1 Introduction

Surveillance is an integral part of maintaining situational awareness, a field of study concerned with the perception of the environment critical to decision makers in complex, dynamic scenarios, that often arises in civil and military operations. Situational awareness arises in many applications, such as crime prevention, wildlife monitoring, traffic monitoring, and industrial processes. Government and law enforcement agencies utilize relevant surveillance technologies for maintaining social control, recognizing and monitoring threats, and preventing criminal activities from taking place. Target tracking constitutes a special class of surveillance problems, which refers to the problem of motion planning for a mobile observer in order to track a target in an environment containing obstacles.

With the emergence of computer vision, visibility-based target tracking has received a lot of attention and interest in the research community. In this work, we address such a problem, where we assume that the observer is equipped with a vision sensor for tracking the target. The environment contains an obstacle that occludes the view of the target from the observer. The goal of the observer is to maintain a persistent line-of-sight with the target. In order to account for worst-case scenarios, we assume that the interaction between the observer and the target is adversarial in nature. This gives rise to a pursuit-evasion game between the two agents. Some real-world applications of this problem are in surveillance, wildlife monitoring, and the next-best-view problem. Reference [1] provides numerous other applications of this problem.

In [1], a geometric technique was used to provide a sufficient condition for the target to break the line-of-sight with the observer in an environment containing polygonal obstacles. Additionally, a sufficient condition was provided for the observer to maintain a persistent line-of-sight. Based on these conditions, a polynomial-time algorithm was given to approximate the set of initial positions from which the observer can track the target forever. The aforementioned results have been extended in [1] to address the case when there is a circular obstacle in the environment. In [2], the authors proposed a computationally and memory-efficient algorithm for constructing the players’ optimal paths for the static surveillance-evasion game, in which the pursuer initially chooses its control for the remaining time, assuming that the evader would always counter with its best control. In [3, 4], differential game theory was used to provide the regular construction of the optimal trajectories for an equivalent game of degree near the terminal manifolds for a general environment containing polygonal obstacles. Based on the method of singular characteristics [5], researchers have encountered singular surfaces in pursuit-evasion games related to pursuit and capture [6]. In [7], an algorithm was presented to construct the dispersal surfaces [8] for the problem addressed in [3].

In the past, there have been numerous efforts to provide a solution to the aforementioned problem. However, a complete solution is yet not available for general environments. Given this state of knowledge, we explore in this paper the possibility of finding a complete solution to the problem for a simple environment. We consider the problem of a pursuer tracking an evader in an environment containing a circular obstacle. Initially, the problem is formulated as a game of kind in order to investigate strategies for the pursuer to track the evader indefinitely. We use Isaacs’ techniques [9] to conclude that there is a possibility that the pursuer might never achieve the aforementioned objective. Next, we use Isaacs’ conditions to compute the optimal strategy for the pursuer to maximize the time for which it can track the evader. The overall objective, in this work, is to investigate structural properties of the optimal strategy of the pursuer in simple environments and to shed light for the future in the construction of a feasible strategy for the pursuer in general environments.

The organization of the paper is as follows. In Sect. 2, the problem is formulated, and in Sect. 3, the solution technique is described. In Sect. 4, the controls of the players along the barrier are derived from the property of its semipermeability, and the retrogressive path equations for the normals to the barrier are computed. In Sect. 5, the optimal controls of the players in the escape set are provided, based on our previous work in [10]. In Sect. 6, we present our conclusions along with some suggestions for future research.

2 Problem Formulation

In this section, we formulate the problem under consideration. Referring to Fig. 1, consider a disk of radius a in the plane, with the observer and the target viewed as pursuer (P) and evader (E), respectively.

Fig. 1
figure 1

The figure shows a circular obstacle of radius a

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Both players are assumed to be simple kinematic agents, whose motions are governed by the following equations:

$$\begin{aligned} \dot{\mathbf{y}}=\mathbf{u}_{1},\quad \mathbf{y}\in K_{U};\quad \dot{\mathbf{z}}=\mathbf{u}_{2},\quad \mathbf{z}\in K_{V}, \end{aligned}$$

where

$$\begin{aligned} K_{U}=\{{\mathbf{y}}:\mathbf{y}\in \mathbb {R}^{2} \text { s.t. }\Vert \mathbf{y}\Vert _{2}>a\},\quad K_{V}=\{{\mathbf{z}}:\mathbf{z}\in \mathbb {R}^{2} \text { s.t. }\Vert \mathbf{z}\Vert _{2}>a\}, \end{aligned}$$

where \(\Vert \cdot \Vert _2\) is the Euclidean norm in \(\mathbb {R}^{2}\). Here, \(K_{U}\) and \(K_{V}\) capture the constraint that the players remain outside the disk of radius a, centered at the origin. Let \(\mathbf{x:=(y,z)}^{T}\) and \(f(\mathbf{u}_{1},\mathbf{u}_{2}):=(\mathbf{u}_{1},\mathbf{u}_{2})^{T}\). The controls \(\mathbf{u}_{1}(\cdot )\) and \(\mathbf{u}_{2}(\cdot )\) are measurable mappings, described by

$$\begin{aligned} \mathbf{u}_{1}(\cdot ):\mathbb {R}\rightarrow U,\quad U=B_{1}{(0,0)};\quad \mathbf{u}_{2}(\cdot ):\mathbb {R}\rightarrow V,\quad V=B_{\nu }{(0,0)}, \end{aligned}$$

where \(B_{r}(q)\) is a ball of radius r with center q and \(\nu \) is a parameter which represents the maximum speed of the evader. The controls \(\mathbf{u}_\mathbf{1}\) and \(\mathbf{u}_\mathbf{2}\) can also be expressed in polar coordinates with components of the velocities in radial and tangential directions, as shown in Fig.1. The radial and tangential components of the pursuer’s velocity are denoted by \(u_{r_p}\) and \(u_{\theta _p}\), respectively. The radial and tangential components of the evader’s velocity are denoted by \(u_{r_e}\) and \(u_{\theta _e}\), respectively.

The objective of the pursuer is to maintain a line-of-sight with the evader at all times. The objective of the evader is to break the line-of-sight with the pursuer in finite time. The game terminates when the line-of-sight between the pursuer and the evader is broken. Given this setting, we formulate the following two problems:

  1. 1.

    Problem 2.1: The objective of the pursuer is to maintain a line-of-sight with the evader forever. The objective of the evader is to break the line-of-sight in finite time. What are the strategies for the players?

  2. 2.

    Problem 2.2: The objective of the pursuer is to maximize the time for which it can maintain a line-of-sight with the evader. The objective of the evader is to break the line-of-sight in minimum amount of time. What are the optimal strategies of the players?

The pursuer wins the game if it succeeds in its objective, or else the evader wins. The first formulation above leads to what is called a game of kind, and the second formulation leads to a game of degree. Our goal is to determine the outcome of the two games, given the initial positions of the pursuer and the evader. Since the evader always wins from any given initial position of the players for \(\nu >1\), we only consider the case \(\nu \le 1\). The winning strategy of the evader for \(\nu >1\) consists in moving along the boundary of the obstacle with its maximum speed either clockwise or anticlockwise.

In next section, we present a reformulation of the problem in reduced coordinates.

3 Dimensionality Reduction

The dimension of the game formulated in the previous section is \(\mathbb {R}^4\). By exploiting the symmetry of the problem, its dimensionality can be reduced. Formulating the game in \(\mathbb {R}^3\) helps to visualize the solution in the game space, which is not possible in \(\mathbb {R}^4\).

In order to do so, we formulate the problem in polar coordinates. We express the positions of the players in relative coordinates. Let the polar coordinates of the pursuer and the evader be denoted by \((r_{p},\theta _{p})\) and \((r_{e},\theta _{e})\), respectively. Instead, we can use the following relative coordinates to define the state of the game: \(R=r_{p},r=r_{e},\phi =(\theta _{e}-\theta _{p})\). The equations of motion of the two players in relative coordinates are given by the following:

$$\begin{aligned} f_{R}=\dot{R}=u_{r_{p}};\quad f_{r}=\dot{r}=u_{r_{e}};\quad f_{\phi }=\dot{\phi }=\frac{u_{\theta _{e}}}{r}-\frac{u_{\theta _{p}}}{R}, \end{aligned}$$
(1)

where (\(u_{r_{p}},u_{\theta _{p}}\)) and (\(u_{r_{e}},u_{\theta _{e}}\)) are the radial and tangential components of the velocities of the pursuer and the evader, respectively, and satisfy the following constraints:

$$\begin{aligned} u^{2}_{r_{p}}+u^{2}_{\theta _{p}}\le 1;\quad u^{2}_{r_{e}}+u^{2}_{\theta _{e}}\le \nu ^{2}. \end{aligned}$$
(2)

Based on the problem formulation, the game terminates when the line-of-sight between the pursuer and the evader intersects with the circular obstacle. Additionally, we assume that if the line joining the pursuer and the evader is tangent to the obstacle, visibility is broken. Therefore, the boundary of the terminal manifold is given by the set of states for which the line-of-sight between the pursuer and the evader is tangent to the circular obstacle, which is given by the following equation:

$$\begin{aligned} F(R,r,\phi )=\cos \phi -\frac{a^2}{Rr}+\sqrt{\left( 1-\frac{a^2}{r^2}\right) \left( 1-\frac{a^2}{R^2}\right) }=0. \end{aligned}$$
(3)

The Usable Part (UP) is characterized by the set of all terminal states from which the evader can guarantee termination irrespective of the pursuer’s choice of controls. This is described by the following inequality:

$$\begin{aligned} \frac{r^{2}}{\nu ^{2}}-R^{2}\le a^{2}\left( \frac{1}{\nu ^{2}}-1\right) . \end{aligned}$$
(4)

Therefore, the Boundary of the Usable Part (BUP) of the terminal manifold is given by the following equation:

$$\begin{aligned} \frac{r^{2}}{\nu ^{2}}-R^{2}= a^{2}\left( \frac{1}{\nu ^{2}}-1\right) . \end{aligned}$$
(5)
Fig. 2
figure 2

The figure shows the terminal manifold

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Figure 2 shows the terminal manifold along with the BUP for various speeds of the evader when \(a=5\). There are two regions marked \(``+''\) and \(``-''\), separated by the terminal manifold. If the state of the system is in \(``+''\), i.e., between the two terminal manifolds, then the game has not yet reached termination. If the state of the system is in the region marked \(``-''\), then the game has already terminated. In each case, the UP of the terminal manifold lies to the right of the BUP.

4 Game of Kind

In this section, we address Problem 2.1 introduced in Sect. 2. In a game of kind, the entire state space is partitioned into two disjoint regions. Although the problem considered in this paper is different from the classical capture–pursuit games that have been thoroughly studied, we borrow existing notions of capture and escape set and modify their definitions slightly to address our current scenario. We define the capture set as the set of initial conditions, from which the pursuer can maintain a line-of-sight with the evader at all times. Complementary to the capture set, we define the escape set as the set of initial conditions of the players from which the evader can break the line-of-sight with the pursuer in finite time. Therefore, contrary to the standard notion of termination in capture–pursuit games, the evader wins the game if the final state of the players lies on the terminal manifold. The semipermeable surface that covers the escape set is termed the barrier.

In [9], Isaacs studied several games of kind and presented an elaborate description of different kinds of barriers that might occur in such games. Here, our specific interest lies in the construction of natural barriers that terminate at the BUP [8, 11] of the terminal manifold. In [12], a similar technique was used to analyze a game between a team of UAVs and a mobile aerial jammer. In that game, the characterization of the barrier was restricted to a set of ordinary differential equations due to high dimensionality of the state space of the problem. Since the state space of the game, in this paper, has been reduced to three dimensions, the barrier can be conveniently illustrated in the state space. In next section, we derive the control policies of the players on the barrier.

4.1 Controls Policies of the Players on the Barrier

In this section, we derive the control policies of the players on the barrier based on the property of semipermeability. Let \(\lambda =[\lambda _{R}\quad \lambda _{r}\quad \lambda _{\phi }]^{T}\) denote the normal to the semipermeable surface passing through the point \((R,r,\phi )\). The following proposition states the main result:

Proposition 4.1

The control policies of the players on the barrier are given by the following expressions:

$$\begin{aligned} (u^{*}_{r_p},u^{*}_{\theta _{p}})= & {} \left( \frac{\lambda _{R}}{\sqrt{\lambda ^{2}_{R}+\frac{\lambda ^{2}_{\phi }}{R^{2}}}}, -\frac{\frac{\lambda _{\phi }}{R}}{\sqrt{\lambda ^{2}_{R}+ \frac{\lambda ^{2}_{\phi }}{R^{2}}}} \right) , \end{aligned}$$
(6)
$$\begin{aligned} (u^{*}_{r_e},u^{*}_{\theta _{e}})= & {} \left( -\nu \frac{\lambda _{r}}{\sqrt{\lambda ^{2}_{r}+\frac{\lambda ^{2}_{\phi }}{r^{2}}}}, -\nu \frac{\frac{\lambda _{\phi }}{r}}{\sqrt{\lambda ^{2}_{r} +\frac{\lambda ^{2}_{\phi }}{r^{2}}}}\right) . \end{aligned}$$
(7)

Proof

Based on our assumption that the barrier is embedded in a family of semipermeable surfaces [9], we obtain the control policies of the players on the barrier, as follows:

$$\begin{aligned} (u^{*}_{r_p},u^{*}_{\theta _{p}},u^{*}_{r_e},u^{*}_{\theta _{e}})= \arg \min _{u_{r_e},u_{\theta _e}}\left\{ \lambda _{\phi }\frac{u_{\theta _{e}}}{r} -\lambda _{r}u_{r_{e}}\right\} +\arg \max _{u_{r_p},u_{\theta _p}}\left\{ \lambda _{R}u_{r_{p}} -\lambda _{\phi }\frac{u_{\theta _{p}}}{R}\right\} . \end{aligned}$$

The proposition then follows from the above minimax problem, noting that the Hamiltonian is separable in the four variables. \(\square \)

For a point on the hypersurface, there can be two normals that point in opposite directions. In this paper, the normal to the barrier is chosen in a direction which points toward the capture set. Note that the expression for the optimal control policy of each player at a point on the barrier is a function of the components of the normal vector. In order to compute the optimal control policies as functions of the state, we obtain the retrogressive path equations for the components of the normal to the barrier along the trajectories of the players on the barrier.

The retrogressive path equations for the normals along the trajectories constituting the barrier are given by the following equation [9]:

$$\begin{aligned} \mathring{\lambda }_{x_k}=\displaystyle \sum _{i}\lambda _{x_i}\frac{\partial f_{x_i}}{\partial x_{k}}\quad \forall k=1,2,3, \end{aligned}$$
(8)

where \({\varvec{\lambda }}=[\lambda _{R},\lambda _{r},\lambda _{\phi }]'\) represents the normal vector to the barrier, \(\mathring{}\) denotes the derivative in retrogressive time, and \([x_{1}x_{2}x_{3}]=[R r \phi ]\). In order to solve the above differential equations, we need boundary conditions, and they can be obtained from the terminal manifold. The expression for the terminal manifold is given in (3). From (8), we obtain the following set of differential equations for the evolution of the normal to the barrier, along with the boundary conditions:

$$\begin{aligned} \mathring{\lambda }=\left[ \begin{array}{c}\mathring{\lambda }_{R}\\ \mathring{\lambda }_{r}\\ \mathring{\lambda }_{\phi }\end{array}\right] = \left[ \begin{array}{c} \lambda _{\phi }\frac{u^{*}_{\theta _p}}{R^{2}}\\ -\lambda _{\phi }\frac{u^{*}_{\theta _e}}{r^{2}}\\ 0\end{array}\right] ,\quad \lambda (0)=\left[ \begin{array}{c} \frac{\partial F}{\partial R}\\ \frac{\partial F}{\partial r}\\ \frac{\partial F}{\partial \phi } \end{array}\right] = \left[ \begin{array}{c}\frac{a^{2}}{rR^{2}}\left( \sqrt{\frac{r^2-a^2}{R^2-a^2}}+1\right) \\ \frac{a^{2}}{Rr^{2}}\left( \sqrt{\frac{R^2-a^2}{r^2-a^2}}+1\right) \\ sin\phi \end{array}\right] . \end{aligned}$$
(9)

The barrier of the game corresponds to the semipermeable surface passing through the BUP. Figure 3 shows the barrier of the game for \(\nu =0.8\) along with the termination manifold.

Fig. 3
figure 3

Figure shows the semipermeable surface on the BUP

Full size image

We note that the semipermeable surface emanates from the BUP out of the game space. This is intuitive since the semipermeable surface is tangent to the terminal manifold at BUP. Therefore, for a terminal manifold of positive curvature, the semipermeable surface will be generated out of the game space. Since the curvature of the terminal manifold is positive at all points irrespective of the speed ratio, the semipermeable surface from the terminal manifold will always be generated out of the game space.

Based on the above observation, there is a possibility that the barrier may not exist in the game space, in which case the evader always wins. However, a complete analysis of all the semipermeable surfaces that arise in the game needs to be done before one can arrive at the aforementioned conclusion. In [13], the author has shown the construction of a barrier surface that emanates from the non-usable part of the terminal manifold. However, the construction of the barrier is completely based upon the closed-form solution of the Hamilton–Jacobi–Isaacs equation, which is a formidable task in our case. In [14], the authors have investigated the construction of all possible semipermeable surfaces that can arise in a specific homicidal-chauffeur game. The analysis is in a two-dimensional game space obtained by expressing the coordinates of the evader in the relative frame of the pursuer. In our future work, we are planning to extend this analysis to a three-dimensional space to characterize all semipermeable surfaces that occur in this game.

In next section, we will investigate the optimal control policies of the players in the escape set.

5 Optimal Control Policies of the Players in the Escape Set

In this section, we address Problem 2.1 in Sect. 2. Toward that end, we formulate a time-optimal differential game [15]. More specifically, the objective of the pursuer is to maximize the time for which it can track the evader, and the objective of the evader is to minimize the time by which it can break the line-of-sight.

Let \({\mathbf{u}_\mathbf{1}}\) and \({\mathbf{u}_\mathbf{2}}\) denote the control policies of the pursuer and the evader, respectively. Let \(T_{\mathbf{x}_\mathbf{0}}(\mathbf{{u}_1},\mathbf{{u}_2})\) denote the time of termination of the game when the pursuer and the evader use control policies \(\mathbf{{u}_1}\) and \(\mathbf{{u}_2}\), respectively, and start from an arbitrary initial condition. A pair of strategies \((\mathbf{{u}_1}^{*},\mathbf{{u}_2}^{*})\) for the two players is said to be in saddle-point equilibrium if the following condition holds:

$$\begin{aligned} T_\mathbf{x_{0}}(\mathbf{{u}_1^{*}},\mathbf{{u}_2})\ge T_\mathbf{x_{0}}(\mathbf{{u}^{*}_1},\mathbf{{u}^{*}_2})\ge T_\mathbf{x_{0}}(\mathbf{{u}_1},\mathbf{{u}^{*}_2}). \end{aligned}$$

If such a pair \((\mathbf{{u}^{*}_1},\mathbf{{u}^{*}_2})\) exists, then the function \(T_\mathbf{x_{0}}(\mathbf{{u}^{*}_1},\mathbf{{u}^{*}_2})\) is the value of the game, which we denote by \(J(\mathbf x_{0})\) (called the value function). In this paper, we assume that such a value function exists, and we compute the optimal trajectories of the players. Let \(H(\mathbf{{x}},\nabla J,\mathbf{{u}_1},\mathbf{{u}_2})\) denote the Hamiltonian of the system, given by the following expression:

$$\begin{aligned} H(\mathbf{{x}},\nabla J,\mathbf{{u}_1},\mathbf{{u}_2})=1+J_{R}u_{r_{p}}+J_{r}u_{r_{e}} +J_{\phi }\left( \frac{u_{\theta _{e}}}{r}-\frac{u_{\theta _{p}}}{R}\right) . \end{aligned}$$

Proposition 5.1

Under the assumption that \(J\in C^{2}(\mathbf{{x}})\), the optimal control policies of the players are as follows:

$$\begin{aligned} \small {u^{*}_{r_{e}}=\frac{-J_{r}\nu }{\sqrt{J^{2}_{r}+\frac{J^{2}_{\phi }}{r^{2}}}},\,u^{*}_{\theta _{e}}=\frac{-\frac{J_{\phi }}{r}\nu }{\sqrt{J^{2}_{r}+\frac{J^{2}_{\phi }}{r^{2}}}},\,u^{*}_{r_{p}}=\frac{J_{R}}{\sqrt{J^{2}_{R}+\frac{J^{2}_{\phi }}{R^{2}}}},\, u^{*}_{\theta _{p}}=\frac{-\frac{J_{\phi }}{R}}{\sqrt{J^{2}_{R}+\frac{J^{2}_{\phi }}{R^{2}}}}}. \end{aligned}$$

Proof

Since \(H(\mathbf{{x}},\nabla J,\mathbf{{u}_1},\mathbf{{u}_2})\in C^{2}(\mathbf{{x}})\) and \(H(\mathbf{{x}},\nabla J,\mathbf{{u}_1},\mathbf{{u}_2})\in C^{2}({\nabla J(\mathbf{{x}})})\), we can determine the regular solutions [5] using Isaacs’ approach [9] to arrive at the following optimization problems to solve for the optimal control policies of the players, under the assumption of continuous differentiability of the value function:

  • Pursuer

    $$\begin{aligned} (u^{*}_{r_{p}},u^{*}_{\theta _{p}})=\arg \max _{u_{r_{p}},u_{\theta _{p}}} (J_{R}u_{r_{p}}-u_{\theta _{p}}\frac{J_{\phi }}{R})\text { subject to first inequality in (2).} \end{aligned}$$

  • Evader

    $$\begin{aligned} (u^{*}_{r_{e}},u^{*}_{\theta _{e}})= \arg \min _{u_{r_{e}},u_{\theta _{e}}}(J_{r}u_{r_{e}}+u_{\theta _{e}}\frac{J_{\phi }}{r})\text { subject to second inequality in (2).} \end{aligned}$$

Each of these optimization problems is a quadratic programming problem and admits a unique solution, leading to the optimal control of the pursuer and the evader, as given in the proposition. \(\square \)

In order to complete the construction of the optimal control policies, we have to compute \(\nabla J=(J_{r},J_{R},J_{\phi })\) as a function of the state variables. Substituting the control law into Isaacs’ second condition [p. 67, [9]] leads to the following equation:

$$\begin{aligned} 1+\sqrt{J^{2}_{R}+\left( \frac{J_{\phi }}{R}\right) ^{2}}-\nu \sqrt{J^{2}_{r} +\left( \frac{J_{\phi }}{r}\right) ^{2}}=0. \end{aligned}$$
(10)

The retrogressive path equation [9] is given by the following set of equations:

$$\begin{aligned} \mathring{J_{r}}=-J_{\phi }\frac{u^{*}_{\theta _{e}}}{r^{2}}, \quad \mathring{J_{R}}=J_{\phi }\frac{u^{*}_{\theta _{p}}}{R^{2}}, \quad \mathring{J_{\phi }}=0. \end{aligned}$$

Since \(J\equiv 0\) on the terminal manifold, we obtain the following equations by parameterizing the terminal manifold using coordinates \(s_{1}=r\) and \(s_{2}=R\):

$$\begin{aligned} J_{s_{1}}= & {} J_{r}\underbrace{\frac{\partial r}{\partial s_{1}}}_{1}+J_{R}\underbrace{\frac{\partial R}{\partial s_{1}}}_{0}+J_{\phi }\frac{\partial \phi }{\partial s_{1}}=0\quad \Rightarrow J_{r}=-J_{\phi }\frac{\partial \phi }{\partial s_{1}}=-J_{\phi }\frac{\partial \phi }{\partial r}, \end{aligned}$$
(11)
$$\begin{aligned} J_{s_{2}}= & {} J_{r}\underbrace{\frac{\partial r}{\partial s_{2}}}_{0}+J_{R}\underbrace{\frac{\partial R}{\partial s_{2}}}_{1}+J_{\phi }\frac{\partial \phi }{\partial s_{2}}=0\quad \Rightarrow J_{R}=-J_{\phi }\frac{\partial \phi }{\partial s_{2}}=-J_{\phi }\frac{\partial \phi }{\partial R}. \end{aligned}$$
(12)

Substituting (11) and (12) in (10) leads to the following expression for \(J^{0}_{\phi }\)

$$\begin{aligned} J^{0}_{\phi }=\pm \frac{1}{\left( \sqrt{\left( \frac{\partial \phi }{\partial r}\right) ^{2}+\left( \frac{1}{r}\right) ^{2}}-\sqrt{\left( \frac{\partial \phi }{\partial R}\right) ^{2}+\left( \frac{1}{R}\right) ^{2}}\right) }. \end{aligned}$$

On the termination manifold corresponding to \(\phi \in ]0,\pi [\), J is monotonically decreasing in \(\phi \). Therefore, \(J^{0}_{\phi }<0\). The opposite is true for the termination manifold corresponding to \(\phi \in ]-\pi ,0[\). Figure 4a shows the optimal trajectories in the vicinity of the usable part for an interval of 5 s before termination. They are obtained by integrating the retrogressive path equations and the kinematic equations backward in time.

Fig. 4
figure 4

a Optimal trajectories emanating from UP. b Iso-value surfaces for the game of degree

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Figure 4b depicts the variation of the value of the game as a function of the state. The iso-value surfaces [8] occurring in the game are shown in the figure. Since the value of the game is the time of termination, the set of initial points on an iso-value surface leads to the same time of termination. Since the game terminates in a finite time inside the escape set, the iso-value surfaces appear only inside the escape set. If the players apply their optimal control policies from an initial state inside the escape set, then the state trajectory penetrates through iso-value surfaces of decreasing magnitude.

As far as the singular surface analysis is concerned, we can see that the symmetry of the environment leads to the existence of a dispersal surface [9] in the state space. The plane \(\phi =0\) is the dispersal surface. When the players are on the dispersal surface inside the escape set, there are two ways for the evader to reach the terminal state. If the pursuer has additional knowledge about the instantaneous velocity of the evader, then it can choose the optimal trajectory. Otherwise, the optimal strategy of the pursuer is a mixed strategy in which it can choose either of the trajectories with equal probability, 0.5. A complete analysis of other types of singular surfaces present in the state space is a focus of ongoing research.

6 Conclusions

In this paper, we have addressed the visibility-based target tracking game for the scenario when the environment consists of a circular obstacle. Due to the symmetry of the environment, the dimension of the state space, which is originally four, can be reduced to three. The control policies of the players on possible barrier surfaces have been computed on the basis of semipermeability of the barriers. A standard surface that can be a barrier has been constructed using Isaacs’ techniques. It has been shown that the surface lies outside the game space. Finally, local construction of the optimal controls and trajectories for the pursuer near the termination conditions has been presented when the initial positions of the players lie in the escape set.

An open question for the future is investigation of additional barriers in the game space so as to complete the analysis of the game. Another future direction consists in extending the technique presented here to provide an approximation of the capture set and the escape set for multiple obstacles. Yet another direction would be to analyze the problem for multiple targets and multiple observers.