No regret control for Heat equation with delay and incomplete data
Abstract: The objective of this paper is to study the optimal control of distributed systems with incomplete data.
Particularly control for the heat equation with missing initial condition and delay parameter. The low-regret
control seems to be the best method to solve this kind of problems which characterized by an optimality systems.
Key-Words: Distributed system ; Incomplete data ; Low-regret control ; Optimality system.
Received: October 19, 2021. Revised: October 20, 2022. Accepted: November 20, 2022. Published: December 9, 2022.
1 Statement of problem
This paper is devoted to study the notion of no-regret
control for the heat equation with delay time and miss-
ing data. First, we proof the existence and the unique-
ness of the solution of this problem by the notion of
semigroup. Then, we show the approximate optimal-
ity system of the low-regret control.
Let Ωbe an open bounded set of Rnwith smooth
boundary ∂Ω. Consider the heat equation with
Dirichlet boundary condition, with Q= Ω ×(0, T ),
Σ = ∂Ω×[0, T ]and Qτ=Ω×(0, τ):
Ly (x, t)−By (x, t −τ) = vχω+f, in Q,
y(x, t) = 0, on Σ,
y(x, 0) = y0(x),in Ω,
By (x, t −τ) = g(x, t), in Qτ,
(1)
Where
L=∂
∂t −∆,
yis the variable in the state space H, t is the
time and vis a control function and f∈L2is a source
function,
χωis the characteristic function of ωa
bounded open of Ωand τ > 0is a delay parameter,
The intial data y0∈H,g∈C([0, T ];H),
Bis fixed bounded operator of Hin itself.
We then have a possible state for which , we attach
a cost function given by :
J(v, g) = RT
0kcy (v, g)−ydk2
Hdt +
NRT
0kvk2
Udt,
where
v∈ U =L2(ω)and g∈Gsuch that Gis
non empty closed subspace in L2(Qτ),
yd∈L2(Q), N > 0and c∈ L (V, H)
(V isaHilbertspace).
y=y(v, g)is the unique solution of the problem
(1) in L2(Q).
The idea is to find a solution for the following
inf sup problem :
inf (sup (v, g)) , v ∈ U and g∈G, but Jis not
bounded i.e. sup (v, g) = +∞, g ∈G.
Question
Can we controlled the optimal control
problem (1) by the no regret control ?
Objective : Our goal is to show the existence and
uniqueness of the solution of problem(1) .The main
results are given in section (3.4) ,and introduce the
notion of low-regret control.
Finally, in section (3.6) and (3.7) we give the per-
turbed optimality system and by passing to the limit
in the optimality system associated to the perturbed
problem we obtain the optimality system singular.
2 The existence and the uniqueness
of the solution
To study the existence and uniqueness of the solution
for a heat equation with delay, based on the work of
([20]) and ([36]) .
We consider an operator A= ∆ from Hinto itself
that generates a C0-semigroup (S(t))t≥0that is expo-
nentially stable i.e. there exist two positive constants
Mand wsuch that S(t)≤Mexp (−wt),∀t≥0.
For any initial y0∈H, there exists a unique solu-
tion y∈CL2(Ω ×[0, T ),H)of problem (1) .
Moreover,
y(t) = S(t)y0+Rt
0S(t−s)v(s)χω
+By (s−τ) + f(s)ds,
By (t−τ)=g(t)in (0, τ).
(2)
We use an iterative argument. Namely in
the interval (0, τ),problem (1) can be seen
REZZOUG IMAD, NECIB ABDELHALIM, OUSSAEIF TAKI-EDDINE
Department of Mathematics and Computer Science
University of Oum El Bouaghi, Algeria.
Laboratory of Dynamics Systems and Control
ALGERIA
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DOI: 10.37394/23203.2022.17.53
Rezzoug Imad, Necib Abdelhalim, Oussaeif Taki-Eddine
E-ISSN: 2224-2856
483
Volume 17, 2022
as an inhomogeneous evolution problem :
y0−∆y−g1(t) = vχω+f, inQ,
y= 0, onΣ,
y(0) = y0,inΩ,
where g1(t) = g(t)in (0, τ).
For all y0∈L2(Ω) and v∈
L20, τ;L2(Ω)this problem has a unique
solution y∈CL2(Ω ×(0, τ)) , H
(see"T h.1.2, Ch.6of P azy([20]) ")satisfying :
y(t) = S(t)y0+
Rt
0S(t−s) [v(s)χω+g1(s) + f(s)] ds,
∀t∈[0, τ],
this solution y(t)is obtained, for t∈[0, τ]
therefore on (τ, 2τ),problem (1) can be seen
as an inhomogeneous evolution problem :
y0−∆y−g2(t) = vχω+f, inQ,
y= 0, onΣ,
y(τ) = yτ,
where g2(t) = By (t−τ)in (τ, 2τ).
hence, this problem has a unique solution y∈
CL2(Ω ×[τ, 2τ]) , Hgiven by
y(t) = S(t)yτ+
Rt
0S(t−s) [v(s)χω+g2(s) + f(s)] ds,
∀t∈[τ, 2τ],
by iteration, we obtain a global solution ysatisfy-
ing (2) .
And so, we make sure the existence and the
uniqueness of the solution for each continuous initial
function.
If g∈C(0, T ;H),with Ha Hilbert space, then
the solution y∈C((0, T ) ; H)of problem (1) satis-
fies (2) .
Let be a classical solution of problem (1) and
(S(t))t≥0aC0−semigroup.
For all t∈[0, T ],consider the function
Ψ : (0, t)−→ Hdefined by : Ψ(s) =
S(t−s)y(s),0< s < t.
Since y∈H2(Ω) the function τ−→ S(τ)y(τ)
is differentiable for any τ > 0.
Therefore, Ψis differentiable on (0, t)and we
have : Ψ0(s) = −S0(t−s)y(s) + S(t−s)y0(s)
=−S(t−s) ∆y(s)1+
S(t−s) [∆y(s) + By (s−τ) + v(s)χω+f(s)]
=−S(t−s) ∆y(s) + S(t−s) ∆y(s) +
S(t−s) [By (s−τ) + v(s)χω+f(s)]
=S(t−s) [By (s−τ) + v(s)χω+f(s)] .
By integrating with respect to s, we finds
Rt
0Ψ0(s)ds =Rt
0S(t−s) [By (s−τ) + v(s)χω+f(s)] ds,
so,
Ψ (t)−Ψ (0) =
Rt
0S(t−s) [By (s−τ) + v(s)χω+f(s)] ds,
1S(t) = exp (∆t)
S(t−s) = exp ∆ (t−s)
S(t−s)y=exp ∆ (t−s)y
−S0(t−s)y=−∆exp (t−s)y=−S0(t−s) ∆y
this is equivalent to,
Ψ (t) = Ψ (0) +
Rt
0S(t−s) [By (s−τ) + v(s)χω+f(s)] ds,
as a result, (Ψ (s) = S(t−s)y(s))
S(t−t)y(t) = S(t−0) y(0) +
Rt
0S(t−s) [By (s−τ) + v(s)χω+f(s)] ds,
this implies that, (S(0) = I)
y(t) = S(t)y(0) +
Rt
0S(t−s) [By (s−τ) + v(s)χω+f(s)] ds.
3 No regret control
In this instant, we are interested in the study of the ex-
istence and characterization of no regret control. For
this, we assume that : [0, T ] = ∪[(l−1) τ, lτ ], l =
1, n.
We study the no regret control for t∈(0, τ )of the
following problem :
y0
1−∆y1=vχω+g+f, inQ,
y1= 0, onΣ,
y1(0) = y0, in Ω.
(3)
For a choice of vand g, the problem (3) admits a
unique solution denoted y1(v, g)∈L2(0, τ;V).
For t∈(0, τ)fixed, and for all g∈G, a closed
vector subspace of L2(Q),we pose :
J1(v, g) = Rτ
0kcy1(v, g)−ydk2
H+Nkvk2
Udt,
where
N > 0, C ∈ L L2(0; τ, V ), Hand yd∈
Hfixed.
Now, we study the optimality system (existence
and characterization) of the no regret control of the
problem (3) ,in the case where G6={0},i.e., find
the control for the problem
inf sup (J1(v, g)−J1(0, g)) ,(4)
We introduce low-regret control for developing the
no regret control of problem (3) .low-regret control
which is an approximation of no regret control.
4 low-regret control
In the previous chapter, we define the low-regret con-
trol by relaxed the problem (4) as follows :
inf sup J1(v1, g)−J1(0, g)−γkgk2
G, γ > 0.
(5)
With the low-regret control, we admit the possibil-
ity of making a choice v1controls, the best possible
choice of vis then given by :
starting by a linear case by relaxing the function
J1(v1, g)−J1(0, g) = J1(v1,0)
−J1(0,0) + 2 (S(0, v1), g)V,G ,∀g∈G,
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let be γ > 0a fixed number, then the re-
laxed problem (5) has the following formulation
: inf sup J1(v1,0) −J1(0,0)
+2 (S(0, v1), g)V,G −γkgk2
G!,it is
clear,
sup 2 (S(0, v1), g)V,G −γkgk2
G=
1
γkS(v1)k2
V,
the problem (5) we obtain :
inf J1(v1,0) −J1(0,0) + 1
γkS(v1)k2
V, γ > 0.
(6)
Hence we obtain a standard control optimal.
As for the linear case, (6) also admits a solution
noted uγ, that is the low-regret control.
Let y1(v, g)be the solution to the problem (3) .
Then,
y1(v, g) = y1(v1,0) + y1(0, g)−y1(0,0) ,
with simple calculation, we obtain :
J1(v1,0) −J1(0,0) + 2Ξ = J1(v1, g)−J1(0, g),
(7)
such that
Ξ = Rτ
0(ξ1(v1), y1(0, g)−y1(0,0))V,G dt,
and c∗c(y1(., 0) −y1(0,0)) = ξ1(.).
The adjoint state ξ1(v1)defined by ξ1(v1) =
c∗c(y1(v1,0) −y1(0,0)). So, (7) becomes
J1(v1, g)−J1(0, g) = J1(v1,0)−J1(0,0)+(ξ(v1), g)V,G .
(8)
Where ξ(v1) = S(v1)and ξ1is the solution of prob-
lem :
−ξ0
1−∆ξ1=c∗c(y1(v1,0) −y1(0,0)) , ξ1(τ, v1) =
0.
Hence, the problem (8) becomes for all γ > 0
find uγ∈ U such thatJγ
1(u1γ) = inf Jγ
1(v1), v ∈
U,
where the now cost function is given by
Jγ
1(v1) = J1(v1,0) −J1(0,0) + 1
γkS(v1)k2
V.
5 Approximate optimal system
The low-regret control u1γis characterized by the
unique solution{y1γ, ξ1γ, ρ1γ, p1γ}of the optimality
system :
y0
1γ−∆y1γ=
y1γ(0) =
−ξ0
1γ−∆ξ1γ=
ξ1γ(τ) =
ρ0
1γ−∆ρ1γ=
ρ1γ(τ) =
−p0
1γ−∆p1γ=
p1γ(τ) =
p1γ+Nu1γ=
u1γχω+g1+f,
y0,
c∗c(y1γ−y1(0,0)) ,
0,
0,
1
γξ1γ,
c∗c(y1γ−yd) + c∗cρ1γ,
0,
0, in U.
(9)
The solution u1γsatisfies the first order optimality
conditions gives
J
0γ
1(u1γ) (v1−u1γ)≥0,∀v1∈ U,
where
J
0γ
1(u1γ) (v1−u1γ) =
lim 1
t(Jγ
1(u1γ+t(v1−u1γ)) −Jγ
1(u1γ)) ≥0.
Simple calculations gives
1
t(Jγ
1(u1γ+t(v1−u1γ)) −Jγ
1(u1γ)) =
tkcy1γ(v1−u1γ,0)k2
H
+2 Rτ
0(cy1γ(u1γ,0) −yd, cy1γ(v1−u1γ,0))H×Hdt
+tN kv1−u1γk2
U+
2NRτ
0(u1γ, v1−u1γ)U×U dt
+t
γkS(v1−u1γ)k2
V+
21
γS(u1γ), S(v1−u1γ)G×G.
Make ttend to 0to get
J
0γ
1(u1γ) (v1−u1γ) =
2Rτ
0(cy1γ(u1γ,0) −yd, cy1γ(v1−u1γ,0))H×Hdt
+2NRτ
0(u1γ, v1−u1γ)U×U dt +
21
γS(u1γ), S(v1−u1γ)G×G,
thanks to linearity
y1γ(v1−u1γ,0) = y1γ(v1,0) −y1γ(u1γ,0)
=y1γ(v1−u1γ,0) −y1γ(0,0) .
Then,
J
0γ
1(u1γ) (v1−u1γ) =
2Rτ
0cy1γ(u1γ,0) −yd,
c(y1γ(v1−u1γ,0) −y1γ(0,0)) H×H
dt
+2NRτ
0(u1γ, v1−u1γ)U×U dt +
21
γS(u1γ), S(v1−u1γ)G×G
= 2 Rτ
0c∗c(y1γ(u1γ,0) −y1γ(0,0)) ,
y1γ(v1−u1γ,0) −y1γ(0,0) H×H
dt
+2NRτ
0(u1γ, v1−u1γ)U×U dt +
21
γS(u1γ), S(v1−u1γ)G×G,
we recall the adjoint state defined previ-
ously by ξ1(u1γ) = c∗c(y1(u1γ,0) −y1(0,0))
then 1
γS(u1γ), S(v1−u1γ)G×G=
1
γξ(u1γ), ξ(v1−u1γ)G×G.
6 Singular Optimality system (SOS)
we now establish the optimality system for the no re-
gret control. For this, we need an additional hypothe-
sis.
Hypothesis : Let (ρ1γ, δ1γ)∈L2(0; τ, V )2de-
fined by : ρ0
1γ−∆ρ1γ= 0,
ρ1γ(0)=g1, g1∈G,
and −δ0
1γ−∆δ1γ
δ1γ(τ)
=
=
c∗cρ1γ,
0,and we de-
fended the continuous operator R:F −→ U by
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Rg=δ, such that : kRgkU1≥ckgkF, c > 0, g ∈G.
where Fis a vector subspace of G.
Under the hypothesis, then the no regret control
u1for the system (3) is characterizated by the unique
solution {y1, λ1, ρ1, p1}of the optimality system :
y0
1−∆y1=
y1(0) =
−ξ0
1−∆ξ1=
ξ1(τ) =
ρ0
1−∆ρ1=
ρ1(0) =
−p0
1−∆p1=
p1(τ) =
p1+Nu1=
u1χω+g1+f,
y0,
c∗c(y1−y1(0,0)) ,
0,
0,
λ,
c∗c(y1−yd) + c∗cρ1,
0,
0, in U.
(10)
With λ∈ˆ
Gis the completeness of Gin F, containing
the elements Rg.
By the following theorem : “The unique low-
regret control uγis converge weakly when γtends to
0to the unique no-regret control uin Uad” the unique
low-regret control u1γconverge weakly when γtend
to 0to the unique no regret control u1in the approx-
imation of the optimal system (9) we find the system
(10) .
We repeat the same steps on each time interval
[(l−1) τ, lτ],l= 1, n we obtain a low-regret control
ulγ converge weakly to the unique no regret control
ul.
Finally, the low-regret control of problem (1) is
given by uγ=uiγ ∈ U,∀t∈[(i−1) τ, iτ ], i = 1, n
solution of the following problem :
inf Jγ(v) = inf Jγ
i(vi), t ∈[(i−1) τ, iτ], v ∈
U, i = 1, n,
where
Jγ
i(vi) = Ji(vi,0) −Ji(0,0) + 1
γkS(vi)k2
V,
the low-regret control is characterizated by the
unique solution {yiγ , ξiγ , ρiγ , piγ }of the approximate
optimal system :
y0
iγ −∆y1γ=
yiγ (0) =
−ξ0
iγ −∆ξiγ =
ξiγ (τ) =
ρ0
iγ −∆ρiγ =
ρiγ (τ) =
−p0
iγ −∆piγ =
piγ (τ) =
piγ +Nuiγ =
uiγ χω+ziγ (t, 1) + f,
y0,
c∗c(yiγ −yi(0,0)) ,
0,
0,
1
γξiγ ,
c∗c(yiγ −yd) + c∗cρiγ ,
0,
0, in U.
(11)
The low-regret control uγconverge weakly to the
unique no regret control uwhen γ−→ 0,the no re-
gret control ufor the system (1) ,is characterizated by
the unique solution {y, λ, ρ, p}of the Singular Opti-
mality system (SOS):
y0−∆y
y(0)
−ξ0−∆ξ
ξ1(iτ)
ρ0−∆ρ
ρ(0)
−p0−∆p
p(iτ)
p+Nu
=
=
=
=
=
=
=
=
=
uχω+z(t, 1) + f,
y0,
c∗c(y−y(0,0)) ,
0,
0,
λ,
c∗c(y−yd) + c∗cρ,
0,
0, inU.
(12)
Where y=yi, ρ =ρi, z =zi, p =pi, t ∈
[(i−1) τ, iτ ].
7 Conclusion
This paper is devoted to study the characterization of
the no regret control for the heat equation with delay
and missing data. First, we establish the existence and
the uniqueness of the weak solution for the heat equa-
tion by introducing the notion of semigroup. Then, we
associate the no-regret control by a sequence of low-
regret control defined by a quadratic perturbation. We
show here that the perturbed system which character-
ize the sequence of low-regret control converges to
the no-regret control for which we get a singular opti-
mality system. Finally, we can construct and charac-
terize the no-regret control for the heat equation with
missing data and delay time from the no-regret control
method already established on each interval contained
in this work.
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Acknowledgement
The authors thank the referee for his helpful com-
ments and suggestions.
This work was supported by the Directorate-General
for Scientific Research and Technological Develop-
ment (DGRSDT).
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WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2022.17.53
Rezzoug Imad, Necib Abdelhalim, Oussaeif Taki-Eddine
E-ISSN: 2224-2856
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