# Linear Pursuit Differential Game under Phase Constraint on the State of Evader

Authors: Askar Rakhmanov; Gafurjan Ibragimov; Massimiliano Ferrara
Journal: Discrete Dynamics in Nature and Society
 (2016)

---

## Abstract

We consider a linear pursuit differential game of one pursuer and one evader. Controls of the pursuer and evader are subjected to integral and geometric constraints, respectively. In addition, phase constraint is imposed on the state of evader, whereas pursuer moves throughout the space. We say that pursuit is completed, if inclusiony(t1)-x(t1)∈M is satisfied at some t1>0, where x(t) and y(t) are states of pursuer and evader, respectively, and M is terminal set. Conditions of completion of pursuit in the game from all initial points of players are obtained. Strategy of the pursuer is constructed so that the phase vector of the pursuer first is brought to a given set, and then pursuit is completed.


---

## Body

## 1. Introduction

Pursuit and evasion differential game problems under different type of constraints on controls of players were studied by many researchers, for example, [1–24]. In [11, 16], main approaches of solution of pursuit and evasion differential games were given and fundamental results were obtained. In [2–5, 7, 10, 13, 15, 17–19, 22, 24], various pursuit and evasion differential game problems were investigated under phase constraints. Sufficient conditions of completion of pursuit were obtained in [4, 5, 19], when controls of both players are subjected to geometric constraints [19] or integral constraints [4], and phase constraint is being imposed on the state of the evader. The objectives of the researches [8, 9, 12] are to find value of the game and construct optimal strategies of players. The paper [5] is devoted to discrete pursuit games with phase constraints.The present paper is continuation of [4, 5, 19] and deals with the linear pursuit differential games where pursuit control is subjected to integral constraints and evasion control is subjected to geometric constraint, with phase constraint being imposed on the state of the evader. For such constraints on controls of players, to obtain solution of pursuit problem for all initial states of phase space is complicated because of boundedness of pursuit resources.In the present paper, the following technique is proposed to control the phase vector of the pursuer: first-phase vector of the pursuer is brought to a given bounded, convex set, regardless of the behavior of evading player and then, using the evasion control and initial positions of the players, the pursuit control that ensures the completion of pursuit is constructed.New sufficient conditions of completion of pursuit from all initial positions for a finite time have been obtained and results have been illustrated on a model example. This paper is closely related to [4, 5, 19, 20, 23].


## 2. Statement of Problem

Consider a linear differential game of one pursuer and one evader described by(1)x˙=Ax+u,y˙=By+v,where x,y∈Rn are state vectors of pursuer and evader, respectively, A, B are n×n matrices, and u,v∈Rn are control parameters of pursuer and evader, respectively.The terminal set has the formM=M0+M1, where M0 is a linear subspace of Rn and M1 is a subset of L, which is orthogonal complement of M0 to Rn. It is assumed that relative interior of M0 with respect to L is not empty. Pursuer moves throughout the space, but evader is subjected to state constraint and it moves only within given bounded convex set D.Definition 1.
Measurable functionsu=u(t) and v=v(t), t≥0, that satisfy constraints(2)∫0∞ut2dt≤ρ2,v∈Q⊂Rn,are called admissible controls of pursuer and evader, respectively, where ρ>0 is a given number and Q is a compact subset of Rn.Definition 2.
One says that pursuit can be completed from initial pointsx0,y0∈Rn, y0∈D, y0-x0∉M, for time t1; if, for any admissible control of evader v=v(t), 0≤t≤t1, one can construct an admissible control of pursuer u=u(t), 0≤t≤t1, such that, for solutions x(t), y(t), 0≤t≤t1, of initial value problems,(3)x˙=Ax+ut,x0=x0,y˙=By+vt,y0=y0.Inclusion y(t1)-x(t1)∈M is satisfied and y(t)∈D for all 0≤t≤t1. It is assumed that, to construct pursuit control u(t), it is allowed to use values v(t), x(t), y(t) at any time t≥0.As it is known, ifu=u(t) and v=v(t), t≥0,are any admissible controls of players, then, for the initial value problems (3), we have(4)xt=Ψtx0+∫0tΨt-τuτdτyt=Φty0+∫0tΦt-τvτdτ,where Ψt=exp⁡At, Φt=exp⁡Bt, t≥0.Problem 3.
Find conditions of completion of pursuit in game (1)-(2) from all initial points of players.


## 3. Main Result

Letπ denote the operator of orthogonal projection of Rn onto L. Since int⁡M1≠Ø, set D is convex and bounded, Q is compact set, and then there exist numbers l>0, R>0, σ>0, and vector m∈M1, such that D⊂RS⊂Rn, lS1⊂-m+M1, Q⊂σS, where S and S1 are closed unit balls of spaces Rn and L and centered at the origins of these spaces, respectively. It is clear that, in the case of R≤l, solution of pursuit problem is straightforward, and, therefore, from now on, we assume that R≫l.Assumption 4.
There exist numbersd>0, l1,l2≥0, l1+l2=l, α,1<α≤R/R-l2 and a linear measurable mapping Ft,τ:Rn→Rn in τ, 0≤τ≤t, such that(i)
Ft,τ≤d,0≤τ≤t,(ii)
1/α∫0tπΦt-τ-πΨt-τFt,τQdτ⊂l1S1.Assumption 5.
There exist continuous functionT:(0,ρ)→[0,∞) and number ρ1, 0<ρ1<ρ, such that(i)
ρ-ρ12/Tρ1>d2σ2/α2,(ii)
for each pair of points(x∗,y), y-x∗∉M,y∈D, x∗∈D, there is time t=tx∗,y≤Tρ1 for (5)-m+1απΦty-πΨtx∗∈∫0tπΨt-τρ1tSdτ,wherex∗∈D and this point can be, in particular, the center of the ball of minimal radius containing D.It follows from Assumption5(i) that there is number ε, 0<ε<ρ-ρ1, such that(6)ρ-ρ1-ε2Tρ1>d2σ2α2.Assumption 6.
For anyx∈Rn, there is finite number r=r(x)>0 such that(7)-x∗+Ψrx∈∫0rΨr-τ2ρε-ε2rSdτ.Theorem 7.
Let Assumptions4–6 hold true. Then, pursuit can be completed in game (1)-(2) from all points (x0,y0), y0-x0∉M, y0∈D, for a finite time.Proof.
Let(x0,y0), y0-x0∉M, y0∈D be any point. Consider two cases: (i) x0∈D and (ii) x0∉D. Let x0∈D. According to Assumption 5(ii), for pair (x0,y0), y0-x0∉M, x0∈D, specify time t1=t1(x0,y0)≤T(ρ1), satisfying(8)-m+1απΦt1y0-πΨt1x0∈∫0t1πΨt1-τρ1t1Sdτ.It follows from (8) that there exists measurable selection w(t), 0≤t≤t1, of multivalued mapping πΨ(t1-τ)ρ1/t1S, 0≤τ≤t1, such that(9)-m+1απΦt1y0-πΨt1x0=∫0t1wtdt.Now, we consider (10)wt=πΨt1-tωt,0≤t≤t1,with respect to unknown vector-function ω(t)∈ρ1/t1S, 0≤t≤t1. Then, we can apply Filippov’s Lemma [6] on existence of measurable solution of (10).
Therefore,(11)-m+1απΦt1y0-πΨt1x0=∫0t1πΨt1-tωtdt,with respect to unknown vector-function ω(t)∈ρ1/t1S, 0≤t≤t1, has measurable solution ω=ω10(t), 0≤t≤t1.
Next, assuming thatv=v(t), 0≤t≤t1, is any admissible control of the evader, we construct pursuit control as follows:(12)ut=u1t=1αFt1,tvt+ω10t,0≤t≤t1.Then, using (4) yields (13)1απyt1-πxt1=1απΦt1y0-πΨt1x0+1α∫0t1πΦt1-τ-πΨt1-τFt1,τvτdτ-∫0t1πΨt1-τω10τdτ=1α∫0t1πΦt1-τ-πΨt1-τFt1,τvτdτ+1απΦt1y0-πΨt1x0-∫0t1πΨt1-τω10τdτ.Next, adding vector -m to both sides of (13) and then using Assumption 4(ii) and (12) yield(14)-m+1απyt1-πxt1=-m+1απΦt1y0-πΨt1x0-∫0t1πΨt1-tω10tdt+1α∫0t1πΦt1-τ-πΨt1-τFt1,τvτdτ=1α∫0t1πΦt1-t-πΨt1-tFt1,tvtdt∈1α∫0t1πΦt1-τ-πΨt1-τFt1,τQdτ⊂l1S1.Therefore,(15)-m+1απyt1-πxt1≤l1.Now, we use inclusion y(T)∈D⊂RS, inequality (15), and definition of numbers α, l1, l2 to obtain(16)-m+πyt1-πxt1=πyt1-1απyt1-m+1απyt1-πxt1≤α-1απyt1+-m+1απyt1-πxt1≤α-1αR+l1≤l2+l1=l.Therefore,(17)πyt1-πxt1∈m+lS1⊂M1,which implies that(18)yt1-xt1∈M.Thus, we have proven that, in case (i), game (1)-(2) is completed at time t1.
Let us turn to case (ii). By Assumption6, for point x0∈Rn, the following inclusion is satisfied at some time r1=r1(x0)>0:(19)-x∗+Ψr1x0∈∫0r1Ψr1-τ2ρε-ε2r1Sdτ.Consider(20)-x∗+Ψr1x0=∫0r1Ψr1-τuτdτ,with respect to unknown vector-function u(t)∈2ρε-ε2/r1S, 0≤t≤r1. Using Filippov’s lemma [6], we obtain from inclusion (19) existence of measurable solution of (20). Denote it by u0(t), 0≤t≤r1. Let t=r1, u(t)=-u0(t), 0≤t≤r1, in (3) and use (20) with u(t)=u0(t), 0≤t≤r1, to obtain(21)xr1=x∗-x∗+Ψr1x0-∫0r1Ψr1-τu0τdτ=x∗∈D.Now, take t=r1 as the initial time and let x0=x(r1)=x∗, y0=y(r1). Assume that y0-x0∉M, since game (1)-(2) is completed at t=r1. Then, conditions of case (i) are satisfied and the rest of the proof runs as before. It should be noted that, in case (ii), pursuit is completed at the time r1+t1.
Next, we show admissibility of control (12). Indeed, using the Minkowski inequality gives(22)∫0t1u1t2dt=∫0t11αFt1,tvt+ω10t2dt≤∫0t11αFt1,tvt2dt1/2+∫0t1ω10t2dt1/22.Then, by Assumptions 4(i) and 5, according to ω10(t)∈ρ1/t1S, Q⊂σS, we have, in case (i), that(23)∫0t1u1t2dt≤dσαt1+ρ1t1t12≤dσαTρ1+ρ12<ρ-ρ1+ρ12=ρ2,and, in case (ii),(24)∫0t1u1t2dt≤dσαt1+ρ1t1t12≤dσαTρ1+ρ12<ρ-ε-ρ1+ρ12=ρ-ε2.Since, by construction u0(t)≤ε(2ρ-ε)/r1, 0≤t≤r1, (25)∫0r1u0t2dt≤ε2ρ-ε,∫0r1+t1ut2dt=∫0r1u0t2dt+∫r1r1+t1u1t2dt=ε2ρ-ε+ρ-ε2=ρ2,which implies admissibility of pursuit control in case (ii). Theorem 7 has been proven.Assumption 8.
There exist numbersd1>0, α1, 1<α1≤R/R-l and a linear measurable mapping F1(t,τ):Rn→Rn in τ, 0≤τ≤t, such that(i)
πΨt-τF1t,τ=πΦt-τ,0≤τ≤t,(ii)
Assumptions4(i) and 5 are satisfied at α=α1, d=d1.Theorem 9.
Let Assumptions6 and 8 hold true. Then, pursuit can be completed in game (1)-(2) from any point (x0,y0), y0-x0∉M, y0∈D for a finite time.Proof.
Let(x0,y0), y0-x0∉M, y0∈D be an arbitrary point. As in the proof of Theorem 7, there are two cases: (i) x0∈D, and (ii) x0∉D. Let x0∈D. Using Assumption 5(ii), we define time t2=t2(x0,y0)≤T(ρ1) for point (x0,y0), y0-x0∉M, that satisfies(26)-m+1α1πΦt2y0-πΨt2x0∈∫0t2πΨt2-τρ1t2Sdτ.Consider(27)-m+1α1πΦt2y0-πΨt2x0=∫0t2πΨt2-tωtdt,with respect to unknown vector-function ω(t)∈ρ1/t2S, 0≤t≤t2. Using Filippov’s lemma [6], we obtain from (26) that there exists a measurable solution of (27). Denote it by ω20(t), 0≤t≤t2.
Letv=v(t), 0≤t≤t2, be any admissible control of the evader. For each t∈[0,t2], we construct pursuit control as follows:(28)ut=u2t=1α1Ft2,tvt+ω20t,0≤t≤t2.Then, using Assumption 8(i) and the fact that ω20(t), 0≤t≤t2, is a solution of (27), we obtain, for solution (4) of initial value problem (3) at t=t2, (29)-m+πyt2-πxt2=-m+πΦt2y0-πΨt2x0+∫0t2πΦt2-t-1α1πΨt2-tF1t2,tvtdt-∫0t2πΨt2-tω20tdt=-m+1α1πΦt2y0-πΨt2x0+1-1α1πΦt2y0+∫0t2πΦt2-tvtdt-∫0t2Ψt2-tω20tdt=1-1α1πyt2.Since, by Assumption 8, 1<α1≤R/R-l and y(t2)∈D⊂RS, (29) implies that(30)-m+πyt2-πxt2=1-1α1πyt2∈1-1α1RS⊂lS.Hence,(31)πyt2-πxt2∈M1.Let us turn to case (ii). By Assumption 6, for any point x0∈Rn, there is time r2=r2(x0)>0 such that inclusion (19) is satisfied at r1=r2. The same reasoning as that in the proof of Theorem 7 applies to conclude that x(r2)=x∗∈D. If now t=r2 is taken as the initial time for the game and set x0=x(r2)=x∗, y0=y(r2), then we arrive at case (i), and so there exists t2>0 such that πy(t2)-πx(t2)∈M1. Note that, in case (ii), in order to complete the game, the pursuer has to spend time r2+t2.
Thus, it has been proven that control (28) guarantees completion of pursuit in game (1)-(2) at r2+t2, which proves the theorem.Admissibility of pursuit control (28) can be obtained in much the same way as in the proof of Theorem 7.From Theorems7 and 9, one can easily obtain the following statements, respectively.Corollary 10.
Let, for point(x0,y0), y0-x0∉M, x0,y0∈D, there exists time t3=t3(x0,y0)>0 such that Assumptions 4 and 5 are satisfied at t=t3, x∗=x0, y=y0. Then, pursuit can be completed from point (x0,y0) at time t3.Corollary 11.
Let, for point(x0,y0),y0-x0∉M, x0,y0∈D, there exists time t4=t4(x0,y0)>0 such that Assumption 8 is satisfied at t=t4, x∗=x0, y=y0. Then, pursuit can be completed from point (x0,y0) at time t4.Thus, Corollaries10 and 11 enable us to draw conclusion that if the pursuer is in set D at the initial time, then completion of pursuit is guaranteed without Assumption 6.Example 12.
Consider a simple motion differential game:(32)x˙=u,y˙=v,where x,y,u,v∈Rn,n≥1, pursuit control function is a measurable function subject to integral constraint:(33)∫0∞ut2dt≤ρ2and evasion control function is a measurable function subject to geometric constraint:(34)v≤σ.A phase constraint is imposed to the state of the evader y(35)y∈RS⊂Rn.Differential game (32)–(34) is considered completed if y-x∈lS, l>0; that is, M=lS⊂Rn.
It is not difficult to verify that if(36)ρ>2σR-l,then, for game (32) with different type constraints (33)–(35), all the hypotheses of Assumptions 4–8 are satisfied. Therefore, Theorems 7 and 9 can be applied and they give the same result, namely, if inequality (36) is satisfied, then pursuit is completed in game (32) with constraints (33)–(35) from all initial points y0-x0∉M, y0∈RS, for a finite time. Note that if there is not a state constraint for the evader, then pursuit can be completed in game (32)–(34) only from some points of the phase space.


## 4. Conclusion

We have proposed a technique to solve pursuit problem which consists of two steps: first-step state of the pursuer is brought to a given bounded convex set, regardless of the behavior of evading player and then, in the second step, pursuit is completed.New sufficient conditions of completion of pursuit from all initial positions for a finite time have been obtained. Moreover, results have been illustrated on a model example.Note that Assumptions4, 5, and 8 depend on linear measurable function F(t,τ):Rn→Rn of τ, 0≤τ≤t, to a considerable extent. Therefore, future investigations can be carried out towards the choice of this function to weaken Assumptions 4, 5, and 8.



---

*Source: 1289456-2016-10-10.xml*