Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift

Подождите немного. Документ загружается.

348 T. Hishida

In fact, from

∂

∂τ

−(t−τ )A

v(τ)} = e

−(t−τ)A

P (div F (τ))

in L

(Ω) we ﬁnd

v(t)=e

−(t−σ)A

v(σ)+



−(t−τ)A

P (div F (τ)) dτ (3.4)

for −∞ <σ<t<∞. Since we have (1.3) for r



<s<r<r

by (H1), the

condition (3.2) implies

e

−(t−σ)A

v(σ)

≤ C(t − σ)

−(n/s−n/r)/2

v(σ)

→ 0

as σ →−∞.Thus,(3.3)isconvergentinL

(Ω) and it is the unique solution within

the class (3.2).

Theorem 3.1. Suppose (H1), (H2) and deﬁne the exponent p

by (2.3). Then the

problem (3.1) admits a solution (3.3), which is of class (3.2) for every s ∈ (p

Proof. We ﬁrst show the summability (3.2) of v(t) given by (3.3). Let ϕ ∈ C

∞

0,σ

(Ω).

An integration by parts yields

v(t),ϕ =



∞

e

−τA

P (div F (t − τ)),ϕdτ

= −



∞

F (t − τ), ∇e

−τA

ϕdτ =



∞

(3.5)

Put M =sup

t∈R

F (t)

∞

, then we have sup

t∈R

F (t)

≤ M|K|

1/q

for q ∈ [1, ∞)

by (H2), where |·| denotes the Lebesgue measure. Let s ∈ (p

), then s



∈ (r



where 1/s



+1/s = 1. By (H1) we employ (1.4) to get



≤



F (t −τ)

∇e

−τA

ϕ



dτ ≤ CM|K|

1/s

ϕ



Since (2.3) together with s>p

leads us to



−

=1−

−

> 1 −

−

(3.6)

we can take r ∈ (s



) such that 1/s



−1/r > 1/n,namely,





−



> 1. (3.7)

One can use (1.4) to see from (3.7) that



∞

≤



∞

F (t − τ )

r/(r−1)

∇e

−τA

ϕ

dτ ≤ CM|K|

(r−1)/r

ϕ



Hence we ﬁnd

|v(t),ϕ| ≤ Cϕ



∀ϕ ∈ C

∞

0,σ

(Ω)

which implies (3.2) for every s ∈ (p

). Next, it is easy to see that v(t)given

by (3.3) is actually a solution to (3.1). In fact, for any ﬁxed σ ∈ R,wehave

Large Time Behavior of the Stokes Semigroup 349

the representation (3.4) in L

(Ω),s∈ (p

), for −∞ <σ<t<∞. It follows

from (H2) that P (div F ) is locally H¨older continuous with values in L

(Ω); as a

consequence ([4, Chapter II]), v(t)isofclass

v ∈ C

((σ, ∞); L

(Ω)) ∩ C((σ, ∞); D

(A)) ∩ C([σ, ∞); L

(Ω))

and satisﬁes (3.1) for t ∈ (σ, ∞)inL

(Ω). Since R  σ is arbitrary, v(t)isa

solution to (3.1) for t ∈ R. The proof is complete. 

It is also possible to give another shorter proof by employing some Lorentz

spaces without dividing integral in (3.5). We will mention this although the ele-

mentary proof above by use of less function spaces might be preferred by most of

readers. We deﬁne the solenoidal Lorentz spaces by

q,β

(Ω) = (L

(Ω),L

(Ω))

θ,β

where

1 − θ



<p<q<r<r

, 1 ≤ β ≤∞

and (·, ·)

θ,β

denotes the real interpolation functor. For the interpolation theory,

see for instance Amann [4, Chapter I]. We have the duality relation L

q,1

(Ω)

∗

q/(q−1),∞

(Ω), and C

∞

0,σ

(Ω) is dense in L

q,1

(Ω). By interpolation we immediately

see from (1.4) that

∇e

−tA



r,1

≤ Ct

−(n/q−n/r)/2−1/2

u



q,1

(t>0), (3.8)

for r



<q≤ r<q

,where·

q,β

stands for the norm of L

q,β

(Ω). Following

the idea of Yamazaki [45], we apply interpolation to the sublinear operator u

→

t →∇e

−tA



r,1

for ﬁxed r; then, we can deduce from (3.8) that



∞

∇e

−tA



r,1

dt ≤ Cu



q,1



<q<r<q

−

(3.9)

in spite of pointwise estimate like 1/t of the integrand. Note that the set of pairs

(q, r) satisfying the relation (3.9) is not vacuous because q

> 2 implies 1/r



−

1/q

> 1/n.

Another proof of Theorem 3.1. We will show (3.2) for every s ∈ (p

). Given

s ∈ (p

), we take r ∈ (s



) such that 1/s



− 1/r =1/n, which is possible

because of (3.6) and slightly diﬀerent from (3.7), where 1/s



+1/s =1.Inviewof

(3.5) it follows from (3.9) that

|v(t),ϕ| ≤



∞

F (t − τ)

r/(r−1),∞

∇e

−τA

ϕ

r,1

dτ ≤ CM|K|

(r−1)/r

ϕ



for all ϕ ∈ C

∞

0,σ

(Ω). By duality we obtain

v(t) ∈ L

s,∞

(Ω), ∀t ∈ R;sup

t∈R

v(t)

s,∞

< ∞

for every s ∈ (p

), from which we conclude (3.2) for the same s by interpolation.

The proof is complete. 

350 T. Hishida

We are now in a position to show Theorem 2.3.

ProofofTheorem2.3. When F (t) is periodic in Theorem 3.1, so is the solution

v(t) given by (3.3) with the same period. If, in particular, F is independent of

t,thenv can be regarded as periodic solution with arbitrary period; hence, we

conclude that the solution v ∈ D

(A) obtained in Theorem 3.1 is a steady ﬂow:

Av = P (div F )inL

(Ω). This completes the proof. 

4. Proof of Corollary 2.4

Let Ω be an exterior domain in R

(n ≥ 2) with smooth boundary ∂Ω. We will

show that we cannot avoid the condition r ≤ n for (1.4).

Proof of Corollary 2.4. If (1.4) were correct for 1 <q≤ r<q

with some q

>n≥

2, we would have (H1) for such q

; because we already know the other conditions

with r

= ∞ ([38], [42], [22], [43], [8], [10], [17], [27], [12], [36]). The exponent p

given by (2.3) fulﬁlls p

<n/(n−2) if n ≥ 3, and p

< ∞ if n = 2. Hence Theorem

2.3 implies that the steady problem (1.2) has always a smooth solution v of class

v ∈



n/(n−2)

(Ω)

(n ≥ 3),

(Ω)

for ∃s<∞ (n =2),

(4.1)

for every F ∈ C

∞

(Ω)

n×n

; here, the smoothness of v follows from the regularity

theory for the Stokes system, see for instance [20, Section IV.4]. The summability

(4.1) is, however, possible only in a special situation due to Lemma 4.1 below and

one cannot always have it; see also Remark 4.3. The proof is complete. 

It is well known that the net force exerted on the obstacle by the ﬂuid must

vanish whenever the steady ﬂow decays faster than the Stokes fundamental solu-

tion. This is usually proved by use of the asymptotic representation for |x|→∞of

the steady ﬂow, see Chang and Finn [11] and also Galdi [20, Section V.3]. We here

give another independent proof of the following lemma by using the summability

(4.1) directly.

Lemma 4.1. Let F ∈ C

∞

(Ω)

n×n

. Suppose that v isasmoothsolutionto(1.2) of

class (4.1).Then



∂Ω

ν ·{T (v, π)+F }dσ =0, (4.2)

where ν is the outward unit normal to ∂Ω,

T (v, π)=∇v +(∇v)

−πI (4.3)

is the Cauchy stress tensor and I is the n × n-identity matrix.

Proof. We ﬁx η ∈ C

∞

([0, ∞); [0, 1]) such that η(t)=1(0≤ t ≤ 2/3) and η(t)=

0(t ≥ 5/6). For R>0andx ∈ R

,wesetζ

(x)=η(|x|/R); then

∇



q,R

≤ CR

−2+n/q

(1 ≤ q ≤∞). (4.4)

Large Time Behavior of the Stokes Semigroup 351

For each k ∈{1,...,n},wesetφ

(k)

(x)=ζ

(x)e

,wheree

stands for the unit

vector along x

-axis. Let B

be the Bogovskii operator ([5], [9], [20]) in the annulus

= {x ∈ R

; R/2 < |x| <R} and set

(k)

(x)=φ

(k)

(x) −B

[∂

](x).

Note that B

[∂

] ∈ C

∞

)

and that

∇

[∂

]

q,A

≤ C∇∂



q,A

(1 <q<∞) (4.5)

with some constant C = C(q) > 0 independent of R>0, see Borchers and Sohr

[9, Theorem 2.10]. Since the compatibility condition



∂

dx =



div φ

(k)

dx =



|x|=R/2

−x

R/2

dσ =0

is satisﬁed, we have div ψ

(k)

= 0. Combining (4.4) with (4.5), we ﬁnd

∇

(k)



q,A

≤ CR

−2+n/q

(1 <q<∞). (4.6)

Set

N =(N

,...,N

)



∂Ω

ν ·{T (v, π)+F }dσ

and rewrite the equation (1.2)

div {T (v, π)+F } =0 (x ∈ Ω).

Since ψ

(k)

(x)=0for|x|≥R and ψ

(k)

(x)=e

for |x|≤R/2, we test this equation

with ψ

(k)

to obtain



T (v, π):∇ψ

(k)

for 1 ≤ k ≤ n,whereR>0istakensolargethatF (x)=0for|x|≥R/2, and

T : S =

i,j

for matrices T, S. By integration by parts once more and by

div ψ

(k)

= 0 we ﬁnd

= −



v · Δψ

(k)

dx,

from which together with (4.6) it follows that

|≤Cv

s,A

−2+n(1−1/s)

for arbitrary R>0. When n ≥ 3, we can take s = n/(n −2) by (4.1) to obtain

|≤Cv

n/(n−2),A

→ 0(R →∞).

When n =2,thesolutionv belongs to L

(Ω)

for some s<∞; therefore, for such

s, we ﬁnd

|≤Cv

s,A

−2/s

→ 0(R →∞).

In any case we conclude N = 0, which completes the proof. 

352 T. Hishida

Remark 4.2. Even for the Navier-Stokes ﬂow (n ≥ 3) subject to v =0on∂Ω, the

summablity (4.1) or ∇v ∈ L

n/(n−1)

(Ω)

n×n

necessarily implies the net force condi-

tion (4.2); for further details, see Kozono and Sohr [31], Borchers and Miyakawa

[7], Kozono, Sohr and Yamazaki [32].

Remark 4.3. It is easy to ﬁnd many solutions of (1.2) which do not satisfy (4.2).

Consider, for instance, three-dimensional problem; set u(x)=

8π

(

|x|

), the

ﬁrst column vector of the Stokes fundamental solution, and p(x)=

4π|x|

,where

=(1, 0, 0)

. Assuming 0 ∈ int Ω

,wehave−Δu + ∇p =0, div u =0inΩ.



∂Ω

ν · udσ = 0 there is a vector ﬁeld w ∈ C

∞

(Ω)

such that div w =0in

Ωandw = u on ∂Ω. Setting v = u − w,weseethat(v, p) obeys (1.2) with

F = ∇w ∈ C

∞

(Ω)

3×3

and that



∂Ω

ν ·{T (v, p)+F }dσ = e

The principle is that the solution of (1.2) whose asymptotic rate at inﬁnity is the

same as that of the Stokes fundamental solution must possess non-zero net force

on ∂Ω. For the Navier-Stokes ﬂow as well, it has been proved by Galdi [21] that

ﬂows for which (4.2) happens are rare.

5. Concluding remarks

We conclude this paper with remarks on applications of our idea to some other

interesting problems.

(i) The boundary condition (e

−τA

ϕ)|

∂Ω

= 0 is needed in (3.5) since F |

∂Ω

=0

in general. When F ∈ C

∞

(Ω)

n×n

is replaced by F ∈ C

∞

(Ω)

n×n

, Theorem 2.3

remains valid even for some other boundary conditions provided that the associ-

ated Stokes operator can be deﬁned in an appropriate way. Shibata and Shimizu

[41] proved (1.4) with 1 <q≤ r<∞ (even r = ∞ unless q = ∞) for the Stokes

initial value problem subject to Neumann boundary condition ν · T (u, p)|

∂Ω

in exterior domains (n ≥ 3), where T (u, p) is given by (4.3). It is worth while

noting that there is no restriction on (q, r) due to the null force on ∂Ωandthat

this is consistent with better summability of the steady Stokes ﬂow with Neumann

boundary condition when F ∈ C

∞

(Ω)

n×n

(ii) We have assumed the generation of analytic semigroup in (H1). But

the analyticity itself of the semigroup is not essential in our argument. We think

of the Stokes operator with rotating eﬀect of an obstacle in 3D exterior domains;

actually, the semigroup generated by this operator is no longer analytic ([16], [24]),

nevertheless Shibata and the present author [26] proved the L

-L

estimates, in

particular, (1.4) for 1 <q≤ r ≤ 3(= n). Recently in [15] Farwig and the present

author have derived the asymptotic representation for |x|→∞of the steady

Stokes ﬂow around a rotating obstacle, from which it follows that the summability

(4.1) (with n =3,namelyv ∈ L

(Ω)

) does not always hold. This suggests that

the restriction r ≤ 3 above is unavoidable.

Large Time Behavior of the Stokes Semigroup 353

(iii) Consider the Oseen operator, where translating eﬀect of an obstacle is

taken into account in 3D exterior domains. This operator generates an analytic

semigroup ([38]) and it satisﬁes (1.4) for 1 ≤ q ≤ r ≤ 3(= n), which was proved

by Kobayashi and Shibata [29] (see also [13], [14] for n ≥ 3). It is obvious that our

argument in Section 3 works for the Oseen semigroup as well, so that Theorem 2.3

still remains true. On the other hand, the 3D steady Oseen ﬂow possesses better

summability v ∈ L

(Ω)

,s>2, because it has good decay structure outside wake

region, see [20, Chapter VII]. This suggests, in view of (2.3) with p

=2andn =3,

that the 3D Oseen semigroup may possibly satisfy the gradient estimate (1.4) for

1 <q≤ r<6 (or even r ≤ 6).

References

[1] T. Abe and Y. Shibata, On a resolvent estimate of the Stokes equation on an inﬁnite

layer. II. λ =0case. J. Math. Fluid Mech. 5 (2003), 245–274.

[2] H. Abels, Reduced and Generalized Stokes resolvent equations in asymptotically ﬂat

layers, Part I: Unique solvability. J. Math. Fluid Mech. 7 (2005), 201–222.

[3] H. Abels, Reduced and Generalized Stokes resolvent equations in asymptotically ﬂat

layers, Part II: H

∞

-calculus. J. Math. Fluid Mech. 7 (2005), 223–260.

[4] H. Amann, Linear and Quasilinear Parabolic Problems, Vol. I: Abstract Linear The-

ory. Birkh¨auser, Basel, 1995.

[5] M.E. Bogovski˘ı, Solution of the ﬁrst boundary value problem for the equation of

continuity of an incompressible medium. Soviet Math. Dokl. 20 (1979), 1094–1098.

[6] M.E. Bogovski˘ı, Decomposition of L

(Ω, R

) into the direct sum of subspaces of

solenoidal and potential vector ﬁelds. Soviet Math. Dokl. 33 (1986), 161–165.

[7] W. Borchers and T. Miyakawa, On stability of exterior stationary Navier-Stokes

ﬂows. Acta Math. 174 (1995), 311–382.

[8] W. Borchers and H. Sohr, On the semigroup of the Stokes operator for exterior

domains in L

-spaces. Math. Z. 196 (1987), 415–425.

[9] W. Borchers and H. Sohr, On the equations rot v = g and div u = f with zero

boundary conditions. Hokkaido Math. J. 19 (1990), 67–87.

[10] W. Borchers and W. Varnhorn, On the boundedness of the Stokes semigroup in two-

dimensional exterior domains. Math. Z. 213 (1993), 275–299.

[11] I.-D. Chang and R. Finn, On the solutions of a class of equations occurring in contin-

uum mechanics, with application to the Stokes paradox. Arch.RationalMech.Anal.

7 (1961), 388–401.

[12] W. Dan and Y. Shibata, On the L

-L

estimates of the Stokes semigroup in a two-

dimensional exterior domain. J. Math. Soc. Japan 51 (1999), 181–207.

[13] Y. Enomoto and Y. Shibata, Local energy decay of solutions to the Oseen equations

in the exterior domains. Indiana Univ. Math. J. 53 (2004), 1291–1330.

[14] Y. Enomoto and Y. Shibata, On the rate of decay of the Oseen semigroup in exte-

rior domains and its application to Navier-Stokes equation. J. Math. Fluid Mech. 7

(2005), 339–367.

354 T. Hishida

[15] R. Farwig and T. Hishida, Asymptotic proﬁle of steady Stokes ﬂow around a rotating

obstacle. Manuscripta Math. (to appear); preprint Nr. 2578, TU Darmstadt, 2009.

[16] R. Farwig and J. Neustupa, On the spectrum of a Stokes-type operator arising from

ﬂow around a rotating body. Manuscripta Math. 122 (2007), 419–437.

[17] R. Farwig and H. Sohr, Generalized resolvent estimates for the Stokes system in

bounded and unbounded domains. J. Math. Soc. Japan 46 (1994), 607–643.

[18] R. Farwig and H. Sohr, Helmholtz decomposition and Stokes resolvent system for

aperture domains in L

-spaces. Analysis 16 (1996), 1–26.

[19] H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I. Arch. Rational

Mech. Anal. 16 (1964), 269–315.

[20] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equa-

tions, Vol. I: Linearized Steady Problems. Revised edition, Springer, New York, 1998.

[21] G.P. Galdi, A steady-state exterior Navier-Stokes problem that is not well-posed.

Proc. Amer. Math. Soc. 137 (2009), 679–684.

[22] Y. Giga, Analyticity of the semigroup generated by the Stokes operator in L

spaces.

Math. Z. 178 (1981), 297–329.

[23] Y. Giga and T. Miyakawa, Solutions in L

of the Navier-Stokes initial value problem.

Arch. Rational Mech. Anal. 89 (1985), 267–281.

[24] T. Hishida, An existence theorem for the Navier-Stokes ﬂow in the exterior of a

rotating obstacle. Arch. Rational Mech. Anal. 150 (1999), 307–348.

[25] T. Hishida, The nonstationary Stokes and Navier-Stokes ﬂows through an aperture.

Contributions to Current Challenges in Mathematical Fluid Mechanics, 79–123, Adv.

Math. Fluid Mech., Birkh¨auser, Basel, 2004.

[26] T. Hishida and Y. Shibata, L

-L

estimate of the Stokes operator and Navier-Stokes

ﬂows in the exterior of a rotating obstacle. Arch. Rational Mech. Anal. 193 (2009),

339–421.

[27] H. Iwashita, L

-L

estimates for solutions of the nonstationary Stokes equations in

an exterior domain and the Navier-Stokes initial value problem in L

spaces. Math.

Ann. 285 (1989), 265–288.

[28] T. Kato, Strong L

solutions of the Navier-Stokes equation in R

, with applications

to weak solutions. Math. Z. 187 (1984), 471–480.

[29] T. Kobayashi and Y. Shibata, On the Oseen equation in the three dimensional exte-

rior domains. Math. Ann. 310 (1998), 1–45.

[30] H. Kozono, Global L

-solution and its decay property for the Navier-Stokes equations

in half-space R

. J. Diﬀerential Equations 79 (1989), 79–88.

[31] H. Kozono and H. Sohr, On stationary Navier-Stokes equations in unbounded do-

mains. Ricerche Mat. 42 (1993), 69–86.

[32] H. Kozono, H. Sohr and M. Yamazaki, Representation formula, net force and energy

relation to the stationary Navier-Stokes equations in 3-dimensional exterior domains.

Kyushu J. Math. 51 (1997), 239–260.

[33] T. Kubo, The Stokes and Navier-Stokes equations in an aperture domain. J. Math.

Soc. Japan 59 (2007), 837–859.

Large Time Behavior of the Stokes Semigroup 355

[34] T. Kubo and Y. Shibata, On some properties of solutions to the Stokes equation in the

half-space and perturbed half-space. Dispersive Nonlinear Problems in Mathematical

Physics, 149–220, Quad. Mat. 15, Seconda Univ. Napoli, 2005.

[35] T. Kubo and Y. Shibata, On the Stokes and Navier-Stokes equation in a perturbed

half-space. Adv. Diﬀerential Equations 10 (2005), 695–720.

[36] P. Maremonti and V.A. Solonnikov, On nonstationary Stokes problem in exterior

domains. Ann. Sc. Norm. Sup. Pisa 24 (1997), 395–449.

[37] V.N. Maslennikova and M.E. Bogovski˘ı, Elliptic boundary value problems in un-

bounded domains with noncompact and nonsmooth boundaries. Rend. Sem. Mat.

Fis. Milano 56 (1986), 125–138.

[38] T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exte-

rior domain. Hiroshima Math. J. 12 (1982), 115–140.

[39] T. Miyakawa, The Helmholtz decomposition of vector ﬁelds in some unbounded do-

mains. Math. J. Toyama Univ. 17 (1994), 115–149.

[40] M.-H. Ri and R. Farwig, Existence and exponential stability in L

-spaces of station-

ary Navier-Stokes ﬂows with prescribed ﬂux in inﬁnite cylindrical domains. Math.

Methods Appl. Sci. 30 (2007), 171–199.

[41] Y. Shibata and S. Shimizu, Decay properties of the Stokes semigroup in exterior

domains with Neumann boundary condition. J. Math. Soc. Japan 59 (2007), 1–34.

[42] C.G. Simader and H. Sohr, A new approach to the Helmholtz decomposition and the

Neumann problem in L

-spaces for bounded and exterior domains. Mathematical

Problems Relating to the Navier-Stokes Equation, 1–35, Ser. Adv. Math. Appl. Sci.

11, World Sci. Publ., River Edge, NJ, 1992.

[43] V.A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations.

J. Sov. Math. 8 (1977), 467–529.

[44] S. Ukai, A solution formula for the Stokes equation in R

. Commun. Pure Appl.

Math. 40 (1987), 611–621.

[45] M. Yamazaki, The Navier-Stokes equations in the weak-L

space with time-dependent

external force. Math. Ann. 317 (2000), 635–675.

Toshiaki Hishida

Graduate School of Mathematics

Nagoya University

Nagoya 464-8602, Japan

e-mail: hishida@math.nagoya-u.ac.jp

Progress in Nonlinear Diﬀerential Equations

and Their Applications, Vol. 80, 357–387

 2011 Springer Basel AG

Well-posedness and Exponential Decay

for the Westervelt Equation with

Inhomogeneous Dirichlet Boundary Data

Barbara Kaltenbacher

,IrenaLasiecka

and Slobodan Veljovi´c

Dedicated to Professor Herbert Amann

Abstract. This paper deals with global solvability of Westervelt equation,

which model arises in the context of high intensity ultrasound. The PDE

equations derived are evolutionary quasilinear, potentially degenerate damped

wave equations deﬁned on a bounded domain in R

, n =1, 2, 3.

We prove local and global well-posedness as well as exponential decay

rates for the Westervelt equation with inhomogeneous Dirichlet boundary con-

ditions and small Cauchy data. The local existence proofs are based on an

application of Banach’s ﬁxed point theorem to an appropriate formulation of

these PDEs.

Global well-posedness is shown by exploiting functional analytic proper-

ties enjoyed by the semigroups generated by strongly damped wave equations.

These include analyticity, dissipativity and suitable characterization of frac-

tional powers of the generator – properties that enable the applicability of the

“barrier” method.

The obtained result holds for all times, provided that the Cauchy data

are taken from a suitably small (independent on time) ball characterized by

the parameters of the equation, and the boundary data satisfy some decay

condition.

Mathematics Subject Classiﬁcation (2000). Primary 35L05, 35B40; 35L70.

Keywords. Westervelt equation, quasilinear equations, analytic semigroups,

global well-posedness, decay rates.

1) Partially supported by the Cluster of Excellence SimTech, University of Stuttgart.

2) Partially supported by NSF Grant, DMS 0606682 .

3) Supported by project KA 1778/5-1 of the German Science Foundation and Elitenetzwerk

Bayern (Elite Network of Bavaria) within the International Doctorate Program Identiﬁcation,

Optimization and Control with Applications in Modern Te chnologies.