Uncertain pursuit-evasion game

Abstract

Pursuit-evasion game deals with the situation in which a pursuer tries to catch an evader. Taking into account the subjectivity of the players’ strategies and the fact that the noise of system state does not obey the statistical regularity, this paper employs an uncertain differential equation to describe the dynamics of the pursuit-evasion system, and introduces an uncertain pursuit-evasion game. Within the framework of uncertain differential game theory, a solution for the uncertain pursuit-evasion game is derived via the corresponding Riccati equation. At last, as an application, a target interception problem is proposed.

Appendix A uncertain differential game

In a general uncertain differential game, the player $i\in N=\{1, 2, \ldots , n\}$ optimizes his objective

$$\begin{aligned} \displaystyle \sup \limits _{{{\varvec{u}}}_i}\mathrm{E}\left[ \int _{0}^TR_{i}(t,{\varvec{X}}_t,{{\varvec{u}}}_1,{{\varvec{u}}}_2,\ldots ,{{\varvec{u}}}_n)\mathrm{d}t +W_{i}(T,{\varvec{X}}_T)\right] \end{aligned}$$

(7)

subject to the uncertain state equation

$$\begin{aligned}&\mathrm{d}{\varvec{X}}_t=f(t,{\varvec{X}}_t,{{\varvec{u}}}_1,{{\varvec{u}}}_2,\ldots ,{{\varvec{u}}}_n)\mathrm{d}t\nonumber \\&\qquad \qquad +g(t,{\varvec{X}}_t,{{\varvec{u}}}_1,{{\varvec{u}}}_2,\ldots ,{{\varvec{u}}}_n)\mathrm{d}{\varvec{C}}_t\\&{\varvec{X}}_0={{\varvec{x}}}_0\nonumber \end{aligned}$$

(8)

where $T>0$, $R_i: [0,T]\times \mathfrak {R}^m\times \mathfrak {R}^{r_1} \times \cdots \times \mathfrak {R}^{r_n}\rightarrow \mathfrak {R}$ is an return function, $W_i(T,\cdot ): \mathfrak {R}^m\rightarrow \mathfrak {R}$ is a function of terminal reward at time T, ${\varvec{X}}_t\in \mathfrak {R}^m$ denotes the state variables of game, ${{\varvec{u}}}_i\in \mathfrak {R}^{r_i}$ is the control of player i, the initial state ${{\varvec{x}}}_0$ is given, $\mathrm{E}(\cdot )$ denotes the expectation operator performed at time 0, ${\varvec{C}}_t$ is a k-dimensional Liu process defined on an uncertainty space $(\Gamma , {\mathcal {L}},{\mathcal {M}})$, and suppose that

$$\begin{aligned} f:[0,T]\times \mathfrak {R}^m\times \mathfrak {R}^{r_1} \times \cdots \times \mathfrak {R}^{r_n}\rightarrow \mathfrak {R}^m \end{aligned}$$

and

$$\begin{aligned} g:[0,T]\times \mathfrak {R}^m\times R^{r_1} \times \cdots \times \mathfrak {R}^{r_n}\rightarrow \mathfrak {R}^m\times \mathfrak {R}^k \end{aligned}$$

have continuous partial derivatives, and satisfy linear growth condition and Lipschitz condition.

Each player has perfect observations of the state vector ${\varvec{X}}_t$ at every moment $t\in [0,T]$ and constructs his strategy in the game (7)–(8) as an admissible feedback control of the following type

$$\begin{aligned} {{\varvec{u}}}_i:={{\varvec{u}}}_i(t,{\varvec{X}}_t):[0,T]\times \mathfrak {R}^m\longrightarrow \mathfrak {R}^{r_i}. \end{aligned}$$

We write

$$\begin{aligned} {{\varvec{u}}}_{-i}=\{{{\varvec{u}}}_1,{{\varvec{u}}}_2,\ldots ,{{\varvec{u}}}_{i-1},{{\varvec{u}}}_{i+1},\ldots ,{{\varvec{u}}}_n\}. \end{aligned}$$

A feedback Nash equilibrium of the uncertain differential game (7)–(8) can be defined as follows:

Definition 1

(Yang and Gao (2013)) A set of strategies $\{{{\varvec{u}}}_1^*,{{\varvec{u}}}_2^*,\ldots ,{{\varvec{u}}}_n^*\}$ is called a feedback Nash equilibrium for the n-player uncertain differential game (7)–(8), and ${\varvec{X}}^{*}(s),t\le s\le T\}$ is the corresponding state trajectory, if there exist real-valued functions $V^i(t,{{\varvec{x}}}):[0,T]\times \mathfrak {R}^m\rightarrow \mathfrak {R}$, satisfying the following relations for each $i\in N$

$$\begin{aligned}&V^i(t,{{\varvec{x}}})\\&\quad =\mathrm{E}\left[ \int _{t}^TR_{i}\left( s,{\varvec{X}}^*_s,{{\varvec{u}}}_i^*,{{\varvec{u}}}_{-i}^*\right) \mathrm{d} s +W_{i}\left( T,{\varvec{X}}^*_T\right) \right] \\&\quad \ge \mathrm{E}\left[ \int _{t}^TR_{i}\left( s,{\varvec{X}}^{[i]}_s,{{\varvec{u}}}_i,{{\varvec{u}}}_{-i}^*\right) \mathrm{d} s+W_{i}\left( T,{\varvec{X}}^{[i]}(T)\right) \right] ,\\&V^i(T,{{\varvec{x}}})=W_i(t,{\varvec{X}}_T) \end{aligned}$$

where on the time interval [t, T]:

$$\begin{aligned}&\mathrm{d} {\varvec{X}}^*_s=f\left( s,{\varvec{X}}^*_s,{{\varvec{u}}}_i^*,{{\varvec{u}}}_{-i}^*\right) \mathrm{d} s +g\left( s,{\varvec{X}}^*_s,{{\varvec{u}}}_i^*,{{\varvec{u}}}_{-i}^*\right) \mathrm{d}{\varvec{C}}_s,\\&{\varvec{X}}^*_t={{\varvec{x}}};\\&\mathrm{d} {\varvec{X}}^{[i]}_s=f\left( s,{\varvec{X}}^{[i]}_s,{{\varvec{u}}}_i,{{\varvec{u}}}_{-i}^*\right) \mathrm{d} s +g\left( s,{\varvec{X}}^{[i]}_s,{{\varvec{u}}}_i,{{\varvec{u}}}_{-i}^*\right) \mathrm{d}{\varvec{C}}_s,\\&{\varvec{X}}^{[i]}_t={{\varvec{x}}}. \end{aligned}$$

Theorem 4

(Yang and Gao (2013)) An n-tuple of strategies $\{{{\varvec{u}}}_i^*;i\in N\}$ provides a feedback Nash equilibrium to the n-player uncertain differential game (7)–(8) if there exist real-valued functions $V^i(t,{{\varvec{x}}}):[0,T]\times \mathfrak {R}^m\rightarrow \mathfrak {R},i\in N$, satisfying the partial differential equations

$$\begin{aligned}&-V_t^i(t,{{\varvec{x}}})\\&\quad =\sup _{{{\varvec{u}}}_i}\left\{ R_{i}\left( t,{{\varvec{x}}},{{\varvec{u}}}_i,{{\varvec{u}}}_{-i}^*\right) +\nabla _{{\varvec{x}}} V^i(t,{\varvec{x}})^\tau f\left( t,{{\varvec{x}}},{{\varvec{u}}}_i,{{\varvec{u}}}_{-i}^*\right) \right\} \\&\quad =R_{i}\left( t,{{\varvec{x}}},{{\varvec{u}}}_i^*,{{\varvec{u}}}_{-i}^*\right) +\nabla _{{\varvec{x}}} V^i(t,{\varvec{x}})^\tau f\left( t,{{\varvec{x}}},{{\varvec{u}}}_i^*,{{\varvec{u}}}_{-i}^*\right) ,\\&V^i(T,{\varvec{x}})=W_i(T,{\varvec{x}}) \end{aligned}$$

where $V_t^i(t,{\varvec{x}})$ is the partial derivatives of $V^i(t,{\varvec{x}})$ with respect to t, $\nabla _{{\varvec{x}}}V^i(t,{\varvec{x}})$ is the gradient of $V^i(t,{\varvec{x}})$ with respect to ${\varvec{x}}$, and the superscript $\tau$ denotes transpose.

In a two person zero-sum uncertain differential game, player 1 wants to maximize the index function while player 2 wants to minimize the index function. The performance index function is as follows:

$$\begin{aligned} \begin{aligned} J({{\varvec{u}}}_1,{{\varvec{u}}}_2)=&\mathrm{E}\left\{ \int _{0}^T\left[ {\varvec{X}}^{\tau }_tM{\varvec{X}}_t+2({{\varvec{u}}}_1^{\tau }S_1 +{{\varvec{u}}}_2^{\tau }S_2){\varvec{X}}_t\right. \right. \\&\left. \phantom {\displaystyle \int _{0}^{T}}\left. +{{\varvec{u}}}_1^{\tau }R_{1}{{\varvec{u}}}_1 +{{\varvec{u}}}_2^{\tau }R_{2}{{\varvec{u}}}_2\right] \mathrm{d}t+{\varvec{X}}_T^{\tau }G{\varvec{X}}_T\right\} . \end{aligned} \end{aligned}$$

(9)

And the state equation is

$$\begin{aligned} \mathrm{d}{\varvec{X}}_t=(A{\varvec{X}}_t+B_1{{\varvec{u}}}_1+B_2{{\varvec{u}}}_2)\mathrm{d}t+g(t,{\varvec{X}}_t,{{\varvec{u}}}_1,{{\varvec{u}}}_2)\mathrm{d}{\varvec{C}}_t\nonumber \\ \end{aligned}$$

(10)

with initial condition ${\varvec{X}}_0={\varvec{x}}_0$. Here, $A, B_i$ and $S_i\, (i=1,2)$ are bounded measurable matrix functions on [0, T] with dimensions $m\times m, m\times r_i, r_i\times m$, respectively; M and $R_{i}$ are bounded measurable symmetric matrix functions on [0, T] with dimensions $m\times m, r_i\times r_i$, respectively; G is a $m\times m$ constant symmetric matrix; and $-R_1,R_2$ are positive definite matrices.

Theorem 5

(Yang and Gao (2016)) The two players zero-sum of linear-quadratic uncertain differential game (9)–(10) has a saddle-point Nash equilibrium solution if the following Riccati equation (with the time argument t suppressed) has a solution P(t),

$$\begin{aligned}&{\dot{P}}+A^{\tau }P+PA+M\\&\quad -\sum \limits _{i=1}^2(B_i^{\tau }P+S_i)^{\tau }R_i^{-1} (B_i^{\tau }P+S_i)=0\\&P(T) =G \end{aligned}$$

where P(t) is bounded measureable symmetric matrix function on [0, T] with dimensions $m\times m$. Moreover, saddle point and the optimum value of performance index function are

$$\begin{aligned}&\{{{\varvec{u}}}_i^*=-R_i^{-1}(B_i^{\tau }P+S_i){\varvec{X}}_t; i=1, 2\}, \\&J({{\varvec{u}}}_1^*,{{\varvec{u}}}_2^*)={\varvec{x}}_0^{\tau }P(0){\varvec{x}}_0, \end{aligned}$$

respectively.