j + 1) - y_{1} (j)] + C_{1} y_{1} (j) + {\tilde{h}}_{0} (w (j)) + {\tilde{h}}_{1} (v (j)), \\ y_{2} (j + 1) = y_{2} (j) + A_{2} y_{2} (j) + B_{2} [y_{2} (j + 1) - y_{2} (j)] + C_{2} y_{2} (j) - {\tilde{h}}_{2} (v (j)), \\ j = 0, 1, 2, \dots, Ξ - 1, \end{matrix} \end{matrix}$ (90) where the elements of vector a are all equal to 1. We assume that the vectors and matrices in the problem satisfy appropriate dimensions. The constraint conditions in model (90) are reformulated as follows:

$\begin{matrix} H y (j + 1) = A y (j) + h (w (j)) + \tilde{h} (v (j)), \end{matrix}$ (91) where $\begin{matrix} H = [\begin{matrix} B_{1} & 0 \\ 0 & B_{2} - I_{2} \end{matrix}], \\ A = [\begin{matrix} I_{2} - A_{1} + B_{1} - C_{1} & 0 \\ 0 & - I_{2} - A_{2} + B_{2} - C_{2} \end{matrix}], \\ y (j) = [\begin{matrix} y_{1} (j) \\ y_{2} (j) \end{matrix}], \end{matrix}$ (92) $\begin{matrix} h (w (j)) = [\begin{matrix} {\tilde{b}}_{0}^{w} \\ 0 \end{matrix}] w (j) + [\begin{matrix} {\tilde{c}}_{0}^{w} \\ 0 \end{matrix}] w (j)^{2}, \\ \tilde{h} (v (j)) = [\begin{matrix} {\tilde{b}}_{1}^{v} \\ - {\tilde{b}}_{2}^{v} \end{matrix}] v (j) + [\begin{matrix} {\tilde{c}}_{1}^{v} \\ - {\tilde{c}}_{2}^{v} \end{matrix}] v (j)^{2} . \end{matrix}$ (93)

The game (90) can be further reformulated as follows:

$\begin{matrix} {\begin{matrix} J (y (0), y (Ξ), 0) = max_{\overset{w (j) \in [γ_{w}, η_{w}]}{0 \leq j \leq Ξ - 1}} min_{\overset{v (j) \in [γ_{v}, η_{v}]}{0 \leq j \leq Ξ - 1}} [\sum_{j = 0}^{Ξ} (a^{T}, - a^{T}) y (j)], \\ = min_{\overset{v (j) \in [γ_{v}, η_{v}]}{0 \leq j \leq Ξ - 1}} max_{\overset{w (j) \in [γ_{w}, η_{w}]}{0 \leq j \leq Ξ - 1}} [\sum_{j = 0}^{Ξ} (a^{T}, - a^{T}) y (j)], \\ s . t . \\ H y (j + 1) = A y (j) + h (w (j)) + \tilde{h} (v (j)), \\ j = 0, 1, 2, \dots, Ξ - 1 . \end{matrix} \end{matrix}$ (94)

When ( H , A ) satisfies the regularity condition, the equivalent game is $\begin{matrix} {\begin{matrix} J (y_{1} (0), y_{2} (Ξ), 0) = max_{\overset{w (ξ) \in [γ_{w}, η_{w}]}{0 \leq ξ \leq Ξ - 1}} min_{\overset{v (ξ) \in [γ_{v}, η_{v}]}{0 \leq ξ \leq Ξ - 1}} \tilde{U} ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), w, v, 0) \\ = min_{\overset{v (ξ) \in [γ_{v}, η_{v}]}{0 \leq ξ \leq Ξ - 1}} max_{\overset{w (ξ) \in [γ_{w}, η_{w}]}{0 \leq ξ \leq Ξ - 1}} \tilde{U} ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), w, v, 0) \\ s . t . \\ {\tilde{y}}_{1} (ξ + 1) = A_{1} {\tilde{y}}_{1} (ξ) + b_{1}^{w} w (ξ) + c_{1}^{w} w (ξ)^{2} + b_{1}^{v} v (ξ) + c_{1}^{v} v (ξ)^{2}, \\ {\tilde{y}}_{2} (ξ) = H_{2} {\tilde{y}}_{2} (ξ + 1) - b_{2}^{w} w (ξ) - c_{2}^{w} w (ξ)^{2} - b_{2}^{v} v (ξ) - c_{2}^{v} v (ξ)^{2}, \\ ξ = 0, 1, 2, \dots, Ξ - 1, \end{matrix} \end{matrix}$ (95) with $\tilde{U} ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), w, v, 0) = \sum_{ξ = 0}^{Ξ} φ_{ξ}^{T} [M_{nc, 1} {\tilde{y}}_{1} (ξ) + M_{nc, 2} {\tilde{y}}_{2} (ξ)]$ (96) where $φ_{ξ}^{T} = (a^{T}, - a^{T})$ , $F_{nc} [\begin{matrix} {\tilde{b}}_{0}^{w} \\ 0 \end{matrix}] = [\begin{matrix} b_{1}^{w} \\ b_{2}^{w} \end{matrix}]$ , $F_{nc} [\begin{matrix} {\tilde{c}}_{0}^{w} \\ 0 \end{matrix}] = [\begin{matrix} c_{1}^{w} \\ c_{2}^{w} \end{matrix}]$ , $F_{nc} [\begin{matrix} {\tilde{b}}_{1}^{v} \\ - {\tilde{b}}_{2}^{v} \end{matrix}] = [\begin{matrix} b_{1}^{v} \\ b_{2}^{v} \end{matrix}]$ , $F_{nc} [\begin{matrix} {\tilde{c}}_{1}^{v} \\ - {\tilde{c}}_{2}^{v} \end{matrix}] = [\begin{matrix} c_{1}^{v} \\ c_{2}^{v} \end{matrix}]$ .

Theorem 6.1.The allocation scheme of promotional funds is

$\begin{matrix} w^{*} (ξ) = {\begin{matrix} η_{w}, & if α_{w, ξ} > 0 and β_{w, ξ} = 0, or - \frac{α_{w, ξ}}{2 β_{w, ξ}} \geq η_{w} and β_{w, ξ} < 0, \\ or ∣ γ_{w} + \frac{α_{w, ξ}}{2 β_{w, ξ}} ∣ < ∣ η_{w} + \frac{α_{w, ξ}}{2 β_{w, ξ}} ∣ and β_{w, ξ} > 0, \\ γ_{w}, & if α_{w, ξ} \leq 0 and β_{w, ξ} = 0, or - \frac{α_{w, ξ}}{2 β_{w, ξ}} \leq γ_{w} and β_{w, ξ} < 0, \\ or ∣ γ_{w} + \frac{α_{w, ξ}}{2 β_{w, ξ}} ∣ \geq ∣ η_{w} + \frac{α_{w, ξ}}{2 β_{w, ξ}} ∣ and β_{w, ξ} > 0, \\ - \frac{α_{w, ξ}}{2 β_{w, ξ}}, & if γ_{w} \leq \frac{α_{w, ξ}}{2 β_{w, ξ}} \leq η_{w} and β_{w, ξ} < 0, \end{matrix} \end{matrix}$ (97) if (α_w,ξ, β_w,ξ) ≠ (0, 0); and

$\begin{matrix} w^{*} (ξ) = {\begin{matrix} η_{w}, & if α_{w, ξ}^{1} > 0 and β_{w, ξ}^{1} = 0, or - \frac{α_{w, ξ}^{1}}{2 β_{w, ξ}^{1}} \geq η_{w} and β_{w, ξ}^{1} < 0, \\ or ∣ γ_{w} + \frac{α_{w, ξ}^{1}}{2 β_{w, ξ}^{1}} ∣ < ∣ η_{w} + \frac{α_{w, ξ}^{1}}{2 β_{w, ξ}^{1}} ∣ and β_{w, ξ}^{1} > 0, \\ γ_{w}, & if α_{w, ξ}^{1} \leq 0 and β_{w, ξ}^{1} = 0, or - \frac{α_{w, ξ}^{1}}{2 β_{w, ξ}^{1}} \leq γ_{w} and β_{w, ξ}^{1} < 0, \\ or ∣ γ_{w} + \frac{α_{w, ξ}^{1}}{2 β_{w, ξ}^{1}} ∣ \geq ∣ η_{w} + \frac{α_{w, ξ}^{1}}{2 β_{w, ξ}^{1}} ∣ and β_{w, ξ}^{1} > 0, \\ - \frac{α_{w, ξ}^{1}}{2 β_{w, ξ}^{1}}, & if γ_{w} \leq \frac{α_{w, ξ}^{1}}{2 β_{w, ξ}^{1}} \leq η_{w} and β_{w, ξ}^{1} < 0, \end{matrix} \end{matrix}$ (98) if $(α_{w, ξ}, β_{w, ξ}) = (0, 0), (α_{w, ξ}^{1}, β_{w, ξ}^{1}) \neq (0, 0)$ ;

$\dots$ and

$\begin{matrix} w^{*} (ξ) = {\begin{matrix} η_{w}, & if α_{w, ξ}^{ξ - 1} > 0 and β_{w, ξ}^{ξ - 1} = 0, or - \frac{α_{w, ξ}^{ξ - 1}}{2 β_{w, ξ}^{ξ - 1}} \geq η_{w} and β_{w, ξ}^{ξ - 1} < 0, \\ or ∣ γ_{w} + \frac{α_{w, ξ}^{ξ - 1}}{2 β_{w, ξ}^{ξ - 1}} ∣ < ∣ η_{w} + \frac{α_{w, ξ}^{ξ - 1}}{2 β_{w, ξ}^{ξ - 1}} ∣ and β_{w, ξ}^{ξ - 1} > 0, \\ γ_{w}, & if α_{w, ξ}^{ξ - 1} \leq 0 and β_{w, ξ}^{ξ - 1} = 0, or - \frac{α_{w, ξ}^{ξ - 1}}{2 β_{w, ξ}^{ξ - 1}} \leq γ_{w} and β_{w, ξ}^{ξ - 1} < 0, \\ or ∣ γ_{w} + \frac{α_{w, ξ}^{ξ - 1}}{2 β_{w, ξ}^{ξ - 1}} ∣ \geq ∣ η_{w} + \frac{α_{w, ξ}^{ξ - 1}}{2 β_{w, ξ}^{ξ - 1}} ∣ and β_{w, ξ}^{ξ - 1} > 0, \\ - \frac{α_{w, ξ}^{ξ - 1}}{2 β_{w, ξ}^{ξ - 1}}, & if γ_{w} \leq \frac{α_{w, ξ}^{ξ - 1}}{2 β_{w, ξ}^{ξ - 1}} \leq η_{w} and β_{w, ξ}^{ξ - 1} < 0, \end{matrix} \end{matrix}$ (99) if $(α_{w, ξ}, β_{w, ξ}) = (0, 0), (α_{w, ξ}^{1}, β_{w, ξ}^{1}) = (0, 0), \dots, (α_{w, ξ}^{ξ - 1}, β_{w, ξ}^{ξ - 1}) \neq (0, 0)$ ; and

$\begin{matrix} w^{*} (ξ) = {\begin{matrix} η_{w}, & if α_{w, ξ}^{ξ} > 0 and β_{w, ξ}^{ξ} = 0, or - \frac{α_{w, ξ}^{ξ}}{2 β_{w, ξ}^{ξ}} \geq η_{w} and β_{w, ξ}^{ξ} < 0, \\ or ∣ γ_{w} + \frac{α_{w, ξ}^{ξ}}{2 β_{w, ξ}^{ξ}} ∣ < ∣ η_{w} + \frac{α_{w, ξ}^{ξ}}{2 β_{w, ξ}^{ξ}} ∣ and β_{w, ξ}^{ξ} > 0, \\ γ_{w}, & if α_{w, ξ}^{ξ} \leq 0 and β_{w, ξ}^{ξ} = 0, or - \frac{α_{w, ξ}^{ξ}}{2 β_{w, ξ}^{ξ}} \leq γ_{w} and β_{w, ξ}^{ξ} < 0, \\ or ∣ γ_{w} + \frac{α_{w, ξ}^{ξ}}{2 β_{w, ξ}^{ξ}} ∣ \geq ∣ η_{w} + \frac{α_{w, ξ}^{ξ}}{2 β_{w, ξ}^{ξ}} ∣ and β_{w, ξ}^{ξ} > 0, \\ - \frac{α_{w, ξ}^{ξ}}{2 β_{w, ξ}^{ξ}}, & if γ_{w} \leq \frac{α_{w, ξ}^{ξ}}{2 β_{w, ξ}^{ξ}} \leq η_{w} and β_{w, ξ}^{ξ} < 0, \end{matrix} \end{matrix}$ (100) if $(α_{w, ξ}, β_{w, ξ}) = (0, 0), (α_{w, ξ}^{1}, β_{w, ξ}^{1}) = (0, 0), \dots, (α_{w, ξ}^{ξ - 1}, β_{w, ξ}^{ξ - 1}) = (0, 0)$ ; Furthermore, the allocation strategy of industrial wastewater treatment funds is

$\begin{matrix} v^{*} (ξ) = {\begin{matrix} η_{v}, & if α_{v, ξ} < 0 and β_{v, ξ} = 0, or - \frac{α_{v, ξ}}{2 β_{v, ξ}} \geq γ_{w} and β_{v, ξ} < 0, \\ or ∣ γ_{v} + \frac{α_{v, ξ}}{2 β_{v, ξ}} ∣ \geq ∣ η_{v} + \frac{α_{v, ξ}}{2 β_{v, ξ}} ∣ and β_{v, ξ} > 0, \\ γ_{v}, & if α_{v, ξ} > 0 and β_{v, ξ} = 0, or - \frac{α_{v, ξ}}{2 β_{v, ξ}} \geq η_{v} and β_{v, ξ} < 0, \\ or ∣ γ_{v} + \frac{α_{v, ξ}}{2 β_{v, ξ}} ∣ < ∣ η_{v} + \frac{α_{v, ξ}}{2 β_{v, ξ}} ∣ and β_{v, ξ} > 0, \\ - \frac{α_{v, ξ}}{2 β_{v, ξ}}, & if γ_{v} \leq \frac{α_{v, ξ}}{2 β_{v, ξ}} \leq η_{v} and β_{v, ξ} > 0, \end{matrix} \end{matrix}$ (101) if (α_v,ξ, β_v,ξ) ≠ (0, 0); and

$\begin{matrix} v^{*} (ξ) = {\begin{matrix} η_{v}, & if α_{v, ξ}^{1} < 0 and β_{v, ξ}^{1} = 0, or - \frac{α_{v, ξ}^{1}}{2 β_{v, ξ}^{1}} \geq γ_{w} and β_{v, ξ}^{1} < 0, \\ or ∣ γ_{v} + \frac{α_{v, ξ}^{1}}{2 β_{v, ξ}^{1}} ∣ \geq ∣ η_{v} + \frac{α_{v, ξ}^{1}}{2 β_{v, ξ}^{1}} ∣ and β_{v, ξ}^{1} > 0, \\ γ_{v}, & if α_{v, ξ}^{1} > 0 and β_{v, ξ}^{1} = 0, or - \frac{α_{v, ξ}^{1}}{2 β_{v, ξ}^{1}} \geq η_{v} and β_{v, ξ} < 0, \\ or ∣ γ_{v} + \frac{α_{v, ξ}^{1}}{2 β_{v, ξ}^{1}} ∣ < ∣ η_{v} + \frac{α_{v, ξ}^{1}}{2 β_{v, ξ}^{1}} ∣ and β_{v, ξ}^{1} > 0, \\ - \frac{α_{v, ξ}^{1}}{2 β_{v, ξ}^{1}}, & if γ_{v} \leq \frac{α_{v, ξ}^{1}}{2 β_{v, ξ}^{1}} \leq η_{v} and β_{v, ξ}^{1} > 0, \end{matrix} \end{matrix}$ (102) if $(α_{v, ξ}, β_{v, ξ}) = (0, 0), (α_{v, ξ}^{1}, β_{v, ξ}^{1}) \neq (0, 0)$ ; $\dots$ and

$\begin{matrix} v^{*} (ξ) = {\begin{matrix} η_{v}, & if α_{v, ξ}^{ξ - 1} < 0 and β_{v, ξ}^{ξ - 1} = 0, or - \frac{α_{v, ξ}^{ξ - 1}}{2 β_{v, ξ}^{ξ - 1}} \geq γ_{w} and β_{v, ξ}^{ξ - 1} < 0, \\ or ∣ γ_{v} + \frac{α_{v, ξ}^{ξ - 1}}{2 β_{v, ξ}^{ξ - 1}} ∣ \geq ∣ η_{v} + \frac{α_{v, ξ}^{ξ - 1}}{2 β_{v, ξ}^{ξ - 1}} ∣ and β_{v, ξ}^{ξ - 1} > 0, \\ γ_{v}, & if α_{v, ξ}^{ξ - 1} > 0 and β_{v, ξ}^{ξ - 1} = 0, or - \frac{α_{v, ξ}^{ξ - 1}}{2 β_{v, ξ}^{ξ - 1}} \geq η_{v} and β_{v, ξ} < 0, \\ or ∣ γ_{v} + \frac{α_{v, ξ}^{ξ - 1}}{2 β_{v, ξ}^{ξ - 1}} ∣ < ∣ η_{v} + \frac{α_{v, ξ}^{ξ - 1}}{2 β_{v, ξ}^{ξ - 1}} ∣ and β_{v, ξ}^{ξ - 1} > 0, \\ - \frac{α_{v, ξ}^{ξ - 1}}{2 β_{v, ξ}^{ξ - 1}}, & if γ_{v} \leq \frac{α_{v, ξ}^{ξ - 1}}{2 β_{v, ξ}^{ξ - 1}} \leq η_{v} and β_{v, ξ}^{ξ - 1} > 0, \end{matrix} \end{matrix}$ (103) if $(α_{v, ξ}, β_{v, ξ}) = (0, 0), (α_{v, ξ}^{1}, β_{v, ξ}^{1}) = (0, 0), \dots, (α_{v, ξ}^{ξ - 1}, β_{v, ξ}^{ξ - 1}) \neq (0, 0)$ ;

$\begin{matrix} v^{*} (ξ) = {\begin{matrix} η_{v}, & if α_{v, ξ}^{ξ} < 0 and β_{v, ξ}^{ξ} = 0, or - \frac{α_{v, ξ}^{ξ}}{2 β_{v, ξ}^{ξ}} \geq γ_{w} and β_{v, ξ}^{ξ} < 0, \\ or ∣ γ_{v} + \frac{α_{v, ξ}^{ξ}}{2 β_{v, ξ}^{ξ}} ∣ \geq ∣ η_{v} + \frac{α_{v, ξ}^{ξ}}{2 β_{v, ξ}^{ξ}} ∣ and β_{v, ξ}^{ξ} > 0, \\ γ_{v}, & if α_{v, ξ}^{ξ} > 0 and β_{v, ξ}^{ξ} = 0, or - \frac{α_{v, ξ}^{ξ}}{2 β_{v, ξ}^{ξ}} \geq η_{v} and β_{v, ξ} < 0, \\ or ∣ γ_{v} + \frac{α_{v, ξ}^{ξ}}{2 β_{v, ξ}^{ξ}} ∣ < ∣ η_{v} + \frac{α_{v, ξ}^{ξ}}{2 β_{v, ξ}^{ξ}} ∣ and β_{v, ξ}^{ξ} > 0, \\ - \frac{α_{v, ξ}^{ξ}}{2 β_{v, ξ}^{ξ}}, & if γ_{v} \leq \frac{α_{v, ξ}^{ξ}}{2 β_{v, ξ}^{ξ}} \leq η_{v} and β_{v, ξ}^{ξ} > 0, \end{matrix} \end{matrix}$ (104) if $(α_{v, ξ}, β_{v, ξ}) = (0, 0), (α_{v, ξ}^{1}, β_{v, ξ}^{1}) = (0, 0), \dots, (α_{v, ξ}^{ξ - 1}, β_{v, ξ}^{ξ - 1}) = (0, 0)$ , where $α_{w, ξ} = r_{ξ + 1}^{T} b_{1}^{w} - φ_{ξ}^{T} M_{nc, 2} b_{2}^{w}, β_{w, ξ} = r_{ξ + 1}^{T} c_{1}^{w} - φ_{ξ}^{T} M_{nc, 2} c_{2}^{w},$ (105) $α_{v, ξ} = r_{ξ + 1}^{T} b_{1}^{v} - φ_{ξ}^{T} M_{nc, 2} b_{2}^{v}, β_{v, ξ} = r_{ξ + 1}^{T} c_{1}^{v} - φ_{ξ}^{T} M_{nc, 2} c_{2}^{v},$ (106) and $α_{ξ}^{l} = - φ_{ξ - l}^{T} M_{nc, 2} H_{2} b_{2}, β_{ξ}^{l} = - φ_{ξ - l}^{T} M_{nc, 2} H_{2} c_{2},, r_{ξ}^{T} = φ_{ξ}^{T} M_{nc, 1} + r_{ξ + 1}^{T} A_{1},$ (107) for l = 1, 2, ⋯ , ξ, ξ = 0, 1, 2, ⋯ , Ξ - 1, and $r_{Ξ}^{T} = φ_{Ξ}^{T} M_{nc, 1}$ . The optimal value is $\begin{matrix} J ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), 0) \\ = r_{0}^{T} {\tilde{y}}_{1} (0) + s_{0}^{T} {\tilde{y}}_{2} (Ξ) + \sum_{τ = ξ + 1}^{Ξ - 1} \sum_{ξ = 1}^{τ - ξ} [α_{w, τ}^{l} w^{*} (τ) + β_{w, τ}^{l} w^{*} (τ)^{2} + α_{v, τ}^{l} v^{*} (τ) + β_{v, τ}^{l} v^{*} (τ)^{2}] + \sum_{τ = 0}^{Ξ - 1} (σ_{w, τ} + σ_{v, τ}), \end{matrix}$ (108) where $c s_{ξ} = φ_{ξ}^{T} M_{nc, 2} H_{2}^{Ξ - ξ} + s_{ξ + 1}^{T},$ (109) and $σ_{w, τ} = α_{w, ξ} w^{*} (ξ) + β_{w, ξ} w^{*} (ξ)^{2}, σ_{v, τ} = α_{v, ξ} v^{*} (ξ) + β_{v, ξ} v^{*} (ξ)^{2},$ (110) for ξ = 0, 1, 2, ⋯ , Ξ - 1, with $s_{Ξ}^{T} = φ_{Ξ}^{T} M_{nc, 2}$

Refer to the proof of Theorem 6.1 in Appendix.

Remark 6.1. When the matrix pair ( H , A ) fulfills the criteria of regularity and absence of impulses, the game (94) corresponds to: $\begin{matrix} {\begin{matrix} J (y_{1} (0), y_{2} (Ξ), 0) = max_{\overset{w (ξ) \in [γ_{w}, η_{w}]}{0 \leq ξ \leq Ξ - 1}} min_{\overset{v (ξ) \in [γ_{v}, η_{v}]}{0 \leq ξ \leq Ξ - 1}} \tilde{U} ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), w, v, 0) \\ = min_{\overset{v (ξ) \in [γ_{v}, η_{v}]}{0 \leq ξ \leq Ξ - 1}} max_{\overset{w (ξ) \in [γ_{w}, η_{w}]}{0 \leq ξ \leq Ξ - 1}} \tilde{U} ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), w, v, 0) \\ s . t . \\ {\tilde{y}}_{1} (ξ + 1) = A_{1} {\tilde{y}}_{1} (ξ) + b_{1}^{w} w (ξ) + c_{1}^{w} w (ξ)^{2} + b_{1}^{v} v (ξ) + c_{1}^{v} v (ξ)^{2}, \\ {\tilde{y}}_{2} (ξ) = - b_{2}^{w} w (ξ) - c_{2}^{w} w (ξ)^{2} - b_{2}^{v} v (ξ) - c_{2}^{v} v (ξ)^{2}, \\ ξ = 0, 1, 2, \dots, Ξ - 1 . \end{matrix} \end{matrix}$ (111) The results in the Theorem 6.1 degenerate to: The allocation scheme of promotional funds is

$\begin{matrix} w^{*} (ξ) = {\begin{matrix} η_{w}, & if α_{w, ξ} > 0 and β_{w, ξ} = 0, or - \frac{α_{w, ξ}}{2 β_{w, ξ}} \geq η_{w} and β_{w, ξ} < 0, \\ or ∣ γ_{w} + \frac{α_{w, ξ}}{2 β_{w, ξ}} ∣ < ∣ η_{w} + \frac{α_{w, ξ}}{2 β_{w, ξ}} ∣ and β_{w, ξ} > 0, \\ γ_{w}, & if α_{w, ξ} \leq 0 and β_{w, ξ} = 0, or - \frac{α_{w, ξ}}{2 β_{w, ξ}} \leq γ_{w} and β_{w, ξ} < 0, \\ or ∣ γ_{w} + \frac{α_{w, ξ}}{2 β_{w, ξ}} ∣ \geq ∣ η_{w} + \frac{α_{w, ξ}}{2 β_{w, ξ}} ∣ and β_{w, ξ} > 0, \\ - \frac{α_{w, ξ}}{2 β_{w, ξ}}, & if γ_{w} \leq \frac{α_{w, ξ}}{2 β_{w, ξ}} \leq η_{w} and β_{w, ξ} < 0, \\ undetermined, & if α_{w, ξ} = 0 and β_{w, ξ} = 0, \end{matrix} \end{matrix}$ (112)

Furthermore, the allocation strategy of industrial wastewater treatment funds is

$\begin{matrix} v^{*} (ξ) = {\begin{matrix} η_{v}, & if α_{v, ξ} < 0 and β_{v, ξ} = 0, or - \frac{α_{v, ξ}}{2 β_{v, ξ}} \geq γ_{w} and β_{v, ξ} < 0, \\ or ∣ γ_{v} + \frac{α_{v, ξ}}{2 β_{v, ξ}} ∣ \geq ∣ η_{v} + \frac{α_{v, ξ}}{2 β_{v, ξ}} ∣ and β_{v, ξ} > 0, \\ γ_{v}, & if α_{v, ξ} > 0 and β_{v, ξ} = 0, or - \frac{α_{v, ξ}}{2 β_{v, ξ}} \geq η_{v} and β_{v, ξ} < 0, \\ or ∣ γ_{v} + \frac{α_{v, ξ}}{2 β_{v, ξ}} ∣ < ∣ η_{v} + \frac{α_{v, ξ}}{2 β_{v, ξ}} ∣ and β_{v, ξ} > 0, \\ - \frac{α_{v, ξ}}{2 β_{v, ξ}}, & if γ_{v} \leq \frac{α_{v, ξ}}{2 β_{v, ξ}} \leq η_{v} and β_{v, ξ} > 0, \\ undetermined, & if α_{v, ξ} = 0 and β_{v, ξ} = 0, \end{matrix} \end{matrix}$ (113) for ξ = 0, 1, 2, ⋯ , Ξ - 1, and $r_{Ξ}^{T} = φ_{Ξ}^{T} M_{nc, 1}$ . And the optimal value is

$\begin{matrix} J ({\tilde{y}}_{1} (0), {\tilde{y}}_{2} (Ξ), 0) = r_{0}^{T} {\tilde{y}}_{1} (0) \\ + s_{0}^{T} {\tilde{y}}_{2} (Ξ) + \sum_{τ = 0}^{Ξ - 1} (σ_{w, τ} + σ_{v, τ}), \end{matrix}$ (114) where

$\begin{matrix} s_{ξ} = φ_{ξ}^{T} M_{nc, 2} H_{2}^{Ξ - ξ} + s_{ξ + 1}^{T}, σ_{w, τ} \\ = α_{w, ξ} w^{*} (ξ) + β_{w, ξ} w^{*} (ξ)^{2}, \\ σ_{v, τ} = α_{v, ξ} v^{*} (ξ) + β_{v, ξ} v^{*} (ξ)^{2}, \end{matrix}$ (115) for ξ = 0, 1, 2, ⋯ , Ξ - 1, with $s_{Ξ}^{T} = φ_{Ξ}^{T} M_{nc, 2}$ .

7 Conclusions

A NS essentially represents a singular system that satisfies regularity conditions but not impulse-free conditions. Such a system can be transformed into equivalent subsystems, thus establishing a two-player zero-sum game problem based on these subsystems. The resolution of these ZSGs involves recurrence equations derived from Bellman’s principle of optimality and insights from singular control systems. This approach allows us to analyze discrete-time two-player ZSGs, determining equilibrium controls and optimal values. Firstly, we investigated ZSGs constrained by linear NSs. By leveraging equivalent transformation techniques and recurrence equations, we derived analytical expressions for saddle-point solutions and optimal values. An algorithm was developed to compute equilibrium outcomes, and an illustrative example showcased the method’s effectiveness in solving a detailed problem involving a noncausal system. Secondly, we extended this methodology to address environmental management challenges. We employed recurrence equations to address the zero-sum environmental management game with NSs and quadratic control constraints. This yielded strategies for allocating pollution control funds and promotional expenses. This research initiates a comprehensive exploration of ZSGs under NSs, opening the door for further investigations. For linear NSs, the study of linear-quadratic ZSGs can be pursued. In the case of nonlinear NSs, the development of simulation algorithms for numerical solutions to nonlinear ZSGs subject to NSs holds potential.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Footnotes

Acknowledgment

This study is backed by the National Natural Science Foundation of China (Grant No. 62003158), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 23KJB110013).

Appendix

Proof. Theorem 3.1 indicates that $J ({\tilde{y}}_{1} (Ξ), {\tilde{y}}_{2} (Ξ), Ξ) = r_{Ξ}^{T} {\tilde{y}}_{1} (Ξ) + s_{Ξ}^{T} {\tilde{y}}_{2} (Ξ)$ where $r_{Ξ}^{T} = φ_{Ξ}^{T} M_{nc, 1}$ , $s_{Ξ}^{T} = φ_{Ξ}^{T} M_{nc, 2}$ . For ξ = Ξ - 1, we have $\begin{matrix} J ({\tilde{y}}_{1} (Ξ - 1), {\tilde{y}}_{2} (Ξ), Ξ - 1) \\ = max_{w (Ξ - 1) \in [γ_{w}, η_{w}]} min_{v (Ξ - 1) \in [γ_{v}, η_{v}]} {φ_{Ξ - 1}^{T} [M_{nc, 1} {\tilde{y}}_{1} (Ξ - 1) \\ + M_{nc, 2} {\tilde{y}}_{2} (Ξ - 1)] + r_{Ξ}^{T} {\tilde{y}}_{1} (Ξ) + s_{Ξ}^{T} {\tilde{y}}_{2} (Ξ)} \\ = max_{w (Ξ - 1) \in [γ_{w}, η_{w}]} min_{v (Ξ - 1) \in [γ_{v}, η_{v}]} {φ_{Ξ - 1}^{T} [M_{nc, 1} {\tilde{y}}_{1} (Ξ - 1) \\ + M_{nc, 2} (H_{2} {\tilde{y}}_{2} (Ξ) - b_{2}^{w} w (Ξ - 1) - c_{2}^{w} w (Ξ - 1)^{2} \\ - b_{2}^{v} v (Ξ - 1) - c_{2}^{v} v (Ξ - 1)^{2})] + r_{Ξ}^{T} (A_{1} {\tilde{y}}_{1} (Ξ - 1) \\ + b_{1}^{w} w (Ξ - 1) + c_{1}^{w} w (Ξ - 1)^{2} + b_{1}^{v} v (Ξ - 1) \\ + c_{1}^{v} v (Ξ - 1)^{2}) + s_{Ξ}^{T} {\tilde{y}}_{2} (Ξ)} \\ = (φ_{Ξ - 1}^{T} M_{nc, 1} + r_{Ξ}^{T} A_{1}) {\tilde{y}}_{1} (Ξ - 1) \\ + (φ_{Ξ - 1}^{T} M_{nc, 2} H_{2} + s_{Ξ}^{T}) {\tilde{y}}_{2} (Ξ) \end{matrix}$

(116) $\begin{matrix} + max_{w (Ξ - 1) \in [γ_{w}, η_{w}]} [α_{w, Ξ - 1} w (Ξ - 1) \\ + β_{w, Ξ - 1} w (Ξ - 1)^{2}] + min_{v (Ξ - 1) \in [γ_{v}, η_{v}]} [α_{v, Ξ - 1} v (Ξ - 1) \\ + β_{v, Ξ - 1} v (Ξ - 1)^{2}] \end{matrix}$ where (117) $\begin{matrix} α_{w, Ξ - 1} = r_{Ξ}^{T} b_{1}^{w} - φ_{Ξ - 1}^{T} M_{nc, 2} b_{2}^{w}, \\ β_{w, Ξ - 1} = r_{Ξ}^{T} c_{1}^{w} - φ_{Ξ - 1}^{T} M_{nc, 2} c_{2}^{w}, \end{matrix}$ (118) $\begin{matrix} α_{v, Ξ - 1} = r_{Ξ}^{T} b_{1}^{v} - φ_{Ξ - 1}^{T} M_{nc, 2} b_{2}^{v}, \\ β_{v, Ξ - 1} = r_{Ξ}^{T} c_{1}^{v} - φ_{Ξ - 1}^{T} M_{nc, 2} c_{2}^{v} . \end{matrix}$ Next, we need to find the solutions of

(119) $\begin{matrix} max_{w (Ξ - 1) \in [γ_{w}, η_{w}]} [α_{w, Ξ - 1} w (Ξ - 1) + β_{w, Ξ - 1} w (Ξ - 1)^{2}] \\ and min_{v (Ξ - 1) \in [γ_{v}, η_{v}]} [α_{v, Ξ - 1} v (Ξ - 1) + β_{v, Ξ - 1} v (Ξ - 1)^{2}] . \end{matrix}$ The maximum point w^* (Ξ - 1) of $max_{w (Ξ - 1) \in [γ_{w}, η_{w}]} [α_{w, Ξ - 1} w (Ξ - 1) + β_{w, Ξ - 1} w (Ξ - 1)^{2}]$ are summarized below:

(120) $\begin{matrix} w^{*} (Ξ - 1) = {\begin{matrix} η_{w}, & if α_{w, Ξ - 1} > 0 and β_{w, Ξ - 1} = 0, or - \frac{α_{w, Ξ - 1}}{2 β_{w, Ξ - 1}} \geq η_{w} and β_{w, Ξ - 1} < 0, \\ or ∣ γ_{w} + \frac{α_{w, Ξ - 1}}{2 β_{w, Ξ - 1}} ∣ < ∣ η_{w} + \frac{α_{w, Ξ - 1}}{2 β_{w, Ξ - 1}} ∣ and β_{w, Ξ - 1} > 0, \\ γ_{w}, & if α_{w, Ξ - 1} \leq 0 and β_{w, Ξ - 1} = 0, or - \frac{α_{w, Ξ - 1}}{2 β_{w, Ξ - 1}} \leq γ_{w} and β_{w, Ξ - 1} < 0, \\ or ∣ γ_{w} + \frac{α_{w, Ξ - 1}}{2 β_{w, Ξ - 1}} ∣ \geq ∣ η_{w} + \frac{α_{w, Ξ - 1}}{2 β_{w, Ξ - 1}} ∣ and β_{w, Ξ - 1} > 0, \\ - \frac{α_{w, Ξ - 1}}{2 β_{w, Ξ - 1}}, & if γ_{w} \leq \frac{α_{w, Ξ - 1}}{2 β_{w, Ξ - 1}} \leq η_{w} and β_{w, Ξ - 1} < 0, \\ undetermined, & if α_{w, Ξ - 1} = 0 and β_{w, Ξ - 1} = 0 . \end{matrix} \end{matrix}$

The minimum point v^* (Ξ - 1) of $min_{v (Ξ - 1) \in [γ_{v}, η_{v}]} [α_{v, Ξ - 1} v (Ξ - 1) + β_{v, Ξ - 1} v (Ξ - 1)^{2}]$ are summarized below:

(121) $\begin{matrix} v^{*} (Ξ - 1) = {\begin{matrix} η_{v}, & if α_{v, Ξ - 1} < 0 and β_{v, Ξ - 1} = 0, or - \frac{α_{v, Ξ - 1}}{2 β_{v, Ξ - 1}} \geq γ_{w} and β_{v, Ξ - 1} < 0, \\ or ∣ γ_{v} + \frac{α_{v, Ξ - 1}}{2 β_{v, Ξ - 1}} ∣ \geq ∣ η_{v} + \frac{α_{v, Ξ - 1}}{2 β_{v, Ξ - 1}} ∣ and β_{v, Ξ - 1} > 0, \\ γ_{v}, & if α_{v, Ξ - 1} > 0 and β_{v, Ξ - 1} = 0, or - \frac{α_{v, Ξ - 1}}{2 β_{v, Ξ - 1}} \geq η_{v} and β_{v, Ξ - 1} < 0, \\ or ∣ γ_{v} + \frac{α_{v, Ξ - 1}}{2 β_{v, Ξ - 1}} ∣ < ∣ η_{v} + \frac{α_{v, Ξ - 1}}{2 β_{v, Ξ - 1}} ∣ and β_{v, Ξ - 1} > 0, \\ - \frac{α_{v, Ξ - 1}}{2 β_{v, Ξ - 1}}, & if γ_{v} \leq \frac{α_{v, Ξ - 1}}{2 β_{v, Ξ - 1}} \leq η_{v} and β_{v, Ξ - 1} > 0, \\ undetermined, & if α_{v, Ξ - 1} = 0 and β_{v, Ξ - 1} = 0, \end{matrix} \end{matrix}$ Denote σ_w,Ξ-1 = α_w,Ξ-1w^* (Ξ - 1) + β_w,Ξ-1w^* (Ξ - 1) ², and σ_v,Ξ-1 = α_v,Ξ-1v^* (Ξ - 1) + β_v,Ξ-1v^* (Ξ - 1) ². The corresponding optimal value is

(122) $\begin{matrix} J ({\tilde{y}}_{1} (Ξ - 1), {\tilde{y}}_{2} (Ξ), Ξ - 1) = r_{Ξ - 1}^{T} {\tilde{y}}_{1} (Ξ - 1) + s_{Ξ - 1}^{T} {\tilde{y}}_{2} (Ξ) + σ_{w, Ξ - 1} + σ_{v, Ξ - 1}, \end{matrix}$

where

(123) $r_{Ξ - 1}^{T} = φ_{Ξ - 1}^{T} M_{nc, 1} + r_{Ξ}^{T} A_{1}, s_{Ξ - 1}^{T} = φ_{Ξ - 1}^{T} M_{nc, 2} H_{2} + s_{Ξ}^{T} .$

For ξ = Ξ - 2, we obtain

(124) $\begin{matrix} J ({\tilde{y}}_{1} (Ξ - 2), {\tilde{y}}_{2} (Ξ), Ξ - 2) \\ = max_{w (Ξ - 2) \in [γ_{w}, η_{w}]} min_{v (Ξ - 2) \in [γ_{v}, η_{v}]} {φ_{Ξ - 2}^{T} [M_{nc, 1} {\tilde{y}}_{1} (Ξ - 2) + M_{nc, 2} [H_{2}^{2} {\tilde{y}}_{2} (Ξ) - b_{2}^{w} w (Ξ - 2) - c_{2}^{w} w (Ξ - 2)^{2} \\ - b_{2}^{v} v (Ξ - 2) - c_{2}^{v} v (Ξ - 2)^{2} - H_{2} (b_{2}^{w} w^{*} (Ξ - 1) + c_{2}^{w} w^{*} (Ξ - 1)^{2} + b_{2}^{v} v^{*} (Ξ - 1) + c_{2}^{v} v^{*} (Ξ - 1)^{2})]] \\ + r_{Ξ - 1}^{T} [A_{1} {\tilde{y}}_{1} (Ξ - 2) + b_{1}^{w} w (Ξ - 2) + c_{1}^{w} w (Ξ - 2)^{2} \\ + b_{1}^{v} v (Ξ - 2) + c_{1}^{v} v (Ξ - 2)^{2}] + s_{Ξ - 1}^{T} {\tilde{y}}_{2} (Ξ) + σ_{w, Ξ - 1} + σ_{v, Ξ - 1}} \\ = (φ_{Ξ - 2}^{T} M_{nc, 1} + r_{Ξ - 1}^{T} A_{1}) {\tilde{y}}_{1} (Ξ - 2) + (φ_{Ξ - 2}^{T} M_{nc, 2} H_{2}^{2} + s_{Ξ - 1}^{T}) {\tilde{y}}_{2} (Ξ) + σ_{w, Ξ - 1} + σ_{v, Ξ - 1} \\ - φ_{Ξ - 2}^{T} M_{nc, 2} H_{2} [b_{2}^{w} w^{*} (Ξ - 1) + c_{2}^{w} w^{*} (Ξ - 1)^{2}] - φ_{Ξ - 2}^{T} M_{nc, 2} H_{2} [b_{2}^{v} v^{*} (Ξ - 1) + c_{2}^{v} v^{*} (Ξ - 1)^{2}] \\ + max_{w (Ξ - 2) \in [γ_{w}, η_{w}]} [α_{w, Ξ - 2} w (Ξ - 2) + β_{w, Ξ - 2} w (Ξ - 2)^{2}] \\ + min_{v (Ξ - 2) \in [γ_{v}, η_{v}]} [α_{v, Ξ - 2} v (Ξ - 2) + β_{v, Ξ - 2} v (Ξ - 2)^{2}] \end{matrix}$ where (125) $α_{w, Ξ - 2} = r_{Ξ - 1}^{T} b_{1}^{w} - φ_{Ξ - 2}^{T} M_{nc, 2} b_{2}^{w}, β_{w, Ξ - 2} = r_{Ξ - 1}^{T} c_{1}^{w} - φ_{Ξ - 2}^{T} M_{nc, 2} c_{2}^{w},$ (126) $α_{v, Ξ - 2} = r_{Ξ - 1}^{T} b_{1}^{v} - φ_{Ξ - 2}^{T} M_{nc, 2} b_{2}^{v}, β_{v, Ξ - 2} = r_{Ξ - 1}^{T} c_{1}^{v} - φ_{Ξ - 2}^{T} M_{nc, 2} c_{2}^{v}$ Similarly, we have

(127) $\begin{matrix} w^{*} (Ξ - 2) = {\begin{matrix} η_{w}, & if α_{w, Ξ - 2} > 0 and β_{w, Ξ - 2} = 0, or - \frac{α_{w, Ξ - 2}}{2 β_{w, Ξ - 2}} \geq η_{w} and β_{w, Ξ - 2} < 0, \\ or ∣ γ_{w} + \frac{α_{w, Ξ - 2}}{2 β_{w, Ξ - 2}} ∣ < ∣ η_{w} + \frac{α_{w, Ξ - 2}}{2 β_{w, Ξ - 2}} ∣ and β_{w, Ξ - 2} > 0, \\ γ_{w}, & if α_{w, Ξ - 2} \leq 0 and β_{w, Ξ - 2} = 0, or - \frac{α_{w, Ξ - 2}}{2 β_{w, Ξ - 2}} \leq γ_{w} and β_{w, Ξ - 2} < 0, \\ or ∣ γ_{w} + \frac{α_{w, Ξ - 2}}{2 β_{w, Ξ - 2}} ∣ \geq ∣ η_{w} + \frac{α_{w, Ξ - 2}}{2 β_{w, Ξ - 2}} ∣ and β_{w, Ξ - 2} > 0, \\ - \frac{α_{w, Ξ - 2}}{2 β_{w, Ξ - 2}}, & if γ_{w} \leq \frac{α_{w, Ξ - 2}}{2 β_{w, Ξ - 2}} \leq η_{w} and β_{w, Ξ - 2} < 0, \\ undetermined, & if α_{w, Ξ - 2} = 0 and β_{w, Ξ - 2} = 0 . \end{matrix} \end{matrix}$ and

(128) $\begin{matrix} v^{*} (Ξ - 2) = {\begin{matrix} η_{v}, & if α_{v, Ξ - 2} < 0 and β_{v, Ξ - 2} = 0, or - \frac{α_{v, Ξ - 2}}{2 β_{v, Ξ - 2}} \geq γ_{w} and β_{v, Ξ - 2} < 0, \\ or ∣ γ_{v} + \frac{α_{v, Ξ - 2}}{2 β_{v, Ξ - 2}} ∣ \geq ∣ η_{v} + \frac{α_{v, Ξ - 2}}{2 β_{v, Ξ - 2}} ∣ and β_{v, Ξ - 2} > 0, \\ γ_{v}, & if α_{v, Ξ - 2} > 0 and β_{v, Ξ - 2} = 0, or - \frac{α_{v, Ξ - 2}}{2 β_{v, Ξ - 2}} \geq η_{v} and β_{v, Ξ - 2} < 0, \\ or ∣ γ_{v} + \frac{α_{v, Ξ - 2}}{2 β_{v, Ξ - 2}} ∣ < ∣ η_{v} + \frac{α_{v, Ξ - 2}}{2 β_{v, Ξ - 2}} ∣ and β_{v, Ξ - 2} > 0, \\ - \frac{α_{v, Ξ - 2}}{2 β_{v, Ξ - 2}}, & if γ_{v} \leq \frac{α_{v, Ξ - 2}}{2 β_{v, Ξ - 2}} \leq η_{v} and β_{v, Ξ - 2} > 0, \\ undetermined, & if α_{v, Ξ - 2} = 0 and β_{v, Ξ - 2} = 0, \end{matrix} \end{matrix}$ where

(129) $σ_{w, Ξ - 2} = α_{w, Ξ - 2} w^{*} (Ξ - 2) + β_{w, Ξ - 2} w^{*} (Ξ - 2)^{2}, σ_{v, Ξ - 2} = α_{v, Ξ - 2} v^{*} (Ξ - 2) + β_{v, Ξ - 2} v^{*} (Ξ - 2)^{2} .$ In order to maximize the expression $α_{w, Ξ - 1}^{1} w^{*} (Ξ - 1) + β_{w, Ξ - 1}^{1} w^{*} (Ξ - 1)^{2}$ , and the minimum of the expression $α_{v, Ξ - 1}^{1} v^{*} (Ξ - 1) + β_{v, Ξ - 1}^{1} v^{*} (Ξ - 1)^{2}$ where $α_{w, Ξ - 1}^{1} = - φ_{Ξ - 2}^{T} M_{nc, 2} H_{2} b_{2}^{w}$ , $β_{w, Ξ - 1}^{1} = - φ_{Ξ - 2}^{T} M_{nc, 2} H_{2} c_{2}^{w}$ , $α_{v, Ξ - 1}^{1} = - φ_{Ξ - 2}^{T} M_{nc, 2} H_{2} b_{2}^{v}$ , $β_{v, Ξ - 1}^{1} = - φ_{Ξ - 2}^{T} M_{nc, 2} H_{2} c_{2}^{v}$ . The updated control w^* (Ξ - 1) is

(130) $\begin{matrix} w^{*} (Ξ - 1) = {\begin{matrix} η_{w}, & if α_{w, Ξ - 1} > 0 and β_{w, Ξ - 1} = 0, or - \frac{α_{w, Ξ - 1}}{2 β_{w, Ξ - 1}} \geq η_{w} and β_{w, Ξ - 1} < 0, \\ or ∣ γ_{w} + \frac{α_{w, Ξ - 1}}{2 β_{w, Ξ - 1}} ∣ < ∣ η_{w} + \frac{α_{w, Ξ - 1}}{2 β_{w, Ξ - 1}} ∣ and β_{w, Ξ - 1} > 0, \\ γ_{w}, & if α_{w, Ξ - 1} \leq 0 and β_{w, Ξ - 1} = 0, or - \frac{α_{w, Ξ - 1}}{2 β_{w, Ξ - 1}} \leq γ_{w} and β_{w, Ξ - 1} < 0, \\ or ∣ γ_{w} + \frac{α_{w, Ξ - 1}}{2 β_{w, Ξ - 1}} ∣ \geq ∣ η_{w} + \frac{α_{w, Ξ - 1}}{2 β_{w, Ξ - 1}} ∣ and β_{w, Ξ - 1} > 0, \\ - \frac{α_{w, Ξ - 1}}{2 β_{w, Ξ - 1}}, & if γ_{w} \leq \frac{α_{w, Ξ - 1}}{2 β_{w, Ξ - 1}} \leq η_{w} and β_{w, Ξ - 1} < 0, \end{matrix} \end{matrix}$ if (α_w,Ξ-1, β_w,Ξ-1) ≠ (0, 0); and

(131) $\begin{matrix} w^{*} (Ξ - 1) = {\begin{matrix} η_{w}, & if α_{w, Ξ - 1}^{1} > 0 and β_{w, Ξ - 1}^{1} = 0, or - \frac{α_{w, Ξ - 1}^{1}}{2 β_{w, Ξ - 1}^{1}} \geq η_{w} and β_{w, Ξ - 1}^{1} < 0, \\ or ∣ γ_{w} + \frac{α_{w, Ξ - 1}^{1}}{2 β_{w, Ξ - 1}^{1}} ∣ < ∣ η_{w} + \frac{α_{w, Ξ - 1}^{1}}{2 β_{w, Ξ - 1}^{1}} ∣ and β_{w, Ξ - 1}^{1} > 0, \\ γ_{w}, & if α_{w, Ξ - 1}^{1} \leq 0 and β_{w, Ξ - 1}^{1} = 0, or - \frac{α_{w, Ξ - 1}^{1}}{2 β_{w, Ξ - 1}^{1}} \leq γ_{w} and β_{w, Ξ - 1}^{1} < 0, \\ or ∣ γ_{w} + \frac{α_{w, Ξ - 1}^{1}}{2 β_{w, Ξ - 1}^{1}} ∣ \geq ∣ η_{w} + \frac{α_{w, Ξ - 1}^{1}}{2 β_{w, Ξ - 1}^{1}} ∣ and β_{w, Ξ - 1}^{1} > 0, \\ - \frac{α_{w, Ξ - 1}^{1}}{2 β_{w, Ξ - 1}^{1}}, & if γ_{w} \leq \frac{α_{w, Ξ - 1}^{1}}{2 β_{w, Ξ - 1}^{1}} \leq η_{w} and β_{w, Ξ - 1}^{1} < 0, \\ undetermined, & if α_{w, Ξ - 1}^{1} = 0 and β_{w, Ξ - 1}^{1} = 0 . \end{matrix} \end{matrix}$ if $(α_{w, Ξ - 1}^{1}, β_{w, Ξ - 1}^{1}) = (0, 0)$ . And the new control v^* (Ξ - 1) is

(132) $\begin{matrix} v^{*} (Ξ - 1) = {\begin{matrix} η_{v}, & if α_{v, Ξ - 1} < 0 and β_{v, Ξ - 1} = 0, or - \frac{α_{v, Ξ - 1}}{2 β_{v, Ξ - 1}} \geq γ_{w} and β_{v, Ξ - 1} < 0, \\ or ∣ γ_{v} + \frac{α_{v, Ξ - 1}}{2 β_{v, Ξ - 1}} ∣ \geq ∣ η_{v} + \frac{α_{v, Ξ - 1}}{2 β_{v, Ξ - 1}} ∣ and β_{v, Ξ - 1} > 0, \\ γ_{v}, & if α_{v, Ξ - 1} > 0 and β_{v, Ξ - 1} = 0, or - \frac{α_{v, Ξ - 1}}{2 β_{v, Ξ - 1}} \geq η_{v} and β_{v, Ξ - 1} < 0, \\ or ∣ γ_{v} + \frac{α_{v, Ξ - 1}}{2 β_{v, Ξ - 1}} ∣ < ∣ η_{v} + \frac{α_{v, Ξ - 1}}{2 β_{v, Ξ - 1}} ∣ and β_{v, Ξ - 1} > 0, \\ - \frac{α_{v, Ξ - 1}}{2 β_{v, Ξ - 1}}, & if γ_{v} \leq \frac{α_{v, Ξ - 1}}{2 β_{v, Ξ - 1}} \leq η_{v} and β_{v, Ξ - 1} > 0, \end{matrix} \end{matrix}$ if (α_v,Ξ-1, β_v,Ξ-1) ≠ (0, 0); and

(133) $\begin{matrix} v^{*} (Ξ - 1) = {\begin{matrix} η_{v}, & if α_{v, Ξ - 1}^{1} < 0 and β_{v, Ξ - 1}^{1} = 0, or - \frac{α_{v, Ξ - 1}^{1}}{2 β_{v, Ξ - 1}^{1}} \geq γ_{w} and β_{v, Ξ - 1}^{1} < 0, \\ or ∣ γ_{v} + \frac{α_{v, Ξ - 1}^{1}}{2 β_{v, Ξ - 1}^{1}} ∣ \geq ∣ η_{v} + \frac{α_{v, Ξ - 1}^{1}}{2 β_{v, Ξ - 1}^{1}} ∣ and β_{v, Ξ - 1}^{1} > 0, \\ γ_{v}, & if α_{v, Ξ - 1}^{1} > 0 and β_{v, Ξ - 1}^{1} = 0, or - \frac{α_{v, Ξ - 1}^{1}}{2 β_{v, Ξ - 1}^{1}} \geq η_{v} and β_{v, Ξ - 1}^{1} < 0, \\ or ∣ γ_{v} + \frac{α_{v, Ξ - 1}^{1}}{2 β_{v, Ξ - 1}^{1}} ∣ < ∣ η_{v} + \frac{α_{v, Ξ - 1}^{1}}{2 β_{v, Ξ - 1}^{1}} ∣ and β_{v, Ξ - 1}^{1} > 0, \\ - \frac{α_{v, Ξ - 1}^{1}}{2 β_{v, Ξ - 1}^{1}}, & if γ_{v} \leq \frac{α_{v, Ξ - 1}^{1}}{2 β_{v, Ξ - 1}^{1}} \leq η_{v} and β_{v, Ξ - 1}^{1} > 0, \\ undetermined, & if α_{v, Ξ - 1}^{1} = 0 and β_{v, Ξ - 1}^{1} = 0, \end{matrix} \end{matrix}$ if $(α_{v, Ξ - 1}^{1}, β_{v, Ξ - 1}^{1}) = (0, 0)$ . The corresponding optimal value is

(134) $\begin{matrix} J ({\tilde{y}}_{1} (Ξ - 2), {\tilde{y}}_{2} (Ξ), Ξ - 2) & = r_{Ξ - 2}^{T} {\tilde{y}}_{1} (Ξ - 2) + s_{Ξ - 2}^{T} {\tilde{y}}_{2} (Ξ) + α_{w, Ξ - 1}^{1} w^{*} (Ξ - 1) + β_{w, Ξ - 1}^{1} w^{*} (Ξ - 1)^{2} \\ + α_{v, Ξ - 1}^{1} v^{*} (Ξ - 1) + β_{v, Ξ - 1}^{1} v^{*} (Ξ - 1)^{2} + \sum_{τ = Ξ - 2}^{Ξ - 1} (σ_{w, τ} + σ_{v, τ}), \end{matrix}$ where (135) $r_{Ξ - 2}^{T} = φ_{Ξ - 2}^{T} M_{nc, 1} + r_{Ξ - 1}^{T} A_{1}, s_{Ξ - 2}^{T} = φ_{Ξ - 2}^{T} M_{nc, 2} H_{2}^{2} + s_{Ξ - 1}^{T},$ (136) $σ_{w, Ξ - 2} = α_{w, Ξ - 2} w^{*} (τ - 2) + β_{w, Ξ - 2} w^{*} (τ - 2)^{2}, σ_{v, Ξ - 2} = α_{v, Ξ - 2} v^{*} (τ - 2) + β_{v, Ξ - 2} v^{*} (τ - 2)^{2} .$ By induction, the theorem is proved.

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