Homi idal hau�eur game: history and modern studiesValerii S. Patsko∗Institute of Mathemati s and Me hani sS. Kovalevskaya str. 16, 620219Ekaterinburg, Russiapatsko�imm.uran.ruVarvara L. TurovaTe hni al University of Muni hBoltzmannstr. 3, 85747Gar hing, Germanyturova�ma.tum.deAbstract

The paper gives a survey of publications related to the homicidalchauffeur differential game.Key words. Time-optimal di�erential games, homi idal hau�eurgame, level sets of the value fun tion, singular lines.AMS Subje t Classi� ations. Primary 49N70, 49N75; Se ondary93B40.1. Introdu tion�Homi idal hau�eur� game was suggested and des ribed by RufusIsaa s in the report [13℄ for the RAND Corporation in 1951. A detaileddes ription of the problem was given in his book �Di�erential games� pub-lished in 1965. In this problem, a � ar� whose radius of turn is boundedfrom below and the magnitude of the linear velo ity is onstant pursues anon-inertia �pedestrian� whose velo ity does not ex eed some given value.The names � ar�, �pedestrian� and �homi idal hau�eur� turned out to bevery suitable, even if real obje ts that R. Isaa s meant [7, p. 543℄ were aguided torpedo and an evading from him small ship.The attra tiveness of the game is onne ted not only with its lear ap-plied interpretation but also with the possibility of transition to referen e oordinates, whi h enables to deal with two-dimensional state ve tor. Inthe referen e oordinates, we obtain a di�erential game in the plane. Due to

∗This resear h was supported by RFBR Grants Nos. 06-01-00414 and 07-01-96085

2 V. S. Patsko and V. L. Turovathis, the analysis of the geometry of optimal traje tories and singular linesthat disperse, join, or refra t optimal paths be omes more transparent.The investigation started by R. Isaa s was ontinued by John V. Break-well and Antony W. Merz. They improved Isaa s' method for solvingdi�erential games and revealed new types of singular lines for problemsin the plane. A systemati des ription of the solution stru ture for thehomi idal hau�eur game depending on the parameters of the problem ispresented in the PhD thesis by A. Merz supervised by J. Breakwell. Thework performed by A. Merz seems to be fantasti , and his thesis, to ouropinion, is the best resear h among those devoted to on rete model gameproblems.Our paper is an appre iation of the invaluable ontribution made by thethree outstanding s ientists: R. Isaa s, J. Breakwell, and A. Merz to thedi�erential game theory. Thanks to the help of Ellen Sara Isaa s, JohnAlexander Breakwell, and Antony Willitz Merz we have an opportunity topresent formerly unpublished photographs (Figs. 1�6).The signi� an e of the homi idal hau�eur game is also that it stimulat-ed the appearan e of other problems with the same dynami equations asin the lassi statement, but with di�erent obje tives of the players. The

Figure 1: Left picture: Rufus Isaacs (about 1932-1936). Right picture: Roseand Rufus Isaacs with the daughter Ellen in Hartford, Connecticut before Isaacswent to Notre Dame University in about 1945.

Homi idal hau�eur game: history and modern studies 3

Figure 2: Rose and Rufus Isaacs embarking on a cruise in their 40s or 50s.

Figure 3: Rufus Isaacs at his retirement party, 1979.

4 V. S. Patsko and V. L. Turova

Figure 4: John Breakwell at a Stanford graduation.

Figure 5: John Breakwell (April 1987).

Homi idal hau�eur game: history and modern studies 5

Figure 6: Antony Merz (March 2008).most famous among them, is the surveillan e-evasion problem onsideredin papers by J. Breakwell, J. Lewin, and G. Olsder.Very interesting variant of the homi idal hau�eur game is investigatedin the papers by P. Cardaliaquet, M. Quin ampoix, and P. Saint-Pierre.The obje tives of the players are usual ones, whereas the onstraint on the ontrol of the evader depends on the distan e between him and pursuer.We also onsider a statement where the pursuer is reinfor ed: he be- omes more agile.The des ription of the above mentioned problems in the paper is a - ompanied by the presentation of numeri al results for the omputationof level sets of the value fun tion being performed using an algorithmdeveloped by the authors.In the last se tion of the paper, some works using the homi idal hauf-feur game as a test example for omputational methods are mentioned.Also, the two-target homi idal hau�eur game is noted as a very interest-ing problem for the numeri al investigation.2. Classi statement by R. Isaa sDenote the players by the letters P and E. The dynami s read

6 V. S. Patsko and V. L. TurovaP : xp = w sin θ E : xe = v1

yp = w cos θ ye = v2

θ = wu/R, |u| ≤ 1 v = (v1, v2)′, |v| ≤ ρ.

(1)Here w is the magnitude of linear velo ity, R is the minimum radius of turn.By normalizing the time and geometri oordinates, one an a hieve thatw = 1, R = 1. As a result, in the dimensionless oordinates, the dynami shave the form

P : xp = sin θ E : xe = v1

yp = cos θ ye = v2

θ = u, |u| ≤ 1 v = (v1, v2)′, |v| ≤ ν.

(2)Choosing the origin of the referen e system at the position of playerP and dire ting y-axis along P 's velo ity ve tor, one arrives [14℄ at thefollowing system

x = −yu + vx

y = xu − 1 + vy

|u| ≤ 1, v = (vx, vy)′, |v| ≤ ν.(3)The obje tive of player P having ontrol u at his disposal is, as soonas possible, to bring the state ve tor to the target set M being a ir leof radius r with the enter at the origin. The se ond player whi h steersusing ontrol v strives to prevent this. The ontrols are onstru ted basedon a feedba k law.One an see that the des ription of the problem ontains two indepen-dent parameters ν and r.R. Isaa s investigated the problem for some parameters values using hismethod for solving di�erential games. The basis of the method is the ba k-ward omputation of hara teristi s for an appropriate partial di�erentialequation. First, some primary region is �lled out with regular hara teris-ti s, then se ondary region is �lled out, and so on. The �nal hara teristi sin the plane of state variables oin ide with optimal traje tories.As it was noted, the homi idal hau�eur game was �rst des ribed byR. Isaa s in his report of 1951. The title page of this report is given inFig. 7.Figure 8A shows a drawing from the book [14℄ by R. Isaa s. The so-lution is symmetri with respe t to the verti al axis. The upper part ofthe verti al axis is a singular line. Forward time optimal traje tories meet

Homi idal hau�eur game: history and modern studies 7this line at some angle and then go along it towards the target set M .A ording to the terminology by R. Isaa s, the line is alled universal.The part of the verti al axis adjoining the target set from below is also auniversal singular line. Optimal traje tories go down along it. The rest ofthe verti al axis below this universal part is dispersal: two optimal pathsemanate from every point of it. On the barrier line B, the value fun tionis dis ontinuous. The side of the barrier line where the value of the gameis smaller will be alled positive. The opposite side is negative.The equivo al singular line emanates tangentially from the terminalpoint of the barrier (Fig. 8B). It separates two regular regions. Optimaltraje tories that ome to the equivo al urve split into two paths: the �rstone goes along the urve, and the se ond one leaves it and omes to theregular region on the right (optimal traje tories in this region are shownin Fig. 8A).

Figure 7: Title page of the first report [13] by R. Isaacs for the RAND Corpo-ration.

8 V. S. Patsko and V. L. TurovaA B

Figure 8: Pictures by R. Isaacs from [14] explaining the solution to the homicidalchauffeur game.The equivo al urve is des ribed through a di�erential equation whi h an not be integrated expli itly. Therefore, any expli it des ription of thevalue fun tion in the region between the equivo al and barrier lines isabsent. The most di� ult for the investigation is the �rear� part (Fig. 8B,shaded region) denoted by R. Isaa s with a question. He ould not obtaina solution for this region.Figure 9 shows level sets W (τ) = {(x, y) : V (x, y) ≤ τ} of the valuefun tion V (x, y) for ν = 0.3, r = 0.3. The numeri al results presented inFig. 9 and in subsequent �gures are obtained using the algorithm [29℄ bythe authors of the paper. The lines on the boundary of the sets W (τ),τ > 0, onsisting of points (x, y) where the equality V (x, y) = τ holds,will be alled fronts (iso hrones). Ba kward onstru tion of the fronts,beginning from the boundary of the target set, onstitutes the basis of thealgorithm. A spe ial omputer program for the visualization graphs of thevalue fun tion in time-optimal di�erential games has been developed byV.L.Averbukh and O.A.Pykhteev [2℄.The omputation for Fig. 9 is done with the time step ∆ = 0.01 till thetime τf = 10.3. The output step for fronts is δ = 0.2. Figure 10 presentsthe graph of the value fun tion. The value fun tion is dis ontinuous on thebarrier line and on a part of the boundary of the target set. In the ase onsidered, the value fun tion is smooth in the above mentioned rear region.

Homi idal hau�eur game: history and modern studies 9

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M

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Figure 9: Level sets of the value function for the classical problem; game param-eters ν = 0.3 and r = 0.3; backward computation is done till the time τf = 10.3with the time step ∆ = 0.01, output step for fronts δ = 0.2.

Figure 10: Graph of the value function; ν = 0.3, r = 0.3.

10 V. S. Patsko and V. L. Turova3. Investigations by J.V. Breakwell and A.W. MerzJohn V. Breakwell and Antony W. Merz ontinued investigation of thehomi idal hau�eur game in the setting by R. Isaa s. These results arepartly and very brie�y des ribed in the papers [6,23℄. A omplete solutionis obtained by A. Merz in his PhD thesis [22℄ at Stanford University. Thetitle page of the thesis is shown in Fig. 11.A. Merz divided two-dimensional parameter spa e into 20 subregions.He investigated the qualitative stru ture of optimal paths and the typeof singular lines for every subregion. All types of singular urves (dis-persal, universal, equivo al, and swit h lines) des ribed by R. Isaa s fordi�erential games in the plane appear in the homi idal hau�eur game for ertain values of parameters. In the thesis, A. Merz suggested to distin tsome subtypes of singular lines and onsider them separately. Namely,he introdu ed the notion of fo al singular lines whi h are universal ones,but with tangential approa h of optimal paths. The value fun tion isnon-di�erentiable on the fo al lines.

Figure 11: The title page of the PhD thesis by A. Merz.

Homi idal hau�eur game: history and modern studies 11Figure 12 presents a pi ture and a table from the thesis by A. Merz thatdemonstrate the partition of two-dimensional parameter spa e into subre-gions with ertain system of singular lines (A. Merz used symbols γ, β forthe notation of parameters. He alled singular lines as ex eptional lines).The thesis ontains many pi tures explaining the type of singular linesand the stru ture of optimal paths. By studying them, one an easilydete t tenden ies in the behavior of the solution depending on the hangeof the parameters.In Figure 13, the stru ture of optimal paths in that part of the plane thatadjoins the negative side of the barrier is shown for the parameters orre-sponding to subregion IIe. This is the rear part denoted by R. Isaa s witha question sign. For subregion IIe, very ompli ated situation takes pla e.Symbol PDL denotes the dispersal line ontrolled by player P . Two op-timal traje tories emanate from every point of this line. Player P ontrolsthe hoi e of the side to whi h traje tories ome down. Singular urve SE(the swit h envelope) is spe i�ed as follows. Optimal traje tories approa hit tangentially. Then one traje tory goes along this urve, and the other(equivalent) one leaves it at some angle. Therefore, line SE is similar to anequivo al singular line. The thesis ontains arguments a ording to whi hthe swit h envelope should be better onsidered as an individual type ofsingular line.Symbol FL denotes the fo al line. The dotted urves mark boundariesof level sets (in other words, iso hrones or fronts) of the value fun tion.The value fun tion is not di�erentiable on the line omposed of the

Figure 12: Decomposition of two-dimensional parameter space into subregions.

12 V. S. Patsko and V. L. Turova

Figure 13: Structure of optimal paths in the rear part for subregion IIe. urves PDL, SE, FL, and SE.The authors of this paper undertook many e�orts to ompute valuefun tions for parameters from subregion IIe. But it was not su eeded,sin e we ould not obtain orner points that must present on fronts to thenegative side of the barrier. One of possible explanations to this failure an be the following: the e�e t is so subtle that it an not be dete tedeven for very �ne dis retizations. The omputation of level sets of the val-ue fun tion for the subregions where the solution stru ture hanges veryrapidly, dependent on the parameters, an be onsidered as a hallengefor di�erential game numeri al methods being presently developed bydi�erent s ienti� teams.Figure 14 demonstrates omputation results for the ase where frontshave orner points in the rear region. However, the values of parameters orrespond not to subregion IIe but to subregion IId. For the latter ase,singular urve SE remains but fo al line FL disappears.

Homi idal hau�eur game: history and modern studies 13

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Figure 14: Level sets of the value function for parameters from subregion IId;ν = 0.7, r = 0.3; τf = 35.94, ∆ = 0.006, δ = 0.12.For some subregions of parameters, barrier lines on whi h the valuefun tion is dis ontinuous disappear. A. Merz des ribed a very interestingtransformation of the barrier line into two lose to ea h other dispersal urves of players P and E. In this ase, there exist both optimal pathsthat go up and those that go down along the boundary of the target set.The investigation of su h a phenomenon is of great theoreti al interest.Figure 15 presents a pi ture from the thesis by A. Merz that orrespondsto subregion IV (A. Merz as well as R. Isaa s used the symbol ϕ forthe notation of the ontrol of player P . In this paper, the orrespondingnotation is u). Numeri ally onstru ted level sets of the value fun tion areshown in Fig. 16. When examining Fig. 16, it might seem that some bar-rier line exists. But this is not true. We have exa tly the ase like the oneshown in Fig. 15. In Fig. 17, an enlarged fragment of Fig. 16 is given. The urve onsisting of fronts' orner points above the a umulation regionof fronts is the dispersal line of player E. The urve omposed of ornerpoints below the a umulation region is the dispersal line of player P . Thevalue fun tion is ontinuous in the a umulation region. To see where (inthe onsidered part of the plane) the point of a maximal value of the gameis lo ated, additional fronts are shown. The point of the maximal valuehas oordinates x = 1.1, y = 0.92. The value fun tion at this point is equalto 24.22. In Figure 18, one more fragment of the omputation results isgiven (the s ale of y-axis is enlarged with respe t to that of x-axis).

14 V. S. Patsko and V. L. Turova

Figure 15: Structure of optimal trajectories in subregion IVc.

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1 2 3 4 5 6

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Figure 16: Level sets of the value function; ν = 0.7, r = 1.2; τf = 24.1,∆ = 0.005, δ = 0.1.

Homi idal hau�eur game: history and modern studies 15

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Figure 17: Enlarged fragment of Fig. 16; τf = 24.22. Output step for frontsclose to the time τf is decreased up to δ = 0.005.

1.1

1.105

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x

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Figure 18: Enlarged fragment of Fig. 16.

16 V. S. Patsko and V. L. Turova4. Surveillan e-evasion gameIn the PhD thesis by Josef Lewin [18℄ (performed as well under the su-pervision of J. Breakwell), in the joint paper by J. Breakwell and J. Lewin[19℄, and also in the paper by J. Lewin and Geert J. Olsder [20℄, bothdynami s and onstraints on the ontrols of the players are the same asin Isaa s' setting but the obje tives of the players di�er from those in the lassi statement. Namely, player E tries to de rease the time of rea hingthe target set M by the state ve tor, whereas player P strives to in reasethat time. In the �rst and se ond works, the target set is the omplement(with respe t to the plane) of an open ir le entered at the origin. In thethird publi ation, the target set is the omplement of an open one withthe apex at the origin.The meaning related to the original ontext on erning two movingvehi les is the following: player E tries, as soon as possible, to es apefrom some dete tion zone atta hed to the geometri position of player P ,whereas player P strives to keep his opponent in the dete tion zone aslong as possible. Su h a problem was alled the surveillan e-evasion game.To solve it, J. Breakwell, J. Lewin, and G. Olsder used Isaa s' method.One pi ture from the thesis by J. Lewin is shown in Fig. 19A, and onepi ture from the paper by J. Lewin and G. Olsder is given in Fig. 19B.In the surveillan e-evasion game with the oni target set, examples oftransition from �nite values of the game to in�nite values are of interestand an be easily onstru ted.Figure 20 shows level sets of the value fun tion for �ve values of parame-ter α whi h spe i�es the semi-angle of the non onvex oni dete tion zone.Sin e the solution to the problem is symmetri with respe t to y-axis,only the right half-plane is shown for four of �ve �gures. The pi tures areordered from greater to smaller α.In the �rst pi ture, the value fun tion is �nite in the set that adjoins thetarget one and is bounded by the urve a′b′cba. This set is �lled out withthe fronts (iso hrones). The value fun tion is zero within the target set.Outside the union of the target set and the set �lled out with the fronts,the value fun tion is in�nite.In the third pi ture, a situation of the a umulation of fronts is present-ed. Here, the value fun tion is in�nite on the line fe and �nite on the ar ea. The value fun tion has a �nite dis ontinuity on the ar be. The graph ofthe value fun tion orresponding to the third pi ture is shown in Fig. 21A.The se ond pi ture demonstrates a transition ase from the �rst to thethird pi ture.

Homi idal hau�eur game: history and modern studies 17A

B

Figure 19: (A) Picture from the PhD thesis by J. Lewin. Detection zone is acircle. (B) Picture from the paper by J. Lewin and G. Olsder. Detection zone isa convex cone.

18 V. S. Patsko and V. L. Turova

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Figure 20: Surveillance-evasion game. Change of the front structure dependingon the semi-angle α of the nonconvex detection cone; ν = 0.588, ∆ = 0.017,δ = 0.17.In the �fth pi ture, the fronts propagate slowly to the right and �ll out(outside the target set) the right half-plane as the ba kward time τ goesto in�nity. Figure 21B gives a graph of the value fun tion for this ase.

Homi idal hau�eur game: history and modern studies 19A B

Figure 21: Value function in the surveillance-evasion game. (A) ν = 0.588,α = 130◦, (B) ν = 0.588, α = 121◦.The fourth pi ture shows a transition ase between the third and �fthpi tures.5. A ousti gameLet us return to problems where player P minimizes and playerE maximizes the time of rea hing the target set M . In papers [8,9℄,Pierre Cardaliaguet, Mar Quin ampoix, and Patri k Saint-Pierre have onsidered an �a ousti � variant of the homi idal hau�eur problem sug-gested by Pierre Bernhard [4℄. It is supposed that the onstraint ν on the ontrol of player E depends on the state (x, y). Namely,

ν(x, y) = ν∗ min{

1,√

x2 + y2/s}

, s > 0.Here, ν∗ and s are the parameters of the problem.The applied aspe t of the a ousti game: obje t E should not be veryloud if the distan e between him and obje t P be omes less than a givenvalue s.P. Cardaliaguet, M. Quin ampoix, and P. Saint-Pierre investigated thea ousti problem using their own method for numeri al solving of di�er-ential games whi h goes ba k to the viability theory [1℄. It was revealedthat one an hoose the values of parameters in su h a way that the set ofstates where the value fun tion is �nite will ontain a hole in whi h pointsthe value fun tion is in�nite. Espe ially easy su h a ase an be obtainedwhen the target set is a re tangle stret hed along the horizontal axis.

20 V. S. Patsko and V. L. Turova

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Figure 22: Level sets of the value function in the acoustic problem; ν∗ = 1.5,s = 0.9375; ∆ = 0.00625, δ = 0.0625.

yx

Figure 23: Graph of the value function in the acoustic problem; ν∗ = 1.5,s = 0.9375.Figures 22 and 23 demonstrate an example of the a ousti problem withthe hole. The level sets of the value fun tion and the graph of the value

Homi idal hau�eur game: history and modern studies 21fun tion are shown. The value of the game is in�nite outside the set �lledout with the fronts. An exa t theoreti al des ription of the arising holeand the omputation (both analyti al and numeri al) of the value fun tionnear the boundary of the hole seems to be very ompli ated problem.6. Game with a more agile player PThe model of dynami s of player P in Isaa s' setting is the simplestone among those used in mathemati al publi ations for the des ription ofthe ar motion (or the air raft motion in the horizontal plane). In thismodel, the traje tories are urves of bounded urvature. In the paper [21℄by Andrey A. Markov published in 1889, four problems related to the op-timization over the urves with bounded urvature have been onsidered.The �rst problem (Fig. 24) an be interpreted as a time-optimal ontrolproblem where a ar has the dynami s of player P . Similar interpretation an be given to the main theorem (Fig. 25) of the paper by Lester E. Du-bins published in 1957. The name � ar� is not used neither by A. Markov,nor by L. Dubins. A. Markov mentioned problems of railway onstru tion.In modern works on theoreti al roboti s [17℄, an obje t with the lassi aldynami s of player P is alled �Dubins' ar�.The next in omplexity is the ar model from the paper byJames A. Reeds and Lawren e A. Shepp [32℄:xp = w sin θyp = w cos θ

θ = u, |u| ≤ 1, |w| ≤ 1.The ontrol u determines the angular velo ity of motion. The ontrolw is responsible for the instantaneous hange of the linear velo ity mag-nitude. In parti ular, the ar an instantaneously hange the dire tion ofmotion to the opposite one. A non-inertia hange of the linear velo itymagnitude is a mathemati al idealization. But, iting [32, p. 373℄, �forslowly moving vehi les, su h as arts, this seems like a reasonable ompro-mise to a hieve tra tability�.It is natural to onsider problems where the range for hanging the on-trol w is [a, 1]. Here, a ∈ [−1, 1] is the parameter of the problem. If a = 1,Dubins' ar is obtained. For a = −1, one arrives at Reeds-Shepp's ar.Let us repla e in (2) the lassi ar by a more agile ar. Using the trans-formation to the referen e oordinates, we obtain

22 V. S. Patsko and V. L. Turova

Figure 24: Fragment of the first page of the paper by A. Markov.

Figure 25: Two fragments of the paper by L. Dubins.

Homi idal hau�eur game: history and modern studies 23x = −yu + vx

y = xu − w + vy

|u| ≤ 1, w ∈ [a, 1], v = (vx, vy)′, |v| ≤ ν.(4)Player P is responsible for the ontrols u and w, player E steers withthe ontrol v.Note that J. Breakwell and J. Lewin investigated the surveillan e-evasiongame [18,19℄ with the ir ular dete tion zone in the assumption that, atevery time instant, player P either moves with the unit linear velo ity orremains immovable. Therefore, they a tually onsidered dynami s like (4)with a = 0.The homi idal hau�eur game where player P ontrols the ar whi h isable to hange his linear velo ity magnitude instantaneously was onsid-ered by the authors of this paper in [30℄. The dependen e of the solutionon the parameter a spe ifying the left end of the onstraint to the linearvelo ity magnitude was investigated numeri ally.Figure 26 shows the level sets of the value fun tion for a = −0.1,

ν = 0.3, r = 0.3. The omputation is done ba kward in time till τf = 4.89.Pre isely this value of the game orresponds to the last outer front andto the last inner front adjoining to the lower part of the boundary of thetarget ir le M . The front stru ture is well seen in Fig. 27 showing an en-larged fragment of Fig. 26. One an see a nontrivial hara ter of hangingthe fronts near the lower border of the a umulation region. The valuefun tion is dis ontinuous on the ar dhc. It is also dis ontinuous outsideM on two short barrier lines emanating tangentially from the boundary ofM . The right barrier is denoted by ce.When solving time-optimal di�erential games of the homi idal hau�eurtype (with dis ontinuous value fun tion), the most di� ult task is the onstru tion of optimal (or ε-optimal) strategies of the players. Let usdemonstrate su h a onstru tion using the last example.We onstru t ε-optimal strategies using the extremal aiming pro edure[15,16℄. The omputed ontrol remains un hanged during the next stepof the dis rete ontrol s heme. The step of the ontrol pro edure is amodeling parameter. The strategy of player P (E) is de�ned using theextremal shift to the nearest point (extremal repulsion from the nearestpoint) of the orresponding front. If the traje tory omes to a pres ribedlayer atta hed to the positive (negative) side of the dis ontinuity line ofthe value fun tion, then a ontrol whi h pushes away from the dis ontinu-ity line is utilized.

24 V. S. Patsko and V. L. Turova

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Figure 26: Level sets of the value function in the homicidal chauffeur game withmore agile pursuer; a = −0.1, ν = 0.3, r = 0.3; τf = 4.89, ∆ = 0.002, δ = 0.05.

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Figure 27: Enlarged fragment of Fig. 26.

Homi idal hau�eur game: history and modern studies 25A B C

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Figure 28: Homicidal chauffeur game with more agile pursuer. Simulation re-sults for optimal motions. (A) Initial point a = (0.3,−0.4). (B) Initial pointb = (0.29, 0.1). (C) Enlarged fragment of the trajectory from the point b.Let us hoose two initial points a = (0.3,−0.4) and b = (0.29, 0.1).The �rst point is lo ated in the right half-plane below the front a umu-lation region, the se ond one is lose to the barrier line on its negativeside. The values of the game in the points a and b are V (a) = 4.446 andV (b) = 2.012, respe tively.In Figure 28, the traje tories for ε-optimal strategies of the players areshown. The time step of the ontrol pro edure is 0.01. We obtain that thetime of rea hing the target set M is equal to 4.230 for the point a and1.860 for the point b. Figure 28C demonstrates an enlarged fragment ofthe traje tory emanating from the initial point b. One an see a slidingmode along the negative side of the barrier.Figure 29 presents traje tories for non-optimal behavior of player E andoptimal a tion of player P . The ontrol of player E is omputed using arandom number generator (random hoi e of verti es of the polygon ap-proximating the ir le onstraint of player E). The rea hing time is 2.590for the point a and 0.300 for the point b. One an see how the se ondtraje tory penetrates the barrier line. In this ase, the value of the game al ulated along the traje tory drops jump-wise.

26 V. S. Patsko and V. L. TurovaA B

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Figure 29: Homicidal chauffeur game with more agile pursuer. Optimal behaviorof player P and random action of player E. (A) Initial point a = (0.3,−0.4). (B)Initial point b = (0.29, 0.1).7. Homi idal hau�eur game as a test examplePresently, numeri al methods and algorithms for solving antagonisti di�erential games are intensively developed. Often, the homi idal hauf-feur game is used as a test or demonstration example. Some of thesepapers are [3,25,26,27,29,31℄. In the referen e oordinates, the game isof the se ond order in the phase variable. Therefore, one an apply bothgeneral algorithms and algorithms taking into a ount the spe i� s of theplane. The non-triviality of the dynami s is in that the ontrol u entersthe right hand side of the two-dimensional ontrol system as a fa tor bythe state variables, and that the onstraint on the ontrol v an depend onthe phase state. Moreover, the ontrol of player P an be two-dimensional,as it is in the modi� ation dis ussed in Se tion 6, and the target set anbe non onvex like in the problem from Se tion 4.Along with the antagonisti statements of the homi idal hau�eurproblem, some lose but non-antagonisti settings are known as being ofgreat interest for the numeri al investigation. In this onne tion we notethe two-target homi idal hau�eur game [12℄ with players P and E, ea hattempting to drive the state into his target set without being �rst drivento the target set of his opponent. For the �rst time, two-target di�erentialgames were introdu ed in [5℄. The applied interpretation of su h games an be a dog�ght between two air rafts or ships [10,24,28℄.

Homi idal hau�eur game: history and modern studies 278. Con lusionIsaa s' homi idal hau�eur game is important for appli ations and o�ersa wide �eld for mathemati al resear h. A omplete solution is obtained byA. Merz in his PhD thesis performed under the supervision of J. Breakwell.In this paper, some statements of the problem with modi�ed obje tives ofthe players or more ompli ated des ription of dynami s are onsidered to-gether with the lassi statement. The dis ussion is a ompanied by om-putation results on the onstru tion of level sets of the value fun tion.Bibliography[1℄ Aubin J.-P. Viability theory. Basel: Birkh�auser. 1991.[2℄ Averbukh V. L., Ismagilov T.R., Patsko V. S., Pykhteev O.A.,Turova V. L. Visualization of value fun tion in time-optimal di�er-ential games. In A. Handlovi� ov�a, M. Komorn�ikov�a, K. Mikula,D. �Sev� ovi� (eds.), Algoritmy 2000, 15th Conferen e on S ienti� Computing. Vysoke Tatry � Podbanske, Slovakia, September 10�15,2000. P. 207�216.[3℄ Bardi M., Fal one M., Soravia P. Numeri al methods for pursuit-evasion games via vis osity solutions. In M. Bardi, T. E. S. Raghavan,T. Parthasarathy (eds.), Sto hasti and Di�erential Games: Theoryand Numeri al Methods, Annals of the Int. So . of Dynami Games.Boston: Birkh�auser. 1999. Vol. 4. P. 105�175.[4℄ Bernhard P., Larrouturou B. Etude de la barriere pour un probleme defuite optimale dans le plan. Rapport de Re her he. Sophia-Antipolis:INRIA. 1989.[5℄ Blaqui�ere A., G�erard F., Leitmann G. Quantitative and qualitativedi�erential games. New York: A ademi Press. 1969.[6℄ Breakwell J. V., Merz A.W. Toward a omplete solution of the homi i-dal hau�eur game. Pro . of the 1st Int. Conf. on the Theory and Ap-pli ation of Di�erential Games, Amherst, Massa husetts, 1969. P. III-1�III-5.[7℄ Breitner M. The genesis of di�erential games in light of Isaa s' ontri-butions. J. Opt. Theory Appl. 2005. Vol. 124(3). P. 523�559 .[8℄ Cardaliaguet P., Quin ampoix M., Saint-Pierre P. Numeri al methodsfor optimal ontrol and di�erential games. Ceremade CNRS URA 749.University of Paris - Dauphine. 1995.

28 V. S. Patsko and V. L. Turova[9℄ Cardaliaguet P., Quin ampoix M., Saint-Pierre P. Set-valued numer-i al analysis for optimal ontrol and di�erential games. In M. Bardi,T. E. S. Raghavan, and T. Parthasarathy (eds.), Sto hasti and Di�er-ential Games: Theory and Numeri al Methods, Annals of the Int. So .of Dynami Games. Boston: Birkh�auser. 1999. Vol. 4. P. 177�247.[10℄ Davidovitz A., Shinar J. Two-target game model of an air ombatwith �re-and-forget all-aspe t missiles. J. Opt. Theory Appl. 1989.Vol. 63(2). P. 133�165.[11℄ Dubins L. E. On urves of minimal length with a onstraint on aver-age urvature and with pres ribed initial and terminal positions andtangents. Amer. J. Math. 1957. Vol. 79. P. 497�516.[12℄ Getz W.M., Pa hter M. Two-target pursuit-evasion di�erential gamesin the plane. J. Opt. Theory Appl. 1981. Vol. 34(3). P. 383�403.[13℄ Isaa s R. Games of pursuit. S ienti� report of the RAND Corpora-tion. Santa Moni a. 1951.[14℄ Isaa s R. Di�erential games. NY: John Wiley. 1965.[15℄ Krasovskii N.N., Control of a dynami system. The minimum problemof a guaranteed result. Mos ow: Nauka. 1985 (in Russian).[16℄ Krasovskii N.N., Subbotin A. I. Game-theoreti al ontrol problems.NY: Springer. 1988.[17℄ Laumond J.-P. (ed.) Robot motion planning and ontrol. Le t. Notesin Contr. and Inform. S i. Vol. 229. NY: Springer. 1998.[18℄ Lewin J. De oy in pursuit-evasion games: PhD thesis. Stanford Uni-versity, 1973.[19℄ Lewin J., Breakwell J. V. The surveillan e-evasion game of degree. J.Opt. Theory Appl. 1975. Vol. 16(3�4). P. 339�353.[20℄ Lewin J., Olsder G. J. Coni surveillan e evasion. J. Opt. Theory Appl.1979. Vol. 27(1). P. 107�125.[21℄ Markov A.A. Some examples of the solution of a spe ial kind of prob-lem on greatest and least quantities. Soobs enija Charkovskogo matem-ati eskogo obs estva. 1889. Vol. 2, 1(5, 6). P. 250�276 (in Russian).[22℄ Merz A.W. The homi idal hau�eur � a di�erential game: PhD thesis.Stanford University, 1971.[23℄ Merz A.W. The homi idal hau�eur. AIAA Journal. 1974. Vol. 12(3).P. 259�260.[24℄ Merz A.W. To pursue or to evade � that is the question. J. of Guid-

Homi idal hau�eur game: history and modern studies 29an e, Control, and Dynami s. 1985. Vol. 8(2). P. 161�166.[25℄ Meyer A., Breitner M. H., Kriesell M. A pi tured memorandum onsynthesis phenomena o urring in the homi idal hau�eur game. In:G. Martin-Herran, G. Za our (eds.), Pro eedings of the Fifth Inter-national ISDG Workshop. International So iety of Dynami Games.Segovia 2005, P. 17�32.[26℄ Mikhalev D.K., Ushakov V.N. Two algorithms for approximate on-stru tion of the set of positional absorption in the game problem ofpursuit. Autom. and Remote Contr. 2007. Vol. 68(11). P. 2056�2070.[27℄ Mit hell I. Appli ation of level set methods to ontrol and rea habilityproblems in ontinuous and hybrid systems: PhD Thesis. StanfordUniversity, 2002.[28℄ Olsder G. J., Breakwell J.W. Role determination in aerial dog�ght.Int. J. of Game Theory. 1974. Vol. 3. P. 47�66.[29℄ Patsko V. S., Turova V. L. Level sets of the value fun tion in di�eren-tial games with the homi idal hau�eur dynami s. Int. Game TheoryReview. 2001. Vol. 3(1). P. 67�112.[30℄ Patsko V. S., Turova V. L. Numeri al investigation of the value fun -tion for the homi idal hau�eur problem with a more agile pursuer. In:P. Bernhard, V. Gaitsgory, O. Pourtallier (eds.), Advan es in Dynam-i Games and Their Appli ations: Analyti al and Numeri al Develop-ments, Annals of the Int. So . of Dynami Games. Boston: Birkh�auser.2008. Vol. 10. P. 235�262 (to appear).[31℄ Raivio T. , Ehtamo H. On numeri al solution of a lass of pursuit-evasion games. In J.A. Filar, K. Mizukami and V. Gaitsgory (eds.),Advan es in Dynami Games and Appli ations, Annals of the Int. So .of Dynami Games. Boston: Birkh�auser. 2000. Vol. 5. P. 177�192.[32℄ Reeds J.A., Shepp L.A. Optimal paths for a ar that goes both for-wards and ba kwards. Pa i� J. Math. 1990. Vol. 145(2). P. 367�393.