Optimal control in parabolic singular perturbated problem with obstacle
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Інститут прикладної математики і механіки НАН України
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irk-123456789-1692882020-06-10T01:26:31Z Optimal control in parabolic singular perturbated problem with obstacle Kapustyan, V.Y. 1999 Article Optimal control in parabolic singular perturbated problem with obstacle / V.Y. Kapustyan // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 162-167. — Бібліогр.: 7 назв. — англ. 0236-0497 http://dspace.nbuv.gov.ua/handle/123456789/169288 en Нелинейные граничные задачи Інститут прикладної математики і механіки НАН України |
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English |
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Article |
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Kapustyan, V.Y. |
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Kapustyan, V.Y. Optimal control in parabolic singular perturbated problem with obstacle Нелинейные граничные задачи |
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Kapustyan, V.Y. |
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Kapustyan, V.Y. |
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Optimal control in parabolic singular perturbated problem with obstacle |
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Optimal control in parabolic singular perturbated problem with obstacle |
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Optimal control in parabolic singular perturbated problem with obstacle |
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Optimal control in parabolic singular perturbated problem with obstacle |
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Optimal control in parabolic singular perturbated problem with obstacle |
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optimal control in parabolic singular perturbated problem with obstacle |
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Інститут прикладної математики і механіки НАН України |
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1999 |
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http://dspace.nbuv.gov.ua/handle/123456789/169288 |
| citation_txt |
Optimal control in parabolic singular perturbated problem with obstacle / V.Y. Kapustyan // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 162-167. — Бібліогр.: 7 назв. — англ. |
| series |
Нелинейные граничные задачи |
| work_keys_str_mv |
AT kapustyanvy optimalcontrolinparabolicsingularperturbatedproblemwithobstacle |
| first_indexed |
2025-07-15T04:02:41Z |
| last_indexed |
2025-07-15T04:02:41Z |
| _version_ |
1837684137582919680 |
| fulltext |
OPTIMAL CONTROL IN PARABOLIC SINGULAR
PERTURBATED PROBLEM WITH OBSTACLE.
c© Vladimir Y. Kapustyan
1. OPTIMALITY CONDITIONS.
Consider such optimal control problem with an obstacle: to find u(t) ∈ U = {v :
v(t) ∈ L2(0, T ), | v(t) |≤ ξ for a.e. t ∈ [0, T ]} such that
I(v) =
1
2
T∫
0
(
∫
Ω
(y(x, t)− z(x))2dx + νv2(t))dt → min, (1)
where y(x, t) is the solution of variational inequality of parabolic type in [1-2]
(yt(x, t)− ε24y(x, t)− g(x)v(t))(y(x, t)− ψ(x)) = 0a.e.inQ
yt(x, t)− ε24y(x, t)− g(x)v(t) ≥ 0,
y(x, t) ≥ ψ(x) a.e. in Q, (2)
y(x, 0) = y0(x), a.e. in Ω, y(x, t) = 0, a.e. in Σ;
here Q = Ω × (0, T ), Σ = ∂Ω × (0, T ), Ω ∈ Rn–has compact closure and smooth
(from C∞) (n−1)-dimensional boundary ∂Ω, z(x) ∈ L2(Ω), g(x) ∈ Lq(Ω), y0(x) ∈
W
2−2/q,q
0 (Ω), ψ(x) ∈ H2(Ω), ψ(x) ≤ 0 a.e. on ∂Ω, y0 ≥ ψ(x) a.e. in Ω, q >max(n,2),
0 < ε ¿ 1, ν =const> 0, 4 is the Laplace operator.
The problem (1)-(2) has at least one solution u. Let (y, u) be an pair from the problem
(1)-(2). Then ([1]) there exists a function p ∈ L2(0, T ; H1(Ω))
⋂
BV ([0, T ]; Y ∗), Y =
Hs(Ω)
⋂
H1(Ω), s > n/2 which satisfies the following equations:
−pt − ε24p = y(x, t)− z(x)a.e.in{(x, t) : y(x, t) > ψ(x)},
p(x, t) = 0, a.e. inΣ;
p(x, t)(g(x)u(t) + ε24y) = 0a.e.in{y = ψ}, (3)
p(x, T ) = 0a.e.inΩ,
u(t) =
−ξ, (g, p(·, t))− νξ > 0,
−ν−1(g, p(·, t)),
ξ, (g, p(·, t)) + νξ < 0,
(4)
where (g, p(·, t)) =
∫
Ω
g(x)p(x, t)dx.
2. FORMAL ASYMPTOTICS.
We shall find the outer [6] decomposition of the solution of (2)-(4) in the form
ȳ(x, t) =
∞∑
i=0
εiȳi(x, t); p̄(x, t) =
∞∑
i=0
εip̄i(x, t). (5)
ASSUMPTION 1. Suppose z(x), y0(x), ψ(x), 0 ≤ g(x) ∈ C∞(Ω)• Zero components
of the decomposition (5) are defined like the solution of the problem
(ȳ0t
(x, t)− g(x)ū0(t))(ȳ0(x, t)− ψ(x)) = 0 in Q,
ȳ0t
(x, t)− g(x)ū0(t) ≥ 0, ȳ0(x, t) ≥ ψ(x) in Q,
ȳ0(x, 0) = y0(x) in Ω; (6)
−p̄0t
= ȳ0(x, t)− z(x) in {(x, t) : ȳ0(x, t) > ψ(x)},
p̄0(x, t)g(x)ū0(t) = 0 a.e. in {ȳ0 = ψ},
p̄0(x, T ) = 0 in Ω,
ū0(t) =
−ξ, (g, p̄0(·, t))− νξ > 0,
−ν−1(g, p̄0(·, t)),
ξ, (g, p̄0(·, t)) + νξ < 0.
(7)
Introduce sets
Q0 = {(x, t) : y(x, t) = ψ(x)a.e.}, Q+ = {(x, t) : y(x, t) > ψ(x)a.e.}, (8)
Q = Q0
⋃
Q+, Q0
⋂
Q+ = ∅,
where y(x, t) is the solution of (2)-(4).
Then Q̄0, Q̄+ are their zeroth-order approximations.
ASSUMPTION 2. Suppose that p(x, t) = 0 in Q0 and let Q0 be a cylinder in
Rn+1 and its base be two-connected domain (Ω\Ω0). Suppose that the outer boundary
is equivalent to ∂Ω and the inner boundary (∂Ω) possesses properties of the outer
boundary •
Then the correlations are fulfilled in Q̄+
(x ∈ Ω0, t ∈ T 0
1 ) :
{
˙̄y0 = −g(x)ξ,
− ˙̄p0 = ȳ0 − z, (g, p̄0)0 − νξ > 0;
(9)
(x ∈ Ω0, t ∈ T 0
2 ) :
{
˙̄y0 = g(x)ξ,
− ˙̄p0 = ȳ0 − z, (g, p̄0)0 + νξ < 0;
(10)
(x ∈ Ω0, t ∈ T 0
3 ) :
{
˙̄y0 = −ν−1g(x)(g, p̄0)0,
− ˙̄p0 = ȳ0 − z,
(11)
T =
3⋃
i=1
T 0
i , T 0
i
⋂
T 0
j = ∅, i 6= j,
where (·, ·) denotes the scalar products by Ω̄0.
Let’s go over to the problems for (g, ỹ0)0, (g, p̄0)0 for the definition of zero components
of the control switching moments
t ∈ T 0
1 :
{
(g, ˙̃y0)0 = − ‖ g ‖20 ξ,
−(g, ˙̄p0)0 = (g, ỹ0)0, (g, p̄0)0 − νξ > 0;
(12)
t ∈ T 0
2 :
{
(g, ˙̃y0)0 =‖ g ‖20 ξ,
−(g, ˙̄p0)0 = (g, ỹ0)0, (g, p̄0)0 + νξ < 0;
(13)
t ∈ T 0
3 :
{
(g, ˙̃y0)0 = −ν−1 ‖ g ‖20 (g, p̄0)0,
−(g, ˙̄p0)0 = (g, ỹ0)0, ỹ0 = ȳ0 − z.
(14)
The problems (12)-(14) may be solved by dint of phase picture [3] which allows to define
the control structure. Then two cases are possible: 1) phase point ((g, ỹ0)0, (g, p̄0))
doesn’t go on bounds, i.e. it belongs to set P1 = {0 < (g, p̄0)0 ≤ νξ, ν−1/2 ‖ g ‖0
(g, p̄0)0 < (g, ỹ0)0 < ∞} ⋃{−νξ ≤ (g, p̄0)0 < 0, −∞ < (g, ỹ0)0 < ν−1/2× × ‖ g ‖0
(g, p̄0)0}; 2) phase point goes on bounds, i.e. it belongs to set P2 = {(g, p̄0)0 ≥ νξ,
ν−1/2 ‖ g ‖0 (g, p̄0)0 ≤ (g, ỹ0)0 < ∞} ⋃{−νξ > (g, p̄0)0, −∞ < (g, ỹ0)0 ≤ ν−1/2
‖ g ‖0 (g, p̄0)0}. In the case 1) the solution has an appearance:
(g, ỹ0)0 = (y0 − z, g)0ch((ν−1/2 ‖ g ‖0 ×
×(T − t))(ch(ν−1/2 ‖ g ‖0 T ))−1,
(g, p̄0)0 = ν1/2 ‖ g ‖−1
0 (y0 − z, g)0×
×sh(ν−1/2 ‖ g ‖0 (T − t))××(ch(ν−1/2 ‖ g ‖0 T ))−1
(15)
on condition that initial data satisfy the inclusion
{(y0 − z, g)0, ν1/2 ‖ g ‖−1
0 (y0 − z, g)0th(ν−1/2 ‖ g ‖0 T )} ∈ P1. (16)
In the case 2) systems (12)-(13) have the solution in 0 ≤ t ≤ τ0 (τ0 is the moments of
descent of control from limitation)
{
(g, ỹ0) = ∓ξ ‖ g ‖20 t + (g, y0 − z)0,
(g, p̄0) = ±ξ(ν − 1/2 ‖ g ‖20 [τ2
0 − t2]) + (g, y0 − z0)0(τ0 − t).
(17)
On the segment t ∈ (τ0, T ] system (14) has the solution
(g, ỹ0)0 = ±ξν1/2 ‖ g ‖0 ch(ν−1/2 ‖ g ‖0 (T − t))×
×(sh(ν−1/2 ‖ g ‖0 (T − τ0)))−1,
(g, p̄0)0 = ±ξνsh(ν−1/2 ‖ g ‖0 ×
×(T − t))(sh(ν−1/2 ‖ g ‖0 (T − τ0)))−1.
(18)
Let’s regard futher for definition that the condition
{(y0 − z, g)0, ξ(ν − 1/2 ‖ g ‖20 τ2
0 ) + (g, y0 − z)0τ0} ∈ P2
⋂
{(g, p̄0)0 > νξ, ν−1/2 ‖ g ‖0 (g, p̄0)0 ≤ (g, ỹ0)0 < ∞},
(19)
which guarantees uniqueness of the solution of equation
−ξ ‖ g ‖0 (ν1/2cth(ν−1/2 ‖ g ‖0 (T − τ0))+ ‖ g ‖0 τ0) = (g, y0 − z)0, (20)
is fulfilled. Thus in case 1) the couple (ȳ0(x, t), p̄0(x, t)) is fined from (11). In particular,
ȳ0(x, t) = y0(x)− g(x) ‖ g ‖−2
0 (y0 − z, g)0(ch(ν−1/2 ‖ g ‖0 T )−
−ch(ν−1/2 ‖ g ‖0 (T − t)))(ch(ν−1/2 ‖ g ‖0 T ))−1.
(21)
The question about choice of domain Ω̄0 is solved this way. Let Ω̃0 be a set from
Ω, which satisfies the condition y0(x) > ψ(x) and suppose that for any x ∈ Ω̃0 the
inequality
ȳ0(x, T ) > ψ(x) (22)
is fulfilled, at that the function ch(ν−1/2 ‖ g ‖0 T )− ch(ν−1/2 ‖ g ‖0 (T − t)) increases
monotonically. Systems (9)-(11) are the conditions of optimality in the optimal control
problem: find ū0(t) ∈ U such that
I0(v) =
1
2
T∫
0
(
∫
Ω̄0
(ȳ0(x, t)− z(x))2dx+
+νv2(t))dt → min
by bounds
˙̄y0(x, t) = g(x)v(t), ȳ0(x, 0) = y0(x).
Let ˜̃Ω0 be a system of expanded sets which belong to Ω and contain Ω̃0 (boundaries of
the indicated sets have the properties of the boundary ∂Ω). Then Ω̄0 is the solution of
the optimization problem
1
2
T∫
0
(
∫
˜̃Ω0
(ȳ0(x, t)− z(x))2dx +
∫
Ω/ ˜̃Ω0
(ψ(x)− z(x))2dx+
+ν(
∫
˜̃Ω0
g(x)p̄0(x, t))2)dt → min
(23)
by bound (16),(22), ȳ0(x, t) is given by the representation (21) and scalar products (·, ·)0
are calculated by ˜̃Ω0 in all terms.
In case 2) the solution of (9),(11), continuous for t ∈ [0, T ] and smooth for x ∈ Ω̄0, is
given by the couple (ȳ0(x, t), p̄0(x, t)). In particular, for t ∈ [0, τ0]
ȳ0(x, t) = −ξg(x)t + y0(x),
for t ∈ (τ0, T ]
ȳ0(x, t) = −ξg(x)τ0 + y0(x) + ξν1/2 ‖ g ‖−1
0 ×
×g(x)(sh(ν−1/2 ‖ g ‖0 (T − τ0)))−1(ch(ν−1/2×
× ‖ g ‖0 (T − t))− ch(ν−1/2 ‖ g ‖0 (T − τ0))).
(24)
The question about choice of domain Ω̄0 is solved by analogy with preceding case with
next changes: the function (23) is minimized by bounds (19),(20),(22) and ȳ0(x, t) is
given by representation (24). Let’s supplement the solutions (ȳ0(x, t), p̄0(x, t)) on ∂Ω̄0
by following boundary layer functions ỹ0(t̄, s, t), p̃0(t̄, s, t) [5,7].
Thereby the solution of (6)-(7) is constructed completely, i.e. zeroth components of
decomposition (5) are fined.
ASSUMPTION 3. Suppose the problem’s data such that the moment of the control
switching τ0 ∈ (0, T ) exists and ∂Ω̄0 = ∂Ω •
Let τ be a moment of the control descent from the bound of the initial problem.
Let’s to find it in the form of an asymptotic series
τ =
∞∑
j=0
εjτj .
The algorithm of the specification of the control switching moment is constructed in [4].
THEOREM. Let’s suppose that the assumptions 1-3 are true and (19) takes place.
Then the next inequalities hold
||grad(y − y(N))||L2(Q) + ||grad(p − p(N))||L2(Q) ≤ CεN ,
||y − y(N)||L2(Q) + ||p − p(N)||L2(Q) ≤ CεN+1,
||u − u(N)||L2(0,T ) ≤ CεN+1, |I(u) − I(u(N))| ≤ Cε2(N+1),
where
τN =
N∑
i=0
εiτi, y(N)(x, t) =
N∑
j=0
(ȳj(x, t) + ỹj(t̄, s, t))εj ,
p(N)(x, t) =
N∑
j=0
(p̄j(x, t) + p̃j(t̄, s, t))εj ,
u(N)(t) =
{ −ξ, 0 ≤ t ≤ τN ,
−ν−1 (g, p(N)(., t)), τN ≤ t ≤ T.
References
1. Barbu V., Optimal control 0f variation inequalities, Pitman, London, 1984.
2. Barbu V., Analysis and control of nonlinear infinite dimensional systems, Academic Presspubl,
Ins, 1993.
3. Boltyansky V.G., Mathematical methods of optimal control, Nauka, Moscow.
4. Kapustyan V. Y., Asymptotics of locally boundedcontrol in optimal parabolic problems., Ukr. math.
J. 48,N1, (1996), 50-56..
5. Kapustyan V. Y., Asymptotics of control in optimal singular pertyrbated parabolic problems. Global
bounds on control., Docl. AN (Russia), 333, N4, (1993), 428-431..
6. Nazarov S. A., Asymptotic solution of variation inequalittes for linear operator with small param-
eter by senior derivatives., Izv. AN USSR. Series of math., 54, N4, (1990), 754-773..
7. Vasil’eva A. B. and Butuzov V. F., Asymptotic methods in singular perturbations theory., Vysshaya
Shkola, Moscow., 1990.
8 - a / 43, Pisargevsky str.,
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