PURSUIT IN THE HALF PLANE:
THE CASE OF EQUAL CHANCES
S¸TEFAN MIRIC ˘A
We use some recent concepts and results in the theory of Differential Games to ob- tain the complete and theoretically justified solution of R. Isaacs’ differential game ofpursuit in the half-planein the singular case of “equal chances”, in which the two players have the same maximal speed. A pair of admissible feedback strategies and the corresponding value function are obtained using a certain refinement of Cauchy’s Method of Characteristics for nonsmoker (“stratified”) Hamilton-Jacobi equations while their relative optimality is proved using the verification theorem for locally-Lipschitz value functions. Moreover, one uses additional “ad hoc” ar- guments to prove that in the case of approximate capture the problem has a value function on a restricted subset while in the case of exact capture it is proved the existence of an evading feedback strategy and, therefore, the non-existence of the value function, and one provides a justification to R. Isaacs’ remarks on the “Perpetual Dilemma”. Not yet considered in the literature, the solution of this problem illustrates several important theoretical aspects such as the possibly restricted domain of the value function, the existence of evading and/or blocking feedback strategies and the possible nature of restricted problems.
AMS 2000 Subject Classification: 49N70; 49N75; 49N35; 49N90; 91A23.
Key words: differential game, pursuit in the half-plane, dynamic programming, optimal feedback strategies, value function.
1. INTRODUCTION
The aim of this paper is to use a general procedure of Dynamic Program- ming type (that, in principle, is applicable to many other problems), to obtain the complete and rigorous solution of Isaacs’ [6] differential game of pursuit in the half-plane (“pursuit to the wall”) in the singular case of “equal chances”, in which the two players have the same maximal speed.
We recall that in the regular case of a faster pursuer (which guaran- tees capture), this problem has been considered rather heuristically, as Exam- ple 6.4.1 in Isaacs [6] and also Example 4, Section 1.4.4 in Blaqui`ere and Leitmann [4], where, based on some lax concepts of feedback strategies and value functions, as the ultimate argument for optimality one wrongly uses the
REV. ROUMAINE MATH. PURES APPL.,52(2007),4, 429–457
elementary verification Theorem 4.4.1 in Isaacs [6], since in this case the value function it isnot differentiable at the points on the horizontal axis. A theore- tically correct solution of this problem is given in Miric˘a [15] using the same general procedure as in the present paper.
To the best author’s knowledge, the rather singular case of equal chances of this problem has not yet been considered in the literature, probably since the existent concepts and results could not allow a credible mathematical solution.
Let us note here that the discussion in Mertz [8] of the similar “differential game of two identical cars” may hardly be taken as a rigorous mathematical solution of this very complicated problem.
In this paper, as in the regular case in Miric˘a [15], we use the recent concepts and results in Miric˘a [12], [13], summarized as an “User’s Guide”
in Miric˘a [14]; thus, we first use certain extensions of Cauchy’s Method of Characteristics to nonsmooth Hamilton-Jacobi equations to obtain a general- ized characteristic flow, and then we use finite-dimensional optimization pro- cedures to identify the extremal possible value functions and corresponding feedback strategies. Next, in each of the two distinct cases of approximate captureand, respectively,exact capture we use an appropriateverification the- orem to prove the required relative optimality and, finally, we use additional ad hocarguments to identify the nature of restricted problems whose solutions have been obtained.
For the reader’s convenience and the sake of completeness, we recall most of the needed notation, definitions and results. However, for a better understanding of the general procedure and for its possible use in the study of other problems, one may follow the “User’s Guide” in Miric˘a [14].
Besides the aim of testing on a non-trivial example the theoretical con- cepts and results in Miric˘a [12], [13], [14], this work is also intended to suggest an alternative approach to most of the latest theoretical and computational developments (e.g. Bardi and Capuzzo-Dolcetta [1], Bardi et al. [2], Basar and Olsder [3], Cardaliaguet et al [4], Patsko and Turova [17], etc., and the references therein) in which thevalue function, in a multitude of different def- initions, is taken as the main (if notthe only) characteristic of the differential game though it is not clear,how the knowledge of the value function(especially in numerical form) may help any of the two players to play the game, if not optimally, at least admissible, from one’s point of view.
In fact, as pointed out by several authors, the only reasonably way of acting (optimally) in a differential game is that of providing each player a previously calculated feedback strategysince, however “theoretically tempting”, the so called NA (nonanticipative) strategies and the corresponding VREK value functions, even if explicitly described (which, unfortunately, happens only in trivial cases), may hardly be used by the players “to play the game”.
However, though our approach is different from that using NA-strate- gies and VREK value functions, theexplicit evading and/or blocking feedback strategies described in Propositions 8.2, 8.3, 9.2 below, may provide coun- terexamplesto some recent existence results for Pursuit-Evasion differential games at least in the case the dynamics are not separated; this problem re- mains a matter of further study.
2. STATEMENT OF THE PROBLEM AND IDENTIFICATION OF DATA
As an idealization of some “real-life” problems, in Isaacs [6], Exam- ple 6.4.1, the following problem is stated.
Problem2.1 (“practical” problem). Player U (the “Pursuer”) is allowed to move in the half-plane (0,∞)×R along trajectories ξ(.) with velocities ξ(t) =u(t)∈Br(0) for some given r >0 whereBr(0) :={x∈R2; x ≤r} is the ball of radius r and x :=
(x1)2+ (x2)2 is the Euclidean norm;
on the other hand, player V (the “Evader”) is allowed to move on the line {0} ×R along trajectories η(.) = (0, η2(.)) with velocities η(t) = (0, v(t)) ∈ {0} ×[−1,1] and the capture time is defined as the first moment t1 > 0 for which
(2.1) ξ(t)−η(t)> ∀t∈[0, t1), ξ(t1)−η(t1)=
for some given ≥0. Further, for each initial position,ξ0 =ξ(0)∈(0,∞)× R, η0 = η(0) ∈ {0} ×R, player U looks for a control mapping u(.) that minimizes the capture timet1 while player Vlooks for a control mappingv(.) thatmaximizes the capture time and, possibly, toavoid capture.
Using Isaacs’ [6] reduced coordinatesx(t) :=ξ(t)−η(t) that describe the relative motion of the two players, we obtain the following vague formulation of a standard autonomous differential game.
Problem2.2 (DGA-vague formulation). Find
(2.2) inf
u(.)∈Uα
sup
v(.)∈Vα
C(y;u(.), v(.)) ∀y∈Y0, subject to
(2.3) C(y;u(.), v(.)) :=g(x(t1)) + t1
0 f0(x(t), u(t), v(t))dt, y∈Y0, (2.4) x(t) =f(x(t), u(t), v(t)) a.e.(0, t1), x(0) =y,
(2.5) u(t)∈U(x(t)), v(t)∈V(x(t)) a.e.(0, t1), (u(.), v(.))∈Pα,
(2.6) x(.) := (x(.), x0(.))∈Ωα, x0(t) :=
t
0 f0(x(s), u(s), v(s))ds, (2.7) x(t)∈Y0 ∀t∈[0, t1), x(t1)∈Y1,
defined, in our case, by theparameters ≥0, r >0 and by thedata below:
(2.8)
Y0 :={x= (x1, x2)∈(0,∞)×R;x2 > 2}, Y1 =Y1() :=
{x∈(0,∞)×R;x=} if >0 {(0,0)} if= 0, f(x, u, v) :=u+ (0, v), f0(x, u, v) := 1, u∈U(x)≡Br(0), v∈V(x)≡V := [−1,1], g(ξ)≡0, Pα =P1,Ωα= Ω1,
where P1 = U1× V1 is the (largest) class of measurable admissible controls (u(.), v(.)) and Ω1 is the corresponding class of absolutely continuous (hence Lipschitzian, in our case)admissible trajectories.
Thetheoretical algorithmin Miric˘a [14] we are going to use, solve actually the following, more precisely formulated problem.
Problem2.3 (DGA-accurate formulation). Given the data of Problem2.2, find the feedback strategiesU(x)⊂U(x), V(x)⊂V(x), x∈Y0⊆Y0with the following properties.
(A) The pair (U(.),V(.)) is admissible, to mean that for any y ∈ Y0 the set Ωα(y) of (preferably all) trajectories xy(.) ∈ Ωα(y) of the differential inclusion
(2.9) x ∈f(x,U(x),V(x)), x(0) =y∈Y0, that satisfy the constraints
(2.10) xy(t)∈Y0 ∀t∈[0,t1), xy(t1)∈Y1 ⊆Y1, t1=t1(xy(.))>0, is not empty. Moreover, ifPα(y), y∈Y0are the corresponding sets (uy(.), vy(.))
∈Pα ofcontrol mappings that satisfy
(2.11) xy(t) =f(xy(t), uy(t), vy(t)), uy(t)∈U(xy(t)), vy(t)∈V(xy(t)) a.e.(0,t1),
thenthere exists the associated value function defined by
(2.12)
W0(y) :=C(y;uy(.), vy(.))∀(uy(.), vy(.))∈Pα(y), y∈Y0, W(y) :=
W0(y) ify∈Y0
g(y) ify∈Y1 :={xy(t1);xy(.)∈Ωα(y), y∈Y0},
i.e. ifPα(y) contains more than one element then one has
C(y;u1(.), v1(.)) =C(y;u2(.), v2(.))∀(uj(.), vj(.))∈Pα(y), j = 1,2.
(B) The feedback strategies U(.),V(.) arerelatively (or bilaterally) optimalfor the restriction DGA|Y0, to mean that
(2.13) W0(y) =WV(y) := inf
u(.),v(.)C(y;u(.), v(.)) ∀y∈Y0, subject to
(2.14) x(t) =f(x(t), u(t), v(t)), u(t)∈U(x(t)), v(t)∈V(x(t)) a.e., (2.15) x(0) =y, x(t)∈Y0 ∀t∈[0, t1), x(t1)∈Y1,
and also
(2.16) W0(y) =WU(y) := sup
u(.),v(.)C(y;u(.), v(.)) ∀y∈Y0, subject to (2.15) and to
(2.17) x(t) =f(x(t), u(t), v(t)), u(t)∈U(x(t)), v(t)∈V(x(t)) a.e., i.e. if PlayerV (the “Evader”)chooses the strategyV(.), thenU(.) is the best choice for player U and, symmetrically, if Player U (the “Pursuer”) chooses the strategy U(.), then V(.) is the best choice for player V, both properties being considered in the class of admissible feedback strategies (U1(.), V1(.)) that generate trajectories x(.) in the subset Y0, i.e., satisfying the restricted state constraintsin (2.15).
(C) EitherY0=Y0(hence DGA|Y0 = DGA) or the (strict) subsetY0⊂Y0 is FU(.)-invariant and also FV(.)-invariant (i.e. with respect to the control systems in (2.17) and (2.14), respectively) to mean that for any y ∈ Y0 any trajectory x(.) of each of the differential systems
(2.18) x∈FU(x) :=f(x,U(x),V(x)), x∈FV(x) :=f(x, U(x),V(x)), x(0) =y, stays inY0 to mean that x(t)∈Y0 ∀t≥0.
Remark 2.1. We first note that if condition (C) is not satisfied (i.e.
the strict subset Y0 ⊂ Y0 is not invariant with respect to one of the systems in (2.17), (2.14)), then unless additional “ad hoc” arguments are provided, (U(.),V(.),W(.)) cannot be considered as a satisfactory solution of Problem DG|A on the subset Y0 since a large number of possibilities may occur when the problem is considered on the whole setY0 of initial states. Some of these possibilities are illustrated by the partial solutions in Section 5 and also by the results in Sections 8, 9 below.
From the multitude of possible characteristics of the partial solutions (U(.),V(.),W(.)) on Y0 =Y0 we mention the following one.
(C.1). The feedback strategy U(.) of player U achieves the capture on the subset Y0, to mean that for any y ∈ Y0, any trajectory x(.) of the differential system in (2.18) is admissible in the sense of (2.10), i.e., there existst1>0 such that
(2.19) x(t)∈Y0 ∀t∈[0, t1), x(t1)∈Y1.
(C.2). Y0 = Y0 and there exists an “evading” feedback strategy Ve(.) :Y0\Y0 → P(V) such that the differential system
(2.20) x ∈f(x, U(x),Ve(x)), x(0) =y∈Y0\Y0,
does not have any admissible trajectoryx(.) satisfying (2.7).
If condition (C.2) is satisfied, then one may define the global feedback strategyV1(.) by
(2.21) V1(y) :=
V(y) ify∈Y0 Ve(y) ify∈Y0\Y0 and the correspondingunilateral value functionin (2.13) by
(2.22) WV
1(y) =
W0(y) ify∈Y0 inf∅= +∞ ify∈Y0\Y0.
One may also note that if Condition (C.2) holds, then Isaacs [6] transforms Problem 2.2 into a qualitative game of kindfor the “Evader” V, ignoring the possibility of the existence of a blocking feedback strategy Ub(.) of the “Pur- suer”, defined by the fact that the problems in (2.17) do not have admissible trajectories, hence player U achives the worst quantitative performance for his opponent:
WU
b(y) := sup
u(.),v(.)C(y;u(.), v(.)) = sup∅=−∞ ∀y∈Y0,
but which, however, apparently contradicts hisqualitative aimofcapturing the Evader.
(C.3). The strict subsetY0 ⊂Y0 is invariant only with respect to one of the systems in (2.17), (2.14); in this case only one of the corresponding properties in (2.13), (2.16) is satisfied and, therefore, one can say that U(.) (resp. V(.)) is anunilaterally optimal feedback strategy.
There may exist, of course, other possibilities, perhaps too many and too complicated to be described in the general case. A reasonable approach of this problem seems that of studying first a number of significant (“real life”)
examples to obtain a list of such cases and, possibly, to discover general math- ematical concepts and results to tackle this type of problems. Anyway, besides a certain theoretical interest, a partial solution (U(.),V(.),W(.)) defined on a non-invariant subsetY0 ⊂Y0, when found, may be used as an “indication”
of the existence of possible pathological cases of non-existence of the value function as defined in Problem 2.3.
On the other hand, as already noted in Miric˘a [13], [14], the same prob- lems may be formulated in the class of limiting KS (Krassovskii-Subbotin) trajectoriesand also in the class of limiting Euler trajectories which, at least from the point of view of existence, seem more suitable for the (generally) discontinuous differential systems in (2.9), (2.14), (2.17) than the classical AC (absolutely continuous) trajectories.
Perhaps one should point out that in our approach anadmissible trajec- tory(very clearly defined in Problem 2.3) is an AC solution of some differential inclusion which reachesthe targetY1 in finite time and for which the cost func- tional (“payoff“) in (2.3) is well defined. This approach is clearly different from other approaches in which one takes (rather “artificially”) +∞ for the payoff if the trajectory never reaches the target.
Remark2.2. Concerning Problem 2.2, we note that a more realistic but more complicated case is that in which the capture is allowed also at the points (0, ), (0,−) on the vertical axis. In this case, the terminal set in (2.8) should be replaced by
Y10 :=Y1∪ {(0, ),(0,−)},
which is no longer a differentiable manifold; however, it is a stratified set in the sense of Definition 2.1 in Miric˘a [13], [14].
On the other hand, while the solution of Problem 2.3 obviously depends on the values of the parameters ≥ 0, r >0, only certain relative positions will provide qualitatively different solutions. From this point of view, a very special case is that of the“exact capture”, that is,
(2.23) = 0, Y0:= (0,∞)×R, Y1:={(0,0)}
heuristically discussed in Isaacs [6] as producing a “Perpetual Dilemma”.
In the case of “approximate capture”, the parameter >0 can be taken sufficiently small such that 0< <min{r,1}, but the solution of Problem 2.3 may be qualitatively different in each of the cases
(2.24) (I) : 0< <1< r; (II) : 0< < r= 1; (III) : 0< < r <1 from which, only the first one, that guarantees capture, has been considered in Isaacs [6] and in Blaqui`ere and Leitmann [4].
In what follows we shall first apply the theoretical algorithm in Miric˘a [14] to Isaacs’ [6] singular problem in case (II) of“equal chances and ap- proximate capture”, then we shall briefly discuss the case of the exact capturein (2.23) under the same condition, r= 1.
3. HAMILTONIANS AND THE SET OF TRANSVERSALITY TERMINAL POINTS
The “pseudo-Hamiltonian” H(x, p, u, v) := p, f(x, u, v)+f0(x, u, v) is given in our case byH(x, p, u, v) =p, u+p2v+ 1 hence, using the well known fact that inf{p, u; u∈B1(0)}=−p, Isaacs’ Hamiltonian
H(x, p) := min
u∈Umax
v∈V H(x, p, u, v) = max
v∈V min
u∈UH(x, p, u, v)
as well as the corresponding extremal values of the control parameters turn out to be defined onZ :=Y0×(R2\ {0}) by
(3.1)
H(x, p) =−p+|p2|+ 1, U(p) ={u(p)}, u(p) := p−1.p, V(p) = sign(p2) =
{1} ifp2 >0 {−1} ifp2 <0 V = [−1,1] if p2 = 0, also defined by the properties
(3.2) H(x, p) =H(x, p, u, v) ∀u∈U(p), v∈V(p).
The value p = (0,0) ∈ R2 has been eliminated as “not interesting” in the operations to follow.
Next, in order to compute the set of terminal transversality values (3.3) Z1S :={(ξ, q)∈Y1×R2; H(ξ, q) = 0, q, w=Dg(ξ).w ∀w∈TξY1}, we parameterize the terminal manifoldY1 in (2.8) as
(3.4) Y1=
(cosθ,sinθ); θ∈
−π 2,π
2
, so that itstangent spacesare defined by
TξY1={(−sinθ,cosθ).θ; θ∈R} ifξ=(cosθ,sinθ)∈Y1. Therefore, sinceg(ξ)≡0, a point (ξ, q)∈Z1S is fully characterized by
ξ =(cosθ,sinθ), −q1sinθ+q2cosθ= 0, −q+|q2|+ 1 = 0, whence
q1=± 1
1− |sinθ|.cosθ, q2=q1tanθ,
so that the set of transversality terminal values is given by (3.5) Z1S =
(cosθ,sinθ),± 1
1− |sinθ|(cosθ,sinθ)
; θ∈
−π 2,π
2
.
4. THE GENERALIZED HAMILTONIAN AND CHARACTERISTIC FLOW
The HamiltonianH(., .) in (3.1) is obviouslyCω(i.e. analytically) finitely- stratified by the stratificationSH :={Z+, Z−, Z0} defined by
(4.1) Z+:={(x, p)∈Z; p2 >0}, Z− :={(x, p)∈Z; p2<0}, Z0 :={(x, p)∈Z; p1= 0, p2 = 0}
since the restrictions H±(., .) :=H(., .)|Z±, H0(., .) :=H(., .)|Z0, given by (4.2) H±(x, p) =−p ±p2+ 1, (x, p)∈Z±, H0(x, p) =−|p1|+ 1, are analytical functions.
In order to compute the stratified Hamiltonian field d#SH(x, p) :=
(x, p)∈T(x,p)Z; x ∈f(x,U(x, p),V(x, p)), x, p − p, x=DH(x, p).(x, p) ∀(x, p)∈T(x,p)Z
,
we first note that on theopen (i.e. 4-dimensional) strataZ± ∈ SH it coincides with theclassical Hamiltonian vector fields
(4.3) d#SH(x, p) ={(∂H±
∂p (x, p),−∂H±
∂x (x, p))} ∀(x, p)∈Z±.
Next, since the tangent spaces to the 3-dimensional manifold Z0 in (4.1) are given by
T(x,p)Z0 ={(x, p)∈R2×R2; p2 = 0}
and DH0(x, p).(x, p) = −p1sign(p1), a vector (x, p) ∈ d#SH0(x, p) is fully characterized by
p2 = 0, x∈f(x,U(p),V(p)), x1p1−p1x1 =−p1sign(p1) ∀p1, x1 ∈R.
It follows that at each point (x, p) in thesingular stratum Z0 in (4.1) one has
(4.4)
d#SH0(x, p) ={((−sign(p1), v),(0,0)); v∈V = [−1,1]}, sign(t) :=
1 ift >0
−1 ift <0 0 ift= 0.
We recall that a generalized Hamiltonian flow X∗(., .) = (X(., .), P(., .)) is described by a (parameterized) set of solutionsX∗(., a), a= (z, λ), z = (ξ, q)∈ Z1S, λ∈Λ(z) of theHamiltonian inclusion
(4.5) (x, p)∈d#SH(x, p), (x(0), p(0)) =z= (ξ, q)∈Z1S, that satisfy the conditions
(4.6)
H(X(t, a), P(t, a)) = 0, X(t, a)∈Y0 ∀t∈I0(a) = (t−(0),0), X(t, a) =f(X(t, a), ua(t), va(t)) a.e., (ua(.), va(.))∈Pα, ua(t)∈U(X∗(t, a)), va(t)∈V(X∗(t, a)) a.e., a= (z, λ).
We also recall that for each (t, a)∈B0 (i.e. t∈I0(a), a∈A) the Hamiltonian flow defines on the interval [0,−t] the control and, respectively, trajectory (4.7) ut,a(s) :=ua(t+s), vt,a(s) :=va(t+s), xt,a(s) :=X(t+s, a), that are admissible with respect to the initial point y = X(t, a) ∈ Y0, for which the value of the cost functional in (2.3) is given by the function V(., .) defined by
V(t, a) :=g(ξ) + t
0 P(s, a), X(s, a)ds ifa= (ξ, q, λ)∈A,
and which, together with the Hamiltonian flow, defines thegeneralized charac- teristic flow C∗(., .) = (X∗(., .), V(., .)). It follows from (2.8) that in our case the functionV(., .) is given by
(4.8) V(t, a) =−t, (t, a)∈B :={(t, a); a∈A, t∈I(a) := (t−(a),0]}. Due to the structure in (4.3)–(4.4) of the Hamiltonian field, we shall construct the Hamiltonian flow separately, on each of the strataZ±, Z0 in (4.1).
The Hamiltonian flow on the stratum Z+. On the stratum Z+ in (4.1) (on whichp2 >0) the differential inclusion in (4.5) coincides with thesmooth Hamiltonian system
(4.9)
x1=−pp1 , x1(0) =cosθ;
x2=−pp2 + 1, x2(0) =sinθ,
p= (0,0), p(0) =q±(θ) :=±1−|1sinθ|(cosθ,sinθ), θ ∈(−π2,π2), whose solutions automatically satisfy the first condition in (4.6) (sinceH+(., .) is a first integral of this system). However, to be admissible, a solutionX∗(.) = (X(.), P(.)) should also satisfy the condition X∗(t) ∈ Z+, i.e. P2(t) > 0
∀t∈(t−X∗,0) for some t−X∗ <0 and the second condition in (4.6) that can be expressed as
(4.10) α(t) := (X1(t))2+ (X2(t))2−2 >0 ∀t∈(t−X∗,0).
Elementary computations and arguments show thatonly the valuesθ∈(0,π2), q = 1−sin1 θ(cosθ,sinθ)) are admissible and the components of the partial Hamiltonian flow X+∗(., .) on the stratum Z+ in (4.1) are defined on B+ :=
(−∞,0]×(0,π2) as
(4.11) X+(t, θ) = ((−t) cosθ,(−t) sinθ+t),
P+(t, θ) =q+(θ) := 1−sin1 θ(cosθ,sinθ), (t, θ)∈B+.
Let us note here that geometrically, the trajectoriesX+(., θ) are thehalf-lines given by the equations
(4.12) x2 = sinθ−1
cosθ x1+, x1 ≥cosθ, θ∈(0,π2), and cover the domainY+:=Y0+∪Y1+ defined by
(4.13) Y0+:=X+(B0+) ={(x1, x2)∈Y0; x2∈(−x1+, )}, Y1+:={(cosθ,sinθ); θ∈(0,π2)},
whereB0+= (−∞,0)×(0,π2).
The Hamiltonian flow on the stratum Z−. On the stratum Z− in (4.1) (on whichp2 <0) the differential inclusion in (4.5) coincides with thesmooth Hamiltonian system
(4.14)
x1=−pp1 , x1(0) =cosθ, θ∈(−π2,π2), x2=−pp2 −1, x2(0) =sinθ,
p = (0,0), p(0) =q±(θ) :=±1−|1sinθ|(cosθ,sinθ),
whose solutions automatically satisfy the first condition in (4.6) (H−(., .) is a first integral). The same type of computations and arguments as above show that the Hamiltonian flow on the symmetric stratum Z− is defined on B−:= (−∞,0]×(−π2,0) by
(4.15) X−(t, θ) = ((−t) cosθ,(−t) sinθ−t), t≤0, P−(t, θ) = 1+sin1 θ(cosθ,sinθ), (t, θ)∈B−.
As in the other case, geometrically, the trajectoriesX−(., θ) are thehalf-lines given by the equations
(4.16) x2= sinθ+ 1
cosθ x1−, x1≥cosθ, θ∈(−π2,0), and cover the domainY−:=Y0−∪Y1− defined by
(4.17) Y0−:=X−(B0−) ={(x1, x2)∈Y0; x2∈(−, x1−)}, Y1−:={(cosθ,sinθ); θ∈(−π2,0)},
whereB0−= (−∞,0)×(−π2,0).
Thus, the smooth Hamiltonian systems in (4.9), (4.14) generate the smooth characteristic flowsC±∗(., .) = (X±∗(., .), V(., .)) given explicitly in (4.8), (4.11), (4.15) and which, according to well known classical results (e.g. Miric˘a [9], [10], [16], etc.) satisfy the basic differential relation
(4.18) DV(t, a).(t, a) =P(t, a), DX±(t, a).(t, a).
The Hamiltonian flow on the singular stratum Z0. In contrast with the previous cases, on the singular stratum Z0 in (4.1) (on which p2 = 0) the differential inclusion in (4.5) coincides with theproper differential inclusion
(4.19)
x1=−sign(p1), x1(0) =cosθ, θ∈(−π2,π2), x2∈V = [−1,1], x2(0) =sinθ,
p = (0,0), p(0) =q±(θ) :=±1−|1sinθ|(cosθ,sinθ),
that for each terminal point (ξ, q) ∈ Z1S has an infinite number of solutions given by the set V of measurable functions v(.) : (−∞,0] → V for which x2(t) = v(t) ∈ [−1,1] a.e. ((−∞,0)). The first condition in (4.6) is easily verified but for the second one, only the valuesθ= 0, q= (1,0) are admissible and generate the trajectories
X1(t) =−t+, X2(t) = t
0 v(s)ds∈[t,−t] ∀t≤0, v(.)∈ V. Choosing the constant control mappings v(t) = λ∈[−1,1] ∀t∈ (−∞,0], we obtain theparameterized Hamiltonian flow
(4.20) X0(t;λ) := (−t+, λt), P0(t;λ) := (1,0), (t, λ)∈B0, which is defined on B0 := (−∞,0]×[−1,1] and, as one can verify directly, together with the function V(., .) in (4.8), also satisfies the basic differential relations in (4.18). Moreover, the trajectories X0(., λ), λ∈V in (4.20) cover the domainY0 :=Y00∪Y10, where Y10 :={(,0)} and Y00 is defined by
(4.21) Y00:=X0(B00) ={x∈Y0;x1 > , x2∈[−x1, x1−]}. 5. SMOOTH “PARTIAL” SOLUTIONS
As indicated in the theoretical algorithm in Miric˘a [14], the natural candi- dates for value functions and optimal strategies in Problem 2.3 are the extreme
ones, defined by
(5.1)
W0m(y) := inf
X(t,a)=yV(t, a), W0M(y) := sup
X(t,a)=yV(t, a), Bm(y) :={(t, a) ∈B0; X(t, a) =y, V(t, a) =W0m(y)}, BM(y) :={(t, a)∈B0; X(t, a) =y, V(t, a) =W0M(y)}, Um(y) :=U(Bm(y)), Vm(y) :=V(Bm(y)),
UM(y) :=U(BM(y)), VM(y) :=V(BM(y)),
U¯(t, a) :={u(t);u(.) ∈U¯(a)}, V¯(t, a) :={v(t);v(.)∈V¯(a)}, where ¯U(a), V¯(a) are the sets of control mappings that satisfy (4.6). Let us note that
U¯(t, a)⊆U(X∗(t, a)), V¯(t, a)⊆V(X∗(t, a)) ∀(t, a)∈B0,
and also that ifX(., .) isinvertible at(t, a)∈B0 with inverseB0(y) :=X−1(y), then one has
(5.2) W0m(y) =W0M(y) =V(B0(y)),
Bm(y) =BM(y) =B0(y) := (X(., .))−1(y).
moreover, it follows from (4.18) that if, in addition, the function W0(.) = W0m(.) =W0M(.) is differentiable at a pointy ∈ Int(Y0) then its derivative is given by
(5.3) DW0(y) =P(y) :=P(B0(y)) and satisfies the relations
(5.4) DW0(y).f(y, u, v) +f0(y, u, v) = 0∀u∈U(y), v∈V(y), U(y) :=U(y,P(y)), V(y) :=V(y,P(y)),
and U(.), V(.) are the corresponding candidates for optimal feedback strate- gies. Moreover, it follows from (3.2) and (4.6) that in this caseW0(.) satisfies Isaacs’ basic equation
(5.5)
u∈U(y)min max
v∈V(y)[DW0(y).f(y, u, v) +f0(y, u, v] =
= max
v∈V(y) min
u∈U(y)[DW0(y).f(y, u, v) +f0(y, y, v)] = 0.
Due to these relations, the computations and the arguments to follow can be significantly simplified if the characteristic flow may be split into a finite col- lection of smooth “invertible” characteristic flowsCj∗(., .), so that the marginal characteristic value functions in (5.1) can be represented as
(5.6) W0m(y) = min
1≤j≤kW0j(y), W0M(y) = max
1≤j≤kW0j(y) ∀y∈Y0,
where W0j(.) :Y0j ⊆ Y0 → R are differentiable functions of the form in (5.2) satisfying relations of the forms in (5.3)–(5.5) and which characterize partial solutions (Uj(.),Vj(.)) of the original problem DGA; however, they may be considered complete solutions to the restrictions DGA|Yj of Problem 2.3 to the subsetsY0j ⊆Y0 (see Remark 2.1).
In what follows we shall prove that in the particular case of Problem 2.2, the extreme solutions in (5.1) can be expressed as in (5.6). The main “ingre- dient” is the following quasi-elementary result.
Lemma5.1. (i) The mappingX+(., .) : B+→ Y+ defined in (4.11) is a diffeomorphism whose inverseB+(.) is given by
(5.7)
B+(x) := (t+(x),θ+(x)), t+(x) := (x1)2+ (x2)2−()2 2(x2−) , θ+(x) := arctanx2−t+(x)
x1 , x= (x1, x2)∈Y+.
(ii)Symmetrically, the mapping X−(., .) :B− →Y− defined in (4.15) is a diffeomorphism whose inverseB−(.) is given by
(5.8)
B−(x) := (t−(x),θ−(x)), t−(x) :=−(x1)2+ (x2)2−()2 2(x2+) , θ−(x) := arctanx2+t−(x)
x1 , x= (x1, x2)∈Y−.
(iii) The mapping X00(., .) : B00 := Int(B0) → Y00 defined in (4.20) is a C1-diffeomorphism whose inverseB00(.) is given by
(5.9) B00(x) := (t0(x),λ0(x)), t0(x) :=−x1, λ0(x) := x2 −x1 and lim
x→(,0)t0(x) = 0.
Proof. (i). Ifx = (x1, x2)∈Y0+ then it follows from (4.11) that a point (t, θ)∈B0+:= (−∞,0)×(0,π2) for whichX+(t, θ) =x is characterized by the relations
cosθ= x1
−t, sinθ= x2−t
−t , tanθ= x2−t x1 .
Hence from the identity (cosθ)2+ (sinθ)2−1≡0 it follows that somet <0 is the only root of the equation (−t)2−(x1)2−(x2−t)2 = 0, which turns out to bet+(x) in (5.7). The fact that t+(x) and θ+(x) are analytic functions is obvious.
(ii). The proof of this symmetric statement is entirely similar so we may omit it. Let us note here that it follows from the relations above that the
symmetric functions in (5.8) can equivalently be given by
(5.10) t−(x) =t+(x1,−x2), θ−(x) =−θ+(x1,−x2) ∀x∈Y−. (iii). By the formulae in (4.20), this statement follows from elementary computations and arguments.
The results in Lemma 5.1 show that the characteristic flows C±∗(., .), C0∗(., .) described in Section 4 are invertible in the sense of (5.2) and define the smooth partial proper value functions
(5.11)
W0±(x) :=−t±(x) = (x1)2+ (x2)2−2
2(∓x2) , x∈Y0±, W00(x) :=−t0(x) =x1−, x∈Y00,
which can be naturally extended by W±(ξ) := 0 ∀ξ ∈ Y1±, W0(ξ) := 0
∀ξ ∈ Y10 := {(,0)} to the corresponding terminal sets defined in (4.13), (4.17), (4.21), respectively.
Moreover, the corresponding feedback strategies in (5.4) are given by (5.12)
u±(x) :=−(cosθ±(x),sinθ±(x)), v±(x) :=±1, x∈Y0±,
u0(x) :=u(P 0(B00(x))) = (−1,0), v0(x) :=λ(x) = x2 −x1, where the domainsY0±, Y00 are the sets defined in (4.13), (4.17), (4.21), res- pectively.
Corollary 5.1. (i) The functions W0±(.), W00(.) defined in (5.11) are smooth solutions of Isaacs’ equation in (5.5) (defined by the data in (2.8)) on their corresponding domains Y0±, Y00. Moreover, each of them is the value function in the sense of (2.12) of the corresponding feedback strategies (u±(.),v±(.)),(u0(.),v0(.)).
(ii) The feedback strategies (u±(.),v±(.)), (u0(.),v0(.)) in (5.12) are op- timal in the sense of (2.13), (2.16) for the restrictions DGA|Y0±, DGA|Y00, respectively, of the differential game in Problem2.3.
Proof. (i). The fact that W±(.) in (5.11) are smooth solutions of (5.5) (on their domains, Y0±) follows from Lemma 5.1 and the classical theory of smooth Hamilton-Jacobi equations (e.g. Miric˘a [9], [10], [16], etc.) using the basic differential relations in (4.18). On the other hand, the fact that the extension ofW0(.) on the open set (,∞)×R has the same property follows by direct inspection.
(ii) The optimality follows from statement (i) and the elementary veri- fication theorem (e.g. Isaacs [6]), Theorem 4.4.1, Miric˘a [13], Theorem 5.1, etc.).
Remark 5.1. According to Remark 2.1, the smooth partial solutions in Corollary 5.1 above are not satisfactory since none of them has property (C) in Problem 2.3: none of the domainsY0±, Y00 coincide withY0 and none of them isinvariant with respect to the control system in (2.14)–(2.17). Therefore, we have to look for optimal feedback strategies that are defined on larger subsets ofY0 that satisfy condition (C) in Problem 2.3.
6. EXTREME VALUE FUNCTIONS AND STRATEGIES Due to the results in Sections 4, 5 above, the extreme value functions in (5.1) are given by
(6.1) W0m(x) = min{W0+(x), W0−(x), W00(x)},
W0M(x) = max{W0+(x), W0−(x), W00(x)}, x∈Y0, whereW0±(.), W00(.) are the functions defined in (5.11).
Explicit expressions of these functions are obtained in the following quasi elementary result.
Proposition6.1. The functionsW0±(.), W00(.) defined in(5.11), (5.7), (5.8) satisfy the relations
(6.2) W0+(x)−W0−(x) =x2(x1)2+ (x2)2−2
2−(x2)2 ∀x∈Y0+∩Y0−, (6.3) W0+(x)−W00(x)>0 ∀x∈Y0+∩Y00,
(6.4) W0−(x)−W00(x)>0 ∀x∈Y0−∩Y00, and therefore the extreme value functions in(6.1) are given by
(6.5) W0m(x) =
W0+(x) ifx∈Y0+\Y00 W00(x) ifx∈Y00 W0−(x) ifx∈Y0−\Y00
(6.6) W0M(x) =
W0+(x) ifx∈Y0+,+:={x∈Y0; x2 ∈(0, )}, W0+=W0−(x) ifx∈Y00,0 := (,∞)× {0},
W0−(x) ifx∈Y0−,−:={x∈Y0; x2 ∈(−,0)}. Moreover, the functionsW0m(.),W0M(.) as well as their natural extensions to Ym:=Y0m∪Y1, YM :=Y0M∪Y1, respectively, defined by
(6.7) Wm(x) :=
W0m(x) ifx∈Y0m
0 ifx∈Y1, WM(x) :=
W0M(x) ifx∈Y0M 0 ifx∈Y1
areC1-stratified and have the following additional regularity properties:
(i) the maximal value function WM(.) is locally-Lipschitz but not diffe- rentiable at the points(x1,0)∈Y00,0 (x1> );
(ii)the minimal value functionWm(.)is only lower semicontinuous(l.s.c.) withdiscontinuity pointsin the subsets
(6.8) Y00,+,1 :={x= (x1, x1−); x1 ∈(,2)}, Y00,−,1 :={x= (x1,−x1+); x1 ∈(,2)}.
Proof. It follows from the definitions in (5.11), (5.7), (5.8) and elementary computations that at the pointsx∈Y0+∩Y0− given by
(6.9) Y0+∩Y0−=Y0M :={(x1, x2)∈Y0; x2∈(−, )} one has the formula in (6.2).
To prove the inequality in (6.3) we first note that it follows from (5.11), (5.7) that forx∈Y0+∩Y00 given by
Y0+∩Y00={(x1, x2)∈Y0; x1 ∈(,2), x2 ∈(−x1, x1−)}∪
∪{x∈Y0; x1 ≥2, x2∈(−x1, )} one has
W0+(x)−W00(x) = (x1+x2−)2
2(−x2) >0 ∀x∈Y0+∩Y00, and the inequality in (6.3) is proved.
The inequality in (6.4) follows, obviously, in the same way from (5.11) and (5.10).
SinceW0±(x)>0∀x∈Y0±, the formula in (6.5) follows from (6.2) while the formula in (6.6) follows obviously from (6.3), (6.4).
Next, it follows from the classical theory of smooth Hamiltonian systems and, in this case also from the formulas in (5.11), that the functionsW0±(.), W00(.) are differentiable, with derivatives
(6.10) DW0±(x) = x1
∓x2)±1
2,± (x1)2 2(∓x2)2
, DW00(x) =P0(x) :=P0(B0(x)) = (1,0).
The functionW0M(.) in (6.6) is locally-Lipschitz since the derivativesDW0±(x) remain bounded as x → ξ ∈ Y1± and also as x → y = (y1,0) ∈ Y00,0, but it is not differentiable at the later points since DW0+(y1,0) = (y1 + 12,0) = (y1 −12,0) =DW0−(y1,0), as one may easily verify using the formulas above.
On the other hand, the minimal function Wm(.) in (6.5) is l.s.c. at each point but discontinuous at the points in (6.8) since if, for instance, x =
