Detectability in Games of Pursuit Evasion with Antagonizing Players Shuzhi Sam Ge, Hongbin Ma and Kai-Yew Lum

Abstract— In reality, the games of pursuit and evasion actually involve antagonizing players, rather than the commonly studied pursuit and evasion games with pre-fixed pursuers or evaders. In these practical scenarios, the game process can normally be divided into three stages: detection, attack and engagement stages. In such games, the roles of players are symmetric and each one is to search and attack the other one. This key point distinguishes the games of pursuit and evasion with antagonizing players from most existing pursuitevasion games. In this paper, a basic framwork for such games is established first by introducing several essential concepts including vision zone, range of view, field of view, etc. Then, a fundamental and important concept, detectability, is presented under the assumptions that (i) each player has a limited range vision zone and (ii) the players would follow their own predefined trajectories until one player could detect its opponent. At the detection stage, to demonstrate concepts of detectability and related concepts, a simple yet typical case is investigated in this contribution, where two players are moving along two straight lines with constant speeds and each player has a circular vision zone. Sufficient and necessary conditions for all possible cases of detectability are given under several natural assumptions, which consequently yields a complete analysis. Index Terms— Pursuit-evasion game, Detectability, Limited range of view, Predefined trajectory

I. I NTRODUCTION Pursuit-evasion (short for PE throughout this paper) games have been extensively studied because of their wide applications, such as combat games [1], missile guidance [2], hideand-seek game [3], art gallery guarding [4], etc. Existing works on PE games (see [1], [5]–[8] and the references therein) usually assume that players have only fixed roles as pursuers or evaders throughout the whole games. But in many practical cases, the games of pursuit and evasion actually involve antagonizing players, where the players are dual-role players, i.e., they could be pursuers or evaders depending on the situations and progression of the games. Although there are extensive studies on PE games in the literature, to the best knowledge of the authors, PE games with dual-role players are seldom addressed though there are many practical applications. Another limitation of most existing work on PE games is the assumption that all players can accesss complete This research is funded by Defence Science & Technology Agency POD 513242. Shuzhi Sam Ge is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576; the Interactive Digital Media Institute, National University of Singapore, Singapore 117576. Email: [email protected] Hongbin Ma and Kai-Yew Lum are with the Temasek Laboratories, National University of Singapore, Singapore 117508. Email: {tslmh,tsllumky}@nus.edu.sg

information of their opponents, especially in classical differential games (e.g. [1], [7]–[12]), where motion of players is described by a group of differential equations and each player can design its own control law by utilizing the states of the other player. Hence, each player can obtain perfect information of its opponent in some sense, which is obviously not practical in many cases. In recent years, to address the problem of incomplete information, several PE games different from classical differential games have been proposed and studied in the literature. In [13], a visibility-based pursuit-evasion problem in a polygon environment F is investigated, and this problem has the following settings: the evader is unpredictable, has unknown initial position and can move arbitrarily fast; and the pursuers have 360 degrees visibility because every pursuer can “see” any point in F which makes the line segment linked to position of the pursuer lie in F . Obviously, this concept of visibility is ideal in the sense that the pursuer has the ability to see objects infinitely far away from itself. Other similar concepts, such as k-searcher [14] (searcher equipped with k flashlights) and φ-searcher [15] (searcher equipped with a limited field of view sensor), have also been introduced in the literature. All these searchers discussed in the literature are capable to see unlimited range, and they cannot penetrate obstacles. Hence, there are still few quantitative research on PE games concerning players with limited range of view. Search theory ([16], [17] and references therein) is another area related with PE games, in which the searcher has only incomplete information on the objects to be searched. As stated in [18], “The main distinction consists of the fact that in pursuit problems, the pursuer has complete information about the location of the evader at each instant of time, while the incomplete information available to the searching object is typical.” Search problems are different from PE problems in their formulations because in search problems, a probability distribution of the searching object is usually assumed and the problems are aimed at maximizing the probability of successful finding of the searching object. Besides search problems of static objects, several problems on dynamical search for objects are surveyed in [18], where the incomplete information on the possibilities of trajectories is one key issue to be considered. The issues above motivate the study of this research. To avoid the possible ambiguity with existing PE games, this type of games is called the games of pursuit and evasion with antagonizing players (PEAP) through this paper. In the PEAP games, the roles of players are symmetric and each player has possibility to attack and disable its opponent. Based

on observations of many practical situations, the studies of PEAP games are important, challenging yet practically useful. The differences between the players lie in their resources/equipments such as their types of sensors and weapons. For such practical games, players have limited range of view, finite speed of travel, etc. The games involve roughly three stages: detection, attack and engagement stages. In the detection stage, both players are in searching or detecting mode and they cannot see their opponents yet. In the attack stage (the second stage) one player can see its opponent and launches the attack. In the last stage, players on both sides could see their opponents, and engage in the dog fight. In this paper, we shall first lay the basic framework of the PEAP games by introducing a few essential concepts including vision zone, field of view, range of view, and then introduce the fundamental concept of “detectability” in the detection stage. In this stage, each player follows its own predefined trajectory and it could not see the other one until the other one enters its own vision zone. To demonstrate the concept clearly, a simple yet typical PEAP game with two players is used to show the concepts through detailed analysis. The remainder of this paper is organized as follows: In Section II, we give several essential definitions including field of view, range of view and vision zone, which play foundational roles in this paper. Problem formulation is given in Section III, where three possible stages in the process of PEAP games are described, fundamental concepts of “detectability” are defined and several necessary assumptions are presented. Then in Section IV, for the detection stage of a typical planar PEAP game with only two players, sufficient and necessary conditions for the detectability are given, which are verified further by several simple examples, and consequently we can give complete analysis for the proposed concepts. Finally, some concluding remarks are drawn in Section V.

Definition 2.2: (Field of view) The field of view of Player i at position P , denoted by Fi (P ), is the set of directions in which it can see its opponent. Definition 2.3: (Range of view) The range of view of Player i at position P , denoted by Ri (P ), is the maximum distance that it can see its opponent. The concept of field of view has been used in previous researches on planar PE games. In fact, several typical concepts of searchers mentioned earlier, including k-searcher (k = 1, 2, ∞) and φ-searcher, can be regarded as special cases of field of view: φ-searcher is a player with field of view of angle φ, ∞-searcher is a player with 360-degree field of view and the field of view of a k-searcher has exactly k directions. However, those concepts assume that the searchers have unlimited range of view, which are impractical in any real games. In our study, limited range of view is a major concern and we want to quantitatively explore basic problems and concepts for PE games with antagonizing players. For different system configuration, there would be many different types of vision zone, range of view, field of view, etc. For example, circular vision zone and sectorial vision zone as defined below are two common ones in some practical planar PEAP games. Definition 2.4: Circular vision zone of Player i at position Pi (t) is defined as a circle centered at Pi (t) with fixed radius Ri . Definition 2.5: Sectorial vision zone of Player i at position Pi (t) is defined as a sector centered at Pi (t) with radius Ri and angle φi . Its base direction is defined as the center one of all directions in Fi (Pi (t)). For convenience of discussion, the PEAP games with m players in one group (G1 ) and n players in another group (G2 ) will be denoted as PEAP(m,n) . When Group Gk (k = 1 or k = 2) has only one player, the player in Group Gk will be refered to “Player k” directly. A PEAP game with two antagonizing players is denoted by PEAP(1,1) .

II. P RELIMINARIES

III. P ROBLEM F ORMULATION

Throughout this paper, we adopt the following conventions: [·]τ denotes the transpose of vector or matrix; arctan(x, y) denotes the arctangent angle in the correct quadrant determined by the coordination (x, y); sign(x) denotes the sign of x, taking value 1 (x > 0), −1 (x > 0) or 0 (x = 0); Rn denotes the space of n-dimensional real vectors; and R[2π] denotes the quotient space of R over [0, 2π), i.e. x ∼ x + 2kπ for any integer k. For convenience of discussion, we first present several basic definitions for the pursuit-evasion games in Rn . Let d Pi (t) ∈ Rn be the position vecPi (t) ∈ Rn and Vi (t) = dt tor and the velocity vector of Player i at time t, respectively. Definition 2.1: (Vision zone) The vision zone (or V -zone) of Player i at position P , denoted by Vi (P ), is the set of positions at which its opponent can be seen by this player, i.e. Player i can observe the position of its opponent. The following concepts are important to determine the size and the shape of a vision zone:

For PEAP game, generally speaking, the game process may be normally divided into three possible stages: S1: Initially no player can observe the opponent, but the players search their opponents until at least one player can observe the opponent. S2: In this stage, the player (say A) who observes the opponent (say B) first can take the initiative and launch attack on its opponent before its opponent can counterattack. S3: In this stage, players from both sides could see each other and try their best to attack the other by choosing its own actions (including trajectory, velocity, etc.). Note that not all the stages listed above are necessary. For example, player B may have no chance to counterattack because player A has captured player B in Stage S2. For the convenience, the following assumptions are made. Assumption 3.1: Each player has a limited vision zone at any time.

y

Ri

θi Oi

x

o

(a) Circular vision zone

y

Ri

φi θi

Oi o

x (b) Sectorial vision zone

Fig. 1. Examples of vision zones: (a) Circular vision zone has 360-degree field of view and limited range of view with radius Ri . It is unnecessary to specify its base direction because of its axial symmetry w.r.t. all directions. (b)Sectorial vision zone has limited field of view with angle φi and limited range of view with radius Ri . Whether it is biased depends on the relationship between its base direction and Player i’s heading direction.

Assumption 3.2: Initially each player does not lie in the vision zone of the opponent. Assumption 3.3: Player i has a predefined fixed trajectory γi ⊂ Rn , and it will follow γi exactly until the opponent enters its vision zone. Assumption 3.4: Once a player finds the opponent in its vision zone, it will take the initiative to launch an attack. Remark 3.1: Most of these assumptions are natural in many practical games. These assumptions imply that the roles of players are symmetric. The assumption on initial conditions helps us exclude many trivial cases in which at least one player has been in the vision zone of its opponent initially. The last assumption means that the player who sees the opponent first has more chances to win the game. Based on Assumptions 3.1—3.4, we are ready to introduce the following fundamental concept of “detectability”: Definition 3.1: (Detectability in PEAP(1,1) ) In a PEAP game with two players, Player 1 is able to detect Player 2, if there exists a time T > t0 such that 1) P2 (T ) ∈ V1 (P1 (T )), and 2) P1 (t) 6∈ V2 (P2 (t)), ∀t ∈ [t0 , T ). The time T is called detection time. In this case, we say also Player 2 can be detected by Player 1 at time T . Definition 3.2: In a PEAP game with two players, Player 1 is unable to detect Player 2, if P2 (t) 6∈ V1 (P1 (t)) for any time t ≥ t0 .

Based on the concept of detectability in PEAP(1,1) , we can define the following concept of detectability in PEAP(m,1) , where Group G1 has m players and Group G2 has only one player (Player 2). Definition 3.3: (Detectability in PEAP(m,1) ) In the game PEAP(m,1) , Group G1 is able to detect Player 2, if there exists a player in Group G1 , such that this player is able to detect Player 2. Definition 3.4: In the game PEAP(m,1) , Group G1 is unable to detect Player 2, if each player in Group G1 is unable to detect Player 2. Definition 3.5: (k-Detectability in PEAP(m,n) ) In the game PEAP(m,n) , Group G1 is able to detect k players in Group G2 , if there exists k players in Group G2 such that each of them can be detected by Group G1 . Definition 3.6: In the game PEAP(m,n) , Group G1 is unable to detect Group G2 , if each player in Group G2 cannot be detected by Group G1 . The concept of “detectability” is crucial for each player because it will gain the initiative if it is able to detect the opponent. In this paper, we will only concern, for several simple examples, whether one player is able to detect the opponent and how long the detection time will be. How the game proceeds after detection will be discussed in [20]. Now we consider PEAP(1,1) . Obviously, when the initial position P2 (0) of Player 2 varies in Rn , the trajectory γ2 of Player 2 varies consequently, which forms a bundle of curves. Thus, when the parameters (denoted as Θ below) of the game and the initial position P1 (0) ∈ Rn of Player 1 are fixed, initial position P2 (0) ∈ Rn determines whether Player 1 is able to detect Player 2. Then, we can give the following concept of detectable area for Player 1: Definition 3.7: (Detectable area in PEAP(1,1) ) In a PEAP game with two players, when the parameters of the game and the initial position P1 (0) of Player 1 are fixed, the set of initial positions P2 (0) of Player 2, in which Player 2 can be detected by Player 1 at some time, is called detectable area for Player 1, denoted by D1 (P1 (0); Θ), where Θ represents all parameters in this game. Remark 3.2: For a player in the game, it is useful to have a global view on its detectable area, which clearly marks those positions where its opponent can be detected by itself. The illustration of detectable area can also be utilized to study how the detectability relates to the parameters of the game. Based on the definitions given above, for Stage S1, naturally, it is a main task to find out when one player is able to detect its opponent and what is the detectable area of one player. IV. C OMPLETE A NALYSIS ON D ETECTABILITY FOR TWO PLAYERS

In this section, a typical PEAP game with only two players is used to demonstrate the concepts through detailed analysis under the assumption that both players have circular vision zones and follow straight lines with constant speeds until one player could be detected by its opponent. This section will

be organized into four parts: firstly, Theorem 4.1 is presented to give sufficient and necessary conditions for the concepts of detectability; then, several simple examples are given to verify Theorem 4.1; later, we discuss the effects of changes in parameters based on Theorem 4.1; finally, we present a theorem on detectable area. In this game, for i = 1, 2, let [xi (0), yi (0)]τ , θi , vi denote the initial position, heading direction and speed of Player i, respectively. Then we obtain that Vi (t) = [vi cos θi , vi sin θi ]τ and xi (t)

= xi (0) + vi t cos θi

yi (t)

= yi (0) + vi t sin θi .

yR

D C

O2

xR

R1

R2 d0

γR O1

Let R1 , R2 be the radii of circular vision zones of Player 1 and Player 2, respectively. Thus, this game can be described by 6 parameters, and the parameter vector Θ of the game can be taken as Θ = [R1 , R2 , v1 , v2 , θ1 , θ2 ]τ .

R1

A. Sufficient and Necessary Conditions The following main theorem gives sufficient and necessary conditions for the concept of detectability of two players in the PEAP game. Theorem 4.1: Consider two players equipped with circular vision zones, whose radii are R1 and R2 , respectively. Let

y

x1 (0) − x2 (0)

xR (0)

=

yR (0)

vR

= y1 (0) − y2 (0) q 2 (0) x2R (0) + yR = q = v12 + v22 − 2v1 v2 cos(θ1 − θ2 )

γR

=

arctan(v1 cos θ1 − v2 cos θ2 , v1 sin θ1 − v2 sin θ2 )

D

=

|xR (0) sin γR − yR (0) cos γR |

C

=

−(xR (0) cos γR + yR (0) sin γR )

d0

Fig. 2. Motion of Player 1 relative to Player 2

then under the assumptions given early, we have 1) If vR = 0 or C < 0 or D > max(R1 , R2 ), then both two players cannot be detected by the opposite one. 2) Otherwise, we have the following conclusions: (a) If R1 = R2 ≥ D, then both players are able to detect the opposite one and the detection time p C − R12 − D2 T = . vR (b) If R1 > R2 and R1 ≥ D, then Player 1 is able to detect Player 2 and the detection time p C − R12 − D2 T = . vR (c) If R2 > R1 and R2 ≥ D, then Player 2 is able to detect Player 1 and the detection time p C − R22 − D2 T = . vR Proof: See [19].

θ2

O2 R2

O1

θ1

R1 o

x

Fig. 3. Both players have same speeds and heading directions

B. Several Simple Examples We give several examples to verify Theorem 4.1. Example 4.1: (Both players have same speeds and heading directions) As shown in Fig. 3, in this case, v1 = v2 and θ1 = θ2 , which is equivalent with vR = 0. Obviously, the players cannot detect their opponents because the position of Player 1 relative to Player 2 is fixed and initially each player does not lie in the vision zone of its opponent. Example 4.2: (Player 1 moving away from Player 2) As shown in Fig. 4, in this case Player 1 and Player 2 are moving along the same line but in opposite direction. In fact, the heading directions, θ1 and θ2 , have a phase difference π, then, obviously, we obtain vR = v1 + v2 and γR = θ1 . Simple calculation yields that C = −d0 < 0, thus both two players cannot detect their opponents according to Theorem 4.1. Example 4.3: (Player 1 moving towards Player 2) As shown in Fig. 5, this case is similar to the last case except that two players are moving towards each other. In this case, we have vR = v1 + v2 and γR = θ1 , but C = d0 > 0. Noting also tan γR = tan θ1 = xyRR(0) (0) , we know that D = |xR (0) sin γR − yR (0) cos γR | = 0. Thus, by Theorem

O1

y

O2

R1

θ2 O2 R2

d0

yR

xR

y

R2

θ1

R1 O1

x

x

o

Fig. 4. Player 1 moving away from Player 2 Fig. 6. Another simple example

θ2 θ1

y O1

xR

O2 R2 yR

R1 o

x Fig. 5. Player 1 moving towards Player 2

4.1, whether Player 1 is able to detect Player 2 completely depends on the relationship between R1 and R2 . If R1 > R2 , then Player 1 is able to detect Player 2; and if R2 > R1 , then Player 2 is able to detect Player 1; and if R1 = R2 , then both players can be detected by each other. These results are consistent with our intuitive knowledge. Example 4.4: (Another simple example) As shown in Fig. 6, in this √ example, we take v1 = v2 = 1, θ1 = π2 , θ2 = 0, R1 = 2, R2 = 1, x1 (0) = y( 0) = 0, x2 (0) = −4, y2 (0) =√4. Then, by√Theorem 4.1, xR (0) = 4, yR (0) = −4, d0 = 4 2, vR = 2, γR = arctan(−1, 1) = 3π 4 , D = |xR (0) sin γR − yR√ (0) cos γR | = 0, C = −(xR (0) cos γR + yR (0) sin γR ) = 4 2. Then, we conclude that Player 1 is able to detect Player 2 since R1 > R2 > D and C > 0. The detection time is p √ √ C − R12 − D2 4 2− 2 √ T = = = 3. vR 2

(iii) If vR = 0 or C < 0 or R1 < max(R2 , D), then Player 1 is unable to detect Player 2. (iv) In the case where Player 1 is unable to detect Player 2, to make Player 1 be able to detect Player 2 without changing parameters of Player 2 and their initial conditions, Player 1 should (a) adjust v1 so as to make vR > 0 and C ≥ 0 when vR = 0 or C < 0; (b) increase R1 so as to make R1 ≥ max(R2 , D); (c) adjust v1 and/or θ1 to decrease D such that R1 ≥ max(R2 , D) in case of R1 ≥ R2 . Remark 4.1: In (a), how to adjust v1 depends on the sign of ∂v∂ 1 C. By simple calculation, we obtain ∂ C ∂v1

Summarizing the results of Theorem 4.1, we can draw the following conclusions: (i) If R1 ≥ max(R2 , D), vR > 0 and C ≥ 0 , then Player 1 is able to detect Player 2. (ii) In the case where Player √ 1 is able to detect Player 2, C− R12 −D 2 the detection time is T = . vR

[xR (0) sin γR − yR (0) cos γR ] ·

= D0 ·

∂ C ∂v1

v1 − v2 cos(θ1 − θ2 ) vR

where vR and γR are defined in Theorem 4.1 and D0 = xR (0) sin γR − yR (0) cos γR . Thus, if D0 [v1 − v2 cos(θ1 − θ2 )] > 0, Player 1 should increase v1 ; otherwise, Player 1 should decrease v1 . Remark 4.2: Noticing that D = xR (0) sin γR − yR (0) cos γR just depends on γR , which is a relative complex function of v1 and θ1 . How to adjust v1 and θ1 so as to decrease D in (c)? Since D = sign(D0 )D0 , we obtain that ∂D ∂v1

Now, we check this result. In fact, [x1 (T ), y1 (T )]τ = [0, 3]τ , [x2√ (T ), y2 (T )]τ = [−1, 4]τ , hence their distance at time T is 2, which is just the radius of vision zone of Player 1. Note also that, at time T , Player 2 cannot see Player 1 yet. C. Discussions on Parameters

=

and ∂D ∂θ1

∂D0 ∂v1

=

sign(D0 )

=

sign(D0 )[xR (0) cos γR + yR (0) sin γR ]

=

sign(D0 )C

=

sign(D0 )

=

sign(D0 )[xR (0) cos γR + yR (0) sin γR ]

∂γR ∂v1

v2 sin(θ1 − θ2 ) (v1 cos θ1 − v2 cos θ2 )2

∂D0 ∂θ1

= − sign(D0 )C

v1 (v1 − v2 ) (v1 cos θ1 − v2 cos θ2 )2

∂γR ∂θ1

V. C ONCLUSION

C1

R2

I1 nI

y

gio Re

γR

O2

n gio Re

A1

D1

C0

R1 O1

I ion o g Re

II

C2

γR

nI A2 gio e R

I2

x

Fig. 7. Detectable area of Player 1

Thus, when R1 ≥ R2 and C > 0, to decrease D without changing R1 , Player 1 can • •

increase (decrease) v1 if D0 sin(θ1 − θ2 ) < 0 (> 0); or increase (decrease) θ1 if D0 (v1 − v2 ) < 0 (> 0).

D. Detectable Area Based on Theorem 4.1, the detectable area of Player 1 is explicitly given in the following theorem: Theorem 4.2: Assume that v1 , v2 , R1 , R2 , θ1 , θ2 and (x1 (0), y1 (0)) are given, which satisfy (1) R1 > R2 ; (2) either v1 6= v2 or θ1 6= θ2 . The detectable area D1 of Player 1 is a domain enclosed by the following curves/lines (see Fig. 7): ∆

C0

=

C1

=

C2

=

∆

∆

{(x, y)|w ∈ [γR − π2 , γR + π2 ] : x(w) = x1 (0) + R1 cos w y(w) = y1 (0) + R1 sin w} {(x, y)|w ∈ [0, ∞) : x(w) = [x1 (0) − R1 sin γR ] + w cos γR y(w) = [y1 (0) + R1 cos γR ] + w sin γR } {(x, y)|w ∈ [0, ∞) : x(w) = [x1 (0) + R1 sin γR ] + w cos γR y(w) = [y1 (0) − R1 cos γR ] + w sin γR }

(1) where γR is defined in Theorem 4.1. Proof: See [20]. Remark 4.3: Theorem 4.2 is easy to understand intuitively. Consider one special case — Player 2 does not move. In this case, the detectable area of Player 1 is just the belt-like area with width 2R1 in front of itself, towarding its heading direction. This fact is consistent with our intuition that one can clear a belt-like area when one moves ahead. Remark 4.4: The detectable area of Player 1 given in Theorem 4.2 is very simple. This is partially because we only consider the most simple case. Generally, calculation of detectable area in other cases will be much difficult.

Based on observations of some practical PE games, games of pursuit and evasion with antagonizing players (PEAP) have been proposed and studied in this paper. The PEAP games may usually be divided into three different stages because each player has only limited vision zone. Such games are essentially different from traditional PE games because of two unique characteristics: firstly, each player has possibility to search and attack its opponent and thus roles of players are symmetric; secondly, players have some limitations in their resources/equipments, e.g. sensors with limited range of view. We have laid a foundational framework for PEAP games in this paper by introducing several essential concepts such as vision zone, range of view, and field of view. Then, in the framework established, for the detection stage (Stage S1), basic problems on “detectability” have been formulated mathematically and several related concepts, such as detection time and detectable area, have been also defined, all of which quantitatively describe the ability of players in detecting their opponents. R EFERENCES [1] R. Isaacs, Differential Games. New York: Wiley, 1965. [2] V. Turetsky and J. Shinar, “Missile guidance laws based on pursuitevasion game formulations,” Automatica, vol. 39, pp. 607–618, 2003. [3] S. M. LaValle, Planning Algorithms. Cambridge, U.K.: Cambridge University Press, 2006, available at http://planning.cs.uiuc.edu/. [4] J. O. Rourke, Art gallery theorems and algorithms. U.K.: Oxford University Press, 1987. [5] O. H´ajek, Pursuit Games. New York: Academic Press, 1975. [6] S. Gal, Search Games. New York: Academic Press, 1980. [7] Y. Yavin and M. Pachter, Pursuit-Evasion Differential Games. Oxford, U.K.: Pergamon, 1987. [8] L. A. Petrosjan, Differential Games of Pursuit. Singapore: World Scientific, 1993. [9] J. Yong, “On differential evasion games,” SIAM Journal on Control & Optimization, vol. 26, no. 1, pp. 1–22, Jan. 1988. [10] ——, “On differential pursuit games,” SIAM Journal on Control & Optimization, vol. 26, no. 2, pp. 478–495, Mar. 1988. [11] L. S. Zaremba, “Differential games reducible to optimal control problems,” in Proceedings IEEE Conference Decision & Control, Tampa, FL, Dec. 1989, pp. 2449–2450. [12] K. Haji-Ghassemi, “On differential games of fixed duration with phase coordinate restrictions on one player,” SIAM Journal on Control & Optimization, vol. 28, no. 3, pp. 624–652, May 1990. [13] L. J. Guibas, J. C. Latombe, S. M. LaValle, D. Lin, and R. Motwani, “A visibility-based pursuit-evasion problem,” International Journal of Computational Geometry and Applications, vol. 9, no. 4/5, pp. 471–493, 1999. [Online]. Available: citeseer.ist.psu.edu/guibas96visibilitybased.html [14] I. Suzuki and M. Yamashita, “Searching for a mobile intruder in a polygonal region,” SIAM Journal on Computing, vol. 21, no. 5, pp. 863–888, Oct. 1992. [15] B. P. Gerkey, S. Thrun, and G. Gordon, “Visibility-based pursuitevasion with limited field of view,” in Proc. of the National Conference on Artificial Intelligence (AAAI 2004), San Jose, California, July 2004. [16] L. D. Stone, Theory of Optimal Search. New York; San Franciso; London: Academic Press, 1975. [17] R. Ahlswede and I. Wegener, Search Problems. Chichester; New York; Brisbane; Toronto; Singapore: John Wiley and Sons, 1987. [18] L. G. Chkhartishvili and E. V. Shikin, “Geometry of dynamical search for objects,” Journal of Mathematical Sciences, vol. 110, no. 2, pp. 2508–2527, 2002. [19] H. B. Ma, S. S. Ge, and K. Y. Lum, “Attackability in games of pursuit evasion with antagonizing players,” submitted to IFAC 2008. [20] S. S. Ge, H. B. Ma, and K. Y. Lum, “Detectability in games of pursuit evasion with antagonizing players,” submitted to Automatica.