Evolution Equations and Applications

Evolution Equations and Applications

Chapter One

Preamble of the Study

In this section, we recall some de nitions and results from linear functional analysis 

De nition 1.1.1 Let X be a linear space over a  eld K, where K holds either for R or C. A mapping k.k: X −→ R is called a norm provided that the following conditions hold: 

kxk≥ 0 for all x ∈ X, and kxk= 0 ⇔ x = 0 

kλxk= |λ|kxk, for all λ ∈ K, x ∈ X 

 kx + yk≤ kxk+kyk, for arbitrary x, y ∈ X. 

If X is a linear space and k.k is a norm on X, then the pair (X, k.k) is called a normed linear space over K. 

Should no ambiguity arise about the norm, we simply abbreviate this pair by saying that X is a normed linear space over K. 

Example . Let X = C([0, 1]) be the space of all real-valued continuous functions on [0, 1]. Each of the following expressions de nes on the vector space C([0, 1]) a norm which is in common use. 

CHAPTER TWO

ABSTRACT LINEAR EVOLUTION EQUATIONS 

Most often signicant external forces a sect the evolution of a process. Let X be a real Banach space. In this chapter, we discuss the non-homogeneous Cauchy problem 

 U0(t) = AU(t) + f(t), t > 0 

U(0) = U0 

where A : D(A) ⊂ X −→ X is a given linear operator and f : [0, ∞) −→ X is a given function of the time variable only. This equation is called linear evolution equation. Basically we shall study the existence and uniqueness of solutions of the above problem, imposing di erent conditions on f. 

 Linear Evolution Equations in  nite dimensional spaces: Well Posedness 

In this section, we examine the linear Cauchy problem in the  nite dimensional case. In this case we identify the linear operator A with an N × N matrix. We shall see that if the forcing term is continuous, then there is existence and uniqueness of the solution. So consider the following initial value problem (I.V.P) : 

 U0(t) = AU(t) + f(t), t > 0 

U(0) = U0(2.1) 

where A∈ MN (R), f : [0, ∞) −→ RN and U0 ∈ RN . We shall formulate the theory for (2.1). We have 

U0(s) = AU(s) + f(s) 

⇐⇒ U0(s) − AU(s) = f(s), 

⇐⇒ e−sA(U0(s) − AU(s)) = e−sAf(s) 

⇐⇒ dds (e−sAU(s)) = e−sAf(s),since,dds (e−sA) = −Ae−sA. 

⇐⇒ R t0dds (e−sAU(s))ds =R t0e−sAf(s)ds ⇐⇒ e−tAU(t) − U(0) = R t0e−sAf(s)ds It implies that 

U(t) = etAU0 + 

Z t 0 

e(t−s)Af(s)ds (2.2) 

De nition 2.1.1 Let U : [0, T] −→ RN be a function 

2. a) U is a classical solution of (1) if 
3. i) U is continuous on [0, T] 
4. ii) U is di erentiable on (0, T] 

iii) U satis es (2.1) 

2. b) U is a mild solution of (1) if 
3. i) U is continuous on [0, T] 
4. ii) U is given by (2), ∀t ∈ [0, T] . 

Theorem 2.1.1 (Existence and Uniqueness) Let T > 0 and suppose that f ∈ C([0, T]; RN ). Then (2.1) has a unique classical solution on [0, T] given by (2.2). 

Proof: 

Existence 

Let U be given by (2.2). From the continuity of f we have that the map t 7→R t0e(t−s)Af(s)ds and also the map t 7→ etAU0 is continuous. therefore U given by (2.2) is continuous as the sum of two continuous functions. Moreover 

U0(t) = AetAU0 + f(t) + A 

Z t 0 

e(t−s)Af(s)ds 

= A(etAU0 + 

Z t 0 

e(t−s)Af(s)ds) + f(t) 

= AU(t) + f(t) 

Also U(0) = e0AU0 = e0U0 = U0 

Thus U is a classical solution of (2.2). 

Uniqueness: 

Suppose that U and V are both classical solutions of (2.1). Then de ne Z : [0, T] −→ RN by Z(t) = U(t) − V (t) .Then Z is continuous and di erentiable on [0,T] and (0,T] respectively as the sum of two continuous and di erentiable functions. Moreover 

Z0(t) = U0(t) − V0(t) 

= A(U(t) − V (t)) 

= AZ(t) 

So Z(t) = etAZ0 but Z0 = Z(0) = 0, thus Z(t) = 0, ∀t ∈ [0, T] and therefore U = V proving uniqueness and completing the proof of the theorem. 

Continuous dependence on the given data: 

Consider the following perturbed system from (2.1). 

 V0(t) = AV (t) + g(t), t > 0 

V (0) = V0(2.3) 

where A is the matrix given in (2.1). 

We are hopeful that the di erence between the solutions U and V of (2.1) and (2.3), respectively, in the sense of the supnorm on C([0, T]; RN ) , for any time interval [0,T] can be controlled by making the error terms su ciently small. In this case we say that the solution depends continuously on the given data. We summarize this in the following proposition. 

₦3,000.00 – DOWNLOAD CHAPTERS 1-5 PROCEED TO DOWNLOAD Added to cart

 

CHAPTER THREE

SEMI-LINEAR EVOLUTION EQUATIONS 

Introduction 

In this chapter we study another class of evolution equations in which the forcing term depends on the state of the system at some time t. We consider the following Cauchy problem:  u0(t) = Au(t) + f(t, u(t)), t > 0 

u(0) = u0(3.1) 

where A is the in nitesimal generator of a C0-semigroup denoted by {etA, t ≥ 0} and f : [0, T] × X → X is continuous. 

In the linear case we need the forcing term to just be continuous to guarantee the existence of a mild solution. But in this present case, we will require more than continuity on f to have existence of a solution, as we can see in the following example. 

Example: In (3.1) above, let A = 0 and X = C0 the Banach space of all real-valued sequences u = {ξn}∞n=1 with limn→∞ ξn = 0 and kuk = supn≥1|ξn|. De ne the function 

f : X → X by f(u) = {|ξn|12 + n−1}∞n=1, u = {ξn}∞n=1 ∈ X. 

The continuity of the function ξ 7→ ξ12 for ξ ≥ 0 and the de nition of the norm on X imply that f is continous on X. But the initial value problem

References

• Parmar DN, Mehta BB (2014) Face Recognition Methods and Applications.
• NEC Corporation of America (2013) NeoFace reveal advanced criminal investigative solution using face recognition technology
• Dessimoz D, Champod C (2018) Linkages between biometrics and forensic science. In: Handbook of biometrics. Springer, US
• Saini M, Kapoor AK (2016) Biometrics in Forensic Identification: Applications and Challenges. J Forensic Med 1: 108. doi:10.4172 /2472-1026 1000108 Page 3 of 6 J Forensic Med ISSN:2472-1026 Volume 1 • Issue 2 • 1000108
• Zhang D, Kong WK, You J (2003) Online palmprint identifcation- 3attern Analysis and Machine Intelligence. IEEE Transactions 25: 1041-1050
• Counter PB (2015) Invisible Biometrics Month: 4 Unique Applications of Voice Biometrics.
• Gallagher R (2012) Watch your tongue: Law enforcement speech recognition system stores millions of voices
• Delac K, Grgic M (2014) A survey of biometric recognition methods. In: Electronics in Marine. Proceedings Elmar 2014. 46th International
  Symposium. IEEE
• Jain AK, Ross A, Prabhakar S (2014) An introduction to biometric recognition. Circuits and Systems for Video Technology. IEEE Transactions 14: 4-20
• Prabhakar S, Pankanti S, Jain AK (2013) Biometric recognition: Security and privacy concerns. IEEE Security and Privacy 33-42.
• Saini M, Kapoor AK (2014) Estimation of ethnicity from handwriting. IEEE Security and Privacy 33-42.

₦3,000.00 – DOWNLOAD CHAPTERS 1-5 PROCEED TO DOWNLOAD Added to cart