- Email: [email protected]
Asymptotics for Singularly Perturbed Differential Games
Copyright © IFAC Nonsmooth and Discontinuous Problems of Control and Optimization, Chelyabinsk, Russia, 1998
ASYMPTOTICS FOR SINGULARLY PERTURBED DIFFERENTIAL GAMES
N.N.Subbotina Institute of Mathematics and Mechanics, Ural Branch Russ. A cad. Sci., S.Kovalevskoi str.,16, 620219 Ekaterinburg, Russia e-mail: [email protected]
Abstract: Singularly perturbed differential games with "fast" and "slow" motions and the Bolza type payoff functionals are considered. Sufficient conditions are obtained for the value functions of the games to converge to the value function of the asymptotic unperturbed game as a parameter of singularity £ tends to o. Copyright @1998 IFAC Keywords: Attractors, characteristic equations, invariance, minimax techniques, non-linear systems, perturbation analysis
1. INTRODUCTION
The considered games have the fixed end time and a pay-off functional of the Bolza type
Mathematical models of dynamical systems with "fast" and "slow" motions occur quite frequently in various problems of economics, engineering, mechanics, biology and other applications.
(J
+
The paper deals with singularly perturbed differential games Gc whose dynamics are described in the following way
9c (r,x(r), y(r), u(r),v(r»dr
(3)
t
where (x(·), y(.»: [t,O] H R!' X Rk are the trajectories of the equations (1) started at the point (x(t),y(t» = (x,y), t E [0,0] under controls
:i: = !,,(t,x,y,u,v),
£iJ = kc(t, x, y, u, v) + hc(t, x, u, v, a, (3)
J
°
(u(·),v(·),a(·),{3(·»: [t,O]
(1)
H
P x Q x A x B.
The first player has the aim to minimized the payoff functional and the second player tries to maximized 'y".
where £ > 0 is a small parameter, time t E [0,0] (x, y) E R!' X R k is the phase vector, x is the "slow" variable, y is the "fast" one. These" attributes" underline a difference between dynamics of variables x and y. The velocity iJ is of order 1/£. Hence y can rapidly vary with respect to time.
The paper deals with conditions under which the value functions WE (t, x, y) of the singularly perturbed games Gc converge to the value function w(t, x) of an unperturbed game G, as £ -!. o. The unperturbed game G is the asymptotics for Gc.
It is assumed that controls of the first player (u, a) and controls of the second player (v, (3) are restricted with the constraints
Researches of singularly perturbed control problems are wide represented in literature, see, for example, (Kokotovic, 1984, Bensoussan, 1989 ) and references within. The classical approach of obtaining asymptotics (Tikhonov, 1952; Vasil'eva and Butuzov, 1973; O'Malley, 1974) contains
(2) where P, Q, A, B are compacta.
43
+ gE:(t,x,y,u,v) + Ije(kE:(t,x,y,u,v),q) + + l/e(hE:(t,x,u,v,a,,B),q)] = max min [(r(t,x,y,u,v),p) + vEQ.f3EB uEP.aEA
• equating E to 0 in relations (1)-(3); • solving the asymptotic algebraic equation in formulae (1) relative to y and • substituting the expression for y into the asymptotic equation in relations (1) and the asymptotic pay-off functional (3).
+ gE:(t,x,y,u,v) + l/e(kE:(t,x,y,u,v),q) +
The other well known approach is a decomposition of the problem to • an optimization problem for " fast" variables under fixed "slow" ones and • an asymptotic control problem for "slow" variables taking into account results of previous calculations having been undepended on "fast" variables. See to get acquainted with (Pervozvansky and Gaitsgory, 1988) and references within.
+ l/E(hE:(t,x,u,v,a,,B),q)]
The equation (4) is called the Isaacs equation (Isaacs R" 1965) for the singularly perturbed differential game GE: (1)-(3). It is singularly perturbed because it contains terms with coefficients l/E where E is a small parameter. These terms are positive homogeneous with respect to impulse "fast" variables q = DywE:. Definitions of minimax solutions to the first-order PDE will be essentially used in the paper to obtain sufficient conditions of convergence wE:(t,x,y) as E .j.. O. Hence, let us consider this concept. It should be mentioned that the notion of a minimax solution to the first-order PDE may be defined in a variety of ways; among other things, one can use the tools of nonsmooth analysis: directional derivatives, tangent or contingent cones, su~ and super-differentials (the definitions and proof of equivalence of all the definitions can be found, for example, in (Subbotin, 1995)).
In the paper an different approach based on existence of attractors in the subspace of" fast" variables is presented. Here main results are o~ tained due to constructions of the minimax s0lutions to the first-order PDEs, namely, to the Isaacs-Bellman equations (Subbotin 1980, 1991, 1995). The approach is close to two mentioned above ones considering convergence problems in the whole of " fast" variable subspace.
The present paper deals with the definition that can be viewed as a generalization of the classical method of characteristics. In this definition the characteristic system is replaced by a family of "characteristic" differential inclusions. The graph of the minimax solution is weakly invariant with respect to these differential inclusions. The role of the characteristic inclusions to the Cauchy pro~ lem pE: (4)-(5) can play inclusions which define u-stability and v-stability properties of the value function. Let us remain the definition.
2. PRELIMINARIES
2.1 Value function as minimax solution of the Isaacs equation
Let 8 be a nonempty set and ME: a multivalued mapping
It is known that the value function wE: (t, x, y) of the game GE: is the minimax (Subbotin, 1980, 1991, 1995) or/and viscosity (Crandall and Lions, 1983; Crandall et al., 1984) generalized solution
[0,8] x Rn H
to the following Cauchy problem pE:
+ HE:(t, x, y, DxwE:, DywE:)
= 0,
(4)
tE[O,B), XER n , YER k ; wE:(B,x,y) = erE: (x), x ERn, yE R k ,
(5)
min
max [(r(t,x,y,u,v),p)
X
Rk
X
8 3 (t,x,y,s)
ME:(t,x,y,s) C Rn
X
Rk
X
H
R
(7)
The pair (8, ME:) is called a characteristic complex (or, briefly, a complex) to the singularly perturbed problem pE: (4)-(5) if the following conditions hold: lOa) for any (t,x,y) E [0,8] X ~ X R k and s E 8 the set ME:(t,x,y,s) = {j,d,g} C ~ X R k X R is nonempty, convex and closed. Components (I,d,g) E ME:(t,x,y,s) satisfy the inequalities
where vectors p = D",wE: , q = DywE: are gradients of wE: (t, x, y) relative to x and y, namely, D",wE: = (8wE: j8Xl, ..., 8wE: j8x n ), DywE: = (8wE:j8Yl, ... ,8wE:/Byk) and HE:(t,x,y,p,q) is the Hamiltonian uEP,aEA vEQ.f3EB
(6)
HE:(t,x, Y,P, q)
Two mentioned approaches consider convergence problems in the whole of "fast" variables su~ space. When a convergence problem in optimal control theory for singularly perturbed data and state constraints on "fast" variables is under consideration it makes sense to consider the convergence only within the constraints. In this case controllability conditions for" fast" variables play an essential role. References on this topic one can find in (Bagagiolo et al., 1997 j Bardi and Capuz~ Dolcetta, 1997).
8wE: /at
=
+
44
Ilfll
~ JLE:(l
+ IIxll + lIyll);
Ildll
~ JLE:(l
+ Ilxll + lIyll)j
is called an upper (lower) minimax solution of the singularly perturbed Hamilton-Jacobi equation (4) if the following invariance conditions hold: • for any (to,xO,YO,zo) Eepiue ((to,xO,YO,zo) E hypo u e ), s = s· E S· (s = s. E S.) a trajectory (x(,),y(·),z(·)) E Sol (to,XO,YO,ZO,s) exists such that (r,x(r), y(r), z(r)) E epi u e ((r, x(r), y(r), z(r)) E hypo u e ) for all r E [to,O].
°
where J-Le > are constant. The multivalued mappings (t,x,y) H Me(t,x,y,s) are upper semicontinuous for all s E S; 2°a) for any (t, x, y) E [0,0] x R" X Rk and (P, q) E
R"xRk
+ l/c(d,q) -
maxmin{(f,p) sES
g: (j,d,g) E
Here (S·, Me) E ct(He) ((S.,Ne) E CJ.(He)). The symbols epi u e , hypo u e and gr u e denote the epigraph, the hypograph and the graph of the function ue(t, x, y) respectively, Le. the sets
E Me(t,x,y,s)} = He(t,x,y,p,q);
2° b) for any (t, x, y) E [0,0] x R" X R k and (P, q) E
R"xRk minmax{ (f,p) sES
+ l/c(d, q) -
epi u e = {(t,x,y,z): z
g: (j, d,g) E
~
ue(t,x,y)},
hypo u e = {(t,x,y,z): z 2: ue(t,x,y)}, E Me(t,x,y,s)} = He(t,x,y,p,q)
gr u e
The symbols 11 . 11, and (-,.) denote the Euclidean norms and inner products respectively.
A pair (S·, Me) ((S., Me)) is called an upper (lower) characteristic complex if conditions 1 0 a) and 2° a) ( 1° a) and 2° b) ) are satisfied. The set of upper (lower) characteristic complexes (S·, Me) ((S., Me)) will be denoted by Ct(He) ( CJ.(He)).
It should be noted that the definitions of minimax and upper (lower) minimax solutions do not depend on which of complexes (S, Me) E C(He) and (S·, Me) E ct(He) ((S., Me) E cJ.(He)) are utilized.
S· = Q x B 3 s· = (v·, (3.), =
k (t, x, y, P, v·)
co {r(t,x,y,P,v·),
+ he(t, x, P, v·, A, (3.),
_ge(t,x,y,P,v·)};
S.
=P
k e (t,
2.2 Nonsmooth analysis tools
(10)
Let us choose among wide variety of nonsmooth analysis tools those ones are applicable in the constructions below. These are the following notions of proximal calculus (see Clarke et aI, 1998).
x A 3 s. = (u.,a.),
Me(t,x,y,s.)
= co {r(t,x,y,u.,Q),
Let S be a nonempty closed set in a finite dimensional space. Suppose that x is a point not lying in S. Suppose further that there exist a point s in S whose distance to x is minimal. Then s is called a closest point or a projection of x onto S.
x, y, u., Q) + he (t, x, u., Q, a., B), _ge(t,x,y,u.,Q)};
(11)
The symbol co D denotes the closed convex hull of the set D.
The vector x - s determines what is called a proximal normal direction to S at s. Any nonnegative multiply n = >.(x - s), >. 2: 0, of such a vector is called a proximal normal (or a P-normal) to S at s. The set of all n obtainable in this manner is termed the proximal normal cone to S at s, and is denoted by Nf(s). Note that for any point s belonging to the interior of the set S the proximal normal cone is a singleton. It consists of the unique null element.
For any (S·, Me) E Ct(He) ((S., Me) E CJ.(He)) and s = s· E S· (s = s. E S.) the symbol Sol (to, xo, Yo, zo, s) will denote the set of absolutely continuous functions (x('), y(.), z(·)) : [0,0] H R" X R k X R, that satisfy the condition (x(to), y(to), z(to)) = (xo, Yo, zo) and the upper (lower) characteristic differential inclusion
(x(t), cy(t), i(t)) = Me(t, x(t), y(t), s)
= ue(t, x, y)}.
It is known that for the minimax solution the following invariance condition holds. • For any (to, Xo, yo, zo) E gr u e , sE S, (S, Me) E C(He) a trajectory (x(·),y(·), z(.)) E Sol (to,xo, Yo, zo, s) exists such that (r, x(r), y(r), z(r)) E gr u e for all r E [to,O].
It is easy to check that the following complexes can be considered as upper and lower characteristic ones for the Isaacs equation (4)
e
z
Definition II. A continuous function [0,0] x R" X R k 3 (t, x) H ue(t, x, y) E R is called a minimax solution of the singularly perturbed HamiltonJacobi equation (4) iff it is an upper minimax solution and a lower minimax solution simultaneously.
The set of all characteristic complexes (S, Me) will be denoted by the symbol C(He).
Me(t,x,y,s·)
= {(t, x, y, z):
(12)
Definition I. A lower (upper) semicontinuous function [0,0] x R" X R k 3 (t, x) H ue(t, x, y) E R
The other important thing is the notion of strong invariance. Let S be a nonempty closed set in
45
kE:(t,x,y,u,v)+ hE:(t,x,u,v,a,/3)=O};
R x R!' and its section at the moment t be denoted by St. Let F be a multivalued mapping R x R!' to the subsets of R!'. Consider a differential inclusion
x(t) E F(t,x(t)),
t E [to, 9]
Y';p = Y':p(t,x,v·,/3·) =
U
(13)
Yi~ = Yi~(t, x, u., a.)
(17)
'Vy E Yi~(t, x, s.): Ilyll
(18)
r
E (0, J1-E:];
R!', s· E S·, s. E S. the following Lipschitz conditions are valid
~
KE:(lt' - t"l
< -
+ IIx' - x"I!),
(19)
dist (Yi~(t'x's.), Yi~(t",x",s.)) ~ ~
KE:(lt' - t"l
°
+ IIx' - x"II),
(20)
where KE: > are constant and the symbol dist (y l , y2) denotes the Hausdorff distance between the sets yl and y2; 7) the maps (t,x) H y';p(t,x,v·,/3·), (t,x) H Yi~(t, x, u., a.) are upper semicontinuous for any (v·,f3*) = s· E S· = Q x B, (u., a.) = s. E S. =
+ Ilx' - x"lI + IIY' - y"II);
Px A; 8) for any compacta D, IfJ, Do : D ::) [0,9] x Rn and
~
+ IIx' - x"ll + IIY' - y"II);
Rk
~
::) DO::)
U
U
YE:(to, Xo, s·),
E:E[O,I) (to.xo)ED.s·ES·
x"II); Rk::)Do ::)
" y" ,u, v ) I ~ l9 E: (t' ,x",y ,u, v ) - 9 E: (t" ,x, ~ Le(lt' - t"l + IIx' - x"lI + IIY' - y"II);
U
U
yE:(to,xo,s.)
E:E[O,lj (to.:Z:o)ED,s.ES. there exist numbers ",E: inequalities are valid
k
4) for any (t, x, y) E [O,OJ x R!' X R , (P, q) E Rn X R k the Isaacs condition (6) is valid. It is known (Krasovskii and Subbotin, 1974, 1985; et al.), that conditions 1)-4) guarantee existence of the value functions wE:(t, x, y) in the games GE: for any e E (0,9]. To provide a convergence pro{r erty for the value functions wE: (t, x, y) as e tends to zero the following constructions and assumptions are added. Y;~
r
+ Ilxll)'
dist (yE: (t'x's·) ' u yE:p(t" x" s·)) up "
11 hE: (t', x', u, v, a, /3) - hE: (t", x", U, v, a, /3) 11
Let sets YE:, Y';p,
are constant,
~ XE:(l
6) for any (t', x') E [0,9] x R!', (t", x") E [0,9] x
+ Ilxll + Ilyll);
+ Ilx' -
(16)
'VyEY':p(t,x,s·): lIyll~xe(l+llxl!),
where
1If"(t', x', y', u, v) - f" (t", x", y", u, v) 11 ~
LE: /2(lt' - t"l
yE:(t, x, u., v, a., /3);
5) for any (t,x) E [0,9] x R!', s· E S·, s. E S. the sets ye, y';p(t,x,s·), Yi~(t,x,s.) are nonempty and restricted
Note that constants J1-E: are the same as in condition lOa); 3) for any compactum C E [0,9] x R!' X R k and any u E P, v E Q, a E A, /3 E B there exist constants LE: > 0, NE: E (0, LE: /2) such that
~
=
Assume that
IIkE:(t, x, y, u,v)1I ~ ve(l + Ilxll + Ilyll); IIh e(t, x, u, v, a, /3)11 ~ J1-E: /2(1 + IIxll); IgE:(t,x,y,u,v)1 ~ J1-E:(1 + IIxll + lIyll);
~ NE:(lt' - t"l
(15)
'VEQ.t3EB
It is assumed for data of singularly perturbed games Ge (1)-(3) to satisfy the following conditions 1) functions J€(t,x,y,u,v), gE:(t,x,y,u,v), ~(x), kE:(t, X, y, u, v), hE:(t, x, u, v, a, /3), are continuous relative to all variables and parameter e when t E [0,9], x E R!', yE R k , u E P, v E Q, a E A, /3 E B, e E [0, I]; 2) there exist constants J1-E: > 0, vE: E (0, J1-E: /2) such that
IIkE:(t',x',y',u,v) - kE:(t",x",y",u,v)11
U
=
2.3 Basic assumptions and main result
~ L"(lt' - t"l
yE:(t,x,u,v·,a,/3·);
uEP,aEA
Definition Ill. The pair (S, F) is said to be strongly invariant if every trajectory x(·) of (13) on [to, 0], to E [0,0] for which x(to) E Sto is such that x(t) E St for all t E [to,9].
1IJ€(t,x,y,u,v)11 ~ J1-E:(1
(14)
~ -",E:dist
>
°such that the following
2(y., Y':p(t, x, s·))
(21)
f/: y';p(t, x, s·), yO E Y';p(t, x, s·), s· E S·, (t,x) E Dx exp(J1-E:(9-to)(1+XE:+2~)), ~y = max 1Iy' - y"lIj
for any y.
y'EDO ,y"EDO
analogously,
define via the relations
max(y. - Yo, kE: (t, x, y., u., v) - kE: (t, x, Yo, u., v))
ye = yE:(t,x,u,v,a,/3) = {y:
tlEQ
46
:5 -KEdist 2(y., y/~(t, x, s.»
where aO(x)
(22)
for any y. rt. Yi~(t,x,s.), Yo E Yi~(t,x,s.), s. E S., (t,x) E D x exp(J,LE(O - to)(1 + ~ + 2do,y» , do,y = y 'EDm~ED 1Iy' - y"II, o.y 0 ( the symbol dist (y, Y) denotes the distance be-
The main result of the paper is the following
Theorem 1. Let conditions 1) -11) in the singularly perturbed differential games GE (1) -(3) be satisfied. Then on any compacturn C C [O,OJ x ~ X R k the value function wE(t,x,y) converges uniformly to the value function w( t, x) of the following unperturbed game G
tween point y and set Y, i.e. dist (y, Y) = inf
y'EY
Ily -
y'lI
= lim~(x). E.j.O
);
9) for any yO E Y';p(t, x, s·), s· E S· and a P-
± E fl(t,x, yD(t,x,u,v,a,f3),u,v),
normal n·(yO) to the set Y';p(t,x,s·) at the point yO the equality holds
(t,x) E [0,0] x Rnj (u,a) E P x A,
+ hE(t, x, u, v·, a, f3.»)
'Yto,xo(x(·») E aO(x(O))
=
°
max
(J,g) E
co
min
s.=(u. ,a.)ES.
min[(J,s)-g:
(33)
The unperturbed game G (31)-(33) is called the asymptotics for the singularly perturbed differential games GE (1)-(3).
(25)
The value function w( t, x) is the unique minimax solution of the problem P (29),(30).
max[(J,s)-g:
{r(t,x,y/~(t,x,s.),u.,Q),
- gE (t, x, y/~(t, x, s.), u., Q)},
gO(r,x(r), yD(r,
The functions fl (.), gO (. ), aO (.) are obtained from the data of the singularly perturbed games GE if e: = 0. The sets yO(t, x, u, v, a, (3) are the limits of the sets YE(t, x, u, v, a, (3) in the Hausdorff metric as e: .j. 0.
(J, g) E co {r (t, x, Y';p(t, x, s·), P, v·), Hio(t,x,s) =
J
where (x(·): [t,O] H ~ is a trajectory of the differential inclusion (31) started at the point x(to) = Xo, to E [0,0] under controls (u(·),v(·),a(·),f3(·»: [to, 0] HP x Q x A x B.
(24)
- gE (t, x, Y';p(t, x, s·), P, v·)},
+
x(r), u(r), v(r), a(r), f3(r)), u(r), v(r))dr
Let us introduce the "upper" H;'p and "lower" Hio Hamiltonians via the formulae s'=(v' .f3·)ES·
(32)
to
10) constants J,LE, v E, ~, K E, LE, NE, KE depend continuously on parameter e: E [0,1J;
H~p(t,x,s)=
(v,f3) E Q x Bj (J
(23)
= 0;
analogously, for any Yo E Yi~(t, x, s.), s. E S. and a P-normal n.(yo) to the set Yi~(t,x,s.) at the point Yo the equality holds
+ hE(t,x,u.,v,a.,f3))
(31)
3. SUFFICIENT CONDITIONS FOR CONVERGENCY OF THE VALUE FUNCTIONS
(26)
11) Assume that IH~p(t,x,s) -
where 8(e:) .j.
Hio(t,x,s)l:5 8(e:),
(27)
The Theorem I provides sufficient conditions for convergency of the value functions to the singularly perturbed differential games. Its proof is based on the results of the papers (Subbotina, 1996, 1998) where convergence of the minimax solutions to the singularly perturbed HamiltonJacobi equations (4) was studed.
(28)
The Hamiltonians for the Isaacs equations arising in the differential games theory have the special form (6). Also the specific form of characteristic complexes (10) and (11) will be used below to construct sufficient conditions for convergence of the value functions of the differential games with fast and slow motions. These complexes were wide utilized in researches on u-stability and v-stability properties of the value functions in the differential
°as e: .j. 0.
Let us denote by HO(t, x, s) the limits
HO(t,x,s) = limH~p(t,x,s) = E.j.O lim Hfo(t, x, s)
E.j.O
which will be considered as the Hamiltonian in the unperturbed Cauchy problem P
t
E [0,0),
x ERn,
w(O,x) = aO(x),
x ERn,
(30)
47
games theory (see Krasovskii and Subbotin, 1974, 1985). 3.1 Properties of the sets Y';p,
and define
Yo
YoII
lIyo -
Yi~
= dist (Yo, Y:p(to, Xo, s*) =
+he (t,x e(t),P,v·,A,f3·)}, yO(to)
(34)
as c: t in the Hausdorff metric. The sets Y,?p(t, x, s*), Yi~(t, x, s.), yO(t, x, 1.£, v, a, (3) are compacta and the maps H
(t,x) (t,x)
Y.?p(t,x,v·, 13*),
H
Yi~(t,x,u.,a.)
c:ye (t) E
ye (to)
Conditions 9) imply (see Subbotin, 1995; Clarke et aI., 1997) that for any continuous bounded function x(·): [O,e] H nn, s* E S*, s. E S., c: > the sets Y';p(t, x(t), s·) are strongly invariant with respect to the differential inclusions
dist (ye (t), Y:p(t, xe(t), s·» ~ Ilye (t) - yO (t) 11; dllif(t) - yO(t)11 dt
y
- k e (t, xe(t), yO(t), 1.£, v·»
Using these relations and the condition 8) one can obtain that
C:Yo (t) E c-o {k e (t, x(t), yo(t), 1.£., Q) + he(t,x(t),u*,Q,a.,B)},
= 2( e(t)_ O(t) dif(t) _ dyO(t»
uEP
(35)
and sets Yi~(t, x(t), s.) are strongly invariant with respect to the differential inclusions
+
2
y 'dt dt e ~ 2/C: max(ye (t) - yO (t), k (t, xe(t), ye (t), 1.£, v*)-
+
+ h e(t,x(t),P,v*,A,f3*)},
= Yo
Hence the following estimates are valid
°
co {ke(t, x(t), yO(t), P, v*)
(38)
And let the terms he undepended relative to y in the right-hand sides of the inclusions (37) and (38) coincide.
= P x A;
efl(t) E
co {k e(t, XC (t), ye (t), P, v·)+
+he(t,xe(t),P,v·,A,p·)},
are uniformly Lipschitz continuous for any (v·,f3·) = s· E S· = Qx B, (u.,a.) = s. E
S.
= iio
Remember that the dynamics of if (t) describes in the follows way
yO(t,x,u.,v*,a.,{3*), H
(37)
The strong invariance property of the sets Y:p(t,xe(t),s·) with respect to the inclusion (37) implies that yO(t) E Y';p(t,xe(t), s·) for all t E [to, 0].
Y:p(t,x,v·,f3·) ~ r',?p(t, x, v*, 13·)
(t,x)
°
c:yo (t) E co {k e(t, XC (t), yO (t), P, v·)+
ye(t,x,u,v,a,f3) ~ yO(t,x,u,v,a,f3)
°
do>
Consider a solution of the differential inclusion
It is easy to see from the conditions 1), 5), 7), 10) that for any (t,x) E [0,0] x nn, s· = (v·,f3·) E S· = Q x B, s. = (1.£., a.) E S. = P x A the sets ye, Y';p, Yi~ are compacta and
Yi~(t,x,u.,a.) ~ ~(t,x,u.,a.)
E Y:p(to,x,s·)
(36)
-J t
• It means that all trajectories of the differential inclusions started at the points on the corresponding sets stay in these sets all the time up to 0.
2;e dist (ye (r), Y:p(r,xe(r),s·»dr;
to
In order to be specific below important properties will be obtained for the upper sets Y';p and the upper characteristic inclusions (10). All the conclusions are valid for the lower sets Yi~ and the lower characteristic inclusions too.
dist (ye(t), Y';p(t,xe(t), s·»
~ doe- ".< (t-to) ~ do
~
(39)
So the fast components if(t) of the upper characteristics (38) go fast to the corresponding sets Y';p (t, XC (t), s*). Analogous property holds for the lower characteristics and the corresponding lower sets Yi~.
Let xe(·),if(·),ze(.) : [to,e] H nn x R k x R,(xe(to),if(to),ze(to» = (XO,YO,Zo) be a solution of the upper characteristic inclusion (10) corresponding to a parameter s· = (v·, f3*) E S· , i.e. (xe(-),if(·),ze(.» E Sol (to,xO,YO, Zo,s·). Let us estimate dist 2 (if (t), Y';p(t,xe(t),s·).
Hence, the sets Y';p, Yi~ play roles of attractors for the considered characteristic inclusions. The conditions 8) and 9) imply that any doneighbourhood (do > 0) of the sets is strongly invariant relative to the corresponding characteristic inclusions.
Let us choose
48
Let Xo ER", Zo E R. inclusions
At last let us show that the following relation takes place for any (t, x) E [0,0] x R", s· E S·, s. E S.
(Xi(t),~(t»
(40)
Consider the differential
E Fi(t,xi(t), Zi(t»
(Xi(tO), Zi(tO» = (xo, Zo), It is known (Subbotin, 1991, 1995) that the analogous conditions are valid for the right-hand side sets in the characteristic inclusions, namely
= M€(t,x,Y):f:
for all (t, x, Y, s*, s., e) E [0,0] x R" S. x (0,1].
(41) X
Rk
X
S·
1.
For
X
II x I(t) (42)
(x€ (to), y€ (to), z€(to» = (xo, Yo, Zo)
(43)
J
(48)
J t
II zl(t) - Z2(t) 11
From the above properties of the sets Y';p' Yi~ it follows that for any Tf > 0 there exist e(Tf) > 0, 6(e) > 0, (6(c» .!. 0 as c .!. 0) such that for anye :::; e(Tf), r 2: to + 6(e) the inclusion holds
n {Y:p(r, x€(r), s.) + TfB}
dist (FI(r,
x2(t)11 :::;
to
It is easy to see (Filippov, 1985) that there exists a solution of the inclusion.
{(Y:p(r,x€(r),s·)+TfB}
solution
t
(x€(t),q/(t),z€(t» = M€(t,x(t),y(t»
E
any
E Sol I (to, Xo, Zo) there exists a solution (X2(·),Z2(·» E So12(to,xo,Zo) such that the following estimates take place for all t E [to,O] (Xl (.), Zl (.»
Consider the differential inclusion
y€(r)
(47)
i = 1,2.
The set of all solutions (XiO, Zi(·» of the differential inclusion (46), (47) with number "i" will be denote as Sol i(tO,XO, Zo). The following proposition is valid (see Filippov, 1985).
Proposition
0;
t E [to, 0], (46)
:::;
dist (FI(r,
to
Let (to, xo, Yo) E D x Do, Zo E RI, e E (0,1] and Y';p(t, x, s·), s· E S·, (S,M€) be defined by the relations (15), (10). Fix a solution (x€(·),if(·),z€(·» E Sol (to,xO,YO, Zo,s·), s· E S·, Le.
n (44)
where the symbol B denotes the unit ball in R k •
(x€ (t), ql (t), z€ (t» E
Choosing Tfi .!. 0, ei = e(Tfi) .!. 0, 6i = 6(ei) .!. 0 one can obtain in the limit the last required property
M~p(t,
x€ (t), y€ (t), s·)
(x€(to), y€(to), z€(to» = (xo, Yo, Zo) For the function if (.) : [to, B] H R k let us construct the multivalued mapping
(t,x) 3.2. Proof of the main result.
H
Yo"(t,x,s·)
Yo"(t,x,s·)
To prove the Theorem I. it will be shown that the conditions 8),9),11) are analogous to the conditions 3 and 4 in the paper (Subbotina, 1996) and guarantee convergence w€(t,x,y) to w(t,x).
=
{YO
E
C
Y';p(t,x,s·)
Y';p(t,x,s·):
dist (y€(t), Y:p(t,x,s·» = lIy€(t)-Yoll}:f:
0 (50)
One can easy check that for any s· the mapping (t,x) H Y
For the sake of definity all the constructions below will be done for the upper characteristic complexes and their attractors. The similar results for the lower characteristics and attractors one can easy obtain in the same way.
- g€(t,x,Yo"(t,x,s·),s·)} The following fact of the theory of differential inclusions will be useful for the forthcoming estimates.
(51)
Consider the following differential inclusion corresponding to the mapping (51)
Let (t, x, z) H Fi(t,x, z) C R" x R: [to, 0] x R" x RH 2 R "xR, i = 1,2 - to be two multivalued mappings which have convex, compact values. For the mappings are assumed to be upper semicontinuous.
(x6(t),z6(t» E
co {r(t,x6(t), Yo"(t,x6(t),s·),s·)
- g€ (t, x6(t), 10€ (t, x6(t), s·), s·)} x6(to) = Xo,
49
z6(to) = Zo
(52)
(53)
Taking into account the assumed Lipschitz conditions and properties of the operation "dist " one can continue the inequalities (60).
According to the theory of differential inclusions (see Filippov, 1985) a solution of (52), (53) exists for t E [to, 0]0 Denote the set of all solutions (Xb(o), za(o» of (52), (53) by the symbol Sol g(to, Xo, zo, s")o The symbol Sol E(to, Xo, zo, s.. ) will be denoted the set of all solutions (x E(-), ZE(-) of the differential inclusion
IIxE(t)-x~(t)1I
xE(to)
= Xo,
ZE(tO)
= ZO
JLE{lIxE(r)-x~(r)II+llyE(r)to
(xE(t), ZE(t» E co {r(t,xE(t), Y:l'(t, xE(t), s"), s"), - gE (t, xE(t), Y:l'(t, xE(t), s'"), s.. )}
t
::;
-y~(r)ll}dr
(54)
J£E{llxE(r)-x~(r)lI+dist t
::;
(yE(r),
to
(55)
The inclusion is obvious t
Sol g(to,xo, zo, s.. ) C Sol E(to,xO' ZO, s"o)
Y:l'(r, x~(r), s")}dr ::;
(56)
E
£E{(1 + v E)lI xE (r)-
to
For the chosen trajectory (xE(o),!/(o),ZE(o» E Sol (to, Xo, Yo, ZO, s.. ) let us estimate the distance between (xE(t), ZE(t» and (xE(t), ZE(t)) which is a point of the trajectory (x E(-), ZE(o» E Sol E(to,xO,zo,s"), closest to (XE(-),ZE(o)). Here
t
J
- xg(r)11
+ 2dist (yE (r), Y:l'(r, xE(r), s"»)}dr (59)
Using the exponential estimates (39) for the distance between the fast components of the characteristics and the corresponding attractors and also the Gronwall inequality (see Warga, 1976) the relations (59) are completing
[to,O].
It is follows from the definitions that the men-
tioned distance is not more than the distance between (x E(t), ZE(t» and (xQ(t), zQ(t») which is a point on the trajectory (xb(·) , za(o) E Sol g(to, Xo, ZO, s) closest to (x E(0), ZE ('»)0
J 8
IlxE(t) - xg(t)11 ::;
L E(l
+ vE)llxE(r)_
to
Using the Proposition 1. one can obtain the following relations
t
J
x~(r)lldr
IIx E(r) -
IIx E(t) -
::;
x~(t)11
to
:; J
+ 2LE(1
to
x [(t - to)
t
:; J
maxllr(r,xE(r), yE(r),u,v")uEP
to
c
/\,E
+
to)} x
(1- exp{ - :E (t - to)})]
= p(c)
Let us denote by GE(to, r, Xo, Yo, ZO, s") - the projection of the attainability set for the system (12) at the moment r to the subspace of variables x, z and by G~(E)(to, r, Xo, ZO, s·) - a closed p(c) -neighbourhood of the attainability set for the system (54),(55)
(57)
t
IlzE(t) - z6(t)11 ::;
/\,E
(60) One can see that p(c) ..j.. 0 as c ..j.. O. Analogous estimate can be obtain for IIz E(t) - z811.
t
- r (r, x~(r), Y6(r), u, v")lIdr IIz E(t) - ZE (t) 11 ::;
2cdo (1 _exp{ _ /\,Ec (t - to)})
+ v E)-exp{£E(l + vE)(O -
dist (co r(r,xE(r),yE(r),P,v"), co r(r,
xMr), Ycf(r,xMr), P, v", A, ,8"), P,v"»)dr
::;
J
IW(r) - z6(r)lldr <
to
Estimates obtained above for upper (and analogous estimates for lower) characteristic complexes imply that' the following proposition is truth. Proposition 2 For any compacta D, Do from the condition 8) there exist the mappings (0,1] H R+ x R+, c Ho (a(c),p(c), such that a(c) .j.. 0, p(c)..j.. 0 as c.j.. 0, and for any (to, xo) E D, Yo E Do, ZO E R, s' =. s" E S", (s' = s. E S.), c E (0,1], (xE(.),!/(o),ZE(.)) E Sol (to,xO,YO,ZO,s')
:; J t
max IIg E(r,x E(r), yE(r), u, v")uEP
to
- gE(r, x~(r), Y6(r), u, v"»)lIdr
(58)
where Yb(o) : [to, r] Ho Do: t Ho Yb(t) E measurable function satisfying in accordance with (50)
Yo" (t, xb( t), s.. ) is a
'i0
Y,?p( t, x, {3) = Y (t, x,.8)
there exist such (x"('), z,{)) E Sol "(to,xQ, zo, s'), that for r E [to + 8(E:) , 0], the following relations are valid ~
Ilx"(r) - xe(r) 11
p(E:), Ilz"(r) - z.,(r)1I
~
where the symbol Y(t,x,.8)" denotes the closed E: -neighbourhood of the set Y(t, x, .8).
p(E:)
To obtain the lower characteristic complexes and corresponding arrtactors one can change places {3 and a, Band A.
(61) The proposition imply that the situation is close to that which was studed in papers (Subbotina, 1996, 1998). Modifying proofs of these papers the main result of this paper can be successfully proved.
The asymptotic game has the form
=
x(t)
f(t, x(t), Y(t, x(t), a, {3)), a E A, {3 E Bj
3.3 Example
+
k(y) =
if Y
~
0,
if
~
0
'
(75)
where
x, a, {3)
(76)
(64) y
4. CONCLUSION
.8 E B,
(65)
In this paper, differential games with fast and slow motions are considered. There are obtained sufficient conditions for the value functions in singularly perturbed differential games to converge to the value function in the asymptotics game. It was shown that the convergence property is based on the existence of attractors in the subspace of fast variables.
sets A and B are compacta. Let the Isaacs condition holds. The pay-off functional has the form
'Yto,XO'yo(x(')'y(,)) = u(x(O))+ (J
J
+
g(r,x(r), Y(r,x(r),a,{3))dr
Y(t, x, a, {3) =
{ -2y,
a E A,
J to
(63)
where k(y) has the form
-y
(73)
(J
(62)
+ ~(t, x, a,.8)
E:y(t) = k(y)
(72)
'Yto,xo(x(,)) = u(x(O))+
Consider the following singularly perturbed differential game
x(t) = f(t,x(t),y(t)),
(71)
g(r,x(r),y(r))dr
(66)
to
5. ACKNOWLEDGEMENTS
Here the upper characteristic complexes and the corresponding attractors satisfying the assumptions 1)-11) are the following
M~p(t,x,y,(3) =
co {(J(t,x,y), ~ E
g(t,x,y)):
co
1
€"(~
The research was supported by the Russian Fund of Fundamental Researches under Grants N96-01-00219, N96-15-96245 and N97-01-00371.
- y),
~(t,x,A,.8)};
REFERENCES
(67)
Bagagiolo, F., M. Bardi and I. Capuzzo--Dolcetta (1997). A Viscosity Solutions Approach to Some Asymptotic Problems in Optimal Control. In: Partial differential equation methods in' control and shape analysis (Da Prato, G. and J. P. Zolesio. (Ed)), 29-39. Dekker, Inc., New York. Bardi, M. and I. Capuzz
M~p(t,x,(3) = co {(J(t,x,~), g(t,x,~)): ~
E Y(t, x, (3)}
(68)
where
Y(t,x,{3) =
U ~(t,x,a,{3)
(69)
"'EA
and the function
I,
=
{
timal Control and Viscosity Solutions of Hamilton-lacobi-Bellman Equations, Birk-
is defined as
if
~ ~
0,
hauser, Boston. Barron, E. N., L. C. Evans and R. Jensen (1984). Viscosity solutions of Isaacs' equations and differential games with Lipschitz controls. 1. Different. Equat., 53(2), 213-233. Bensoussan, A. (1988). Perturbation Methods in Optimal Control Problems, Wiley-Gautier, New York.
(70) 1/2,
if
~ ~
0
The upper attractors Y';p and Y,?p are obtained by the formulae
51
Clarke, F. H., Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski (1998). Nonsmooth Analysis and Control Theory, Springer-Verlag, New York. Crandall M. G.and P. L. Lions (1983). Viscosity solutions of Hamilton-Jacobi equations. Trans. A mer. Math. Soc., 277, 1-42. Crandall M. G., L. C. Evans and P. L. Lions (1984). Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. A mer. Math. Soc., 282, 487-502. Filippov, A. F. (1985). Differential Equations with Discontinuous Right-hand Side, Nauka, Moscow [in Russian]. Gaitsgory, V. G. (1996). Limit Hamilton-Jac~ bi-Isaacs Equations for Singularly Perturbed Zero-Sum Differential Games. J. Math. Anal. Appl., 202, 862-899. Isaacs, R. (1965). Differential Games. Wiley, New York. Krasovskii, N.N. and A.I. Subbotin (1974). Positional Differential Games. Nauka, Moscow [in Russian]. Krasovskii, N.N. and A.I. Subbotin (1988). Game- Theoretical Control Problems. Springer-Verlag, New York. Kokotovic, P. V. (1984). Applications of singular perturbations techniques to control pro~ lems. SIAM Rev., 26, 501-550. O'Malley, R. E. (1974). Introduction to Singular Perturbations, Academic Press, New York. Pervozvansky, A. A. and V. G. Gaitsgory (1988). Theory of Suboptimal Solutions, Kluwer Academic, Dordrecht. Subbotin, A. I. (1980). Generalization of the main equation of the theory of differential games. Soviet. Math. Dokl., 22(2), 358-362. Subbotin, A. I. (1991) Minimax inequalities and Hamilton-Jacobi equations, Nauka, Moscow [in Russian]. Subbotin, A. I. (1995) Generalized Solutions of First-Order PDEs. The Dynamical Optimization perspective, Birkhauser, Boston. Subbotina, N. N. (1996). Asymptotic properties of minimax solutions of Isaacs-Bellman equations in differential games with fast and slow motions. J. Appl. Maths Mechs, 60(6), 883-890. Subbotina, N. N. (1998). Asymptotics for singularly perturbed Hamilton-Jacobi equations. Prikl. Mat. Mech., [in Russian], (to appear). Tikhonov, A. N. (1952). Systems of differential equations containing small parameters near derivatives. Mat. Sbornik, 31(3), 575-586 [in Russian]. Vasil'eva, A. B. and A. F. Butuzov (1973). Asymptotic Expansions of Solutions to Singularly Perturbed Equations, Nauka, Moscow [in Russian]. Warga, J. (1976). Optimal Control of Differential and Functional Equations, Academic Press, New York.
52
ASYMPTOTICS FOR SINGULARLY PERTURBED DIFFERENTIAL GAMES
N.N.Subbotina Institute of Mathematics and Mechanics, Ural Branch Russ. A cad. Sci., S.Kovalevskoi str.,16, 620219 Ekaterinburg, Russia e-mail: [email protected]
Abstract: Singularly perturbed differential games with "fast" and "slow" motions and the Bolza type payoff functionals are considered. Sufficient conditions are obtained for the value functions of the games to converge to the value function of the asymptotic unperturbed game as a parameter of singularity £ tends to o. Copyright @1998 IFAC Keywords: Attractors, characteristic equations, invariance, minimax techniques, non-linear systems, perturbation analysis
1. INTRODUCTION
The considered games have the fixed end time and a pay-off functional of the Bolza type
Mathematical models of dynamical systems with "fast" and "slow" motions occur quite frequently in various problems of economics, engineering, mechanics, biology and other applications.
(J
+
The paper deals with singularly perturbed differential games Gc whose dynamics are described in the following way
9c (r,x(r), y(r), u(r),v(r»dr
(3)
t
where (x(·), y(.»: [t,O] H R!' X Rk are the trajectories of the equations (1) started at the point (x(t),y(t» = (x,y), t E [0,0] under controls
:i: = !,,(t,x,y,u,v),
£iJ = kc(t, x, y, u, v) + hc(t, x, u, v, a, (3)
J
°
(u(·),v(·),a(·),{3(·»: [t,O]
(1)
H
P x Q x A x B.
The first player has the aim to minimized the payoff functional and the second player tries to maximized 'y".
where £ > 0 is a small parameter, time t E [0,0] (x, y) E R!' X R k is the phase vector, x is the "slow" variable, y is the "fast" one. These" attributes" underline a difference between dynamics of variables x and y. The velocity iJ is of order 1/£. Hence y can rapidly vary with respect to time.
The paper deals with conditions under which the value functions WE (t, x, y) of the singularly perturbed games Gc converge to the value function w(t, x) of an unperturbed game G, as £ -!. o. The unperturbed game G is the asymptotics for Gc.
It is assumed that controls of the first player (u, a) and controls of the second player (v, (3) are restricted with the constraints
Researches of singularly perturbed control problems are wide represented in literature, see, for example, (Kokotovic, 1984, Bensoussan, 1989 ) and references within. The classical approach of obtaining asymptotics (Tikhonov, 1952; Vasil'eva and Butuzov, 1973; O'Malley, 1974) contains
(2) where P, Q, A, B are compacta.
43
+ gE:(t,x,y,u,v) + Ije(kE:(t,x,y,u,v),q) + + l/e(hE:(t,x,u,v,a,,B),q)] = max min [(r(t,x,y,u,v),p) + vEQ.f3EB uEP.aEA
• equating E to 0 in relations (1)-(3); • solving the asymptotic algebraic equation in formulae (1) relative to y and • substituting the expression for y into the asymptotic equation in relations (1) and the asymptotic pay-off functional (3).
+ gE:(t,x,y,u,v) + l/e(kE:(t,x,y,u,v),q) +
The other well known approach is a decomposition of the problem to • an optimization problem for " fast" variables under fixed "slow" ones and • an asymptotic control problem for "slow" variables taking into account results of previous calculations having been undepended on "fast" variables. See to get acquainted with (Pervozvansky and Gaitsgory, 1988) and references within.
+ l/E(hE:(t,x,u,v,a,,B),q)]
The equation (4) is called the Isaacs equation (Isaacs R" 1965) for the singularly perturbed differential game GE: (1)-(3). It is singularly perturbed because it contains terms with coefficients l/E where E is a small parameter. These terms are positive homogeneous with respect to impulse "fast" variables q = DywE:. Definitions of minimax solutions to the first-order PDE will be essentially used in the paper to obtain sufficient conditions of convergence wE:(t,x,y) as E .j.. O. Hence, let us consider this concept. It should be mentioned that the notion of a minimax solution to the first-order PDE may be defined in a variety of ways; among other things, one can use the tools of nonsmooth analysis: directional derivatives, tangent or contingent cones, su~ and super-differentials (the definitions and proof of equivalence of all the definitions can be found, for example, in (Subbotin, 1995)).
In the paper an different approach based on existence of attractors in the subspace of" fast" variables is presented. Here main results are o~ tained due to constructions of the minimax s0lutions to the first-order PDEs, namely, to the Isaacs-Bellman equations (Subbotin 1980, 1991, 1995). The approach is close to two mentioned above ones considering convergence problems in the whole of " fast" variable subspace.
The present paper deals with the definition that can be viewed as a generalization of the classical method of characteristics. In this definition the characteristic system is replaced by a family of "characteristic" differential inclusions. The graph of the minimax solution is weakly invariant with respect to these differential inclusions. The role of the characteristic inclusions to the Cauchy pro~ lem pE: (4)-(5) can play inclusions which define u-stability and v-stability properties of the value function. Let us remain the definition.
2. PRELIMINARIES
2.1 Value function as minimax solution of the Isaacs equation
Let 8 be a nonempty set and ME: a multivalued mapping
It is known that the value function wE: (t, x, y) of the game GE: is the minimax (Subbotin, 1980, 1991, 1995) or/and viscosity (Crandall and Lions, 1983; Crandall et al., 1984) generalized solution
[0,8] x Rn H
to the following Cauchy problem pE:
+ HE:(t, x, y, DxwE:, DywE:)
= 0,
(4)
tE[O,B), XER n , YER k ; wE:(B,x,y) = erE: (x), x ERn, yE R k ,
(5)
min
max [(r(t,x,y,u,v),p)
X
Rk
X
8 3 (t,x,y,s)
ME:(t,x,y,s) C Rn
X
Rk
X
H
R
(7)
The pair (8, ME:) is called a characteristic complex (or, briefly, a complex) to the singularly perturbed problem pE: (4)-(5) if the following conditions hold: lOa) for any (t,x,y) E [0,8] X ~ X R k and s E 8 the set ME:(t,x,y,s) = {j,d,g} C ~ X R k X R is nonempty, convex and closed. Components (I,d,g) E ME:(t,x,y,s) satisfy the inequalities
where vectors p = D",wE: , q = DywE: are gradients of wE: (t, x, y) relative to x and y, namely, D",wE: = (8wE: j8Xl, ..., 8wE: j8x n ), DywE: = (8wE:j8Yl, ... ,8wE:/Byk) and HE:(t,x,y,p,q) is the Hamiltonian uEP,aEA vEQ.f3EB
(6)
HE:(t,x, Y,P, q)
Two mentioned approaches consider convergence problems in the whole of "fast" variables su~ space. When a convergence problem in optimal control theory for singularly perturbed data and state constraints on "fast" variables is under consideration it makes sense to consider the convergence only within the constraints. In this case controllability conditions for" fast" variables play an essential role. References on this topic one can find in (Bagagiolo et al., 1997 j Bardi and Capuz~ Dolcetta, 1997).
8wE: /at
=
+
44
Ilfll
~ JLE:(l
+ IIxll + lIyll);
Ildll
~ JLE:(l
+ Ilxll + lIyll)j
is called an upper (lower) minimax solution of the singularly perturbed Hamilton-Jacobi equation (4) if the following invariance conditions hold: • for any (to,xO,YO,zo) Eepiue ((to,xO,YO,zo) E hypo u e ), s = s· E S· (s = s. E S.) a trajectory (x(,),y(·),z(·)) E Sol (to,XO,YO,ZO,s) exists such that (r,x(r), y(r), z(r)) E epi u e ((r, x(r), y(r), z(r)) E hypo u e ) for all r E [to,O].
°
where J-Le > are constant. The multivalued mappings (t,x,y) H Me(t,x,y,s) are upper semicontinuous for all s E S; 2°a) for any (t, x, y) E [0,0] x R" X Rk and (P, q) E
R"xRk
+ l/c(d,q) -
maxmin{(f,p) sES
g: (j,d,g) E
Here (S·, Me) E ct(He) ((S.,Ne) E CJ.(He)). The symbols epi u e , hypo u e and gr u e denote the epigraph, the hypograph and the graph of the function ue(t, x, y) respectively, Le. the sets
E Me(t,x,y,s)} = He(t,x,y,p,q);
2° b) for any (t, x, y) E [0,0] x R" X R k and (P, q) E
R"xRk minmax{ (f,p) sES
+ l/c(d, q) -
epi u e = {(t,x,y,z): z
g: (j, d,g) E
~
ue(t,x,y)},
hypo u e = {(t,x,y,z): z 2: ue(t,x,y)}, E Me(t,x,y,s)} = He(t,x,y,p,q)
gr u e
The symbols 11 . 11, and (-,.) denote the Euclidean norms and inner products respectively.
A pair (S·, Me) ((S., Me)) is called an upper (lower) characteristic complex if conditions 1 0 a) and 2° a) ( 1° a) and 2° b) ) are satisfied. The set of upper (lower) characteristic complexes (S·, Me) ((S., Me)) will be denoted by Ct(He) ( CJ.(He)).
It should be noted that the definitions of minimax and upper (lower) minimax solutions do not depend on which of complexes (S, Me) E C(He) and (S·, Me) E ct(He) ((S., Me) E cJ.(He)) are utilized.
S· = Q x B 3 s· = (v·, (3.), =
k (t, x, y, P, v·)
co {r(t,x,y,P,v·),
+ he(t, x, P, v·, A, (3.),
_ge(t,x,y,P,v·)};
S.
=P
k e (t,
2.2 Nonsmooth analysis tools
(10)
Let us choose among wide variety of nonsmooth analysis tools those ones are applicable in the constructions below. These are the following notions of proximal calculus (see Clarke et aI, 1998).
x A 3 s. = (u.,a.),
Me(t,x,y,s.)
= co {r(t,x,y,u.,Q),
Let S be a nonempty closed set in a finite dimensional space. Suppose that x is a point not lying in S. Suppose further that there exist a point s in S whose distance to x is minimal. Then s is called a closest point or a projection of x onto S.
x, y, u., Q) + he (t, x, u., Q, a., B), _ge(t,x,y,u.,Q)};
(11)
The symbol co D denotes the closed convex hull of the set D.
The vector x - s determines what is called a proximal normal direction to S at s. Any nonnegative multiply n = >.(x - s), >. 2: 0, of such a vector is called a proximal normal (or a P-normal) to S at s. The set of all n obtainable in this manner is termed the proximal normal cone to S at s, and is denoted by Nf(s). Note that for any point s belonging to the interior of the set S the proximal normal cone is a singleton. It consists of the unique null element.
For any (S·, Me) E Ct(He) ((S., Me) E CJ.(He)) and s = s· E S· (s = s. E S.) the symbol Sol (to, xo, Yo, zo, s) will denote the set of absolutely continuous functions (x('), y(.), z(·)) : [0,0] H R" X R k X R, that satisfy the condition (x(to), y(to), z(to)) = (xo, Yo, zo) and the upper (lower) characteristic differential inclusion
(x(t), cy(t), i(t)) = Me(t, x(t), y(t), s)
= ue(t, x, y)}.
It is known that for the minimax solution the following invariance condition holds. • For any (to, Xo, yo, zo) E gr u e , sE S, (S, Me) E C(He) a trajectory (x(·),y(·), z(.)) E Sol (to,xo, Yo, zo, s) exists such that (r, x(r), y(r), z(r)) E gr u e for all r E [to,O].
It is easy to check that the following complexes can be considered as upper and lower characteristic ones for the Isaacs equation (4)
e
z
Definition II. A continuous function [0,0] x R" X R k 3 (t, x) H ue(t, x, y) E R is called a minimax solution of the singularly perturbed HamiltonJacobi equation (4) iff it is an upper minimax solution and a lower minimax solution simultaneously.
The set of all characteristic complexes (S, Me) will be denoted by the symbol C(He).
Me(t,x,y,s·)
= {(t, x, y, z):
(12)
Definition I. A lower (upper) semicontinuous function [0,0] x R" X R k 3 (t, x) H ue(t, x, y) E R
The other important thing is the notion of strong invariance. Let S be a nonempty closed set in
45
kE:(t,x,y,u,v)+ hE:(t,x,u,v,a,/3)=O};
R x R!' and its section at the moment t be denoted by St. Let F be a multivalued mapping R x R!' to the subsets of R!'. Consider a differential inclusion
x(t) E F(t,x(t)),
t E [to, 9]
Y';p = Y':p(t,x,v·,/3·) =
U
(13)
Yi~ = Yi~(t, x, u., a.)
(17)
'Vy E Yi~(t, x, s.): Ilyll
(18)
r
E (0, J1-E:];
R!', s· E S·, s. E S. the following Lipschitz conditions are valid
~
KE:(lt' - t"l
< -
+ IIx' - x"I!),
(19)
dist (Yi~(t'x's.), Yi~(t",x",s.)) ~ ~
KE:(lt' - t"l
°
+ IIx' - x"II),
(20)
where KE: > are constant and the symbol dist (y l , y2) denotes the Hausdorff distance between the sets yl and y2; 7) the maps (t,x) H y';p(t,x,v·,/3·), (t,x) H Yi~(t, x, u., a.) are upper semicontinuous for any (v·,f3*) = s· E S· = Q x B, (u., a.) = s. E S. =
+ Ilx' - x"lI + IIY' - y"II);
Px A; 8) for any compacta D, IfJ, Do : D ::) [0,9] x Rn and
~
+ IIx' - x"ll + IIY' - y"II);
Rk
~
::) DO::)
U
U
YE:(to, Xo, s·),
E:E[O,I) (to.xo)ED.s·ES·
x"II); Rk::)Do ::)
" y" ,u, v ) I ~ l9 E: (t' ,x",y ,u, v ) - 9 E: (t" ,x, ~ Le(lt' - t"l + IIx' - x"lI + IIY' - y"II);
U
U
yE:(to,xo,s.)
E:E[O,lj (to.:Z:o)ED,s.ES. there exist numbers ",E: inequalities are valid
k
4) for any (t, x, y) E [O,OJ x R!' X R , (P, q) E Rn X R k the Isaacs condition (6) is valid. It is known (Krasovskii and Subbotin, 1974, 1985; et al.), that conditions 1)-4) guarantee existence of the value functions wE:(t, x, y) in the games GE: for any e E (0,9]. To provide a convergence pro{r erty for the value functions wE: (t, x, y) as e tends to zero the following constructions and assumptions are added. Y;~
r
+ Ilxll)'
dist (yE: (t'x's·) ' u yE:p(t" x" s·)) up "
11 hE: (t', x', u, v, a, /3) - hE: (t", x", U, v, a, /3) 11
Let sets YE:, Y';p,
are constant,
~ XE:(l
6) for any (t', x') E [0,9] x R!', (t", x") E [0,9] x
+ Ilxll + Ilyll);
+ Ilx' -
(16)
'VyEY':p(t,x,s·): lIyll~xe(l+llxl!),
where
1If"(t', x', y', u, v) - f" (t", x", y", u, v) 11 ~
LE: /2(lt' - t"l
yE:(t, x, u., v, a., /3);
5) for any (t,x) E [0,9] x R!', s· E S·, s. E S. the sets ye, y';p(t,x,s·), Yi~(t,x,s.) are nonempty and restricted
Note that constants J1-E: are the same as in condition lOa); 3) for any compactum C E [0,9] x R!' X R k and any u E P, v E Q, a E A, /3 E B there exist constants LE: > 0, NE: E (0, LE: /2) such that
~
=
Assume that
IIkE:(t, x, y, u,v)1I ~ ve(l + Ilxll + Ilyll); IIh e(t, x, u, v, a, /3)11 ~ J1-E: /2(1 + IIxll); IgE:(t,x,y,u,v)1 ~ J1-E:(1 + IIxll + lIyll);
~ NE:(lt' - t"l
(15)
'VEQ.t3EB
It is assumed for data of singularly perturbed games Ge (1)-(3) to satisfy the following conditions 1) functions J€(t,x,y,u,v), gE:(t,x,y,u,v), ~(x), kE:(t, X, y, u, v), hE:(t, x, u, v, a, /3), are continuous relative to all variables and parameter e when t E [0,9], x E R!', yE R k , u E P, v E Q, a E A, /3 E B, e E [0, I]; 2) there exist constants J1-E: > 0, vE: E (0, J1-E: /2) such that
IIkE:(t',x',y',u,v) - kE:(t",x",y",u,v)11
U
=
2.3 Basic assumptions and main result
~ L"(lt' - t"l
yE:(t,x,u,v·,a,/3·);
uEP,aEA
Definition Ill. The pair (S, F) is said to be strongly invariant if every trajectory x(·) of (13) on [to, 0], to E [0,0] for which x(to) E Sto is such that x(t) E St for all t E [to,9].
1IJ€(t,x,y,u,v)11 ~ J1-E:(1
(14)
~ -",E:dist
>
°such that the following
2(y., Y':p(t, x, s·))
(21)
f/: y';p(t, x, s·), yO E Y';p(t, x, s·), s· E S·, (t,x) E Dx exp(J1-E:(9-to)(1+XE:+2~)), ~y = max 1Iy' - y"lIj
for any y.
y'EDO ,y"EDO
analogously,
define via the relations
max(y. - Yo, kE: (t, x, y., u., v) - kE: (t, x, Yo, u., v))
ye = yE:(t,x,u,v,a,/3) = {y:
tlEQ
46
:5 -KEdist 2(y., y/~(t, x, s.»
where aO(x)
(22)
for any y. rt. Yi~(t,x,s.), Yo E Yi~(t,x,s.), s. E S., (t,x) E D x exp(J,LE(O - to)(1 + ~ + 2do,y» , do,y = y 'EDm~ED 1Iy' - y"II, o.y 0 ( the symbol dist (y, Y) denotes the distance be-
The main result of the paper is the following
Theorem 1. Let conditions 1) -11) in the singularly perturbed differential games GE (1) -(3) be satisfied. Then on any compacturn C C [O,OJ x ~ X R k the value function wE(t,x,y) converges uniformly to the value function w( t, x) of the following unperturbed game G
tween point y and set Y, i.e. dist (y, Y) = inf
y'EY
Ily -
y'lI
= lim~(x). E.j.O
);
9) for any yO E Y';p(t, x, s·), s· E S· and a P-
± E fl(t,x, yD(t,x,u,v,a,f3),u,v),
normal n·(yO) to the set Y';p(t,x,s·) at the point yO the equality holds
(t,x) E [0,0] x Rnj (u,a) E P x A,
+ hE(t, x, u, v·, a, f3.»)
'Yto,xo(x(·») E aO(x(O))
=
°
max
(J,g) E
co
min
s.=(u. ,a.)ES.
min[(J,s)-g:
(33)
The unperturbed game G (31)-(33) is called the asymptotics for the singularly perturbed differential games GE (1)-(3).
(25)
The value function w( t, x) is the unique minimax solution of the problem P (29),(30).
max[(J,s)-g:
{r(t,x,y/~(t,x,s.),u.,Q),
- gE (t, x, y/~(t, x, s.), u., Q)},
gO(r,x(r), yD(r,
The functions fl (.), gO (. ), aO (.) are obtained from the data of the singularly perturbed games GE if e: = 0. The sets yO(t, x, u, v, a, (3) are the limits of the sets YE(t, x, u, v, a, (3) in the Hausdorff metric as e: .j. 0.
(J, g) E co {r (t, x, Y';p(t, x, s·), P, v·), Hio(t,x,s) =
J
where (x(·): [t,O] H ~ is a trajectory of the differential inclusion (31) started at the point x(to) = Xo, to E [0,0] under controls (u(·),v(·),a(·),f3(·»: [to, 0] HP x Q x A x B.
(24)
- gE (t, x, Y';p(t, x, s·), P, v·)},
+
x(r), u(r), v(r), a(r), f3(r)), u(r), v(r))dr
Let us introduce the "upper" H;'p and "lower" Hio Hamiltonians via the formulae s'=(v' .f3·)ES·
(32)
to
10) constants J,LE, v E, ~, K E, LE, NE, KE depend continuously on parameter e: E [0,1J;
H~p(t,x,s)=
(v,f3) E Q x Bj (J
(23)
= 0;
analogously, for any Yo E Yi~(t, x, s.), s. E S. and a P-normal n.(yo) to the set Yi~(t,x,s.) at the point Yo the equality holds
+ hE(t,x,u.,v,a.,f3))
(31)
3. SUFFICIENT CONDITIONS FOR CONVERGENCY OF THE VALUE FUNCTIONS
(26)
11) Assume that IH~p(t,x,s) -
where 8(e:) .j.
Hio(t,x,s)l:5 8(e:),
(27)
The Theorem I provides sufficient conditions for convergency of the value functions to the singularly perturbed differential games. Its proof is based on the results of the papers (Subbotina, 1996, 1998) where convergence of the minimax solutions to the singularly perturbed HamiltonJacobi equations (4) was studed.
(28)
The Hamiltonians for the Isaacs equations arising in the differential games theory have the special form (6). Also the specific form of characteristic complexes (10) and (11) will be used below to construct sufficient conditions for convergence of the value functions of the differential games with fast and slow motions. These complexes were wide utilized in researches on u-stability and v-stability properties of the value functions in the differential
°as e: .j. 0.
Let us denote by HO(t, x, s) the limits
HO(t,x,s) = limH~p(t,x,s) = E.j.O lim Hfo(t, x, s)
E.j.O
which will be considered as the Hamiltonian in the unperturbed Cauchy problem P
t
E [0,0),
x ERn,
w(O,x) = aO(x),
x ERn,
(30)
47
games theory (see Krasovskii and Subbotin, 1974, 1985). 3.1 Properties of the sets Y';p,
and define
Yo
YoII
lIyo -
Yi~
= dist (Yo, Y:p(to, Xo, s*) =
+he (t,x e(t),P,v·,A,f3·)}, yO(to)
(34)
as c: t in the Hausdorff metric. The sets Y,?p(t, x, s*), Yi~(t, x, s.), yO(t, x, 1.£, v, a, (3) are compacta and the maps H
(t,x) (t,x)
Y.?p(t,x,v·, 13*),
H
Yi~(t,x,u.,a.)
c:ye (t) E
ye (to)
Conditions 9) imply (see Subbotin, 1995; Clarke et aI., 1997) that for any continuous bounded function x(·): [O,e] H nn, s* E S*, s. E S., c: > the sets Y';p(t, x(t), s·) are strongly invariant with respect to the differential inclusions
dist (ye (t), Y:p(t, xe(t), s·» ~ Ilye (t) - yO (t) 11; dllif(t) - yO(t)11 dt
y
- k e (t, xe(t), yO(t), 1.£, v·»
Using these relations and the condition 8) one can obtain that
C:Yo (t) E c-o {k e (t, x(t), yo(t), 1.£., Q) + he(t,x(t),u*,Q,a.,B)},
= 2( e(t)_ O(t) dif(t) _ dyO(t»
uEP
(35)
and sets Yi~(t, x(t), s.) are strongly invariant with respect to the differential inclusions
+
2
y 'dt dt e ~ 2/C: max(ye (t) - yO (t), k (t, xe(t), ye (t), 1.£, v*)-
+
+ h e(t,x(t),P,v*,A,f3*)},
= Yo
Hence the following estimates are valid
°
co {ke(t, x(t), yO(t), P, v*)
(38)
And let the terms he undepended relative to y in the right-hand sides of the inclusions (37) and (38) coincide.
= P x A;
efl(t) E
co {k e(t, XC (t), ye (t), P, v·)+
+he(t,xe(t),P,v·,A,p·)},
are uniformly Lipschitz continuous for any (v·,f3·) = s· E S· = Qx B, (u.,a.) = s. E
S.
= iio
Remember that the dynamics of if (t) describes in the follows way
yO(t,x,u.,v*,a.,{3*), H
(37)
The strong invariance property of the sets Y:p(t,xe(t),s·) with respect to the inclusion (37) implies that yO(t) E Y';p(t,xe(t), s·) for all t E [to, 0].
Y:p(t,x,v·,f3·) ~ r',?p(t, x, v*, 13·)
(t,x)
°
c:yo (t) E co {k e(t, XC (t), yO (t), P, v·)+
ye(t,x,u,v,a,f3) ~ yO(t,x,u,v,a,f3)
°
do>
Consider a solution of the differential inclusion
It is easy to see from the conditions 1), 5), 7), 10) that for any (t,x) E [0,0] x nn, s· = (v·,f3·) E S· = Q x B, s. = (1.£., a.) E S. = P x A the sets ye, Y';p, Yi~ are compacta and
Yi~(t,x,u.,a.) ~ ~(t,x,u.,a.)
E Y:p(to,x,s·)
(36)
-J t
• It means that all trajectories of the differential inclusions started at the points on the corresponding sets stay in these sets all the time up to 0.
2;e dist (ye (r), Y:p(r,xe(r),s·»dr;
to
In order to be specific below important properties will be obtained for the upper sets Y';p and the upper characteristic inclusions (10). All the conclusions are valid for the lower sets Yi~ and the lower characteristic inclusions too.
dist (ye(t), Y';p(t,xe(t), s·»
~ doe- ".< (t-to) ~ do
~
(39)
So the fast components if(t) of the upper characteristics (38) go fast to the corresponding sets Y';p (t, XC (t), s*). Analogous property holds for the lower characteristics and the corresponding lower sets Yi~.
Let xe(·),if(·),ze(.) : [to,e] H nn x R k x R,(xe(to),if(to),ze(to» = (XO,YO,Zo) be a solution of the upper characteristic inclusion (10) corresponding to a parameter s· = (v·, f3*) E S· , i.e. (xe(-),if(·),ze(.» E Sol (to,xO,YO, Zo,s·). Let us estimate dist 2 (if (t), Y';p(t,xe(t),s·).
Hence, the sets Y';p, Yi~ play roles of attractors for the considered characteristic inclusions. The conditions 8) and 9) imply that any doneighbourhood (do > 0) of the sets is strongly invariant relative to the corresponding characteristic inclusions.
Let us choose
48
Let Xo ER", Zo E R. inclusions
At last let us show that the following relation takes place for any (t, x) E [0,0] x R", s· E S·, s. E S.
(Xi(t),~(t»
(40)
Consider the differential
E Fi(t,xi(t), Zi(t»
(Xi(tO), Zi(tO» = (xo, Zo), It is known (Subbotin, 1991, 1995) that the analogous conditions are valid for the right-hand side sets in the characteristic inclusions, namely
= M€(t,x,Y):f:
for all (t, x, Y, s*, s., e) E [0,0] x R" S. x (0,1].
(41) X
Rk
X
S·
1.
For
X
II x I(t) (42)
(x€ (to), y€ (to), z€(to» = (xo, Yo, Zo)
(43)
J
(48)
J t
II zl(t) - Z2(t) 11
From the above properties of the sets Y';p' Yi~ it follows that for any Tf > 0 there exist e(Tf) > 0, 6(e) > 0, (6(c» .!. 0 as c .!. 0) such that for anye :::; e(Tf), r 2: to + 6(e) the inclusion holds
n {Y:p(r, x€(r), s.) + TfB}
dist (FI(r,
x2(t)11 :::;
to
It is easy to see (Filippov, 1985) that there exists a solution of the inclusion.
{(Y:p(r,x€(r),s·)+TfB}
solution
t
(x€(t),q/(t),z€(t» = M€(t,x(t),y(t»
E
any
E Sol I (to, Xo, Zo) there exists a solution (X2(·),Z2(·» E So12(to,xo,Zo) such that the following estimates take place for all t E [to,O] (Xl (.), Zl (.»
Consider the differential inclusion
y€(r)
(47)
i = 1,2.
The set of all solutions (XiO, Zi(·» of the differential inclusion (46), (47) with number "i" will be denote as Sol i(tO,XO, Zo). The following proposition is valid (see Filippov, 1985).
Proposition
0;
t E [to, 0], (46)
:::;
dist (FI(r,
to
Let (to, xo, Yo) E D x Do, Zo E RI, e E (0,1] and Y';p(t, x, s·), s· E S·, (S,M€) be defined by the relations (15), (10). Fix a solution (x€(·),if(·),z€(·» E Sol (to,xO,YO, Zo,s·), s· E S·, Le.
n (44)
where the symbol B denotes the unit ball in R k •
(x€ (t), ql (t), z€ (t» E
Choosing Tfi .!. 0, ei = e(Tfi) .!. 0, 6i = 6(ei) .!. 0 one can obtain in the limit the last required property
M~p(t,
x€ (t), y€ (t), s·)
(x€(to), y€(to), z€(to» = (xo, Yo, Zo) For the function if (.) : [to, B] H R k let us construct the multivalued mapping
(t,x) 3.2. Proof of the main result.
H
Yo"(t,x,s·)
Yo"(t,x,s·)
To prove the Theorem I. it will be shown that the conditions 8),9),11) are analogous to the conditions 3 and 4 in the paper (Subbotina, 1996) and guarantee convergence w€(t,x,y) to w(t,x).
=
{YO
E
C
Y';p(t,x,s·)
Y';p(t,x,s·):
dist (y€(t), Y:p(t,x,s·» = lIy€(t)-Yoll}:f:
0 (50)
One can easy check that for any s· the mapping (t,x) H Y
For the sake of definity all the constructions below will be done for the upper characteristic complexes and their attractors. The similar results for the lower characteristics and attractors one can easy obtain in the same way.
- g€(t,x,Yo"(t,x,s·),s·)} The following fact of the theory of differential inclusions will be useful for the forthcoming estimates.
(51)
Consider the following differential inclusion corresponding to the mapping (51)
Let (t, x, z) H Fi(t,x, z) C R" x R: [to, 0] x R" x RH 2 R "xR, i = 1,2 - to be two multivalued mappings which have convex, compact values. For the mappings are assumed to be upper semicontinuous.
(x6(t),z6(t» E
co {r(t,x6(t), Yo"(t,x6(t),s·),s·)
- g€ (t, x6(t), 10€ (t, x6(t), s·), s·)} x6(to) = Xo,
49
z6(to) = Zo
(52)
(53)
Taking into account the assumed Lipschitz conditions and properties of the operation "dist " one can continue the inequalities (60).
According to the theory of differential inclusions (see Filippov, 1985) a solution of (52), (53) exists for t E [to, 0]0 Denote the set of all solutions (Xb(o), za(o» of (52), (53) by the symbol Sol g(to, Xo, zo, s")o The symbol Sol E(to, Xo, zo, s.. ) will be denoted the set of all solutions (x E(-), ZE(-) of the differential inclusion
IIxE(t)-x~(t)1I
xE(to)
= Xo,
ZE(tO)
= ZO
JLE{lIxE(r)-x~(r)II+llyE(r)to
(xE(t), ZE(t» E co {r(t,xE(t), Y:l'(t, xE(t), s"), s"), - gE (t, xE(t), Y:l'(t, xE(t), s'"), s.. )}
t
::;
-y~(r)ll}dr
(54)
J£E{llxE(r)-x~(r)lI+dist t
::;
(yE(r),
to
(55)
The inclusion is obvious t
Sol g(to,xo, zo, s.. ) C Sol E(to,xO' ZO, s"o)
Y:l'(r, x~(r), s")}dr ::;
(56)
E
£E{(1 + v E)lI xE (r)-
to
For the chosen trajectory (xE(o),!/(o),ZE(o» E Sol (to, Xo, Yo, ZO, s.. ) let us estimate the distance between (xE(t), ZE(t» and (xE(t), ZE(t)) which is a point of the trajectory (x E(-), ZE(o» E Sol E(to,xO,zo,s"), closest to (XE(-),ZE(o)). Here
t
J
- xg(r)11
+ 2dist (yE (r), Y:l'(r, xE(r), s"»)}dr (59)
Using the exponential estimates (39) for the distance between the fast components of the characteristics and the corresponding attractors and also the Gronwall inequality (see Warga, 1976) the relations (59) are completing
[to,O].
It is follows from the definitions that the men-
tioned distance is not more than the distance between (x E(t), ZE(t» and (xQ(t), zQ(t») which is a point on the trajectory (xb(·) , za(o) E Sol g(to, Xo, ZO, s) closest to (x E(0), ZE ('»)0
J 8
IlxE(t) - xg(t)11 ::;
L E(l
+ vE)llxE(r)_
to
Using the Proposition 1. one can obtain the following relations
t
J
x~(r)lldr
IIx E(r) -
IIx E(t) -
::;
x~(t)11
to
:; J
+ 2LE(1
to
x [(t - to)
t
:; J
maxllr(r,xE(r), yE(r),u,v")uEP
to
c
/\,E
+
to)} x
(1- exp{ - :E (t - to)})]
= p(c)
Let us denote by GE(to, r, Xo, Yo, ZO, s") - the projection of the attainability set for the system (12) at the moment r to the subspace of variables x, z and by G~(E)(to, r, Xo, ZO, s·) - a closed p(c) -neighbourhood of the attainability set for the system (54),(55)
(57)
t
IlzE(t) - z6(t)11 ::;
/\,E
(60) One can see that p(c) ..j.. 0 as c ..j.. O. Analogous estimate can be obtain for IIz E(t) - z811.
t
- r (r, x~(r), Y6(r), u, v")lIdr IIz E(t) - ZE (t) 11 ::;
2cdo (1 _exp{ _ /\,Ec (t - to)})
+ v E)-exp{£E(l + vE)(O -
dist (co r(r,xE(r),yE(r),P,v"), co r(r,
xMr), Ycf(r,xMr), P, v", A, ,8"), P,v"»)dr
::;
J
IW(r) - z6(r)lldr <
to
Estimates obtained above for upper (and analogous estimates for lower) characteristic complexes imply that' the following proposition is truth. Proposition 2 For any compacta D, Do from the condition 8) there exist the mappings (0,1] H R+ x R+, c Ho (a(c),p(c), such that a(c) .j.. 0, p(c)..j.. 0 as c.j.. 0, and for any (to, xo) E D, Yo E Do, ZO E R, s' =. s" E S", (s' = s. E S.), c E (0,1], (xE(.),!/(o),ZE(.)) E Sol (to,xO,YO,ZO,s')
:; J t
max IIg E(r,x E(r), yE(r), u, v")uEP
to
- gE(r, x~(r), Y6(r), u, v"»)lIdr
(58)
where Yb(o) : [to, r] Ho Do: t Ho Yb(t) E measurable function satisfying in accordance with (50)
Yo" (t, xb( t), s.. ) is a
'i0
Y,?p( t, x, {3) = Y (t, x,.8)
there exist such (x"('), z,{)) E Sol "(to,xQ, zo, s'), that for r E [to + 8(E:) , 0], the following relations are valid ~
Ilx"(r) - xe(r) 11
p(E:), Ilz"(r) - z.,(r)1I
~
where the symbol Y(t,x,.8)" denotes the closed E: -neighbourhood of the set Y(t, x, .8).
p(E:)
To obtain the lower characteristic complexes and corresponding arrtactors one can change places {3 and a, Band A.
(61) The proposition imply that the situation is close to that which was studed in papers (Subbotina, 1996, 1998). Modifying proofs of these papers the main result of this paper can be successfully proved.
The asymptotic game has the form
=
x(t)
f(t, x(t), Y(t, x(t), a, {3)), a E A, {3 E Bj
3.3 Example
+
k(y) =
if Y
~
0,
if
~
0
'
(75)
where
x, a, {3)
(76)
(64) y
4. CONCLUSION
.8 E B,
(65)
In this paper, differential games with fast and slow motions are considered. There are obtained sufficient conditions for the value functions in singularly perturbed differential games to converge to the value function in the asymptotics game. It was shown that the convergence property is based on the existence of attractors in the subspace of fast variables.
sets A and B are compacta. Let the Isaacs condition holds. The pay-off functional has the form
'Yto,XO'yo(x(')'y(,)) = u(x(O))+ (J
J
+
g(r,x(r), Y(r,x(r),a,{3))dr
Y(t, x, a, {3) =
{ -2y,
a E A,
J to
(63)
where k(y) has the form
-y
(73)
(J
(62)
+ ~(t, x, a,.8)
E:y(t) = k(y)
(72)
'Yto,xo(x(,)) = u(x(O))+
Consider the following singularly perturbed differential game
x(t) = f(t,x(t),y(t)),
(71)
g(r,x(r),y(r))dr
(66)
to
5. ACKNOWLEDGEMENTS
Here the upper characteristic complexes and the corresponding attractors satisfying the assumptions 1)-11) are the following
M~p(t,x,y,(3) =
co {(J(t,x,y), ~ E
g(t,x,y)):
co
1
€"(~
The research was supported by the Russian Fund of Fundamental Researches under Grants N96-01-00219, N96-15-96245 and N97-01-00371.
- y),
~(t,x,A,.8)};
REFERENCES
(67)
Bagagiolo, F., M. Bardi and I. Capuzzo--Dolcetta (1997). A Viscosity Solutions Approach to Some Asymptotic Problems in Optimal Control. In: Partial differential equation methods in' control and shape analysis (Da Prato, G. and J. P. Zolesio. (Ed)), 29-39. Dekker, Inc., New York. Bardi, M. and I. Capuzz
M~p(t,x,(3) = co {(J(t,x,~), g(t,x,~)): ~
E Y(t, x, (3)}
(68)
where
Y(t,x,{3) =
U ~(t,x,a,{3)
(69)
"'EA
and the function
I,
=
{
timal Control and Viscosity Solutions of Hamilton-lacobi-Bellman Equations, Birk-
is defined as
if
~ ~
0,
hauser, Boston. Barron, E. N., L. C. Evans and R. Jensen (1984). Viscosity solutions of Isaacs' equations and differential games with Lipschitz controls. 1. Different. Equat., 53(2), 213-233. Bensoussan, A. (1988). Perturbation Methods in Optimal Control Problems, Wiley-Gautier, New York.
(70) 1/2,
if
~ ~
0
The upper attractors Y';p and Y,?p are obtained by the formulae
51
Clarke, F. H., Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski (1998). Nonsmooth Analysis and Control Theory, Springer-Verlag, New York. Crandall M. G.and P. L. Lions (1983). Viscosity solutions of Hamilton-Jacobi equations. Trans. A mer. Math. Soc., 277, 1-42. Crandall M. G., L. C. Evans and P. L. Lions (1984). Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. A mer. Math. Soc., 282, 487-502. Filippov, A. F. (1985). Differential Equations with Discontinuous Right-hand Side, Nauka, Moscow [in Russian]. Gaitsgory, V. G. (1996). Limit Hamilton-Jac~ bi-Isaacs Equations for Singularly Perturbed Zero-Sum Differential Games. J. Math. Anal. Appl., 202, 862-899. Isaacs, R. (1965). Differential Games. Wiley, New York. Krasovskii, N.N. and A.I. Subbotin (1974). Positional Differential Games. Nauka, Moscow [in Russian]. Krasovskii, N.N. and A.I. Subbotin (1988). Game- Theoretical Control Problems. Springer-Verlag, New York. Kokotovic, P. V. (1984). Applications of singular perturbations techniques to control pro~ lems. SIAM Rev., 26, 501-550. O'Malley, R. E. (1974). Introduction to Singular Perturbations, Academic Press, New York. Pervozvansky, A. A. and V. G. Gaitsgory (1988). Theory of Suboptimal Solutions, Kluwer Academic, Dordrecht. Subbotin, A. I. (1980). Generalization of the main equation of the theory of differential games. Soviet. Math. Dokl., 22(2), 358-362. Subbotin, A. I. (1991) Minimax inequalities and Hamilton-Jacobi equations, Nauka, Moscow [in Russian]. Subbotin, A. I. (1995) Generalized Solutions of First-Order PDEs. The Dynamical Optimization perspective, Birkhauser, Boston. Subbotina, N. N. (1996). Asymptotic properties of minimax solutions of Isaacs-Bellman equations in differential games with fast and slow motions. J. Appl. Maths Mechs, 60(6), 883-890. Subbotina, N. N. (1998). Asymptotics for singularly perturbed Hamilton-Jacobi equations. Prikl. Mat. Mech., [in Russian], (to appear). Tikhonov, A. N. (1952). Systems of differential equations containing small parameters near derivatives. Mat. Sbornik, 31(3), 575-586 [in Russian]. Vasil'eva, A. B. and A. F. Butuzov (1973). Asymptotic Expansions of Solutions to Singularly Perturbed Equations, Nauka, Moscow [in Russian]. Warga, J. (1976). Optimal Control of Differential and Functional Equations, Academic Press, New York.
52