Construction of exact control for a one-dimensional heat equation with delay

Universität Konstanz

Construction of Exact Control for a One-Dimensional Heat Equation with Delay

Denys Ya. Khusainov Michael Pokojovy Elvin I. Azizbayov

Konstanzer Schriften in Mathematik und Informatik Nr. 317, Juli 2013

ISSN 1430-3558

© Fachbereich Mathematik und Statistik Universität Konstanz

Fach D 197, 78457 Konstanz, Germany

Konstanzer Online-Publikations-System (KOPS)

URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-241387

Equation with Delay

Denys Ya. Khusainov

∗

, Mihael Pokojovy

†

, Elvin I. Azizbayov

‡

July21, 2013

Abstrat

Weproveanexatontrollabilityresultforaone-dimensionalheatequationwithdelayinboth

lowerand highestordertermsandnonhomogeneousDirihletboundaryonditions. Moreover,we

giveanexpliit representationoftheontrol funtion steeringthesysteminto agivennal state.

UnderertaindeaypropertiesfororrespondingFourieroeientswhihanbeinterpretedasa

suientlyhigh Sobolev regularityof thedata,bothontrolfuntion andthe solutionare proved

to beregularinthelassialsensebothwithrespettotimeandspaevariables.

1 Introdution

Studying and developing mathematial models to desribe various phenomena in physis, eonomis,

eologyandpopulationdynamis,et.,areoneofentralproblemsofthemodernappliedmathematis

(f. [6℄, [14℄). Integral and dierential equations with lumped and distributed parameters proved to

be a useful and eient tool for suh studies. Whereas evolution equations with lumped parameters

have alreadybeenrather well investigated (see,e.g.,[7 ℄), therestill remaina lotofopen questionsfor

the ase of dynamialsystems withdistributed parameters (p. monographs [12 ℄, [13℄ and referenes

therein).

The sope of the present paper is a linear one-dimensional heat equation in a bounded domain with

disrete delay interms of both lower and highestorders. Reently, an abstrat semigroup treatment

wasproposedfor distributedsystemswithdelays (viz. [3 ℄,[4 ℄). Thoughthis rathergeneral framework

provides good analytialand ontrol-theoretial tools for various delaysenarios, tehnial diulties

may arrive when applying to problems with delayin thehighestorder terms whih have nevertheless

been solved in[5℄for ertainparaboli-type equations.

Another important problem onsistsinobtaining expliitrepresentation formulas for thesolutions to

distributed evolution equations with delay. We refer to [2 ℄, [9℄, [10 ℄, [11 ℄ for details. Suh represen-

tation formulas an then be naturally used to arefully study the solutions, obtain semi-analytial

approximations, addressontrollabilityand optimal ontrol problems,et.

2 Representation of solutions to the heat equation with delay

In [11℄,anonhomogeneous one-dimensional heatequationwithdelay

v _t (x, t) = a ² ₁ v _xx (x, t)+a ² ₂ v _xx (x, t −τ )+b ₁ v _x (x, t)+b ₂ v _x (x, t −τ )+d ₁ v(x, t)+d ₂ v(x, t −τ )+g(x, t),

^(2.1)

∗

DepartmentofCybernetis,KievNationalTarasShevhenkoUniversity,Ukraine

†

DepartmentofMathematisandStatistis,UniversityofKonstanz,Germany

‡

DepartmentofMehanisandMathematis,BakuStateUniversity,Azerbaijan

dened for

0 ≤ x ≤ l

^and

t ≥ 0

⁽

l > 0

^{), wasstudied. The oeients for the phase}derivatives were assumed to be proportional, i.e., theremust exista onstant

µ ∈ R

suhthat

µ = − _2a ^{b 1} 2

1

= − _2a ^{b 2} 2 2

holds

true. A Dirihlet initial boundary valueproblemwithnonhomogeneous initial

v(x, t) = ψ(x, t)

^for

0 ≤ x ≤ l, −τ ≤ t ≤ 0

^(2.2)

and boundaryonditions

v(0, t) = θ ₁ (t), v(l, t) = θ ₂ (t)

^for

t ≥ −τ

^(2.3)

wasonsideredunderan additional ompatibilityondition onthedata:

ψ(0, t) = θ ₁ (t), ψ(l, t) = θ ₂ (t)

^for

t ≥ −τ

Performing thesubstitution

v(x, t) := e ^µx u(x, t)

^with

µ = − b ₁

2a ² ₁ = − b ₂ 2a ² ₂ ,

Equation (2.1 )wastransformed to

u _t (x, t) = a ² ₁ u _xx (x, t) + a ² ₂ u _xx (x, t − τ ) + c ₁ u(x, t) + c ₂ u(x, t) + f (x, t)

^(2.4)

with

c ₁ := d ₁ − b ² ₁

4a ² ₁ , c ₂ := d ₂ − b ² ₂

4a ² ₂ , f (x, t) := e ^−µx g(x, t)

wherebythe initial andboundaryonditions readas

u(x, t) = ϕ(x, t)

^for

0 ≤ x ≤ l, −τ ≤ t ≤ 0, ϕ(x, t) := e ^−µx ψ(x, t)

^(2.5)

and

u(x, 0) = µ ₁ (t), u(l, t) := µ ₂ (t)

^for

− τ < t < 0, µ ₁ (t) := θ ₁ (t), µ 2 (t) := e ^−µl θ ₂ (t),

^(2.6)

respetively.

Following [10 ℄, the delayed exponential funtion

exp _τ (b, ·)

^{wasintrodued.}

Denition 1. For

τ > 0

^,

b ∈ R

(or

b ∈ C

),dene for eah

t ∈ R

:

exp _τ (b, t) :=



 

 

 

 

 

 

 

 

 

 

0, −∞ < t < −τ,

1, −τ ≤ t < 0,

1 + b _1! ^t , 0 ≤ t < τ, 1 + b _1! ^t + b ^{2 ( b−τ)} _2! ² , τ ≤ t < 2τ,

. . . . . .

1 + b _1! ^t + · · · + b ^{k (t−(k−1)τ)}

k

k! , (k − 1)τ ≤ t < kτ,

. . . . . .

(2.7)

See Figure 1for a plotof the delayed exponential funtion.

Using thespeialfuntion giveninEquation (2.7 ),thelassial solutionto theinitial boundary value

problem (2.4 )(2.6 )withdelay anbe representedas

u(x, t) = S ₁ (ϕ, µ 1 , µ ₂ )(x, t) + S ₂ (f, µ 1 , µ ₂ )(x, t) + µ ₁ (t) + ^{(µ 2 (t)−µ} _l ^{1 (t)} x

^(2.8)

−1 −0.5 0 0.5 1 1.5 2 0

1 2 3 4 5 6 7

t ex p τ ( − 1 , t )

τ = 0.01 τ = 0.1 τ = 0.5

−1 −0.5 0 0.5 1 1.5 2

0 1 2 3 4 5 6 7

t ex p τ (0 , t )

τ = 0.01 τ = 0.1 τ = 0.5

−1 −0.5 0 0.5 1 1.5 2

0 1 2 3 4 5 6 7

t ex p τ (1 , t )

τ = 0.01 τ = 0.1 τ = 0.5

Figure1: Delayed exponential funtion

exp _τ (b, ·)

withlinear operators

S ₁ (ϕ, µ 1 , µ ₂ )(x, t) =

∞

X

n=1

e ^{L n (t+τ)} exp _τ (D n , t)Φ n (−τ )+

Z ₀

−τ

e ^{L n (t−s)} exp _τ (D n , t − τ − s)( ˙ Φ n (s) − L _n Φ n (s))ds

sin( ^πn _l x), S ₂ (f, µ ₁ , µ ₂ )(x, t) =

∞

X

n=1

Z t 0

e ^{L n (t−s)} exp _τ (D n , t − τ − s)F n (s)ds

sin( ^πn _l x ,

(2.9)

where

F n (t) = 2 l

Z t 0

f (ξ, t) sin( ^πn _l x)ξdξ + M n (µ ₁ , µ ₂ )(t), Φ n (t) = 2

l Z l

0

ϕ(ξ, t) sin( ^πn _l ξ)dξ + m _n (µ 1 , µ ₂ )(t)

(2.10)

with

M _n (µ ₁ , µ ₂ )(t) = 2 l

Z l 0

− d dt

µ ₁ (t) + ^{µ 2 (t)−µ} _l ^{1 (t)} ξ

+ ^b _l ² (µ ₁ (t − τ ) − µ ₂ (t − τ ))

sin( ^πn _l ξ)ξdξ+

c ₁ m n (µ ₁ , µ ₂ )(t) + c ₂ (µ ₁ , µ)(t − τ ), m _n (µ ₁ , µ ₂ )(t) = 2

l Z l

0

µ ₁ (t) + ξ

l (µ ₂ (t) − µ ₁ (t))

sin( ^πn _l ξ)dξ

and

L n = c ₁ − ^πn _l a ₁ 2

, D _n =

c ₂ − ^πn _l a ₂ 2 e ⁻

c 1 − ( ^πn _l ^{a 1} ) ^{2 τ} .

(2.11)

inEquation (2.9 ). Regarding itsonvergene, thefollowing resultwasshown in[11℄(f. also[10℄).

Theorem 2. For

T > 0

^,

m := ⌈ ^T _τ ⌉

^and

α > 0

^{, let}

F ∈ C ⁰ ([0, l] × [0, T ], R )

^,

Φ, ∂ t Φ, ∂ tt Φ ∈ C ⁰ ([0, l] × [0, T ], R )

^{be suh thattheir Fourier oeients}

F _n

^and

Φ n

^satisfy

n→∞ lim n ^2m+1+δ max

s∈[−τ,0]

|Φ ^′′ _n (s)| + n ² |Φ ^′ _n (s)| + n ⁴ |Φ n (s)|

= 0,

n→∞ lim max

1≤k≤m n ^{2(m−k)+1+δ} max

(k−1)τ≤s≤kτ

|F _n ^′ (s)| + n ² |F n (s)|

= 0.

Undertheseonditions,problem(2.4 )(2.6 )possessesauniquelassialsolution

u ∈ C ⁰ ([0, l]×[0, T ], R )

with

∂ _t u, ∂ _xx u ∈ C ⁰ ([0, l] × [0, T ], R )

^{. Moreover, the funtions}

u

^,

∂ _t u

^{, and}

∂ _xx u

^are represented by uniformlyandabsolutelyonvergentFourierseriesgivenin(2.8)orobtainedbyaterm-wiseappliation

of

∂ t

^or

∂ xx

^{to (2.8 ),} respetively.

Remark 3. Using standard arguments from the ellipti theory, the onditions of Theorem 2 an be

interpreted as arequirement for the data

ϕ

^,

µ ₁

^,

µ ₂

^,

f

^{to belong to ertain Sobolev spaes (f. [1 ℄)of}

funtions withsuiently many weak derivatives (s. [11℄). The larger

T

^and

α

^{are,thesmoother the}

dataare supposedto be.

Representing ofthe solution to the initial boundary value problem(2.4 )(2.6 )intheform (2.8 )is not

alwaysonvenient when the impatoftheinitial andboundary valuesor theinhomogeneityhasto be

treated separately. For our purposes,itis neessarytosplit orrespondingterms into dierent sums.

Expanding therst sumin(2.8) and performing integration byparts, we obtain

S ₁ (ϕ, µ ₁ , µ ₂ )(x, t)

=

∞

X

n=1

e ^{L n (t+τ)} exp _τ (D n , t)Φ n (−τ )

sin( ^πn _l x)+

∞

X

n=1

Z 0

−τ

e ^{L n (t−s)} exp _τ (D n , t − τ − s)( ˙Φ n (s) − L n Φ n )ds

sin( ^πn _l x)

=

∞

X

n=1

e ^{L n (t+τ)} exp _τ (D _n , t)Φ _n (−τ )

sin( ^πn _l x)+

∞

X

n=1

e ^{L n (t−s)} exp _τ (D n , t − τ − s)Φ n (s)

s=0 s=−τ

sin( ^πn _l x)−

∞

X

n=1

Z 0

−τ

−L n e ^{L n (t−s)} exp _τ (D n , t − τ − s)

Φ n (s)ds

sin( ^πn _l x)−

∞

X

n=1

Z 0

−τ

−e ^{L n (t−s)} D _n exp _τ (D _n , t − 2τ − s)

Φ _n (s)ds

sin( ^πn _l x)

=

∞

X

n=1

e ^{L n t} exp _τ (D _n , t − τ )Φ _n (0) + D _n Z 0

−τ

e ^{L n (t−s)} exp _τ (D _n , t − 2τ − s)Φ _n (s)ds

sin( ^πn _l x).

Plugging

Φ n

^{from Equation(2.10 ),weget}

S ₁ (ϕ, µ ₁ , µ ₂ )(x, t)

=

∞

X

n=1

e ^{L n t} exp _τ (D n , t) 2

l Z l

0

ϕ(ξ, 0) sin( ^πn _l ξ)dξ + m _n (µ ₁ , µ ₂ )(0)

sin( ^πn _l x)+

∞

X

n=1

D n

Z ₀

−τ

e ^{L n (t−s)} exp _τ (D n , t−2τ −s) 2

l Z l

0

ϕ(ξ, s) sin( ^πn _l x)dξ+m n (µ ₁ , µ ₂ )(s)

ds

sin( ^πn _l x).

Expanding thesum inEquation(2.9) and pluggin

F _n

^{fromEquation (2.10 )yields}

S ₂ (ϕ, µ 1 , µ ₂ )(x, t)

=

∞

X

n=1

Z t 0

e ^{L n (t−s)} exp _τ (D n , t − τ − s)F n (s)

sin( ^πn _l x)

=

∞

X

n=1

Z t 0

e ^{L n (t−s)} exp _τ (D n , t − τ − s) 2

l Z l

0

f (ξ, s) sin( ^πn _l ξ)dξ+M n (µ ₁ , µ ₂ )

ds

sin( ^πn _l x).

Thus,thesolution

u

^{to the initial boundary value problem(2.4)(2.6)given inEquation (2.8 ) an be}

written asfollows:

u(x, t) =

∞

X

n=1

e ^{L n t} exp _τ (D n , t − τ ) 2

l Z t

0

ϕ(ξ, 0) sin( ^πn _l ξ)

sin( ^πn _l x)+

∞

X

n=1

e ^{L n t} exp _τ (D _n , t)m _n (µ ₁ , µ ₂ )(0)

sin( ^πn _l x)+

∞

X

n=1

D _n

Z 0

−τ

e ^{L n (t−s)} exp _τ (D _n , t − 2τ − s)

2 l

Z l 0

ϕ(ξ, s) sin( ^πn _l ξ)dξ

ds

sin( ^πn _l x)+

∞

X

n=1

D _n

Z 0

−τ

e ^{L n (t−s)} exp _τ (D n , t − 2τ − s)m n (µ ₁ , µ ₂ )(s)ds

sin( ^πn _l x)+

∞

X

n=1

Z t 0

e ^{L n (t−s)} exp _τ (D n , t − τ − s)

2 l

Z l 0

f (ξ, s) sin( ^πn _l ξ)dξ

ds

e ^{− α 2 x} sin( ^πn _l x)+

∞

X

n=1

Z t 0

e ^{L n (t−s)} exp _τ (D _n , t − τ − s)M _n (µ ₁ , µ ₂ )ds

sin( ^πn _l x) + µ ₁ (t) + ^{µ 2 (t)−µ} _l ^{1 (t)} x.

Now, we ollet appropriate terms inthe following three operators therst one depending on the

initial data:

S ˜ ₁ (ϕ)(x, t) =

∞

X

n=1

e ^{L n t} exp _τ (D n , t − τ ) 2

l Z t

0

ϕ(ξ, 0) sin( ^πn _l ξ)

sin( ^πn _l x)+

∞

X

n=1

D _n

Z 0

−τ

e ^{L n (t−s)} exp _τ (D _n , t − 2τ − s)

2 l

Z l 0

ϕ(ξ, s) sin( ^πn _l ξ)dξ

ds

sin( ^πn _l x),

theseond one depending on theboundary data:

S ˜ ₂ (µ ₁ , µ ₂ )(x, t) =

∞

X

n=1

e ^{L n t} exp _τ (D n , t)m n (µ ₁ , µ ₂ )(0)

sin( ^πn _l x)+

∞

X

n=1

D _n

Z 0

−τ

e ^{L n (t−s)} exp _τ (D _n , t − 2τ − s)m _n (µ ₁ , µ ₂ )(s)ds

sin( ^πn _l x)+

∞

X

n=1

Z t 0

e ^{L n (t−s)} exp _τ (D n , t − τ − s)M n (µ 1 , µ ₂ )ds

sin( ^πn _l x) + µ ₁ (t) + ^{µ 2 (t)−µ} _l ^{1 (t)} x,

and thethird one depending onthe inhomogeneity:

S ˜ ₂ (f )(x, t) =

∞

X

n=1

Z t 0

e ^{L n (t−s)} exp _τ (D _n , t − τ − s)

2 l

Z l 0

f(ξ, s) sin( ^πn _l ξ)dξ

ds

sin( ^πn _l x).

Thus,we arrive at

u(x, t) = ˜ S ₁ (ϕ)(x, t) + ˜ S ₂ (µ ₁ , µ ₂ )(x, t) + ˜ S ₂ (f )(x, t) + ^{µ 2 (t)−µ} _l ^{1 (t)} x.

^(2.12)

In this setion, we onsider the following exat ontrollability problem. Given an initial state

ϕ

^and

boundary data

γ ₁ , γ ₂

^{, replae}

f

^{with a ontrol funtion}

U

^{suh that the solution}

u

^to (2.4 )(2.6 ) is steered into agiven nalstale

Ψ

^{at a presribed time}

T > 0

^,i.e.,

u(x, T ) = Ψ(x)

^for

0 ≤ x ≤ l.

^(3.1)

Sineweareinterestedinlassialsolutions,aompatibilityonditionontheboundaryonditionsand

theend state hasto beimposed:

Ψ(0) = µ ₁ (T ), Ψ(l) = µ ₂ (T).

As it follows from the representation formula given in Equation (2.12 ), Equation (3.1 ) is satised if

and only if

S ˜ ₁ (ϕ)(x, T ) + ˜ S ₂ (µ 1 , µ ₂ )(x, T ) + ˜ S ₂ (U )(x, T ) + ^{µ 2 (T )−µ} _l ^{1 (T )} x = Ψ(x)

^for

0 ≤ x ≤ l.

^(3.2)

Weexpandthefuntions

Ψ

^and

x 7→ ^{µ 2 (T )−µ} _l ^{1 (T )} x

^{ontheinterval}

(0, l)

^{intoFourier serieswithrespet}

to the eigenfuntions ofthe orresponding elliptioperator. Equation(2.10 )yields then

Ψ(x) =

∞

X

n=1

Ψ n sin( ^πn _l x)

^with

Ψ n (x) = 2 l

Z l 0

Ψ(ξ) sin( ^πn _l ξ)dξ

^for

n ∈ N , µ ₁ (t) + ^{µ 2 (t)−µ} _l ^{1 (t)} =

∞

X

n=1

m _n (µ ₁ , µ ₂ ) sin( ^πn _l x).

Weassume now

U

^{to alsohave anexpansion inFourier series ofthe form:}

U (x, t) =

∞

X

n=1

U _n (t) sin( ^πn _l x).

^(3.3)

The operator

S ˜ ₃ (U )

^readsthenas

S ˜ ₃ (U)(x, t) =

∞

X

n=1

Z t 0

e ^{L n (t−s)} exp _τ (D n , t − τ − s)U n (s)ds

sin( ^πn _l x).

Thus,theontrollability onditionrewrites as

S ˜ ₁ (ϕ)(x, T ) + ˜ S ₂ (µ ₁ , µ ₂ )(T, x) +

∞

X

n=1

Z T 0

e ^{L n (t−s)} exp _τ (D n , t − τ − s)U n (s)ds

sin( ^πn _l x)+

∞

X

n=1

m n (µ ₁ , µ ₂ )(T ) sin( ^πn _l x) =

∞

X

n=1

Ψ n sin( ^πn _l x).

Denote

s _1n (t) = e ^{L n (t)} exp _τ (D _n , t) 2

l Z l

0

ϕ(ξ, 0) sin( ^πn _l ξ)dξ

+ D _n

Z l 0

e ^{L n (t−s)} exp _τ (D n , t − 2τ − s) 2

l Z l

0

ϕ(ξ, s) sin( ^πn _l ξ)dξ

ds, s _2n (t) =

Z t 0

e ^{L n (t−s)} exp _τ (D n , t − τ − s)M n (µ ₁ , µ ₂ )(s)ds.

There follows thenfrom (3.2) that the ontrollabilityproblem for an arbitrarytime

T > 0

^{redues to}

nding funtions

u n

^{satisfying the following ondition}

∞

X

n=1

(s _1n (T ) + S _2n (T )) sin( ^πn _l x) +

∞

X

n=1

Z t 0

e ^{L n (t−s)} exp _τ (D n , T − τ − s)U n (s)ds

sin( ^πn _l x)+

∞

X

n=1

m _n (µ ₁ , µ ₂ )(T ) sin( ^πn _l x) =

∞

X

n=1

Ψ n sin( ^πn _l x),

whih isinits turnequivalent toa systemof ountably manyFredholmintegral equations oftherst

type:

s _1n (T ) + s _2n (T ) + Z T

0

e ^{L n (T −s)} exp _τ (D n , T − τ − s)U n (s)ds + m _n (µ ₁ , µ ₂ )(T ) = Ψ n

^for

n ∈ N .

^(3.4)

Lemma 4. For

τ > 0

^,

D 6= 0

^{,thereholds for arbitrary}

T > 0 Z T −τ

−τ

exp _τ (D, s)ds = _D ¹ (exp _τ (D, T ) − 1).

Proof. There existsa unique

k ∈ N

suh that

(k − 2)τ ≤ T − τ < (k − 1)τ

^{. Therefore,}

Z T −τ

−τ

exp _τ (D, s)ds = Z 0

−τ

ds + Z τ

0

1 + D _1! ^s ds +

Z 2τ τ

1 + D _1! ^s + D ^{2 ( s−τ)} _2! ² ds+

Z 3τ 2τ

1 + D _1! ^s + D ^{2 ( s−τ)} _2! ² + D ^{3 ( s−2τ)} _3! ³

ds + · · · + Z _(k−1)τ

(k−2)τ

1 + D _1! ^s + D ^{2 ( s−τ)} _2! ² + D ^{3 ( s−2τ)} _3! ³ + · · · + D ^{k (s−(k−2)τ)} _(k−1)! ^k−1 ds.

Performing theintegration, we obtain

Z T −τ

−τ

exp _τ (D, s)ds = _1! ^s

s=0 s=−τ +

s

1! + D ^s _2! ²

s=t s=0 +

s

1! + D ^s _2! ² + D ^{2 ( s−τ)} _3! ³

s=2τ s=τ + s

1! + D ^s _2! ² + D ^{2 (s−τ)} _3! ³ + D ^{3 (s−2τ)} _4! ⁴

s=3τ

s=2τ + · · · + s

1! + D ^s _2! ² + D ^{2 ( s−τ)} _3! ³ + D ^{3 ( s−2τ)} _4! ⁴ + · · · + D ^{k (s−(k−2)τ)}

k

k!

s=T −τ s=(k−2)τ

= _1! ^τ +

τ

1! + D ^τ _2! ²

+

2τ

1! + D ^(2τ) _2! ² + D ^{2 τ} _3! ³

−

τ

1! + D ^τ _2! ² + 3τ

1! + D ^(3τ) _2! ² + D ^{2 (2} _3! ^{τ) 3} + D ^{3 τ} _4! ⁴

−

2τ

1! + D ^(2τ) _2! ² + D ^{2 τ} _3! ³

+ · · · + T −τ

1! + D ^{(T −τ)} _2! ² + D ^{2 ( T −2τ)} _3! ³ + D ^{3 ( T −3τ)} _4! ⁴ + · · · + D ^{k (T −τ−(k−1)τ)}

k +1

(k+1)!

− (k−1)τ

1! + D ^((k−1)τ) _2! ² + D ^{2 (( k−2)τ)} _3! ³ + D ^{3 (( k−3)τ)} _4! ⁴ + · · · + D ^{k−1 τ} _k! ^k

=

(T −τ)

1! + D ^{(T −τ)} _2! ² + D ^{2 ( D−2τ)} _3! ³ + D ^{3 ( T −3τ)} _4! ⁴ + · · · + D ^{k (T −τ−(k−1)τ)} _(k+1)! ^k+1 + _1! ^τ .

Thus,we an write

Z T −τ

−τ

exp _τ (D, s)ds = 1 D

1 + D ^T _1! + D ^{2 ( T −τ)} _2! ² + D ^{3 ( T −3τ)} _3! ³ + D ^{4 ( T −3τ)} _4! ⁴ + · · · + D ^{k+1 ( T} _(k+1)! ^{−kτ) k+1} − 1 .

Thisompletes the proof.

Z T 0

e ^{L n (T −s)} exp _τ (D _n , T − τ − s)U _n (s)ds = R _n (T )

^(3.5)

where

R _n (T) := Ψ n − s _1n (T ) − s _2n (T ) − m _n (µ ₁ , µ ₂ )(T ).

Substituting

t := T − τ − s

^{into (3.5 ),we further obtain}

Z T −τ

−τ

e ^{L n (τ+t)} exp _τ (D n , t)U n (T − τ − t)dt = R _n (T ).

^(3.6)

Welooknow fora solutionof Equation(3.6) intheform

U _n (T − τ − t) = e ^{−L n (τ+t)} A _n (T ),

where

A(T )

^{areonstants dependingon}

T

^{. Plugging this into (3.6 ) yields}

A n (T )

Z T −τ

−τ

exp _τ (D n , t)dt = R n (T ).

Exploiting Equation(3.5 ) fromLemma 4,wean write

A n (T )

D n (exp _τ (D _n , T ) − 1) = R _n (T ).

Thus,we obtain the following Fourier oeientsfor theontrol funtion

U n (t) = e ^{−L n (T −t)} _exp ^{R n (T )D n}

τ (D ⁿ ,T )−1 .

Summarizing the alulations above,we have proved thefollowing statement.

Theorem 5. Let

ϕ

^,

µ ₁

^,

µ ₂

^and

Ψ

^{be suh that theonditions of Theorem 2 are fullled. Then the}

ontrol funtion

U (x, t) =

∞

X

n=1

U _n (t) sin( ^πn _l x)

with

U n (t) = e ^{−L n (T −t)} _exp ^{R n (T )D n}

τ (D ⁿ ,T )−1 ,

R n (T ) = Ψ n − s _1n (T ) − s _2n (T ) − m n (µ ₁ , µ ₂ )(T ), s _1n (t) = e ^{L n t} exp _τ (D n , t)

2 l

Z l 0

ϕ(ξ, 0) sin( ^πn _l ξ)dξ

+ D _n

Z 0

−τ

e ^{L n (t−s)} exp _τ (D n , t − 2τ − s) 2

l Z l

0

ϕ(ξ, s) sin( ^πn _l ξ)dξ)

ds, s _2n (t) =

Z t 0

e ^{L n (t−s)} exp _τ (D _n , t − τ − s)M _n (µ ₁ , µ ₂ )(s)ds

solvestheexat ontrollability problem(2.4 )(2.6 ),(3.1 ).

Weproved anexat ontrollability result inthe lassial settingsfor a one-dimensional heatequation

with delay. For pratial appliations, it wouldthough be desirable to extend these results to a weak

framework as, for example, the one desribed in [5℄ or even beyond it. For

p, q ∈ (1, ∞)

^{, using the}

maximal

L ^p

-regularity property (f. [15℄, [16 ℄) for the ellipti operator in (2.4), the existene of a unique solution to (2.4 )(2.6 )

u ∈ W ^1,p (0, T ), L ^q (0, l)

∩ L ^p (0, T ), W ^2,q (0, l)

for the data

f ∈ L ^p (0, T ), L ^q (0, l)

, ϕ ∈ W ^1,p (−τ, 0), L ^q (0, l)

∩ L ^p (−τ, 0), W ^2,q (0, l) , γ ₁ , γ ₂ ∈ L ^p (0, T ), R

an be deduedfrom[5℄. Using thisfat to verifyontrollabilityfor a larger lassof dataand ontrol

funtions willbea partof our furtherinvestigations.

Referenes

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Aademi Press

[2℄ Azizbayov, E.I., Khusainov, D.Ya. (2012), Solution to a heat equation with delay (in Russian),

BulletinofTarasShevhenko NationalUniversityof Kyiv,Series: Cybernetis, 12,pp.414.

[3℄ Bátkai, A., Piazzera, S.(2001). Semigroups and Linear Partial Dierential Equations with Delay,

Journal ofMathematial Analysisand Appliations, Vol. 264, pp.120

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10A.K.Peters: Wellesley MA

[5℄ Bátkai,A.,Shnaubelt,R.(2004).AsymptotiBehaviourofParaboli ProblemswithDelaysinthe

HighestOrder Derivatives, SemigroupForum, 69(3),pp 369399.

[6℄ Ek, Ch., Garke, H., Knabber, P. (2008). Mathematishe Modellierung. Springer-Verlag Berlin

Heidelberg

[7℄ Gopalsamy, K. (1992). Stability and Osillations in Delay Dierential Equations of Population

Dynamis, Mathematisand ItsAppliations, 74,KluwerAademiPublishers

[8℄ Hale, J.K. (1977). Theory of Funtional Dierential Equations, Applied Mathematial Sienes

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[9℄ Khusainov, D.Ya., Ivanov, A.F., Kovarzh, I.V. (2009). Solution ofone heat equationwith delay,

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Mathematis

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Enylopediaof Mathematisand its Appliations, 74,pp.1423 pp

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Verlag,New York,Berlin,Heidelberg,pp.1467

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BoundaryData in

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