Necessary condition for some singular stochastic control systems with variable delay
The purpose of this paper is to study conditions for the optimality of singular stochastic control systems with variable delay and constraint on the endpoint of state. The necessary condition of optimality for singular systems is obtained.
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irk-123456789-45572009-12-07T12:00:32Z Necessary condition for some singular stochastic control systems with variable delay Nilgun, M. Agayeva, Ch.A. The purpose of this paper is to study conditions for the optimality of singular stochastic control systems with variable delay and constraint on the endpoint of state. The necessary condition of optimality for singular systems is obtained. 2008 Article Necessary condition for some singular stochastic control systems with variable delay / M. Nilgun, Ch.A. Agayeva // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 108–115. — Бібліогр.: 8 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4557 519.21 en Інститут математики НАН України |
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The purpose of this paper is to study conditions for the optimality of singular stochastic control systems with variable delay and constraint on the endpoint of state. The necessary condition of optimality for singular systems is obtained. |
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Article |
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Nilgun, M. Agayeva, Ch.A. |
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Nilgun, M. Agayeva, Ch.A. Necessary condition for some singular stochastic control systems with variable delay |
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Nilgun, M. Agayeva, Ch.A. |
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Nilgun, M. |
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Necessary condition for some singular stochastic control systems with variable delay |
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Necessary condition for some singular stochastic control systems with variable delay |
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Necessary condition for some singular stochastic control systems with variable delay |
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Necessary condition for some singular stochastic control systems with variable delay |
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Necessary condition for some singular stochastic control systems with variable delay |
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necessary condition for some singular stochastic control systems with variable delay |
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Інститут математики НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/4557 |
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Necessary condition for some singular stochastic control systems with variable delay / M. Nilgun, Ch.A. Agayeva // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 108–115. — Бібліогр.: 8 назв.— англ. |
| work_keys_str_mv |
AT nilgunm necessaryconditionforsomesingularstochasticcontrolsystemswithvariabledelay AT agayevacha necessaryconditionforsomesingularstochasticcontrolsystemswithvariabledelay |
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2025-07-02T07:46:20Z |
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2025-07-02T07:46:20Z |
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1836520446401445888 |
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Theory of Stochastic Processes
Vol. 14 (30), no. 2, 2008, pp. 108–115
UDC 519.21
NILGUN MORALI AND AGAYEVA CH. A.
NECESSARY CONDITION FOR SOME SINGULAR
STOCHASTIC CONTROL SYSTEMS WITH VARIABLE DELAY
The purpose of this paper is to study conditions for the optimality of singular sto-
chastic control systems with variable delay and constraint on the endpoint of state.
The necessary condition of optimality for singular systems is obtained.
Introduction
The necessary conditions of optimality for systems which are described by stochastic
differential equations with constant delay are obtained in [1,2] and with variable delay
in [3,4]. In the present paper we consider optimal control problem for stochastic delay
systems with constraint on the state variable. It is obtained new necessary condition of
optimality for a singular systems [5,6].
Statement of problem
Let (Ω, F, P ) be a complete probability space with the filtration {F t : t0 ≤ t ≤ t1}
generated by Wiener process wt and F t = σ(ws; t0 ≤ s ≤ t). L2
F (t0, t1, Rn) – space of
predictable processes xt(ω) such that:
E
∫ t1
t0
|xt|2dt < +∞.
Consider the following stochastic system with variable delay on state:
dxt = g(xt, xt−h(t), ut, t)dt+ f(xt, t)dwt, t ∈ (t0, t1];(1)
xt0 = x0;(2)
xt = Φ(t), t ∈ [t0 − h(t0), t0);(3)
ut(ω) ∈ U∂ ≡ {u(·) ∈ L2
F (t0, t1;Rm) |u(ω) ∈ U ⊂ Rm, a.c.},(4)
where U – nonempty bounded set, Φ(t) – piecewise continuous non-random function.
h(t) ≥ 0 is a continuously differentiable, non-random function such that dh(t)
dt < 1. Let
it is required to minimize the functional in set of admissible controls:
(5) J(u) = E
∫ t1
t0
l(xt, ut, t)dt
at condition
(6) Eq(xt1) = 0.
Let assume that the following requirements are satisfied:
A1. Functions l, g, f and their derivatives are continuous in (x, u, t):
l(x, u, t) : Rn ×Rm × [t0, t1] → R1; f(x, t) : Rn × [t0, t1] → Rn×n
2000 AMS Mathematics Subject Classification. Primary 93E20.
Key words and phrases. Stochastic differential equations with variable delay, optimal control problem,
stochastic control problem, necessary conditions of optimality, admissible controls, singular controls.
108
SINGULAR STOCHASTIC CONTROL SYSTEMS 109
g(x, y, u, t) : Rn ×Rn ×Rm × [t0, t1] → Rn;
A2. l, g, f functions are twice continuously differentiable with respect to (x, y) and
hold the condition of linear growth:
(1 + |x| + |y|)−1(|g(x, y, u, t)| + |gx(x, y, u, t)| + |gy(x, y, u, t)|
+|l(x, u, t)| + |lx(x, u, t)| + |f(x, t)| + |fx(x, t)|) ≤ N ;
A3. Function q(x) : Rm → Rk is twice continuously differentiable and
|q(x)| + |qx(x)| ≤ N(1 + |x|); |qxx(x)| ≤ N.
Introduce following set:
B(xt, ut) = {(kt, u
∗
t ) : dkt = [gx(xt, xt−h(t), ut, t)kt+
+ gy(xt, xt−h(t), ut, t)kt−h(t) + g(xt, xt−h(t), u
∗
t , t)
g(xt, xt−h(t), ut, t)]dt+ [fx(xt, t)kt]dwt, t ∈ (t0, t1];
Eqx(xt1)kt1 = 0; kt = 0, t ∈ [t0 − h(t0), t0]}
At the same time we will consider following additional problem:
(7) J̃(u) = E
∫ t1
0
l(xt, ut, t)dt → min
(8)⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
dxt = g(xt, xt−h(t), ut, t)dt+ f(xt, t)dwt; t ∈ (t0, t1]
xt0 = x0
xt = Φ(t), t ∈ [t0 − h(t0), t0)
dkt = {[gx(xt, xt−h(t), ut, t)k1 + gy(xt, xt−h(t), ut, t)kt−h(t) + g(xt, xt−h(t), u
∗
t , t)−
−g(xt, xt−h(t), ut, t)]dt+ [fx(xt, t)kt]dwt, t ∈ (t0, t1];
kt = 0, t ∈ [t0 − h(t0), t0]}
Eq(xt1 ) = 0
Eqx(xt1 )kt1 = 0
ut(ω) ∈ U∂ ≡ {u(·) ∈ L2
F (t0, t1; Rm) |u(ω) ∈ U ⊂ Rm, a.c.}.
Note that
min
u∈U
J(u) = min
u∈U
J̃(u)
Theorem. Let conditions A1-A3 hold and (x0
t , u
0
t ) is a solution of the problem (1) – (6).
Then exist random processes (ψ1
t , β
1
t ) ∈ L2
F (t0, t1;Rn) × L2
F (t0, t1;Rn×n) and (ψ2
t , β
2
t ) ∈
L2
F (t0, t1;Rn)×L2
F (t0, t1;Rn×n) which are the solutions of the following adjoint systems:
(9)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
dψ1
t = −{Hx(ψ1
t , ψ
2
t , x
0
t , y
0
t , u
0
t , kt, t) +Hy(ψ1
z , ψ
2
z , x
0
z, y
0
z , u
0
z, kz, z)
∣∣
z=s(t)
+
+[g∗x(x0
t , y
0
t , u
∗
t , t) − g∗x(x0
t , y
0
t , u
0
t , t)]ψ
2
t +
+[g∗y(x0
z , y
0
z , u
∗
z, z) − g∗y(x0
z , y
0
z , u
0
z, z)]ψ2
z
∣∣
z=s(t)
s′(t)+}dt+
+β1
t dwt, t0 ≤ t < t1 − h(t1),
dψ1
t = −{Hx(ψ1
t , ψ
2
t , x
0
t , y
0
t , u
0
t , kt, t)+
+[g∗x(x0
t , y
0
t , u
∗
t , t) − g∗x(x0
t , y
0
t , u
0
t , t)]ψ
2
t }dt+ β1
t dwt, t1 − h(t1) ≤ t < t1,
ψ1
t1 = −λ0qx(x0
t1) − λ1qxx(x0
t1 )kt1 .
(10)
⎧⎪⎪⎪⎨⎪⎪⎪⎩
dψ2
t = −{g∗x(x0
t , y
0
t , u
0
t , t)ψ
2
t + f∗
x(x0
t , t)β
2
t + g∗y(x0
z , y
0
z , u
0
z, z)ψ2
z
∣∣
z=s(t)
s′(t)}dt+
+β2
t dwt, t0 ≤ t < t1 − h(t1),
dψ2
t = −[g∗x(x0
t , y
0
t , u
0
t , t)ψ
2
t + f∗
x(x0
t , t)β
2
t ]dt+ β2
t dwt, t1 − h(t1) ≤ t < t1,
ψ2
t1 = −λ0qx(x0
t1),
110 NILGUN MORALI AND AGAYEVA CH. A.
here λ0, λ1 ∈ Rn such that |λ0|2 + |λ1|2, then for any (k1, u
∗
t ) ∈ B(x0
t , u
0
t ) a.e. t ∈ [t0, t1)
fulfills the following:
(11) Hx(ψ1
t , ψ
2
t , x
0
t , y
0
t , u
∗
t , kt, t) −Hy(ψ1
t , ψ
2
t , x
0
t , y
0
t , u
0
t , kt, t) ≤ 0, a.c.,
were
(12)
H(ψ1
t , ψ
2
t , xt, yt, ν, kt, t) = ψ1∗
t g(xt, yt, ν, t) + β1∗
t f(xt, t) + ψ2∗
t gx(xt, yt, ν, t)kt+
+β2∗
t fx(xt, t)kt + ψ2∗
t gy(xt, yt, ν, t)kt−h(t) − l(xt, ut, t).
Proof. For any natural j let’s introduce the following approximating functional:
Jj(u) = Sj
(
E
∫ t1
t0
l(xt, ut, t)dt, Eq(xt1 )
)
=
= min
c∈E
√
|c− 1/j − E
∫ t1
t0
l(xt, ut, t)dt|2 + |Eq(xt1 )|2 + |Eqx(xt1)kt1 |2
E = {x : c ≤ J0}, J0 minimal value of the functional in (1) – (6). V ≡ (U∂ , d) – space of
controls obtained by means of introducing of the following metric:
d(u, ν) = (l ⊗ P ){(t, ω) ∈ [t0, t1] × Ω : νt �= ut},
V – complete metric space [8].
We will used following result:
Lemma 1. Let’s assume, that conditions A1 – A3 hold, un
t – sequence of admissible
controls from V, zn
t = (xn
t , k
n
t ) – sequence of corresponding trajectories of the system (8).
If d(un
t , ut) → 0, n→ ∞, then limn→∞
{
supt0≤t≤t1 E|zn
t −zt|2
}
= 0, where zt = (xt, kt)
is a trajectory corresponding to an admissible control ut.
Due to Lemma 1 and assumptions A1 – A3 implies continuity of the functional Jj :
V → Rn, then according to variation principle of Ekeland we have that it exists a control
uj
t : d(uj
t , u
0
t ) ≤ √
εj and ∀ u ∈ V it is fulfilled: Jj(uj) ≤ Jj(u) + √
εjd(uj , u), εj = 1
j .
This inequality means that (uj
t , x
j
t , k
j
t ) is a solution of the following problem:
(13)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
Ij(u) = Jj(u) + √
εjE
∫ t1
t0
δ(ut, u
j
t)dt → min
dxt = g(xt, yt, νt, t)dt+ f(xt, t)dwt, t ∈ (t0, t1]
xt = Φ(t), t ∈ [t0 − h(t0), t0]
dkt = [gx(xt, yt, ut, t)kt + gy(xt, yt, ut, t)kt−h(t) + g(xt, yt, u
∗
t , t)−
−g(xt, yt, ut, t)]dt+ [fx(xt, t)kt + fy(xt, t)kt−h(t)]dwt, t ∈ (t0, t1];
kt = 0, t ∈ [t0 − h(t0), t0];
ut ∈ U.
Function δ(u, ν) is determined in the following way: δ(u, ν) =
{
0, u = ν
1, u �= ν.
Let uj
t = uj
t + Δuj
t – some admissible control, xj
t = xj
t + Δxj
t and k
j
t = kj
t + Δkj
t
corresponding solution of problem (13).
Let’s use the following identities:
(14)
dΔxj
t = d(xj
t − xj
t ) =
{Δujg(xj
t , y
j
t , u
j
t , t) + gx(xj
t , y
j
t , u
j
t , t)Δx
j
t + gy(xj
t , y
j
t , u
j
t , t)Δy
j
t }dt+
+fx(xj
t , t)Δx
j
tdwt + η1
t , t ∈ (t0, t1],
SINGULAR STOCHASTIC CONTROL SYSTEMS 111
(15)
dΔkj
t = d(kj
t − kj
t ) = {Δujg(xj
t , y
j
t , u
j
t , t) + gx(xj
t , y
j
t , u
j
t , t)Δk
j
t +
+ gy(xj
t , y
j
t , u
j
t , t)Δk
j
t−h(t) + gxx(xj
t , y
j
t , u
j
t , t)Δk
j
t Δxj
t +
gxy(xj
t , y
j
t , u
j
t , t)Δk
j
t Δyj
t + gyx(xj
t , y
j
t , u
j
t , t)Δk
j
t−h(t)Δx
j
t +
+ gyy(xj
t , y
j
t , u
j
t , t)Δk
j
t−h(t)Δy
j
t }dt+ {fx(xj
t , t)Δk
j
t +
+ fxx(xj
t , t)Δk
j
t Δxj
t}dwt + η2
t , t ∈ (t0, t1],
where
Δujg(x, yt, ut, t) = g(x, yt, u
i
t, t) − g(x, yt, u
j
t , t);
η1
t =
{∫ 1
0
[g∗x(xj
t + μΔxj
t , y
j
t , u
j
t , t) − g∗x(xj
t , y
j
t , u
j
t , t)]Δxtdμ+
+
∫ 1
0
[g∗y(xj
t , y
j
t + μΔyj
t , u
j
t , t) − g∗y(xj
t , y
j
t , u
j
t , t)]Δy
j
tdμ
}
dt+
+
∫ 1
0
[f∗
x(xj
t + μΔxj
t , t) − f∗
x(xj
t , t)]Δx
j
tdμdwt;
η2
t =
{
η1
t +
∫ 1
0
[g∗xx(xj
t + μΔxj
t , y
j
t , u
j
t , t) − g∗xx(xj
t , y
j
t , u
j
t , t)]k
j
t Δxj
tdμ+
+
∫ 1
0
[g∗xy(xj
t , y
j
t + μΔyj
t , u
j
t , t) − g∗xy(xj
t , y
j
t , u
j
t , t)]k
j
t Δyj
tdμ
}
dt+
+
∫ 1
0
[g∗yx(xj
t + μΔxj
t , y
j
t , u
j
t , t) − g∗yx(xj
t , y
j
t , u
j
t , t)]k
j
t−h(t)Δx
j
tdμ+
+
∫ 1
0
[g∗yy(xj
t , y
j
t + μΔyj
t , u
j
t , t) − g∗yy(xj
t , y
j
t , u
j
t , t)]k
j
t−h(t)Δy
j
tdμ
}
dt+
+
∫ 1
0
[f∗
xx(xj
t + μΔxj
t , y
j
t , t) − f∗
xx(xj
t , y
j
t , t)]k
j
t Δxj
tdμdwt.
Assume that stochastic differential the vector ψt = (ψ1
t , ψ
2
t ) has following form: dψt =
(α1dt + β1dwt, α2dt + β2dwt). So that the state of system described by (8), then the
vector of increment will be have stochastic differential of form: dΔzj
t = (dΔxj
t , dΔkj
t ),
where dΔxj
t and dΔkj
t satisfy to equation (14), (15) accordingly.
Then by Ito’s formula we have:
d(ψj∗
t Δzj
t ) = dψ1j∗
t Δxj
t + ψ1j∗
t dΔxj
t + dψ2j∗
t Δkj
t + ψ2∗
t Δkj
t +
+{β1j∗
t fx(xj
t , t)Δxt + β2j∗
t [fx(xj
t , t)Δk
j
t − fxx(xj
t , t)Δk
j
t Δxj
t ]}dt.
Increment for functional (7) across admissible control defined following way
(16)
ΔujJ(u) = J(uj) − J(uj) = E
∫ t1
t0
[l(xj
t , u
j
t , t) − l(xj
t , u
j
t , t)]dt =
= E
∫ t1
t0
[Δuj l(xj
t , u
j
t , t) + lx(xj
t , u
j
t , t)Δx
j
t + lxx(xj
t , u
j
t , t)Δk
j
t Δxj
t ]dt+ η2,
where
η2 = E
∫ t1
t0
{
√
εjδ(u
j
t , u
j
t) +
∫ 1
0
[l∗x(xj
t + μΔxj
t , u
j
t , t) − l∗x(xj
t , u
j
t , t)]Δx
j
tdμ+
+
∫ 1
0
[l∗xx(xj
t + μΔxj
t , u
j
t , t) − l∗xx(xj
t , u
j
t , t)]Δk
j
t Δxj
tdμ
}
dt.
Let defined stochastic process ψj
t at the point t1 following way:
ψj
t1 = (ψ1j
t1 , ψ
2j
t1 ) = (−λj
0qx(xj
t1) − λj
1qxx(xj
t1)kt1 ,−λ
j
0qx(xj
t1)).
112 NILGUN MORALI AND AGAYEVA CH. A.
Assume that, the random processes (ψ1j
t , β
1j
t ) ∈ L2
F (t0, t1;Rn) × L2
F (t0, t1;Rn×n) and
(ψ2j
t , β
2j
t ) ∈ L2
F (t0, t1;Rn)×L2
F (t0, t1;Rn×n) are almost certainly unique solutions of the
following adjoin systems [7]:
(17)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
dψ1j
t = −{Hx(ψ1j
t , ψ
2j
t , x
j
t , y
j
t , u
j
t , k
j
t , t)+
+{Hy(ψ1j
z , ψ
2j
z , x
j
z , y
j
z, u
j
z, k
j
z , z)
∣∣∣∣
z=s(t)
s′(t)+
+[g∗x(xj
t , y
j
t , u
∗
t , t) − g∗x(xj
t , y
j
t , u
j
t , t)]ψ
2j
t + [g∗y(xj
z , y
j
z, u
∗
z, z)−
−g∗y(xj
z , y
j
z, u
j
z, z)]ψ2j
z
∣∣∣∣
z=s(t)
s′(t)}dt+ β1j
t dwt, t0 ≤ t < t1 − h(t1),
dψ1j
t = −{Hx(ψ1j
t , ψ
2j
t , x
j
t , y
j
t , u
j
t , k
j
t , t) + [g∗x(xj
t , y
j
t , u
∗
t , t)−
−g∗x(xj
t , y
j
t , u
j
t , t)]ψ
2
t }dt+ β1j
t dwt,
t1 − h(t1) ≤ t < t0
ψ1j
t1 = −λj
0qx(xj
t1) − λj
1qxx(xj
t1 )kt1 .
(18)
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
dψ2j
t = −{g∗x(xj
t , y
j
t , u
j
t , t)ψ
2j
t + f∗
x(xj
t , t)β
2j
t + g∗y(xj
z , y
j
z, u
j
z, z)ψ2j
z
∣∣∣∣
z=s(t)
×
s′(t)}dt+ β2j
t dwt, t0 ≤ t < t1 − h(t1),
dψ2j
t = −[g∗x(xj
t , y
j
t , u
j
t , t)ψ
2j
t + f∗
x(xj
t , t)β
2j
t ]dt+ β2j
t dwt, t1 − h(t1) ≤ t < t1,
ψ2j
t1 = −λj
0qx(xj
t1),
where meet the following requirement:
(19) (λj
0, λ
j
1) = (−Eq(xj
t1)/J0
j ,−Eqx(xj
t1)kj
t1/J
0
j )
and
J0
j =
√
|cj − 1/j − E
∫ t1
t0
l(xj
t , u
j
t , t)dt|2 + |Eq(xj
t1 )|2.
Since ‖(λj
0, λ
j
1)‖ = 1, then we can think that (λj
0, λ
j
1) → (λ0, λ1). It is known that Sj
is a convex function which is Gato-differentiable at point:
(
E
t1∫
t0
l(xj
t , u
j
t , t)dt, Eq(x
j
t1 )
)
.
Then for all c ∈ E :⎛⎝λj
0, c−
1
j
− E
t1∫
t0
l(xj
t , u
j
t , t)dt
⎞⎠ + (λj
0, Eq(x
j
t1)+)(λj
1, Eqx(xj
t1 )kj
t1) ≤ 1
j
.
Since
(20) ψ1j
t1 = −λj
0qx(xj
t1 ) − λj
1qxx(xj
t1), then ψ1j
t1 → ψ1
t1 in L2
F (t0, t1;Rn)
and
(21) ψ2j
t1 = −λj
0qx(xj
t1), then ψ2j
t1 → ψ2
t1 in L2
F (t0, t1;Rn).
Due to (14) and (15) expression (16) may be writing following way:
ΔJ(uj) = −E
∫ t1
t0
dψ1j∗
t Δxt −
∫ t1
t0
dψ2j∗
t Δkt+
+E
∫ t1
t0
[Δuj l(xj
t , u
j
t , t) + lx(xj
t , u
j
t , t)Δx
j
t + lxx(xj
t , u
j
t , t)Δk
j
t Δxj
t ]dt−(22)
SINGULAR STOCHASTIC CONTROL SYSTEMS 113
−E
∫ t1
t0
β1j∗
t fx(xj
t , t)Δx
j
tdt−
−E
∫ t1
t0
β2j∗
t [fx(xj
t , t)Δk
j
t + fxx(xj
t , t)Δk
j
t Δxj
t ]dt+ ηt0,t1 ,
where
ηt0,t1 = η2 + E
∫ t1
t0
{∫ 1
0
ψ1j∗
t (gx(xj
t + μΔxj
t , y
j
t , u
j
t , t)−
−gx(xj
t , y
j
t , u
j
t , t))Δx
j
tdμ+
∫ 1
0
ψ1j∗
t (gy(xj
t , y
j
t + μΔyj
t , u
j
t , t)−
−gy(xj
t , y
j
t , u
j
t , t))Δy
j
tdμ
}
dt+
+E
∫ t1
t0
{∫ 1
0
β1j∗
t (fx(xj
t + μΔxj
t , t) − fx(xj
t , t))Δx
j
tdμ
}
dt+
+E
∫ t1
t0
ψ2j∗
t [(gx(xj
t + μΔxj
t , y
j
t , u
j
t , t) − gx(xj
t , y
j
t , u
j
t , t))Δk
j
t +
+(gy(xj
t , y
j
t + μΔyj
t , u
j
t , t) − gy(xj
t , y
j
t , u
j
t , t))Δk
j
t−h(t)+
+(gxx(xj
t + μΔxj
t , y
j
t , u
j
t , t) − gxx(xj
t , y
j
t , u
j
t , t))Δk
j
t Δxj
t+
+(gxy(xj
t , y
j
t + μΔyj
t , u
j
t , t) − gxx(xj
t , y
j
t , u
j
t , t))Δk
j
t Δyj
t +
+(gyx(xj
t + μΔxj
t , y
j
t , u
j
t , t) − gyx(xj
t , y
j
t , u
j
t , t))Δk
j
t−h(t)Δx
j
t +
+(gyy(xj
t , y
j
t + μΔyj
t , u
j
t , t) − gyy(xj
t , y
j
t , u
j
t , t))Δk
j
t−h(t)Δy
j
t ]dt+
+E
∫ t1
t0
β2j∗
t [(fx(xj
t + μΔxj
t ) − fx(xj
t , t))Δk
j
t +
(23) +(fxx(xj
t + μΔxj
t , t) − fxx(xj
t , t))Δk
j
t Δxj
t ]dt.
Helping simple transformation (22) has following form:
(24)
ΔJ(uj) = −E
∫ t1
t0
[ψ1j∗
t Δujg(xj
t , x
j
t−h(t), u
j
t , t) + ψ2j∗
t Δujgx(xj
t , x
j
t−h(t), u
j
t , t)k
j
t +
+ψ2j∗
t Δujgy(xj
t , x
j
t−h(t), u
j
t , t)k
j
t − Δuj l(xj
t , u
j)t, t)]dt+ ηt0,t1 .
Let
Δuj
t = Δujθ
t,ε =
{ 0, t∈[θ, theta+ ε), ε > 0, θ ∈ [t0, t1)
νj − uj
t , t ∈ [θ, θ + ε), νj ∈ L2(Ω, F θ, P ;Rm)
some special admissible control, ε > 0 enough smaller number . Let xjθ
t,ε – trajectory
corresponding to control ujθ
t,ε = uj
t + Δujθ
t,ε.Then expression (24) may be represented
following way:
ΔJ(uj) = −E
∫ θ+ε
θ
[ψ1j∗
t Δνjg(xj
t , x
j
t−h(t), u
j
t , t) + ψ2j∗
t Δνjgx(xj
t , x
j
t−h(t), u
j
t , t)k
j
t +
+ψ2j∗
t Δνjgy(xj
t , x
j
t−h(t), u
j
t , t)k
j
t−h(t) − Δνj l(xj
t , u
j
t , t)]dt+ ηθ,θ+ε.
We prove the following lemmas by scheme [4]:
Lemma 2. Let’s assume, that conditions A1 – A3 hold. If ε → 0, then E
∣∣∣∣xjθ
t,ε−xj
t
ε
∣∣∣∣2 ≤
N, where xj
t is a trajectory of system (9), xjθ
t,ε – a trajectory corresponding to an admis-
sible control ujθ
t,ε = uj
t + Δujθ
t,ε.
114 NILGUN MORALI AND AGAYEVA CH. A.
Lemma 3. Let’s assume, that conditions A1 – A3 hold. If ε → 0, then E|kjθ
t,ε −
kj
t |2 ≤ Nε, where kj
t is a trajectory of system (9), kjθ
t,ε – a trajectory corresponding to an
admissible control ujθ
t,ε = uj
t + Δujθ
t,ε.
According to lemma 2 and lemma 3 we have: E|zjθ
t,ε − zj
t |2 ≤ Nε, and from expression
(23) we get: ηθ,θ+ε = o(ε). Then for (22) obtained:
(25)
ΔθJ(uj) = −E[ψ1j∗Δνjg(xj
θ, x
j
θ−h(θ), u
j
θ, θ) + ψ2∗
Δνjgx(xj
θ, x
j
θ−h(θ), u
j
θ, θ)kθ+
+ψ2∗
Δνjgy(xj
θ, x
j
θ−h(θ), u
j
θ, θ)kθ−h(θ) − Δνj l(xj
θ, u
j
θ, θ)]ε + o(ε) ≥ 0.
Due to smallest of ε :
E[ψ1j∗
θ Δν′g(xj
θ, x
j
θ−h(θ), u
j
θ, θ) + ψ2∗
θ Δν′gx(xj
θ, x
j
θ−h(θ), u
j
θ, θ)kθ+
+ψ2∗
θ Δν′gy(xj
θ, x
j
θ−h(θ), u
j
θ, θ)kθ−h(θ) − Δνj l(xj
θ, u
j
θ, θ)] ≤ 0,
and arbitrarily of θ ∈ [t0, t1) :
(26) H(ψ1j
t , ψ
2j
t , x
j
t , y
j
t , u
∗
t , k
j
t , t) −H(ψ1j
t , ψ
2j
t , x
j
t , y
j
t , u
j
t , k
j
t , t) ≤ 0 a.c.
Lemma 4. Let (ψ1j
t , ψ
2j
t ) are solutions of systems (17), (18) and (ψ1
t , ψ
2
t ) are solution
of systems (9), (10). Then
E
∫ t1
t0
|ψ1j
t − ψ1
t |2dt+ E
∫ t1
t0
|β1j
t − β1
t |2dt→ 0,
E
∫ t1
t0
|ψ2j
t − ψ2
t |2dt+ E
∫ t1
t0
|β2j
t − β2
t |2dt→ 0, if d(uj
t , ut) → 0, j → ∞.
Proof. According to Ito formula, for s ∈ [t1 − h(t), t1],
E|ψ2j
t1 −ψ2
t1 |
2 −E|ψ2j
s −ψ2
s |2 = 2E
∫ t1
s
[ψ2j
t −ψ2
t ][(g∗x(xj
t , y
j
t , u
j
t , t)− g∗x(x0
t , y
0
t , u
0
t , t))ψ
2j
t +
+g∗x(x0
t , y
0
t , u
0
t , t)(ψ
2j
t − ψ2
t ) + (f∗
x(xj
t , t) − f∗
x(x0
t , t))β
2j
t + f∗
x(x0
t , t)×
×(β2j
t − β2
t )]dt+ E
∫ t1
s
|β2j
t − β2
t |2dt.
Due to assumptions A1 – A2 and using simple transformations for arbitrary positive we
obtain:
E
∫ t1
s
|β2j
t − β2
t |2dt+ E|ψ2j
t − ψ2
t |2 ≤
≤ EN
∫ t1
s
|ψ2j
t − ψ2
t |2dt+ ENα
∫ t1
s
|β2j
t − β2
t |2dt+ E|ψ2j
t1 − ψ2
t1 |
2.
Hence, according to Gronwall inequality we have:
(27) E|ψ2j
s − ψ2
s |2 ≤ DeN(t1−s) a.e. in [t1 − h(t), t1],
where constant D is determined in the following way: D = E|ψ2j
t1 − ψ2j
t1 |2 → 0.
According to Ito formula, for s ∈ [t0, t1 − h(t1)),
E|ψ2j
t1−h(t1)
− ψ2
t1−h(t1)
|2 − E|ψ2j
s − ψ2
s |2 = 2E
∫ t1−h(t1)
s
(ψ2j
t − ψ2
t )[(g∗x(xj
t , y
j
t , u
j
t , t)+
+g∗x(x0
t , y
0
t , u
0
t , t))(ψ
2j
t − ψ2
t ) + (f∗
x(xj
t ) − (f∗
x(x0
t ))β2j
t + f∗
x(x0
t , t)×
×f∗
x(x0
t , t))β
2j
t + f∗
x(x0
t , t)(β
2j
t − β2
t ) + g∗y(xj
z , y
j
z, u
j
z, z)−
−g∗y(x0
z , y
0
z , u
0
z, z))ψ2j
z s
′(t) + g∗y(x0
z , y
0
z , u
0
z, z)(ψ2j
z − ψ2
z)s′(t) + E
∫ t1−h(t1)
s
|β2j
t − β2
t |2dt.
SINGULAR STOCHASTIC CONTROL SYSTEMS 115
In view of assumptions A1 – A2 and the expression (21) we obtain:
E
∫ t1−h(t1)
s
|β2j
t − β2
t |2dt+ E|ψ2j
s − ψ2
s |2 ≤
≤ EN
∫ t1−h(t1)
s
|ψ2j
t − ψ2
t |2dt+ ENα
∫ t1−h(t1)
s
|ψ2j
z − ψ2
z |2dt+
+ENα
∫ t1−h(t1)
s
|β2j
t −β2
t |2dt+ENα
∫ t1−h(t1)
s
|β2j
z −β2
t |2dt+E|ψ2j
t1−h(t1)
−ψ2
t1−h(t1)|2.
Hence using simple transformations we have:
E(1−2Nα)
∫ t1−h(t1)
s
|β2j
t −β2
t |2dt+E|ψ2j
s −ψ2
s |2 ≤ E(N+Nα)
∫ t1−h(t1)
s
|ψ2j
t −ψ2
t |2dt+
+ENα
∫ t1
t1−h(t1)
|ψ2j
t − ψ2
t |2dt+ ENα
∫ t1
t1−h(t1)
|β2j
t − β2
t |2dt+ E|ψ2j
t1−h(t1)
− ψ2
t1−h(t1)
|2.
According to Gronwall inequality:
E|ψ2j
s − ψ2
s |2 ≤ D exp[(N +Nα)(t1 − h(t1))] a.e. in [t0, t1 − h(t1)),
where
D = E|ψ2j
t1−h(t1)−ψ
2
t1−h(t1)
|2 +ENα
∫ t1
t1−h(t1)
|ψ2j
t −ψ2
t |2dt+ENα
∫ t1
t1−h(t1)
|β2j
t −β2
t |2dt.
Due to sufficient smallness of α and from inequality (27) we receive that D → 0.
Thus,
(28) ψ2j
t → ψ2
t in L2
F (t0, t1;Rn), β2j
t → β2
t in L2
F (t0, t1;Rn×n.
Then, similarly way according to assumptions A1 – A2 and expression (28) using simple
transformations we obtain: Thus,
(29) ψj1
t → ψ1
t in L2
F (t0, t1;Rn), βj1
t → β1
t in L2
F (t0, t1;Rn×n.
Lemma 4 is proved.
It is follows from lemma 4 and assumptions A1 – A3 that we can proceed to the limit
in systems (17),(18),(26) and get fulfillment of (9),(10) and (11). Theorem is proved.
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E-mail : agayeva.cherkez@yasar.edu.tr
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