Saddle-Point Equilibrium Strategy for Linear Quadratic Uncertain Stochastic Hybrid Differential Games Based on Subadditive Measures

Abstract

This paper describes a kind of linear quadratic uncertain stochastic hybrid differential game system grounded in the framework of subadditive measures, in which the system dynamics are described by a hybrid differential equation with Wiener–Liu noise and the performance index function is quadratic. Firstly, we introduce the concept of hybrid differential games and establish the Max–Min Lemma for the two-player zero-sum game scenario. Next, we discuss the analysis of saddle-point equilibrium strategies for linear quadratic hybrid differential games, addressing both finite and infinite time horizons. Through the incorporation of a generalized Riccati differential equation (GRDE) and guided by the principles of the Itô–Liu formula, we prove that that solving the GRDE is crucial and serves as both a sufficient and necessary precondition for identifying equilibrium strategies within a finite horizon. In addition, we also acquire the explicit formulations of equilibrium strategies in closed forms, alongside determining the optimal values of the cost function. Through the adoption of a generalized Riccati equation (GRE) and applying a similar approach to that used for the finite horizon case, we establish that the ability to solve the GRE constitutes a sufficient criterion for the emergence of equilibrium strategies in scenarios extending over an infinite horizon. Moreover, we explore the dynamics of a resource extraction problem within a finite horizon and separately delve into an $H_{\infty }$ control problem applicable to an infinite horizon. Finally, we present the conclusions.

1. Introduction

Rufus Isaacs [1] pioneered differential game theory in 1965, and initially applied it to military strategies like pursuit and evasion. This sparked enduring interest in decision-making dynamics. In 1971, Friedman [2] established key theoretical foundations, revealing value and saddle points. Basar and Olsder [3] expanded on this, exploring noncooperative games, enriching our understanding of strategic dynamics. Later, Docker et al. [4] utilized mathematical tools to analyze equilibrium conditions, strengthening the field’s mathematical underpinnings. Overall, research in differential game theory has led to significant advancements in understanding strategic decision making in dynamic systems.

Linear quadratic differential games, a subset of the broader field of differential games, attract both dynamic game theorists and economists exploring policy coordination, resource extraction, and capital accumulation due to their applicability to real-world scenarios and ability to model complex strategic interactions. In 1965, Ho et al. [5] explored pursuit-evasion games, a key example of linear quadratic differential games, revealing foundational insights into strategic dynamics within this framework. Building on this foundation, Starr and Ho in 1969 [6] established a crucial condition for closed-loop strategies’ existence, essential for strategic decision making, utilizing the solution of the Riccati equation to provide valuable insights into dynamic optimal strategies. In 1970, Schmitendorf’s work [7] illuminated a notable aspect of linear quadratic differential games, revealing that the existence of a closed-loop saddle point does not guarantee the presence of an open-loop saddle point, emphasizing the complexities inherent in strategic decision making within dynamic systems. In recent years, scholars have continued to delve into linear quadratic differential games, refining methodologies and uncovering deeper strategic insights, underscoring the enduring relevance and complexity of this specialized area within differential game theory (see, e.g., Bernhard [8], Delfour et al. [9] and Delfour [10]).

In real-world scenarios, the evolution of states in dynamic systems is frequently disrupted by environmental noise, which can infiltrate the state equation or affect players’ observations of the system. When noise follows the Wiener process, stochastic differential equations become essential for characterizing system evolution, thus converting the differential game into a stochastic differential game. Similarly, when noise is shaped by the Liu process, reflecting the uncertainty tied to experts’ belief degrees, uncertain differential equations offer a pivotal means of characterizing system evolution, leading to the emergence of uncertain differential games. Fleming’s seminal contributions [11] in stochastic control paved the way for solving differential game scenarios with stochastic state dynamics. For instance, in 2006, Mou and Yong [12] utilized the Hilbert space method to examine open-loop strategies in stochastic linear quadratic differential games, while Sun and Yong [13] explored both open-loop and closed-loop saddle-point equilibriums in 2014. In uncertain settings, Zhu’s introduction [14] of uncertain optimal control in 2010 facilitated the analysis of differential games with uncertain state dynamics. Yang and Gao [15,16] further advanced this field by proposing uncertain differential games incorporating Liu process noise, establishing conditions for the existence of a feedback Nash equilibrium in 2013 and delving into linear quadratic uncertain differential games in 2016. The integration of chance theory based on subadditive measures addresses systems where both uncertainty and randomness coexist, modeling noise through the Wiener–Liu process and system evolution through uncertain stochastic hybrid differential equations, thus transforming the differential game into a hybrid differential game. Liu’s pioneering work [17] in 2013 laid the foundation for exploring uncertain stochastic hybrid systems [18,19] based on subadditive measures, followed by subsequent research by Fei et al. [20] in 2014, introducing uncertain stochastic hybrid optimal control and the equation of optimality via uncertain stochastic hybrid differential equations. With the help of uncertain stochastic optimal control, there has been a growing body of literature focusing on uncertain stochastic hybrid differential game systems [21,22,23,24]. Unlike the models and methods mentioned earlier, this work is the beginning of linear quadratic uncertain stochastic hybrid differential games in finite and infinite horizons.

The structure of this paper is organized as follows. Section 2 begins by recalling some basic results about the principle of optimality, the HJB equation and the feedback Nash equilibrium of hybrid differential games and so on, and then gives Max–Min Lemma of a two-player zero-sum game, which is essential for our analysis. Section 3 is devoted to the study of the saddle-point equilibrium strategy for linear quadratic hybrid differential games in continuous time. Section 4 presents a resource extraction problem and an $H_{\infty }$ control problem. Finally, Section 5 presents the conclusions.

2. Preliminaries

For any $(k,z)\in [0,T]\times {\Re }^m$, $s\in [k,T]$, and any control variable $D(·)\in \mathcal{U}$, where f, g, and h are the given functions, we consider the state equation,

$$\left\{\begin{array}{c}\mathrm{d}z(s)=f(s,z(s),D(s))ds+g(s,z(s),D(s))d{W}_s\\ +h(s,z(s),D(s))d{C}_s,z(T)=z.\end{array}\right.$$

The corresponding cost functional is

$$J(k,z,D)=E_{CH}\left[{\int }_s^T{\mathsf{\Phi }}_1(s,z(s),D(s))ds+{\mathsf{\Phi }}_2(T,z(T))\right],$$

where $E_{CH}$ is the chance expectation, $W_s$ is a Wiener process and $C_s$ a Liu process. Define the value function $V(k,z)$ by $V(k,z)\stackrel{\Delta }{=}\underset{D\in \mathcal{U}}{inf}J(k,z,D)$.

Theorem 1 

([20]). (Principle of optimality.) For any $(k,z)\in [0,T]\times {\Re }^p$, we have

$$V(k,z)=\underset{D\in \mathcal{U}}{inf}{E}_{CH}\left[{\int }_k^{\widehat{k}}{\mathsf{\Phi }}_1(s,z(s),D(s))ds+V(\widehat{k},z(\widehat{k}))\right],0\le k\le \widehat{k}\le T.$$

Theorem 2 

([20]). (HJB equation.) Let $C([0,T]\times {\Re }^p)$ denote all functions $V(k,z)$ on $[0,T]\times {\Re }^p$ that are continuously differentiable in time k and continuously twice differentiable in z. If $V(·)\in C([0,T]\times {\Re }^p)$, define operators $L(D)V$ by

$$\begin{array}{cc}\hfill L(D)V(k,z)\stackrel{\Delta }{=}& \frac{\partial V}{\partial k}(k,z)+\sum _{\mathrm{k}=1}^p\frac{\partial V}{\partial z_k}(k,z)f_k(k,z,D)\hfill \\ & +\frac{1}{2}\sum _{k=1}^p\sum _{l=1}^p\frac{{\partial }^{2}V}{z_k{z}_l}(k,z)\sum _{i=1}^m{g}_{ki}(k,z,D)g_{li}(k,z,D).\hfill \end{array}$$

Then, V is a solution of the following terminal problem of an HJB equation

$$\underset{D\in \mathcal{U}}{inf}\left\{L(D)V(k,z)+{\mathsf{\Phi }}_1(k,z,D)\right\}=0$$

with the terminal condition $V(T,z)={\mathsf{\Phi }}_2(T,z).$

A differential game is a class of decision problems in which the evolution of the state is described by a differential equation. The players act throughout a time interval $[k_0,T]$ and aim to maximize their payoffs. In the general n-person differential game model, player i optimizes the objective

$$\begin{array}{c}\hfill \underset{u_i}{sup}{\int }_{k_0}^T{R}_i(k,z(k),u_1(k),u_2(k),\cdots ,u_n(k))dk+{\Theta }_i(z(T))\mathrm{for}i\in N=\{1,2,\cdots ,n\}\end{array}$$

(1)

subject to

$$\begin{array}{c}\hfill dz(k)=f(k,z(k),u_1(k),u_2(k),\cdots ,u_n(k))dk,z(k_0)=z_0,\end{array}$$

(2)

where $T>k_0\ge 0,z(k)\in {\Re }^m$ is the state variable, $z_0$ is the given initial state, $u_i\in U_i$ is the control variable of player i, and $U_i$ is a compact metric space. The function $R_i(k,z,u_1,u_2,\cdots ,u_n)$ is the transient payoff function of player i at time $k,$ ${\Theta }_i(·)$ is the terminal reward function of player i at terminal time $T,$ and $f(k,z,u_1,u_2,\cdots ,u_n)$ is a vector function. All functions mentioned are differentiable.

In the true essence of the game, the state evolution is inevitably disturbed by environmental noise. This noise may may occur directly within the state equations or indirectly through the players’ observations of the system’s condition. In a vector-valued n-person uncertain stochastic hybrid differential game model, player i optimizes the objective

$$\begin{array}{cc}\hfill \underset{u_i}{sup}{\mathrm{E}}_{\mathrm{CH}}^{{\mathrm{k}}_0}& \left[{\int }_0^T{R}_i[k,z(k),u_1(k),u_2(k),\dots ,u_n(k)]dk\right.\hfill \\ & \left.\phantom{{\int }_0^T}+{\Theta }_i(z(T))\right],i\in N=\{1,2,\dots ,n\}.\hfill \end{array}$$

(3)

A vector-valued hybrid differential equation, which delineates the evolution of the state and n objective functions (3), provides a more suitable framework for analyzing differential games with uncertain stochastic noise driven by the Wiener–Liu process:

$$\begin{array}{cc}\hfill dz(k)=& f[k,z(k),u_1(k),u_2(k),\cdots ,u_n(k)]dk\hfill \\ & +\sigma [k,z(k),u_1(k),u_2(k),\cdots ,u_n(k)]d{\mathcal{H}}_k\hfill \\ \hfill z(0)=& z_0\hfill \end{array}$$

(4)

where $E_{CH}^{k_0}$ represents the chance expectation operator performed at time $k_0$, $z(k)\in {\Re }^m$ represents the state variables, $u_i\in {\Re }^{l_i}$ is the control of player $i,$ ${\mathcal{H}}_k=(W_k,C_k)$ is an l-dimensional Wiener–Liu process, similar to refs. [18,19], and $z_0$ is the given initial state.

For the subsequent analysis, the following assumptions are presented:

$$f:[0,T]\times {\Re }^m\times {\Re }^{l_1}\times \dots \times {\Re }^{l_n}\to {\Re }^m$$

and

$$\sigma :[0,T]\times {\Re }^m\times {\Re }^{l_1}\times \dots \times {\Re }^{l_n}\to {\Re }^m\times {\Re }^l$$

which satisfy the Lipschitz condition and linear growth condition and possess continuous partial derivatives.

For $k\in [k_0,T]$, the admissible feedback control denotes:

$$u_i(k)=u_i(k,z(k))$$

where

$$u_i(·,·):[0,T]\times {\Re }^m⟶{\Re }^{l_i}.$$

Denote

$$u_{-i}(k,z)=\{u_1(k,z),u_2(k,z),\cdots ,u_{i-1}(k,z),u_{i+1}(k,z),\cdots ,u_n(k,z)\}.$$

A feedback Nash equilibrium of the hybrid differential game (3)–(4) can be defined as follows.

Definition 1. 

A set of strategies $\{u_1^{*}(s,z),u_2^{*}(s,z),\cdots ,u_n^{*}(s,z)\}$ is called a feedback Nash equilibrium for the n-person hybrid differential game (3)–(4), and $\{z^{*}(s),k\le s\le T\}$ is the corresponding state trajectory, if there exist real-valued functions $V^i(k,z):[0,T]\times {\Re }^m\to \Re ,$ satisfying the following relations for each $i\in N:$

$$\begin{array}{cc}\hfill V_i(k,z)& =E_{CH}^{k_0}[{\int }_{k_0}^T{R}_i(k,z^{*}(k),u_1^{*}(k),u_2^{*}(k),\cdots ,u_n^{*}(k))dk+{\Theta }_i^{*}(z(T))\hfill \\ \hfill & \ge E_{CH}^{k_0}[{\int }_{k_0}^T{R}_i(k,z^{[i]}(k),u_1^{[i]}(k),u_2^{[i]}(k),\cdots ,u_n^{[i]}(k))dk+{\Theta }_i^{[i]}(z(T))\hfill \\ \hfill & \forall u_i(·,·)\in [k_0,T]\times {\Re }^m,z(·)\in {\Re }^m;\hfill \\ \hfill V_i(T,z)& ={\Theta }_i(z(T))\hfill \end{array}$$

where on the time interval $[k,T]$:

$$\begin{array}{cc}\hfill d{z}^{*}(s)=& f[s,z^{*}(s),u_i^{*}(s,z^{*}),u_{-i}^{*}(s,z^{*})]ds\hfill \\ & +g[s,z^{*}(s),u_i^{*}(s,z^{*}),u_{-i}^{*}(s,z^{*})]d{\mathcal{H}}_s\hfill \\ \hfill z^{*}(k)=& z;\hfill \\ \hfill d{z}^{[i]}(s)=& f[s,z^{[i]}(s),u_i(s,z^{[i]}),u_{-i}^{*}(s,z^{[i]})]ds\hfill \\ & \hfill +g[s,z^{[i]}(s),u_i(s,z^{[i]}),u_{-i}^{*}(s,z^{[i]})]d{\mathcal{H}}_s\\ \hfill z^{[i]}(k)=& z.\hfill \end{array}$$

Remark 1. 

If an n-pair $u_i^{*}(s,z);i\in N$ establishes a feedback Nash equilibrium for an n-person differential game, as defined in Equations (3)–(4), over the duration $[k_0,T]$, then its restriction to the time interval $[k,T]$ also constitutes a feedback Nash equilibrium, just as Docker et al. [4] described. Importantly, the feedback Nash equilibrium depends solely on the the current state value $z(T)$ and time variable k, not on any prior history (including the initial state $z_0$).

Next, we give sufficient conditions guaranteeing that $u_i(k,z);i\in N$ is a feedback Nash equilibrium for the game (3)–(4).

Lemma 1. 

An n-tuple of strategies $\{u_i^{*}(k,z);i\in N\}$ provides a feedback Nash equilibrium to the n-player uncertain differential game (3)–(4) if there exist real-valued functions $V^i(k,z):[0,T]\times {\Re }^m\to \Re ,i\in N,$ satisfying the partial differential equations:

$$\begin{array}{cc}\hfill -V_k^i(k,z)=& \underset{u_i}{sup}\left\{R_i[k,z(k),u_i(k,z),u_{-i}^{*}(k,z)]\right.\hfill \\ & \left.+{\nabla }_z{V}^i(k,z)f[k,z,u_i(k,z),u_{-i}^{*}(k,z)]+\frac{1}{2}{\nabla }_{zz}{V}^i(k,z){\sigma }^{\tau }\sigma [k,z,u_i(k,z),u_{-i}^{*}(k,z)]\right\}\hfill \\ \hfill =& R_i[k,z,u_i^{*}(k,z),u_{-i}^{*}(k,z)]\hfill \\ & \hfill +{\nabla }_z{V}^i(k,z)f[k,z,u_i^{*}(k,z),u_{-i}^{*}(k,z)]+\frac{1}{2}{\nabla }_{zz}{V}^i(k,z){\sigma }^{\tau }\sigma [k,z,u_i^{*}(k,x),u_{-i}^{*}(k,z)]\\ \hfill V^i(T,z)=& {\Theta }_i(z(T)).\hfill \end{array}$$

Proof. 

The result can be readily derived from Theorem 2 and the definition of the feedback Nash equilibrium. By holding the strategies of all players fixed at their equilibrium choices, except for the ith player’s, we transform the scenario into a hybrid optimal control problem as described by Theorem 2. □

Now, let us delve into The “Max–Min Lemma” of the two-player zero-sum hybrid differential game.

Lemma 2. 

(Max–Min Lemma): A pair of strategies $\{u_i^{*}(k,z)\in {\Re }^{l_i};i=1,2\}$ provides a feedback Nash equilibrium solution (called a saddle-point Nash equilibrium) to the two-player zero-sum of the game (3)–(4) if there exists a real-valued function $V(k,z):[0,T]\times {\Re }^m\to \Re ,$ satisfying the partial differential equations:

$$\begin{array}{cc}\hfill -V_k(k,z)=& \underset{u_1}{max}\underset{u_2}{min}\{R(k,z,u_1,u_2)\hfill \\ & +{\nabla }_{z}V(k,z)f(k,z,u_1,u_2)\}+\frac{1}{2}{\sigma }^{\tau }(k,z,u_1,u_2){\nabla }_{zz}V(k,z)\sigma (k,z,u_1,u_2)\}\hfill \\ \hfill =& \underset{u_2}{min}\underset{u_1}{max}\{R(k,z,u_1,u_2)\hfill \\ & +{\nabla }_{z}V(k,z)f(k,z,u_1,u_2)\}+\frac{1}{2}{\sigma }^{\tau }(k,z,u_1,u_2){\nabla }_{zz}V(k,z)\sigma (k,z,u_1,u_2)\}\hfill \\ \hfill =& R(k,z,u_1^{*},u_2^{*})+{\nabla }_{z}V(k,z)f(k,z,u_1^{*},u_2^{*})+\frac{1}{2}{\sigma }^{\tau }(k,z,u_1^{*},u_2^{*}){\nabla }_{zz}V(k,z)\sigma (k,z,u_1^{*},u_2^{*})\hfill \\ \hfill V(T,z)& =\Theta (z(T)).\hfill \end{array}$$

Proof. 

As a special case of Lemma 1, the result can be easily obtained by taking $n=2,R_1(·)=-R_2(·)\doteq R(·),$ and ${\Theta }_1(·)=-{\Theta }_2(·)\doteq \Theta (·),$ in which case $V^1=-V^2\doteq V$ and the Max–Min Lemma is completed. □

3. Main Results

Notation: in the following, denote by ${\Re }^n$ the set of n-dimensional Euclidean spaces, ${\Re }^{m\times n}$ the set of all $m\times n$ matrices, $S^n$ the set of all real symmetric $n\times n$ matrices, and ${\tilde{S}}^n$ the set of all positive definite $n\times n$ matrices. $P>0$ denotes $P\in {\tilde{S}}^n$, $P^{\tau }$ denotes the transpose of a matrix or vector, $\dot{P}=\frac{dP}{dk}$, and k is the time. For a Hilbert space $\tilde{H}$ and an interval I, let $L^{\infty }(I,\tilde{H})$ be the space of all bounded and measurable functions from I to $\tilde{H}$, that is, $f:I\to \tilde{H}$, $L^{1,\infty }=\{P\in L^{\infty }(I,\tilde{H})|\dot{P}\in L^{\infty }(I,\tilde{H})\}$, where $I=[s,T)$ or $[s,\infty )$. Now, we discuss two-player zero-sum hybrid differential games in finite and infinite horizons.

3.1. The Case of Finite Horizons

Fix $(s,z)\in [k_0,T]\times {\Re }^n$. Let ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ be two standard independent Wiener–Liu processes in the chance space over $[s,T]$ with ${\mathcal{H}}_i(s)=0$ almost surely. Let $U_i[s,T]$ be the set of ${\Re }^l$-valued square integrable processes adapted with the $\sigma$-field generated by ${\mathcal{H}}_i(·),i=1,2,$ respectively. Associated with each $(u_1,u_2)\in \mathcal{U}[s,T]\equiv U_1[s,T]\times U_2[s,T]$ is a quadratic cost $J(u_1,u_2)$. The performance index function is defined as:

$$\begin{array}{c}\hfill J(u_1(·),u_2(·))=E_{CH}\left\{z^{\tau }(T)Nz(T)+\right.{\int }_s^T(z^{\tau }Gz+2{u^{\tau }}_1{L}_{1}z+{u^{\tau }}_1{R}_1{u}_1+2{u^{\tau }}_2{L}_{2}z+{u^{\tau }}_2{R}_2{u}_2)dk\}\end{array}$$

(5)

where z is the solution to the following equation

$$\begin{array}{c}\hfill \left\{\begin{array}{c}dz=[Az+B_1{u}_1(k)+B_2{u}_2(k)]dk\\ +[C_{1}z+D_1{u}_1(k)]d{\mathcal{H}}_1(k)\\ +[C_{2}z+D_2{u}_2(k)]d{\mathcal{H}}_2(k)\\ z(s)=z_0\end{array}\right.\end{array}$$

(6)

and $E\{\}$ symbolizes the chance expectation. $A,B_i,C_i,D_i,G,N,R_i$ and $L_i$ are matrix functions, $i=1,2.$

Assumption 1. 

We assume that $A,B_i,C_i,D_i,G,N,R_i$ and $L_i$ in (5)–(6) satisfy

$$\begin{array}{c}\hfill \left\{\begin{array}{c}A,C_i\in L^{\infty }(I,{\Re }^{n\times n});B_i,D_i,L_i^{\tau }\in L^{\infty }(I,{\Re }^{n\times l}),\\ R_i\in L^{\infty }(I,S^l),i=1,2;G\in L^{\infty }(I,S^n);N\in S^n.\end{array}\right.\end{array}$$

Hybrid differential game problem 1: for the hybrid system described by (6), find the feasible control $({u_1}^{*}(·),{u_2}^{*}(·))\in \mathcal{U}[s,T]$ and ensure that the following holds:

$$J({u_1}^{*}(·),u_2(·))\le J({u_1}^{*}(·),{u_2}^{*}(·))\le J(u_1(·),{u_2}^{*}(·))$$

Theorem 3. 

The two-player zero-sum linear quadratic hybrid differential game (5)–(6) has a saddle-point Nash equilibrium solution if the following differential Riccati equation has a solution $P(k)\in L^{1,\infty }(I,S^n)$

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{P}+A^{\tau }P+PA+{\displaystyle \sum _{i=1}^2}{C_i}^{\tau }P{C}_i+G-{\displaystyle \sum _{i=1}^2}{S_i}^{\tau }(P)R_i^{-1}(P)S_i(P)=0\\ R_1(P)>0,R_2(P)<0\\ R_i(P)K_i(P)=S_i(P),i=1,2\\ P(T)=N\end{array}\right.\end{array}$$

(7)

where $S_i(P)$, $R_i(P)$, and $K_i(P)$ are defined as

$$\left\{\begin{array}{c}{S}_i(P)=B_i^{\tau }{P+D}_i^{\tau }P{C}_i+L_i\\ R_i(P)=R_i{+D}_i^{\tau }P{D}_i\\ K_i(P)=R_i^{-1}(P)S_i(P),i=1,2.\end{array}\right.$$

and the saddle point and the optimum value are

$$({u_1}^{*}(·),{u_2}^{*}(·))=(-K_1(P)z(·),-K_2(P)z(·))$$

and

$$J({u_1}^{*}(·),{u_2}^{*}(·))=z^{\tau }P(s)z.$$

Proof. 

Let $P(k)\in L^{1,\infty }(I,S^n)$ be the solution of Equation (7) and $z(k)$ be the solution of Equation (6) corresponding to control $(u_1(k),u_2(k)).$ Using the fundamental theorem of calculus and Itô–Liu formula in ref. [20], applied to $z^{\tau }(k)p(k)z(k)$, we obtain

$$\begin{array}{cc}\hfill d[z^{\tau }pz]=& [z^{\tau }(\dot{P}+PA+A^{\tau }P+\sum _{i=1}^2{C}_i^{\tau }P{C}_i)z\hfill \\ \hfill & +2u_1^{\tau }(B_1^{\tau }P+D_1^{\tau }P{C}_1)z+u_1^{\tau }{D}_1^{\tau }P{D}_1{u}_1+\hfill \\ \hfill & +2u_2^{\tau }(B_2^{\tau }P+D_2^{\tau }P{C}_2)z+u_2^{\tau }{D}_2^{\tau }P{D}_2{u}_2]dk\hfill \\ \hfill & +\{\cdots \}d{W}_1+\{\cdots \}d{W}_2+\{\cdots \}d{C}_1+\{\cdots \}d{C}_2.\hfill \end{array}$$

Taking integrations on $[s,T]$ and chance expectation, we obtain

$$\begin{array}{cc}\hfill E_{CH}[z^{\tau }(T)P(T)z(T)]=& {z_0}^{\tau }P(s)z_0+E_{CH}{\int }_s^T{z}^{\tau }(\dot{P}+PA+A^{\tau }P+\sum _{i=1}^2{C}_i^{\tau }P{C}_i)zdk\hfill \\ \hfill & +\sum _{i=1}^2{E}_{CH}{\int }_s^T[2u_i^{\tau }(B_i^{\tau }P+D_i^{\tau }P{C}_i)z+u_i^{\tau }{D}_i^{\tau }P{D}_i{u}_i]dk.\hfill \end{array}$$

(8)

Substituting (8) into (5), $J(u_1,u_2)$ can be be reduced to

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill J(u_1(·),u_2(·))& ={z_0}^{\tau }P(s)z_0+E_{CH}\{z^{\tau }(T)(N-P(T))z(T)\}\hfill \\ \hfill & +E_{CH}{\int }_s^T\{[z^{\tau }(\dot{P}+PA+A^{\tau }P+\sum _{i=1}^2{C}_i^{\tau }P{C}_i+G)z]\}dk\hfill \\ \hfill & +E_{CH}{\int }_s^T\sum _{i=1}^2\{[2u_i^{\tau }(B_i^{\tau }P+D_i^{\tau }P{C}_i+L_i)z\hfill \\ \hfill & +u_i^{\tau }(R_i+D_i^{\tau }P{D}_i)u_i]\}dk\hfill \\ \hfill =& {z_0}^{\tau }P(s)z_0+E_{CH}{\int }_s^T\{[z^{\tau }(\dot{P}+PA+A^{\tau }P+\sum _{i=1}^2{C}_i^{\tau }P{C}_i+G)z]dk\hfill \\ \hfill & +\sum _{i=1}^2[2u_i^{\tau }{S}_i(P)z+u_i^{\tau }{R}_i(P)u_i]\}dk.\hfill \end{array}\end{array}$$

While, by the following equality

$$\begin{array}{cc}\hfill & 2u_i^{\tau }{S}_i(P)z+u_i^{\tau }{R}_i(P)u_i\hfill \\ \hfill =& u_i^{\tau }{R}_i(P)R_i{(P)}^{-1}{R}_i(P)u_i+2u_i^{\tau }{S}_i(P)z\hfill \\ \hfill & +z^{\tau }{S}_i^{\tau }(P)R_i^{-1}(P)S_i(P)z-z^{\tau }{S}_i^{\tau }(P)R_i^{-1}(P)S_i(P)z\hfill \\ \hfill =& {[u_i+R_i^{-1}(P)S_i(P)z]}^{\tau }{R}_i(P)[u_i+R_i^{-1}(P)S_i(P)z]\hfill \\ \hfill & -z^{\tau }{S}_i^{\tau }(P)R_i^{-1}(P)S_i(P)z.\hfill \end{array}$$

we can obtain that

$$\begin{array}{cc}\hfill J(u_1(·),u_2(·))=& {z_0}^{\tau }P(s)z_0+E_{CH}{\int }_s^T\{z^{\tau }[\dot{P}+PA+A^{\tau }P+\sum _{i=1}^2{C}_i^{\tau }P{C}_i+G-\hfill \\ \hfill & \sum _{i=1}^2{S}_i^{\tau }(P)R_i^{-1}(P)S_i(P)]zdk+\hfill \\ & +\sum _{i=1}^2{[u_i+R_i^{-1}(P)S_i(P)z]}^{\tau }{R}_i(P)[u_i+R_i^{-1}(P)S_i(P)z]\}dk.\hfill \end{array}$$

So,

$$\begin{array}{cc}\hfill J(u_1(·),u_2(·))=& {z_0}^{\tau }P(s)z_0\hfill \\ \hfill & +\sum _{i=1}^2{E}_{CH}{\int }_s^T{[u_i+R_i^{-1}(P)S_i(P)z]}^{\tau }{R}_i(P)[u_i+R_i^{-1}(P)S_i(P)z]dk.\hfill \end{array}$$

The theorem is proved. □

Theorem 4. 

If linear feedback controls $({u_1}^{*}(·),{u_2}^{*}(·))=(-K_1(P)z(·),-K_2(P)z(·))$ are optimal for the two-player zero-sum linear quadratic hybrid differential game (5)–(6), then the Riccati Equation (7) must have a solution $P(·)$. Moreover, $K_i(P)=R_i^{-1}(P)S_i(P)$.

Proof. 

By Theorem (2), the value function $V(k,z)$ satisfies the HJB equation

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V_k(k,z)& +\underset{u_1}{min}\underset{u_2}{max}\left\{z^{\tau }Gz+2{u^{\tau }}_1{L}_{1}z\right.+{u^{\tau }}_1{R}_1{u}_1+2{u^{\tau }}_2{L}_{2}z+{u^{\tau }}_2{R}_2{u}_2\hfill \\ \hfill & +{[Az(T)+B_1{u}_1(k)+B_2{u}_2(k)]}^{\tau }·V_z(k,z)\hfill \\ \hfill & +\frac{1}{2}{[C_{1}z(T)+D_1{u}_1(k)]}^{\tau }{V}_{zz}(k,z)[C_{1}z(T)+D_1{u}_1(k)]\hfill \\ \hfill & +\frac{1}{2}{[C_{2}z(T)+D_2{u}_2(k)]}^{\tau }{V}_{zz}(k,z)[C_{2}z(T)+D_2{u}_2(k)]\}\hfill \\ \hfill & =0.\hfill \end{array}\end{array}$$

(9)

Take the following value equation

$$\begin{array}{c}\hfill V(k,z)=z^{\tau }Pz.\end{array}$$

(10)

So, substituting (10) into (9), and by the assumption that $({u_1}^{*}(·),{u_2}^{*}(·))=(-K_1(P)z(·),$$-K_2(P)z(·))$, we can obtain that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill 0=& z^{\tau }(\dot{P}+PA+A^{\tau }P+\sum _{i=1}^2{C}_i^{\tau }P{C}_i+G)z\hfill \\ \hfill & +\underset{u_1}{min}\underset{u_2}{max}\{2u_i^{\tau }{S}_i(P)z+u_i^{\tau }{R}_i(P)u_i\}\hfill \\ \hfill =& z^{\tau }(\dot{P}+PA+A^{\tau }P+\sum _{i=1}^2{C}_i^{\tau }P{C}_i+G)z\hfill \\ \hfill & +\underset{u_1}{min}\underset{u_2}{max}\{{[u_i+R_i^{-1}(P)S_i(P)z]}^{\tau }{R}_i(P)[u_i+R_i^{-1}(P)S_i(P)z]\hfill \\ \hfill & -z^{\tau }\sum _{i=1}^2{S}_i^{\tau }(P)R_i^{-1}(P)S_i(P)z\}\hfill \\ \hfill =& z^{\tau }(\dot{P}+PA+A^{\tau }P+\sum _{i=1}^2{C}_i^{\tau }P{C}_i+G-\sum _{i=1}^2{S}_i^{\tau }(P)R_i^{-1}(P)S_i(P))z.\hfill \end{array}\end{array}$$

So, we obtain that $K_i(P)=R_i^{-1}(P)S_i(P)$ and

$$\dot{P}+A^{\tau }P+PA+\sum _{i=1}^2{C_i}^{\tau }P{C}_i+G-\sum _{i=1}^2{S_i}^{\tau }(P)R_i^{-1}(P)S_i(P)=0.$$

The proof is complete. □

3.2. The Case of Infinite Horizons

Fix $(s,z)\in [k_0,\infty )\times {\Re }^n$. Let ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ be two standard independent Wiener–Liu processes in a chance space over $[s,\infty )$ with ${\mathcal{H}}_i(s)=0$ almost surely. Let $U_i[s,\infty )$ be the set of ${\Re }^l$-valued square integrable processes (denote by $L_i^2({\Re }^l)$) adapted with the $\sigma$-field generated by ${\mathcal{H}}_i(·),i=1,2,$ respectively. Associated with each $(u_1,u_2)\in \mathcal{U}[s,\infty )\equiv U_1[s,\infty )\times U_2[s,\infty )$ is a quadratic cost $J_{\infty }(u_1,u_2)$. The performance index function is defined as:

$$\begin{array}{c}\hfill J_{\infty }(u_1(·),u_2(·))=E_{CH}\{{\int }_s^{\infty }[(z^{\tau }Gz+2{u^{\tau }}_1{L}_{1}z+{u^{\tau }}_1{R}_1{u}_1+2{u^{\tau }}_2{L}_{2}z+{u^{\tau }}_2{R}_2{u}_2)]dk\}\end{array}$$

(11)

where z is the solution to the following equation

$$\begin{array}{c}\hfill \left\{\begin{array}{c}dz=[Az+B_1{u}_1(k)+B_2{u}_2(k)]dk\\ +[C_{1}z+D_1{u}_1(k)]d{\mathcal{H}}_1(k)\\ +[C_{2}z+D_2{u}_2(k)]d{\mathcal{H}}_2(k)\\ z(s)=z_0\end{array}\right.\end{array}$$

(12)

and $E\{\}$ represents the expectation of the enclosed uncertain random variable. $A,B_i,C_i,D_i,G,R_i$ and $L_i$ are matrix functions, $i=1,2.$ and they also satisfy Assumption 1.

Hybrid differential game problem 2: for the hybrid system described by the given Formula (11), find the feasible control $({u_1}^{*}(·),{u_2}^{*}(·))\in \mathcal{U}[s,\infty )$ and ensure that the following holds:

$$J_{\infty }({u_1}^{*}(·),u_2(·))\le J_{\infty }({u_1}^{*}(·),{u_2}^{*}(·))\le J_{\infty }(u_1(·),{u_2}^{*}(·)).$$

For this problem, we introduce the following generalized Riccati equation (GRE)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{A}^{\tau }P+PA+{\displaystyle \sum _{i=1}^2}{C_i}^{\tau }P{C}_i+G-{\displaystyle \sum _{i=1}^2}{S_i}^{\tau }(P)R_i^{-1}(P)S_i(P)=0\\ R_1(P)>0,R_2(P)<0\\ R_i(P)K_i(P)=S_i(P),i=1,2\end{array}\right.\end{array}$$

(13)

where $P\in S^n$ is an unknown matrix, and $S_i(P)$, $R_i(P)$, and $K_i(P)$ are defined as

$$\left\{\begin{array}{c}{S}_i(P)=B_i^{\tau }{P+D}_i^{\tau }P{C}_i+L_i\\ R_i(P)=R_i{+D}_i^{\tau }P{D}_i\\ K_i(P)=R_i^{-1}(P)S_i(P),i=1,2.\end{array}\right.$$

Since we are considering the hybrid differential game problem in an infinite horizon, we need the concept of mean-square stabilizability.

Definition 2. 

(i) An open-loop control $(u_1(·),u_2(·))\in L_1^2({\Re }^l)\times L_2^2({\Re }^l)$ is called (mean-square) stabilizing if the corresponding state of (12) with the initial state $z\in {\Re }^n$ satisfies $\underset{k\to \infty }{lim}E\{z^{\tau }(k)z(k)\}=0$. A feedback control $(u_1(·),u_2(·))=(-K_{1}z(·),-K_{2}z(·))$, where $K_1,K_2\in {\Re }^{l\times n}$ is a constant matrix, is called stabilizing if for every initial state $z\in {\Re }^n$, the solution of the following equation

$$\begin{array}{c}\hfill \left\{\begin{array}{c}dz(k)=(A-B_1{K}_1-B_2{K}_2)z(k)dk\\ +(C_1-D_1{K}_1)z(k)d{\mathcal{H}}_1(k)\\ +(C_2-D_2{K}_2)z(k)d{\mathcal{H}}_2(k)\\ z(s)=z_0\end{array}\right.\end{array}$$

(14)

satisfies $\underset{k\to \infty }{lim}{E}_{CH}\{z^{\tau }(k)z(k)\}=0$.

(ii) The system (12) is called (mean-square) stablizable if there exists a mean-square stabilizing feedback control of the form $(u_1(·),u_2(·))=(-K_{1}z(·),-K_{2}z(·))$, where $K_1,K_2$ is a constant matrix.

Mean-square stabilizability is pivotal in this paper. We now introduce the equivalent conditions for verifying stabilizability, both analytically and computationally.

Lemma 3. 

$(u_1(·),u_2(·))=(-K_{1}z(·),-K_{2}z(·))$ is mean-square stabilizable if and only if there exist a matrix K and $U\in S^n,U>0$ such that

$$\begin{array}{cc}\hfill & (A-B_1{K}_1-B_2{K}_2)U^{\tau }+U(A-B_1{K}_1-B_2{K}_2)\hfill \\ \hfill & +\sum _{i=1}^2{(C_1-D_1{K}_1)}^{\tau }U(C_1-D_1{K}_1)<0.\hfill \end{array}$$

(15)

Proof. 

For any $n\times n$ matrix K, define an operator $\mathsf{\Phi }:S^n\to S^n$ by

$$\mathsf{\Phi }(U)=(A+BK)U+U{(A+BK)}^{\tau }+(C+DK)U{(C+DK)}^{\tau }$$

If $z(·)$ satisfies the feedback Equation (14) (under the feedback gain K), then by Itô–Liu’s formula, the matrix $Z(k)=E[z(k)z{(k)}^{\tau }]$ satisfies the differential matrix system $\dot{Z}(k)=\mathsf{\Phi }(z(k))$. Applying the result in ref. [25], we have the equivalence between the mean-square stabilizability and (15). □

Remark 2. 

Lemma 3 gives the equivalent conditions of mean-square stabilizability, which provides a theoretical basis for the hypothesis of Theorem 5.

Theorem 5. 

Suppose the system (14) is mean-square stabilizable. If the GRE (13) exists and $P^{*}\in S^n$, then hybrid differential game problem 2 is solvable; moreover, the saddle point and the optimum value of the performance index function are

$$({u_1}^{*}(·),{u_2}^{*}(·))=(-K_1(P^{*})z(·),-K_2(P^{*})z(·))$$

and

$$J({u_1}^{*}(·),{u_2}^{*}(·))=z^{\tau }{P}^{*}(s)z,$$

that is, $\forall (u_1(·),u_2(·))$ satisfies $({u_1}^{*}(·),u_2(·)),(u_1(·),{u_2}^{*}(·))\in \mathcal{U}[s,\infty )$. Then, we have

$$J_{\infty }({u_1}^{*}(·),u_2(·))\le J_{\infty }({u_1}^{*}(·),{u_2}^{*}(·))\le J_{\infty }(u_1(·),{u_2}^{*}(·))$$

Proof. 

Suppose $(u_1(·),u_2(·))\in \mathcal{U}[s,\infty )$, $z(·)$ is the solution of (14). For $\forall T>s$, denote

$$J(u_1(·),u_2(·))=E_{CH}{\int }_s^T(z^{\tau }Gz+2{u^{\tau }}_1{L}_{1}z+{u^{\tau }}_1{R}_1{u}_1+2{u^{\tau }}_2{L}_{2}z+{u^{\tau }}_2{R}_2{u}_2)dk.$$

Similar to the proof of Theorem 3, we have

$$\begin{array}{cc}\hfill J_T(u_1(·),u_2(·))=& z_0^{\tau }{P}^{*}{z}_0-E_{CH}\{z^{\tau }(T)P^{*}z(T)\}+E_{CH}{\int }_s^T\{z^{\tau }\{P^{*}A+A^{\tau }{P}^{*}\hfill \\ \hfill & +\sum _{i=1}^2[C_i^{\tau }{P}^{*}{C}_i-S_i^{\tau }(P^{*})R_i^{-1}(P^{*})S_i(P^{*})]z+G\hfill \\ \hfill & +\sum _{i=1}^2[z^{\tau }{S}_i^{\tau }(P^{*})R_i^{-1}(P^{*})S_i(P^{*})z+2z^{\tau }{S}_i(P^{*})u_i+u_i^{\tau }{R}_i(P^{*})u_i]\}dk\hfill \\ \hfill =& z_0^{\tau }{P}^{*}{z}_0-E_{CH}\{z^{\tau }(T)P^{*}z(T)\}\hfill \\ \hfill & +\sum _{i=1}^2{E}_{CH}{\int }_s^T{[u_i+K_i(P^{*})z]}^{\tau }\times R_i(P^{*})[u_i+K_i(P^{*})z]dk.\hfill \end{array}$$

Because $(u_1(·),u_2(·))\in \mathcal{U}[s,\infty )$, then $\underset{k\to \infty }{lim}E\{z^{\tau }(k)P^{*}z(k)\}=0$. Therefore,

$$\begin{array}{c}\hfill J_{\infty }(u_1(·),u_2(·))=z_0^{\tau }{P}^{*}{z}_0+\sum _{i=1}^2{E}_{CH}{\int }_s^{\infty }{[u_i+K_i(P^{*})z]}^{\tau }\times R_i(P^{*})[u_i+K_i(P^{*})z]dk.\end{array}$$

The proof is complete. □

Remark 3. 

Unlike refs. [8,9,10] concerning stochastic differential games and refs. [15,16], the models in this paper have a wider range of applications and cover these models as well.

4. Applications

4.1. Uncertain Stochastic Resource Extraction Game

The resource extraction problem is a classic issue in economics, involving economic agents (such as firms or countries) exploiting natural resources. Under certain assumptions, this problem typically aligns with differential game theory. For instance, Jørgensen and Yeung [26] explored a stochastic differential game model applied to a common-property fishery. Similarly, Yang and Gao [16] examined an uncertain differential game model for resource extraction. Assuming the resource dynamics are governed by a hybrid differential equation driven by a Wiener–Liu process, we can then consider studying a uncertain stochastic hybrid differential game model of resource extraction using chance theory.

Consider two companies (resource extractors) that are exploiting a renewable resource (such as fish stocks). The lease for this resource extraction starts at time 0 and ends at time T, where $T>0$. Let $u_i(T)$ represent the amount of resources extracted by company i at time k, with $i=1,2$, where each extractor can control their own extraction quantity. Let $z(k)$ represent the size of the resource stock at time k, with $z(k)>0$, and the equation of resource dynamics is

$$dz(k)=\left[mz(k)-u_1(k)-u_2(k)\right]dk+z(k)d{\mathcal{H}}_k,z(0)=z_0$$

where $m>0$ represents the growth rate of resources, with the initial state $z_0$ being provided. The hybrid process ${\mathcal{H}}_k$ is a one-dimensional Wiener–Liu process that is defined in a chance space

$$\begin{array}{c}\hfill (\Gamma \times \Omega ,\mathcal{L}\otimes \mathcal{F},\mathcal{M}\times P)\end{array}.$$

The performance index function is

$$J(u_1,u_2)=E_{CH}\left\{{\int }_0^T\left[z{(k)}^2-u_1{(k)}^2+u_2{(k)}^2\right]dk+z{(T)}^2\right\}.$$

Extractor 1 endeavors to maximize the value of the performance index function; on the other hand, Extractor 2 is determined to minimize this value.

Drawing upon Theorems 3 and 4, we arrive at the subsequent Riccati differential equation:

$$\left\{\begin{array}{c}\dot{P}+2mP+3=0\hfill \\ P(T)=1.\hfill \end{array}\right.$$

Thus,

$$P(k)=\left[(2m+3)e^{2m(T-k)}-3\right]/(2m).$$

The saddle-point Nash equilibria are

$$\left\{\begin{array}{c}{u}_1^{*}(k)=-\frac{1}{2m}\left[(2m+3)e^{2m(T-k)}-3\right]z(k)\hfill \\ u_2^{*}(k)=\frac{1}{2m}\left[(2m+3)e^{2m(T-k)}-3\right]z(k).\hfill \end{array}\right.$$

The optimum value of the performance index function is

$$J(u_1^{*},u_2^{*})=\frac{1}{2m}\left[(2m+3)e^{2mT}-3\right]z_0^2.$$

By taking $m=1$, we can obtain the dynamic change curve of the resource stock $z(k)$, equilibrium strategy $(u_1^{*},u_2^{*})$ and equilibrium value $J(k,z)$ in Figure 1.

4.2. Uncertain Stochastic $H_{\infty }$ Control

Now, we apply the previous developed theory to solve some problems related to uncertain stochastic $H_{\infty }$ control.

$$\begin{array}{c}\hfill dz(k)=[Az(k)+B_{1}v(k)+B_{2}u(k)]dk+Cz(k)d{\mathcal{H}}_k\end{array}$$

(16)

with the cost functional

$$\begin{array}{c}\hfill J(u,v;z_0)=E_{CH}\{{\int }_0^{\infty }[z{(k)}^{T}Dz(k)-{\gamma }^2|v(k){|}^2]dk\}\end{array}$$

(17)

where the hybrid process ${\mathcal{H}}_k$ is a one-dimensional Wiener–Liu process that is defined in a chance space

$$\begin{array}{c}\hfill (\Gamma \times \Omega ,\mathcal{L}\otimes \mathcal{F},\mathcal{M}\times P)\end{array}.$$

In Equations (16) and (17), $z(k)\in {\Re }^n$ is the state vector, $u(k)\in {\Re }^{m2}$ is the input control and $v(k)\in {\Re }^{m1}$ is the vector of the exogenous disturbances, and in Equation (17), $J(u,v;z_0)$ represents $H_{\infty }$ constraints. The infinite-horizon uncertain stochastic $H_{\infty }$ control of system (16) is parallel with Definition 2 in [27], which can be described as follows.

Definition 3. 

For a given disturbance attenuation level $\gamma >0,$ we can find $u^{*}(k)\times v^{*}(k)\in L^2([0,\infty ),{\Re }^{m2})\times L^2([0,\infty ),{\Re }^{m1})$, such that

(i) $u^{*}(k)$ stabilizes system (16) internally; i.e., when $v(k)=0,u=u^{*}$, the state trajectory of Equation (16) with any initial value $z_0\in {\Re }^n$ satisfies

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{E}_{CH}[z^T(k)z(k)]=0.\end{array}$$

(ii) $|L_{u^{*}}{|}_{\infty }<\infty$ with

$$\begin{array}{c}\hfill |L_{u^{*}}{|}_{\infty }=\underset{v\in L_F^2([0,\infty ),R^{m1}),v\ne 0,u=u^{*},z_0=0}{sup}\frac{{\{E_{CH}{\int }_0^{\infty }z{(k)}^{T}Dz(k)dk\}}^{1/2}}{\{E_{CH}{\int }_0^{\infty }{\left|v(k)\right|}^2{dk\}}^{1/2}}\end{array}$$

In essence, the $H_{\infty }$ control issue, as outlined by Equations (16) and (17), seeks to identify control $u^{*}$ that ensures $J(u^{*})<0$ in the face of any exogenous disturbances $v(t)$. Following the insights in [27], if we conceptualize $u(t)$ and $v(t)$ in the uncertain stochastic $H_{\infty }$ control scenario as dual strategies employed by players $P_1$ and $P_2$ through a game theory lens, this $H_{\infty }$ control challenge transforms into resolving an uncertain stochastic game dilemma. Consequently, it is acknowledged that the infinite-time horizon uncertain stochastic $H_{\infty }$ control issue yields a solution pair. Clearly, the pair $(u^{*},v^{*})$ represents the equilibrium strategy of the saddle point such that

$$\begin{array}{c}\hfill J(u^{*},v)\le J(u^{*},v^{*})\le J(u,v^{*}).\end{array}$$

According to Theorem 5, the following Theorem 6 can be obtained directly.

Theorem 6. 

For system (16), uncertain stochastic control has a pair of solutions $(u^{*},v^{*})$, with $u^{*}=K_{2}z(k)$ and $v^{*}=K_{1}z(k)$, if the following coupled uncertain stochastic algebraic Riccati equation

$$\begin{array}{c}\hfill PA+A^{T}P+C^{T}PC+D-PB{R}^{-1}{B}^{T}P=0\end{array}$$

with

$$\begin{array}{c}\hfill B=(B_1,B_2),R=\left[\begin{array}{cc}-{\gamma }^{2}I& 0\\ 0& I\end{array}\right]\end{array}$$

has a solution $P\in S^n$, where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{K}_1={\gamma }^{-2}{B}_1^{T}P\hfill \\ K_2=-B_2^{T}P\hfill \end{array}\right.\end{array}$$

In this scenario, $u^{*}$ serves as an $H_{\infty }$ control for the system outlined in Equation (16), while $v^{*}$ acts as the associated worst-case disturbance.

Example 1. 

Consider system (16) with the coefficients as follows:

$$A=\left[\begin{array}{cc}2& 4\\ 3& 1\end{array}\right],C=\left[\begin{array}{cc}-2& -4\\ 3& -1\end{array}\right],B_1=\left[\begin{array}{c}-1\\ 1\end{array}\right],B_2=\left[\begin{array}{c}1\\ -1\end{array}\right],D=\left[\begin{array}{cc}2& 1\\ 1& 3\end{array}\right]$$

Setting $\gamma =3$ and solving (16) by using the LMI toolbox, we have

$$\begin{array}{c}\hfill P=\left[\begin{array}{cc}-0.1236& -0.0615\\ -0.0615& -0.1935\end{array}\right],K_1=\left[\begin{array}{c}0.0069\\ -0.0147\end{array}\right],K_2=\left[\begin{array}{c}0.0622\\ -0.1321\end{array}\right].\end{array}$$

Therefore, $H_{\infty }$ control is given by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}u(k)=0.0622z_1(k)-0.1321z_2(k)\hfill \\ v(k)=0.0069z_1(k)-0.0147z_2(k)\hfill \end{array}\right.\end{array}$$

Furthermore, we can obtain the dynamic change curve of system state $(z_1(k),z_2(k))$ and control inputs $(u(k),v(k))$ in Figure 2.

5. Conclusions

This study aims to propose a new type of uncertain stochastic hybrid differential game. The main contribution of this study is the development of a saddle-point equilibrium strategy for linear quadratic uncertain stochastic two-player zero-sum hybrid differential games over both finite and infinite horizons. In future work, we may consider uncertain stochastic hybrid differential games with jumps.