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The calculation of discriminating kernel based on viability kernel and reachability
Advances in Difference Equations volume 2017, Article number: 370 (2017)
Abstract
We discuss the calculation of discriminating kernel for the discretetime dynamic game and continuoustime dynamic game (namely differential game) using the viability kernel and reachable set. For the discretetime dynamic game, we give an approximation of the viability kernel by the maximal reachable set. Then, based on the relationship between viability and discriminating kernels, we propose an algorithm of the discriminating kernel. For the differential game, we compute an underapproximation of the viability kernel by the backward reachable set from a closed target. Then, we put forward an algorithm of the discriminating kernel using the relationship of the discriminating and viability kernels. This means that the victory domain can be computed because it is computed by the discriminating kernel. The novelty is that we give two algorithms of the discriminating kernel for a dynamic game that contains two control variables, not one control variable as in differential inclusion.
Introduction
As an important part of control theory, game theory pours attention into economics, social, political science, and other behavioral sciences. Game theory aims to help us understand situations in which decision makers interact. A dynamic game usually consists of two players, the pursuer and the evader, with conflicting goals. Each player attempts to control the states of the system so as to achieve his goal. Although dynamic games are closely related to optimal control problems, there is a little difference between the two: there is a single control input \(u(t)\) and a single criterion to be optimized in an optimal control problem, and dynamic game theory generalizes this to two control inputs \(u(t)\), \(v(t)\) and two criteria. In [1], quantitative and qualitative differential game problems are discussed using setvalued analysis and viability theory. In the case of a twoplayer differential game, the value function is computed by determining the discriminating kernel for the game. In [2], a twoplayer zerosum differential game with incomplete information on the initial state is investigated. In [3], a twoplayer zerosum differential game with infinitely many initial positions and without Isaacs condition is proposed. By optimal transportation theory and stochastic control, there exists a value of the game with such random strategies. In [4], a bounded discriminating domain for linear pursuitevasion differential game is studied. For a constraint set K, the discriminating kernel \(\operatorname {Disc}(K)\) is the largest subset of the discriminating domain K.
Viability theory is used to study stability, reachability, and dynamic games. The research of such questions for differential inclusions has started with the pioneering works of Aubin [5]. A presentation of viability kernels and capture basins of a target viable in a constrained subset satisfying tangential conditions or duality and normal conditions is provided in [6]. In [7], a method to construct viability kernels is given. In [8], an algorithm suited to the identification of specific trajectories or to the computation of viability kernels associated with delayed dynamics is proposed. In [9], based on the proximal normal cone, the method to verify the viability of approximate viable set for continuoustime and discretetime linear systems is given. The problem of viable controller design is formulated as a problem of linear inequalities. In [10], an algorithm that computes the approximating viability kernel of a discretetime system is proposed. In [11], it is shown that determining the viability of a polytopic set expressed by a convex hull of finitely many points can be transformed into verifying the viability criteria at vertices without the assumption that the input set is a polytope, which is needed in the existing criteria.
Reachability analysis is an essential problem of control systems. The goal of reachability analysis is to compute the set of reachable states in the state space for a given model and a set of initial states. In [12], the notions of maximal and minimal reachability are introduced. The reachability analysis of a linear control system is discussed in [13]. The main contribution is that its sets of initial states and inputs are given by arbitrary convex compact sets represented by their support functions. In [14], an efficient and scalable maximal reachability technique to compute the continual reachable set is introduced. At the same time, an approximation of this set based on ellipsoidal techniques is presented. In [15], a method to compute overapproximations of the reachable set for nonlinear dynamic systems using trajectory piecewise linearized models is proposed. The method makes it possible to analyze highorder nonlinear dynamic systems based on existing methods for reachability analysis of linear dynamic systems.
Reachability analysis and viability theory provide solid frameworks for control system of constrained dynamical systems in a setvalued fashion [5, 16]. In [16], the computation of viability kernels using Lagrangian methods is discussed. In [17], an algorithm for computing the set of reachable states of a continuous dynamic game is discussed. There is a close relationship between constrained reachability [18] and viability theory [19]. The relationship is often discussed in the context of optimal control theory by formulating both viability problems and reachability in terms of the HamiltonJacobi equations [20].
Motivated by the method in [21], the discriminating kernel of the discretetime game and differential game is researched. Firstly, an approximation of the viability kernel for the discrete dynamic game is computed by the maximal reachable set from a closed target. Then, an algorithm of the discriminating kernel is given. Secondly, an underapproximation of the viability kernel for the differential game is computed by the backward reachable set from a closed target. Then, an algorithm of the discriminating kernel is proposed. Finally, using the alternative theorem, the victory domain can be computed. The difference is that we give an algorithm of the discriminating kernel for a discretetime dynamic game or differential game that contains two control variables, not one control variable as in differential inclusion.
The paper is organized as follows. In Section 2, we introduce some basic concepts and notation of reachability and game theory. In Section 3, we discuss the discriminating kernel of a discrete differential game. In Section 4, we study the discriminating kernel of a continuous differential game.
Preliminaries
Consider the following twotarget twoplayer dynamic game:
where the time t ranges over a time domain T, which can be either discrete or continuous, the state variables \(x\in X\subset\mathrm {R}^{n}\), the control variables \(u\in U\), \(d\in W\), and \(U,W\subset \mathrm{R}^{m}\). Let \(S\subset\mathrm{R}^{n}\) be a closed target for player one, acting by u, and let \(L\subset\mathrm{R}^{n}\) be an open target for player two, acting by d. Player one wants the state either to avoid L totally or to reach S before reaching L. Player two wants the state to reach L in finite time without first reaching S. When T is a discrete time domain, system (1) is a discretetime dynamic game; when T is a continuous time domain, system (1) is a continuous differential game.
In the following, we review some preliminaries; for detailed discussions, see the monographs [5, 22–24] and the references therein.
Definition 1
Let x be a point of a closed set \(S\subset\mathrm{R}^{n}\). A proximal normal to S at x is a vector \(p\subset\mathrm{R}^{n}\) satisfying \(d_{S}(x+p)= \Vert p \Vert \), where \(d_{S}(y)\) is the distance between y and S, that is, \(d_{S}(y)=\inf_{s\in S} \Vert ys \Vert \). The set of all such p is denoted by \(NP_{S}(x)\).
Definition 2
A closed set \(D\subset X\) is a discriminating domain for \(f(x,u,d)\) if for arbitrary \(x\in D\) and \(p\in NP_{D}(x)\), we have
where
Definition 3
Suppose that \(H:X\times X\rightarrow R\) is a lower semicontinuous map. Any closed set \(D\subset X\) contains a largest (closed) discriminating domain for f. The set, denoted by \(\operatorname{Disc}_{f}(D)\), is called the discriminating kernel of D for f.
Definition 4
Let \(x\in S\subset\mathrm{R}^{n}\). The tangent cone of S at x is defined by
Definition 5
A setvalued map \(F: \mathrm {R}^{n}\rightarrow2^{{\mathrm{R}^{n}}}\) is called Marchaud if it is upper semicontinuous with convex compact nonempty values and has a linear growth, which means that there exists a constant \(c>0\) such that
A map \(f: X\times U\times W\rightarrow X\) describes a control system. If f is continuous with linear growth, \(U, W\) are nonempty and compact, and for all \(x\in X\), \(F(x)=\bigcup_{u}G(x,u)\) is convex, where \(G(x,u)=\bigcup_{d}f(x,u,d)\), then F is a Marchaud map.
Proposition 1
Suppose that F is Marchaud. Then the differential game (1) is viable on a closed set \(S\subset\mathrm{R}^{n}\) if and only if, for arbitrary \(x\in S\), we have
Proposition 2
A closed set \(S\subset\mathrm {R}^{n}\) is a discriminating domain of (1) if and only if S is viable for the setvalued mapping \(x\rightrightarrows\bigcup_{u\in U}G(x,u)\).
From Propositions 1 and 2 we get the following conclusion.
If F is Marchaud, then the closed set \(D\subset\mathrm{R}^{n}\) is a discriminating domain of (1) if and only if, for arbitrary \(x\in D\), we have
For the interior point of D, we have \(T_{D}(x)=\mathrm{R}^{n}\). So, we just need to distinguish the boundary point of D for (3).
In the following, we discuss the discriminating kernels of the discretetime dynamic and of the differential game.
Discriminating kernel of a discrete system
Consider the following discretetime system:
where the time \(t\in T=[0,\tau]\cap\mathbb{Z}_{+}\). If \(\tau<\infty \), then this problem has a finite horizon; otherwise, it has an infinite horizon.
Now, we discuss the viability kernel \(\operatorname{Viab}_{f(x,u,W)}\) using the maximal reachable set. The maximal reachable set at time t is the set of all initial states \(x_{0}\) for which there exists an input \(u(t)\in U\) such that, for arbitrary \(d(t)\in W\), the trajectories emanating from those states reach S exactly at time t, that is,
According to [25], the viability kernel for discretetime systems can be computed using SaintPierre’s viability kernel algorithm via the following recursive formula, whichgives the finite horizon viability kernel \(S_{k}=\operatorname{Viab}_{[0,k]\cap\mathbb{Z}_{+}}(S)\):
Theorem 1
The sequence of finitehorizon viability kernels \(S_{k}\) can be computed recursively in terms of reach sets as
where \(F(x)=\bigcup_{u}G(x,u)\), \(G(x,u)=\bigcup_{d}f(x,u,d)\), and \(\operatorname{Reach}_{1}^{m}(\cdot)\) is the unit timestep maximal reachable set.
Proof
The constrained difference system (4) can be written as the difference inclusion \(x(t+1)\in F(x(t))\), where \(F(x)=\{ f(x,u,d)\vert u\in U,d\in W\}\). Next, we will prove that \(x\in S_{k+1}\) is equal to \(x\in S_{k}\cap\operatorname{Reach}_{1}^{m}(S_{k})\). By the definition of \(S_{k+1}\), when \(x\in S_{k+1}\), we have \(x\in S_{k}\) and \(S_{k}\cap F(x)\neq\emptyset\). \(S_{k}\cap F(x)\neq \emptyset\) means that there exists y such that \(y\in S_{k}\) and \(y\in F(x)\). Moreover, \(F(x)=\bigcup_{u}G(x,u)\), where \(G(x,u)=\bigcup_{d}f(x,u,d)\), and thus there exists \(u\in U\) such that \(y=f(x,u,d)\) for all \(d\in W\). So, there exists \(u\in U\) such that \(f(x,u,d)\in S_{k}\) for all \(d\in W\). By (5), \(x\in\operatorname {Reach}_{1}^{m}(S_{k})\), and since \(x\in S_{k}\), we have that \(x\in S_{k}\cap\operatorname{Reach}_{1}^{m}(S_{k})\).
This means that \(S_{k+1}=S_{k}\cap\operatorname {Reach}_{1}^{m}(S_{k})\). In the following, we will prove that \(S_{k}\cap\operatorname{Reach}_{1}^{m}(S_{k})=S_{0}\cap\operatorname {Reach}_{1}^{m}(S_{k})\) by induction.
(1) From \(S_{k+1}=S_{k}\cap\operatorname{Reach}_{1}^{m}(S_{k})\) we have \(S_{1}=S_{0}\cap\operatorname{Reach}_{1}^{m}(S_{0})\) and \(S_{2}=S_{1}\cap\operatorname{Reach}_{1}^{m}(S_{1})\). Since \(S_{1}\subset S_{0}\), we get
which means that \(S_{1}\cap\operatorname {Reach}_{1}^{m}(S_{1})=S_{0}\cap\operatorname{Reach}_{1}^{m}(S_{1})\).
(2) Suppose that \(S_{l}=S_{0}\cap\operatorname {Reach}_{1}^{m}(S_{l1})\) is established for \(k=l\). Then
that is, \(S_{k}\cap\operatorname{Reach}_{1}^{m}(S_{k})=S_{0}\cap \operatorname{Reach}_{1}^{m}(S_{k})\), \(k=0,\ldots,n\). This completes the proof of the theorem. □
Proposition 3
The sequence of closed sets \(S_{k}\) defined by (6) converges to \(\operatorname{Disc}_{f}(S)\), that is,
The proof of Proposition 3 can seen from the proof of [1, Proposition 4.8] or [26, Theorem 4].
In the following, we propose an algorithm of the discriminating kernel and give an example to illustrate the algorithm (see Algorithm 1).
Example 1
An example of a finite point set about U:
where \(x\in S=\{(x_{1},x_{2})\vert x_{1}^{2}+x_{2}^{2}\leq2\}\), \(W=\{ 1\}\), \(U=\{1\}\), and \(t\in[0,1]\cap\mathbb{Z}_{+}\).
Now, we have the figures of the process of computing the discriminating kernel.
Figure 1 shows that S0 is the boundary and interior of the circle. In Figure 2, S1 is the intersection of the three circles. S2 is the intersection of the seven circles in Figure 3. In Figure 4, S3 is \(\operatorname{Disc}_{f}(S)\). We find that the intersection of the seven circles is empty, that is, \(S2=\emptyset\), so \(\operatorname{Disc}_{f}(S)=\emptyset\).
Discriminating kernel of continuous system
Viability kernel
Consider the following continuous system:
where the time \(t\in[0,\tau]\subset R^{+}\). If \(\tau<\infty\), then this problem has a finite horizon; otherwise, this problem has an infinite horizon. To guarantee the existence and uniqueness of the solutions to the initial value problem, we suppose that the function f is sufficiently smooth.
Next, we discuss the viability kernel \(\operatorname{Viab}_{f(x,u,W)}\) using the reachability. There are two ways to deal with the reachability problem: computing the forward reachable set and the backward reachable set. Here, we use the backward approach.
Definition 6
The backward reachable set from the closed target K over \([0,\tau]\) is the set of all initial states \(x_{0}\) such that, for all \(d(t)\in W\), \(t\in[0,\tau]\), there exists \(u(t)\in U\), \(t\in[0,\tau]\), for which some \(\varphi(\tau,x_{0},u(t),d)\in K\) are reachable from \(x(\tau)\) along a trajectory satisfying (10), where \(\varphi (s,x_{0},u(t),d)\) denotes the solution of (10), that is,
Definition 6 gives the backward reachable set from the closed target K, which consists of the terminal states over \([0,\tau ]\) with finite horizon τ.
Definition 7
The vector function \(f:\mathrm {R}^{m+n}\rightarrow\mathrm{R}^{n}\) is bounded by \(M>0\) on S in the norm \(\Vert \cdot \Vert :\mathrm{R}^{n}\rightarrow R^{+}\) if for all \(x\in S\), \(u\in U\), and \(d\in W\), we have \(\Vert f(x,u,d) \Vert \leq M\); \(\Vert \cdot \Vert \)distance of a point \(x\in\mathrm{R}^{n}\) from a nonempty set \(S\subset\mathrm{R}^{n}\) is \(\operatorname{dist}_{ \Vert \cdot \Vert }(x,S)=\inf_{s\in S} \Vert xs \Vert \).
Suppose that f is bounded by \(M>0\) on S in the norm \(\Vert \cdot \Vert \). Given a discretization time interval δ, defining an underapproximation of the viability constraint set \(S_{\delta}=\{x\in S\vert\operatorname{dist}_{ \Vert \cdot \Vert }(x,S^{c})\geq\delta M\}\), we underapproximate S by the distance δM because we only consider the state at \(t_{k}=k\delta\). A solution \(x(t)\) of (10) at \(t\in[t_{k},t_{k+1}]\) can travel the distance \(\Vert x(t_{k})x(t) \Vert \leq\int_{t_{k}}^{t} \Vert \dot{x}(\tau) \Vert \,d\tau\leq M(tt_{k})\leq\delta M\) from its initial state \(x(t_{k})\). So, the underapproximation is recursively defined as
\(S_{k}(\delta)\) is an approximation of the finitehorizon viability kernel \(\operatorname{Viab}_{[0,s]}(S)\) for \(s=k\delta\). We claim that, for all \(\delta>0\), \(S_{k}(\delta)\) underapproximates \(\operatorname {Viab}_{[0,k\delta]}(S)\).
Theorem 2
Assume that f is bounded by \(M>0\) on S in the norm \(\Vert \cdot \Vert \). For any time interval δ, the sets \(\{S_{k}(\delta)\}\) satisfy
Proof
The proof is similar to that of [21, Theorem 2]. This completes the proof of the theorem. □
Theorem 3
Assume that f is bounded by \(M>0\) on S in the norm \(\Vert \cdot \Vert \). Then
Proof
The proof is similar to that of [21, Theorem 3]. This completes the proof of the theorem. □
Discriminating kernel
Following [26], we have
where
starting with \(K_{1}=S\).
By [27] the set
which is called the viability kernel of \(K_{n}\) with target K, is a closed set if \(K_{n}\) and K are closed.
From (15), (16), and (17) we have
and so \(\operatorname{Disc}_{f}(S)\) is closed (possibly the empty set).
When W is a finite point set, that is, \(W=\{d_{1},\dots ,d_{l},d_{i}\in\mathrm{R}^{m},l<\infty\}\), the algorithm is as follows.
In Algorithm 2, the termination condition \(S=S_{j0}\), \(j<\infty\), is unlikely to set up in the actual situation. We can give an accuracy \(\varepsilon>0\), and the termination condition turns into \(\Vert SS_{j0} \Vert <\varepsilon\), \(j<\infty\).
Example 2
An example of a finite point set about U:
where \(x\in K=\{(x_{1},x_{2})\vert(x_{1},x_{2})\in[0,1]\times[0,1]\} \), \(U=\{1\}\), and \(t\in[0,1]\).

When \(W=\{1\}\),
$$\begin{gathered} \mathbb{T}=0.2056,\qquad \operatorname {Disc}_{f}(S)=[0,0.4]\times[0,0.4], \\ \mathbb{T}=0.0350, \qquad \operatorname {Disc}_{f}(S)=[0,0.9]\times[0,0.9], \\ \mathbb{T}=0.0020, \qquad \operatorname {Disc}_{f}(S)=[0,0.99]\times[0,0.99]. \end{gathered} $$ 
When \(W=\{2\}\),
$$\begin{gathered} \mathbb{T}=0.1243,\qquad \operatorname {Disc}_{f}(S)=[0,0.4]\times[0,0.4], \\ \mathbb{T}=0.0201, \qquad \operatorname {Disc}_{f}(S)=[0,0.9]\times[0,0.9], \\ \mathbb{T}=0.0014,\qquad \operatorname {Disc}_{f}(S)=[0,0.99]\times[0,0.99]. \end{gathered} $$ 
When \(W=\{1,2\}\),
$$\begin{gathered} \mathbb{T}=0.1243, \qquad \operatorname {Disc}_{f}(S)=[0,0.4]\times[0,0.4], \\ \mathbb{T}=0.0201, \qquad \operatorname {Disc}_{f}(S)=[0,0.9]\times[0,0.9], \\ \mathbb{T}=0.0014, \qquad \operatorname {Disc}_{f}(S)=[0,0.99]\times[0,0.99]. \end{gathered} $$
In the following, we Figures 5, 6, and 7 of the discriminating kernel.
Remark
(Alternative Theorem)
Let \(S=\mathrm{R}^{n}\setminus\Omega\). If \(f(x,u,d)\) is a continuous function in all variables and a Lipschitz function in x, we have:

1.
The victory domain of player 2 is \(\operatorname{Disc}_{f}(S)\).

2.
The victory domain player 1 is \(S\setminus\operatorname{Disc}_{f}(S)\).
If we can compute \(\operatorname{Disc}_{f}(S)\), we can get the victory domains of player 1 and player 2 using Alternative Theorem.
Conclusions
In this paper, we discussed the discriminating kernel of the dynamic game with two targets and two players. On the one hand, we discussed the discriminating kernel of a discretetime dynamic game. Using setvalued analysis and viability theory, we computed an approximation of the viability kernel by the maximal reachable set. Then, we proposed an algorithm of the discriminating kernel. On the other hand, we discussed the discriminating kernel of a differential game. We computed an underapproximation of the viability kernel by the backward reachable set from a closed target. Using the relationship of discriminating kernel and viability kernel, we proposed an algorithm of the discriminating kernel. In the future work, we will consider the calculation of the kernels of a convex polytope, an ellipsoid, and so on.
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Acknowledgements
This work was supported by National Science Foundation of China (11171221), Doctoral Program Foundation of Institutions of Higher Education of China (20123120110004), Natural Science Foundation of Shanghai (14ZR1429200), Innovation Program of Shanghai Municipal Education Commission (15ZZ073), and Key Research Project Plan of Institutions of Higher of Henan Province(17A120010). The authors would like to thank the anonymous referees for their useful comments.
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Han, Y., Gao, Y. The calculation of discriminating kernel based on viability kernel and reachability. Adv Differ Equ 2017, 370 (2017). https://doi.org/10.1186/s1366201714292
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Keywords
 dynamic game
 viability kernel
 discriminating kernel
 nonsmooth analysis
 reachability