study of linear discrete time-varying system

STUDY OF LINEAR DISCRETE TIME-VARYING SYSTEM

By Usman Riaz

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MS MATHEMATICS BACHELOR OF SCIENCE AT DEPARTMENT OF MATHEMATICS UNIVERSITY OF PESHAWAR, PAKISTAN JANUARY 2018

DEPARTMENT OF MATHEMATICS

Dated: January 2018

External Examiner:

Research Supervisor: Dr. Akbar Zada

Examing Committee:

ii

To my beloved Mother and Father

iii

Table of Contents Table of Contents

iv

Acknowledgements

1

Abstract

2

1 Introduction

4

1.1

Switching Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2

Kallman-Rota Inequality . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3

Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2 Preliminary Results

10

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.2

Switched Problem and its Notations . . . . . . . . . . . . . . . . . . .

11

2.3

Evolution Family and CP-Condition . . . . . . . . . . . . . . . . . . .

14

2.3.1

Evolution Family . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.3.2

CP-Condition . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

Gramian Controllability and Notations . . . . . . . . . . . . . . . . .

22

2.4

3 Stability of Linear Switching Problems 3.1

Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iv

24 24

3.2

Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Kallman-Rota Inequality

28 30

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

Kallman-Rota Inequality by Continuous and Discrete Evolution Family 30 4.2.1

4.3

30

Applications of Inequality . . . . . . . . . . . . . . . . . . . .

35

Kallman-Rota Inequality by (r, q)-Resolvent Operators Families . . .

37

4.3.1

41

Applications of Inequality . . . . . . . . . . . . . . . . . . . .

5 Controllability

44

5.1

Controllability of Linear Time-Varying System . . . . . . . . . . . . .

44

5.2

Controllability of Discrete Linear Time-Varying System . . . . . . . .

46

Bibliography

49

v

Acknowledgements All praise is due to Almighty, the most merciful and the most beneficent, who bestowed upon me health, power of communication and opportunity to successfully complete my thesis Countless solution is upon Holy Prophet, the most perfect and a torch of guidance and knowledge for humanity as a whole. I genuinely thank my thesis advisor prof. Dr. Akbar Zada for his valuable assistance, his useful suggestions and infinite patience. Furthermore, may his manner of work and endeavor make him prominent in his field of study. All my achievements are coming true due to my mother, father, brother’s and sister encouragement and prayers. I have no words to express my gratitude to my family. My consistent attention to work was due to my colleagues. I am obliged to all the members of the Department of Mathematics. Furthermore, i would like to express my thankfulness to the University of Peshawar for its strong academic atmosphere and for the friendly environment, from which I have greatly benefited.

1

Abstract The main purpose of this thesis is to study linear discrete systems. First we prove that the solution of the following discrete linear switched problem Yn+1 = B%(n) Yn + eiνn b,

Y(0) = 0

is bounded, for each ν ∈ R, set of real numbers, and each m-vector b, if the switched transition matrix Θ%(N ) (0) is stable. The converse of this statement is true if the N P matrix Fν = Θ%(N ) (j)eiνj is invertible, for each ν ∈ R and each m-vector b then j=1

solution of the discrete linear switched problem is bounded, where % be any switching path and Θ%(n) (·) be the N -periodic switching transition matrix, having complex scalar as entries, generated by complex valued, N -periodic switching sequence of m × m matrices B%(n) . Next we prove the Kallman-Rota inequality by studying the linear system in continuous and discrete sense by approach of evolution semigroup U c = {U c (t, s) : t, s ∈ R+ , t ≥ s} of bounded linear operators on Banach space Ω and discrete evolution semigroup U d = {U d (n, m) : n, m ∈ Z+ , n ≥ m} of bounded linear operators on Banach space Ω, respectively. We obtained applications of inequality as corollaries in continuous and discrete sense. We also present the same inequality for (r, q)-resolvent operators, which arises in the solution of fractional difference equation. In particular, if the β + 1 order fractional difference equations forms α-times family of linear 2

3

and bounded operators having the algebraic generator is A, then we show that the inequality kAxk2 ≤ 8η 2

Γ(α + β + 2)2 kxkkA2 xk, Γ(α + 1)Γ(α + 2β + 3)

holds for all x ∈ D(A2 ). We also present applications of inequality as corollaries. Finally we study the linear time-varying system to prove controllability in continuous sense, by using gramian controllability matrix. We also prove controllability of linear discrete time-varying system, using gramian controllability matrix in discrete sense.

Chapter 1 Introduction This chapter is divided into three sections. In first section, we give the historical background of linear discrete switching problems. In Section 1.2, we write the history of Kallman-Rota inequality and its importance. The last section of this chapter focus on history and uses of controllability of linear time-varying systems.

1.1

Switching Problems

Hybrid dynamic systems contain an important class of switched systems, which consist of a family of continuous-time or discrete-time subsystems and a switching law, which activate the subsystem in the given interval of time. Quite a growing tendency has been seen during the last few decades, in studying switched systems particularly for control theory [1–3] and practice [4–6]. The interest for studying switched systems stems from the evidence that real-world procedures and systems can be modeled as switched systems, containing chemical processes, communication industries, computer disk drives, multiple-model systems, network control, power systems and robotic manufacture and many other fields [7–9]. Switched systems expressed by linear difference or differential equations are known as switched linear systems [10–14]. 4

5

In recent years, research work specially attention on the analysis of dynamic behaviors of switched linear system, such as stability [13–15], controllability, reachability [16–20] and observability [21, 22], etc. The interesting topics on switched systems is a basic issue of the stability analysis. Stability of switched systems has been classified into two classes, stability in arbitrary switching, which is concerned with finding conditions that assure asymptotic stability of a switched system under all possible switching signals and stability under constrained switching, which is concerned with identifying classes of switching signals for which the switched system is asymptotically stable [23]. For the stability analysis problem under arbitrary switching, it is necessary to require that all the subsystems are asymptotically stable. However, even when all the subsystems of a switched system are exponentially stable, it is still possible to construct a divergent trajectory from any initial state for such a switched system. Therefore, in general, the above subsystems stability assumption is not sufficient to assure stability for the switched systems under arbitrary switching, except for some special cases [24–27]. Stability analysis under constrained switching has focused on the concept of slow switching among asymptotically stable systems. A number of papers published on finding appropriate switching strategy in order to stabilize the system, e.g., [28–30]. To characterize systems which are asymptotically stable for any arbitrary switching signal, e.g. [31–34]. In [35, 36] Lie-algebraic conditions implies the existence of a common quadratic Lyapunov-function. Dayawansa and Martin [37] proved uniform asymptotic stability implies the existence of a common Lyapunov function for compact linear poly-systems. Switched systems are defined by assigning a family of matrices {B%(n) }n≥0 on Cm ,

6

where % is the arbitrary switching signals known as switching rule, the switching rule may be either time-dependent or state-dependent. A time-dependent switching rule consists of a constant map %(n) : [0, +∞) → κ, where κ is the index set. It may, or may not, be dependent on the initial state, provided that the family of matrices {B%(n) }n≥0 are forward complete. The trajectories of a switched system with a timedependent switching rule exist for each initial state and are continuable on the whole interval [0, +∞). A state-dependent switching rule, in its simpler version, corresponds to a discontinuous feedback %(t) : Rl → κ. In [38] Bacciotic and Mazz discussed about periodic and near periodic timedependent switching rule. A strongly reduced dependence on the initial state is displayed from the central quality of periodic switching rules. Moreover, they can be deduce in terms of state dependence. In chapter 3, we prove that the stability of the transition switched matrix implies the boundedness of the linear switched problem and for the converse we add an assumption of the invertibility of matrix N P Fν = Θ%(N ) (j)eiνj with the boundedness of the linear switched problem and prove j=1

that the transition switched matrix is stable.

1.2

Kallman-Rota Inequality

In 1952 Hardy, Littlewood and P´olya (see [39], p.187) established a well-known result kfˆ0 k22 ≤ 2kfˆk2 kfˆ00 k2 for any function fˆ on R+ , where fˆ, fˆ0 , fˆ00 ∈ L2 (R+ ). In [40], Kallman and Rota show that kAxk2 ≤ 4kxkkA2 xk, ∀ x ∈ D(A2 ),

(1.2.1)

7

where the infinitesimal generator of a strongly continuous semigroup on a Banach space (Ω, k.k) is A, and x, Ax are in its domain. In [41], Kraljevi´ c and Kurepa extended the above result for strongly continuous and bounded semigroups with constant W > 0, as kAxk2 ≤ 4W 2 kxkkA2 xk, ∀ x ∈ D(A2 ). In [40], it was proved that the Hardy-Littlewood-P´olya inequality guarantee the Kallman-Rota inequality for C0 -semigroups. The optimal constant 4 in the inequality (1.2.1) is obtained some cases and its improvement is a particular importance. For Hilbert spaces, in [42] Goldstein showed that the optimal constant for a C0 -contraction semigroup is 2. In C-Euclidean spaces for analytic semigroups, different optimal constants are obtained by different approaches, (for detail see, [43, 44]). On the other hand, the theory of existence and qualitative properties of fractional difference equations has drawn a great deal of interest, for instance, see [45–48]. Kutter [49] for the first time deal the time differences of fractional order. In 1974, discrete fractional difference operator defined as an infinite series by Diaz and Osler [50]. Fractional calculus was developed by Grey and Zhang [51] for backward difference operator. Miller and Ross [52] defined the solution of a linear difference equation by fractional sum. Recently, by using the idea of Miller and Ross [52], Atici and Eloe [53] introduced the Riemann-Liouville like fractional difference equation, which is helpful to obtain solutions of certain fractional difference equations, such solutions lead to the idea of (r, q)-resolvent operators theory, see [54]. Discrete (r, q)-resolvent operators deal different families of bounded linear operators in a synthesize method, where r and q represent sequences rn and qn such that rn ∈ l1 (Z+ , Ω) and qn ∈ K00 (Z+ , Ω), respectively.

8

Our aims of chapter 4 is to prove Kallman-Rota inequality with the help of evolution family U c , over Banach space Ω and give the same inequality with the help of discrete evolution family U d , over Banach space Ω. We present the same inequality by using certain discrete (r, q)-resolvent families which arises from fractional difference equation, our approach is based on “CP-condition”. We give applications of inequality as corollaries.

1.3

Controllability

In the area of mathematical control theory, we treat with the fundamental rules concealing the analysis and models of control systems. To control an object means to impact its behavior so as to gain a desired objective. Controllability, observability and stabilizability are the central ideas of the modern mathematical control theory. There are important properties of switch system and are of precise status in control theory. At the beginning of the 1960, efficient study of controllability and observability was started, when such a theory was built for both time-invariant and time-varying linear system in the form of state space [55]. The connection between stability, controllability and observability of linear control systems are important. The theory of minimal realization of linear time-invariant control systems contains strong connection of controllability and observability. Between the ideas of controllability and observability, the formal duality exists. In particular, structural controllability/observability, which was first considered by Lin [56], in the case of standard linear time-invariant systems. The idea to characterize the patterns that guarantee controllability of time-invariant systems have been studied by Mayeda and Yamada [57]. The related problem and

9

equivalent characterizations have been investigated in [58–61]. For all these investigation time varying system is under consideration, although, the fact that it is often most natural to suppose that the physical parameters entering the coefficient matrices over time dependent. In last chapter, we prove controllability of linear time-varying system, by using gramian controllability and show controllability of linear discrete time-varying system using gramian controllability in discrete sense.

Chapter 2 Preliminary Results 2.1

Introduction

In this chapter we discussed the discrete switching problem, also present some Lemma’s, which is related to our aims of chapter 3. We present exponential boundedness of evolution family and (r, q)-resolvent operators families, such boundedness of families are helpful for our aims in chapter 4. The linear time-varying systems (in continues and discrete sense), its transition matrix and notations definitions of Gramian controllability in both (continuous and discrete) sense we discus, to prove our main results on controllability in chapter 5. In the following, we introduce few sequence spaces with whom we will deal in the whole thesis. • C00 (R+ , Ω) consisting by all Ω-valued, continuous functions on R+ , such that ¯ ¯ = 0, h(0) = lim h(t) t→∞

endowed with the sup-norm. • Lp (R+ , Ω), 1 ≤ p < ∞ is the usual Lebesgue-Bochner space of all measurable

10

11

˘ : R+ → Ω, which are equal almost everywhere, such that functions h  ˘ p :=  khk

Z∞

 p1 p ˘ kh(s)k ds < ∞.

0

´ such that h(n) ´ • K00 (Z+ , Ω), consisting of all Ω-valued sequences h, gives zero at 0 and at ∞. • l1 (Z+ , Ω) is the space of all sequences Q : Z+ → Ω, such that kQk1 := sup kQ(s)k < ∞. n≥0

• lp (Z+ , Ω), 1 ≤ p ≤ ∞ is the usual Lebesgue-Bochner space of all measurable sequences N : Z+ → Ω, which are equal almost everywhere, such that kN kp :=

n X

! p1 kN (s)kp

< ∞.

s=0

2.2

Switched Problem and its Notations

In this section, we present the linear discrete switched problem, N -periodic switched transition matrix and its notations. Consider linear discrete switched problem Yn+1 = B%(n) Yn + eiνn b, with Y(0) = 0,

(2.2.1)

Let Θ = {Br∗ | r∗ ∈ κ} represents the set of all m × m switching matrices having complex entries, where κ is an arbitrary set of indexes. The solution of linear switched problem (2.2.1) leads to the idea of the switched transition matrix Θ%(n) (·) having complex entries. If the sequence of m × m switching matrices (B%(n) ) is N -periodic

12

i.e. B%(n+N ) = B%(n) , then the switched transition matrix Θ%(n) (s) is N -periodic, and define Θ%(n) (s) =

  B%(n−1) B%(n−2) . . . B%(s) ,

s≤n−1

I,

n = s,



The N -periodic switched transition matrix satisfies the following conditions: • Θ%(n) (n) = I for all n ≥ 0. • Θ%(n) (s)Θ%(s) (m) = Θ%(n) (m) for all n ≥ s ≥ m ≥ 0. • Θ%(n+N ) (m + N ) = Θ%(n) (m) for all n ≥ m ≥ 0. The solution of the linear switched problem (2.2.1) is given by: Yn =

n X

Θ%(n) (s)eiν(s−1) b.

(2.2.2)

s=1

¯ r∗ ∈ C having the An eigenvalue of a switched matrix Br∗ is any complex scalar λ property that for any eigenvalue there be a non-zero vector l ∈ Cm such that Br∗ l = ¯ r∗ l, where r∗ ∈ κ. The spectrum of the switched matrix Br∗ is denoted by σ(Br∗ ), λ consists of all its eigenvalues. The resolvent set of Br∗ is denoted by ρ(Br∗ ), which is the complement in C of σ(Br∗ ). The complex plane is divided into three sets, denoted − as H1 = {ωr∗ ∈ C : |ωr∗ | = 1}, H+ 1 = {ωr∗ ∈ C : |ωr∗ | > 1} and H1 = {ωr∗ ∈ C : S S |ωr∗ | < 1}. Clearly C = H− H1 H+ 1 1.

The m × m switching matrix B%(n) is stable if there exists two constants βˆ > 1 and γ ∈ (0, 1) such that ˆ n, kB%(n) k ≤ βγ or equivalently its spectrum lies in H− 1.

13

Lemma 2.2.1. Let Br∗ be a switched matrix having complex entries of order m × m. If for any switching path % and for a given real number ν, have that sup

kI + eiν Br∗ + · · · + (eiν Br∗ )s k = C(ν) < ∞,

(2.2.3)

s∈{1,2,3,... }

then e−iν belongs to the resolvent set of switched matrix Br∗ . Proof: Suppose e−iν ∈ σ(Br∗ ), where r∗ ∈ κ. Then for any switching path % such that Br∗ ξ˘ = ξ˘ for some non-zero vector ξ˘ ∈ Cm and Brs∗ ξ˘ = ξ˘ for all s ∈ Z+ . Therefore sup kI + eiν Br∗ + · · · + (eiν Br∗ )s k s∈Z+

=

˘ k(I + eiν Br∗ + · · · + (eiν Br∗ )s )ξk ˘ s∈Z+ kξk6 ˘ =0 kξk

≥

sup

sup sup

s∈Z+

˘ skξk = ∞, ˘ kξk

which is contradiction. This complete the proof. Lemma 2.2.2. Let Br∗ be m × m matrix. If for any switching path % and for each real number ν the inequality (2.2.3) holds true then Br∗ is stable. Proof: We know that, (I − eiν Br∗ )(I + eiν Br∗ + · · · + (eiν Br∗ )s−1 ) = I − (eiν Br∗ )s (eiν Br∗ )s = I − (I − eiν Br∗ )(I + eiν Br∗ + · · · + (eiν Br∗ )s−1 ). Taking norm on both sides k(eiν Br∗ )s k = kBrs∗ k ≤ 1 + k(I − eiν Br∗ )kk(I + eiν Br∗ + · · · + (eiν Br∗ )s−1 )k ≤ 1 + (1 + kBr∗ k)C(ν), that is, the switching matrix Br∗ is power bounded. Then the spectral radius R(Br∗ ) is less than or equal to one, for any switching path %. Recall that ¯ r∗ | : λ ¯ r∗ ∈ σ(Br∗ ) | r∗ ∈ κ} = lim kB n∗ k n1 . R(Br∗ ) = sup{|λ r n→∞

14

As consequences σ(Br∗ ) ⊂ H− 1

S

H1 . On the other hand, from Lemma 2.2.1, follows

that each complex number z = e−iν is in the resolvent set of switched matrix Br∗ , where r∗ ∈ κ. Combining these two results, it follows that σ(Br∗ ) is a subset of H− 1.

2.3

Evolution Family and CP-Condition

The section is divided into two subsection. In the first subsection, we used Banach spaces and discuss the strongly continuous semigroup, evolution semigroup, related lemma and also discuss the discrete evolution semigroup and lemma’s. In second subsection, we take a pair like (r, q), which must satisfy the CP-condition, also write definition of CP-condition, example and lemma.

2.3.1

Evolution Family

Let Ω be a real or complex Banach space and L(Ω) the Banach algebra of all linear and bounded operators acting on Ω. The norm in Ω and L(Ω) will be denoted by k · k. Let R+ and Z+ be the set of all positive real number and non-negative integers, respectively. We remind that family T c = {T c (t) : t ≥ 0} ⊂ L(Ω) is called one-parameter semigroup if T c (0) = I and T c (t + s) = T c (t)T c (s) for all t ≥ s ≥ 0. An one-parameter semigroup is called strongly continuous or C0 -semigroup T c if for each x ∈ Ω the maps t → T (t)x are continuous on R+ . For a C0 -semigroup T c , its infinitesimal generator A with the domain D(A) is defined by o n T c (t)x − x =: Ax . D(A) := x ∈ Ω : there exists in Ω, lim t→0 t It is easy to see that if T c = {T c (t) : t ≥ 0} is a strongly continuous semigroup then the family U c = {U c (t, s) := T c (t − s) : t ≥ s ≥ 0} is a exponentially bounded

15

and strongly continuous evolution family. Conversely, if U c is a strongly continuous evolution family and U c (t, s) = U c (t − s, 0) for all t ≥ s ≥ 0 then the family T c := {T c (t) = U c (t, 0) : t ≥ 0} is a strongly continuous one-parameter semigroup. The family U c := {U c (t, s) : t, s ∈ R+ , t ≥ s} is called strongly continuous evolution family of bounded linear operators on Ω, if it satisfies the following properties: • U c (t, t) = I, for all t ∈ R+ . • U c (t, s)U c (s, s0 ) = U c (t, s0 ), for all t ≥ s ≥ s0 , t, s, s0 ∈ R+ . A strongly continuous evolution family is said to be exponentially bounded if there exists wˆ ∈ R and Wwˆ ≥ 1 such that ˆ kU c (t, s)k ≤ Wwˆ ew(t−s) , f orall t ≥ s ∈ R+ ,

(2.3.1)

and uniformly stable if there exists W0 ≥ 1 such that sup kU c (t, s)k ≤ W0 < ∞.

(2.3.2)

t≥s≥0

For each t ≥ 0, the function T c (t)˘ g given by s → (T c (t)˘ g )(s) := U c (s, s − t)˘ g (s − t) : R → Ω

(2.3.3)

belongs to Ω, and the family T c = {T c (t) : t ≥ 0} is an one-parameter semigroup of bounded linear operators acting on Ω. The following lemma is helpful for Theorem 4.2.2. Lemma 2.3.1. If semigroup T c defined in (2.3.3) is strongly continuous. If (A, D(A)) is the infinitesimal generator of T c with its domain then for every gˆ in Ω the following statements are equivalent: (i) ϕ ∈ D(A) and Aϕ(t) = −ˆ g. Rt c (ii) ϕ(t) = U (t, s)ˆ g (s)ds. 0

16

Proof: This Lemma can be shown as in [62]. For sake of completeness we present Rt the details. (i) implies (ii). For each t ≥ 0, T c (t)ϕ − ϕ = T c (s)Aϕds; therefore, 0

ϕ(t) = (T c (t)ϕ)(t) −

Zt


T c (s)Aϕds (t)

0

= (U c (t, 0)ϕ)(t) −

Zt

U c (t, t − s)(Aϕ)(t − s)ds

0

Zt

Zt

c

U (t, t − s)ˆ g (t − s)ds =

= 0

U c (t, τ )ˆ g (τ )dτ.

0

(ii) implies (i). Let t > 0 be fixed. We prove that 1 1 (−T c (t)ϕ + ϕ) = t t

Zt

T c (r)ϕdr.

0

If s ≥ t, we have: 1 1 (−T c (t)ϕ + ϕ)(s) = [−U c (s, s − t)ϕ(s − t) + ϕ(s)] t t Zs Zs−t 1 c = U (s, τ )ϕ(τ )dτ − U c (s, τ )ϕ(τ )dτ ] t 0

=

1 t

Zt

0

U c (s, s − r)ϕ(s − r)dr

0

Zt 1 = ( T c (r)ϕdr)(s). t 0

(2.3.4)

17

If 0 ≤ s < t, we have 1 1 1 (−T c (t)ϕ + ϕ)(s) = ϕ(s) = t t t

Zs

U c (s, τ )ϕ(τ )dτ

0

=

1 t

Zs

U c (s, s − r)ϕ(s − r)dr

0

Zs 1 = ( T c (r)ϕdr)(s) t 0

Zt 1 = ( T c (r)ϕdr)(s). t 0

Passing to the limit as t → 0 in (2.3.4), we get the conclusion (i). Similarly in discrete sense, the family U d := {U d (n, m) : n, m ∈ Z+ , n ≥ m} is called discrete evolution family of bounded linear operators on Ω, if it satisfies the following properties: • U d (n, n) = I, for all n ∈ Z+ . • U d (n, m)U d (m, r) = U d (n, r), for all n ≥ m ≥ r, n, m, r ∈ Z+ . It is well known that the evolution family U d is exponentially bounded, if there exist ς ∈ R+ and Wς ≥ 0 such that, kU d (n, m)k ≤ Wς eς(n−m) , for all n ≥ m ∈ Z+ ,

(2.3.5)

and uniformly stable if there exists W ≥ 0 such that, kU d (n, m)k ≤ W < ∞, for all n ≥ m ∈ Z+ . For more details about discrete evolution families we refer [63–67].

(2.3.6)

18

For every n ∈ Z+ and each f ∈ Ω, the sequence n 7→ (T d (s)f )(n) := U d (n, n − s)f (n − s) : Z+ → Ω

(2.3.7)

and the family T d = {T d (n) : n ∈ Z+ } is an one-parameter discrete semigroup on Ω, if it satisfies the following two conditions • T d (n) = I, for all n ∈ Z+ . • T d (n + m) = T d (n)T d (m), for all n ≥ m ≥ 0, n, m ∈ Z+ . For more detail see [68]. Lemma 2.3.2. The discrete semigroup T d = {T d (n) : n ∈ Z+ } described by (2.3.7) acts on lp (Z+ , Ω). Proof: Let fn be a sequence on the space Ω such that kf (0)kp = 0 = f (0). It can be seen that for all n ∈ Z+ , we have (T d (s)f )(n) := U d (n, n − s)f (n − s) : Z+ → Ω. Taking norm on both sides, we get k(T d (s)f )(n)kp = kU d (n, n − s)f (n − s)kp n X 1 d p p = kU (n, n − s)f (n − s)k s=0

≤

n X

kU d (n, n − s)kp × kf (n − s)kp

s=0

≤

n X

p

p

W kf (n − s)k

 p1

s=0

= W

n X

p

kf (n − s)k

 p1

s=0

= Wkf (n − s)kp < ∞,

 p1

19

belong to lp (Z+ , Ω). Hence proved. The discrete semigroup T d defined by (2.3.7) is called discrete evolution semigroup associated to U d on the space Ω. The “infinitesimal generator” of the discrete semigroup is denoted by A and is define as A := T d (1) − I, it is clear that d

T (n)x − x =

n−1 X

T d (s)Ax,

for all n ∈ Z+ , x ∈ Ω.

(2.3.8)

s=0

The below lemma solve the Theorem 4.2.4 in last chapter. Lemma 2.3.3. Let T d = {T d (n)}n∈Z+ be the evolution semigroup associated to the discrete evolution family U d on the space Ω and let x, f ∈ Ω. The following two statements are equivalent: (i) Ax = −f . n P (ii) x(n) = U d (n, s)f (s) for all n ∈ Z+ . s=0

Proof: (i) ⇒ (ii). For n = 0 the assertion is obvious. Let n ∈ Z+ , n ≥ 1. From (2.3.8) follows: T d (n)x − x =

n−1 X

T d (s)Ax

s=0 n−1 X

= −

(T d (n)x − x)(n) = −

s=0 n−1 X

T d (s)f (T d (s)f )(n)

s=0

x(n) =

d

n−1 X




T (n)x (n) +

= U d (n, 0)x(0) +

s=0 n−1 X

U d (n, n − s)f (n − s)

s=0

=

n X r=0


T d (s)f (n)

U d (n, r)f (r).

20

(ii) ⇒ (i). Let n ≥ 1, successively one has: (Ax)(n) = [(T d (1) − I)x](n) = U d (n, n − 1)x(n − 1) − x(n), using (2.2.2) n−1 X = U d (n, s)f (s) − x(n) s=0

=

n X

U d (n, s)f (s) − U d (n, n)f (n) − x(n)

s=0

= −f (n). Hence proved.

2.3.2

CP-Condition

In this subsection, we collect the (r, q)-resolvent operators, which satisfy the new condition called CP-condition(convolution product condition) and we talk about definition of CP-condition, lemma and example. Let rn ∈ l1 (Z+ , Ω) and qn ∈ K00 (Z+ , Ω), and A is a bounded operator defined on a Banach space Ω. Following [54], a bounded linear operator family (Kn )n≥0 ⊂ L(Ω) is called (r, q)-resolvent family, with algebraic generator A, if the following holds: (i) Kn Ax = AKn x, for all x ∈ D(A); n ≥ 0 and K0 = q0 I. n P (ii) Kn x = qn x + r(n−s) AKs x; for all x ∈ D(A), n ≥ 0. s=0

In addition, if there is some constant η > 1, such that kKn k ≤ ηqn , for all n ≥ 0, then the family (Kn )n≥0 is called exponentially bounded. The usual convolution product r ∗ q is denoted by, (r ∗ q)n :=

n X s=0

r(n−s) q(s) , n > 0.

21

In case rn is a positive sequence a.e.,then ∞ X

∞ X

(r ∗ r)s =

s=0

and

∞ P

r ∗ rs = ∞ if and only if

s=0

!2 rs

,

s=0 ∞ P

rs = ∞.

s=0

Definition 2.3.4. We say that the pair (r, q) satisfies CP-condition if for any µ > 0 there exists nµ > 0 such that µq(nµ ) = (r ∗ r ∗ q)(nµ ). Let gα (n) =

nα−1 Γ(α)

(2.3.9)

for α ≥ 0 and enµ = eµn for µ ∈ Z+ , 1{1...n} is the indicator

function. It is easy to check that (gθ , gϑ ), with θ, ϑ > 0, (e1 , e1 ), (e−1 , e−1 ) and (e1 , e−1 ) satisfies the CP-condition (2.3.9), however the pair (e−1 , e1 ) does not satisfy it, because e−1 ∗ e−1 ∗ e1 =

n X

e−n+s e−s

s=0

n X

e2s , s ≥ 0,

s=0

= ne−n

n X

e2s , s ≥ 0.

s=0

In the following lemma and example, we show some necessary conditions to get pairs (r, q), which satisfy the CP-condition. Lemma 2.3.5. Let rn ∈ l1 (Z+ , Ω) be a positive sequence and qn ∈ K00 (Z+ , Ω). ∞ P rs = ∞, then the pair (r, q) satisfies the (i) If qn > 0, q is decreasing sequence and s=0

CP-condition. ∞ P (ii) The pair (rn , 1{1,...,n} ) satisfies the CP-condition if and only if rs = ∞. s=0

Proof: (i) Fixed µ > 0. We apply Bolzano’s theorem to the sequence g := µq − r ∗ r ∗ q. Note that g(0) > 0 and n X r ∗ r ∗ qn ≥ lim lim r ∗ rn = +∞, n→∞ n→∞ qn s=0

22

and then limn→∞ gn = −∞. We conclude that there exists nµ > 0 such that µq(nµ ) = (r ∗ r ∗ q)(nµ ). (ii) If the pair (rn , 1{1,...,n} ) satisfies the CP-condition, then there exists nµ such that nµ X

r ∗ rs = µ,

s=0

for all µ > 0. We may conclude that

∞ P

r ∗rs = ∞ and then

s=0

∞ P

rs = ∞. The converse

s=0

statement is proven in a similar way. Example 2.3.6. The pairs (gα .eµ , gβ ) and (gα .eµ , 1{1,...,n} ) with α ≥ 0, µ > 0 and 0 < β ≤ 1 satisfies the CP-condition.

2.4

Gramian Controllability and Notations

In this section, we discuss linear time-varying systems, transition matrix and definition and defined some helpful notations for our results of chapter 5. Consider the linear time-varying systems ˙ ˆ ˆ ξ(t) = A(t)ξ(t) + B(t)o(t),

ξ(t0 ) = ξ0 , where t ∈ [t0 , tf ]

(2.4.1)

and its corresponding homogenous linear time-varying systems is ˙ ˆ ξ(t) = A(t)ξ(t),

ξ(t0 ) = ξ0 , where t ∈ [t0 , tf ]

(2.4.2)

ˆ ∈ Rn×n , B(t) ˆ ∈ Rn×m are the matrices, ξ(t) ∈ Rn is the solution of systems where A(t) and o(t) ∈ Rm is the control input function. The solution of homogenous system is ˆ f ) . . . A(t ˆ 0 ) is the transition matrix. The ξ(t) = ℵAˆ(tf , t0 )ξ0 , where ℵAˆ(tf , t0 ) = A(t transition matrix satisfy the following properties, • ℵAˆ(t, t) = I for all t ∈ [t0 , tf ].

23

• ℵAˆ(tf , t)ℵAˆ(t, t0 ) = ℵAˆ(tf , t0 ) for all t ∈ [t0 , tf ]. Now we state helpful definition. Definition 2.4.1. The n × n grammian controllability matrix G(tf , t0 ) for linear time-varying systems is given by Z tf ˆ BˆT (t)ℵTˆ(tf , t)dt, ℵAˆ(tf , t)B(t) G(tf , t0 ) = A t0

By using inverse of the grammian controllability we define a steering control as given in the Theorem 5.1.2. On the other hand, consider the linear discrete time-varying systems ωn+1 = A¯n ωn + B¯n On ,

ωn0 = ω0 ,

(2.4.3)

and its corresponding homogenous linear discrete time-varying systems ωn+1 = A¯n ωn ,

ωn0 = ω0 ,

(2.4.4)

where A¯n ∈ Z n×n , B¯n ∈ Z n×m are the matrices and On ∈ Z m is the control input factor. The solution of homogenous system is ωn = ℵA¯(n, n0 )ω0 , where ℵA¯(n, n0 ) = A¯n−1 . . . A¯n0 is the transition matrix, which satisfy the following properties, • ℵA¯(n, n) = I for all n ∈ Z+ . • ℵA¯(n, s)ℵAˆ(s, n0 ) = ℵA¯(n, n0 ) for all n ≥ s ≥ n0 ,

n ≥ s ≥ n0 ≥ 0.

Now we state definition of grammian controllability in discrete sense. Definition 2.4.2. The n × n grammian controllability matrix G(n, n0 ) for linear discrete time-varying systems is given by G(n, n0 ) =

n X j=n0

ℵA¯(n, j)B¯j B¯jT ℵT A¯(n, j).

Chapter 3 Stability of Linear Switching Problems In this chapter, we discuss the N -periodic switched transition matrix is stable when the solution of the linear switched problem is bounded and show the invertibility of N -periodic switched transition matrix. Also present an example, which shows that invertibility of switched matrix Fν is important.

3.1

Stability Analysis

We present our first main result, in which system is switched system. Theorem 3.1.1. The sequence Yn in (2.2.2) is bounded for any real number ν and any m-vector b if the N -periodic switched transition matrix Θα(N ) (0) is stable. Proof: Let w ∈ {0, 1, 2, . . . , N − 1} and n = kN + w . From (2.2.2) follows: YkN +w =

kN +w X

Θ%(kN +w) (s)eiν(s−1) b.

s=1

For each j ∈ {1, 2, . . . , N }, consider the set Sj = {j, j + N , . . . , j + (k − 1)N }

24

25

and D = {kN + 1, kN + 2, . . . , kN + w}. Then D

N [ [ ( Sj ) = {1, 2, . . . , n}. j=1

Thus YkN +w = e

−iν

N X X

Θ%(kN +w) (s)eiνs b + e−iν

j=1 s∈Sj

= e

−iν

N X k−1 X

X

Θ%(kN +w) (s)eiνs b

s∈Dj

Θ%(kN +w) (j + ρN )eiν(j+ρN ) b

j=1 ρ=0 w X −iν

Θ%(kN +w) (kN + l)eiν(kN +l) b

+e

= e−iν

l=1 N k−1 XX

(k−ρ−1)

Θ%(w) (0)Θ%(N )

(0)Θ%(N ) (j)eiν(j+ρN ) b

j=1 ρ=0 w X −iν

Θ%(w) (l)eiν(kN +l) b.

+e

l=1

Let zν = eiνN and M = Θ%(N ) (0). Denoting

PN

j=1

Θ%(N ) (j)eiνj by Fν , we get

  ykN +w = e−iν Θ%(w) (0) Mk−1 zν0 + Mk−2 zν1 + · · · + M0 zνk−1 Fν% b +e−iν zνk

w X

Θ%(w) (l)eiνl b.

l=1

By the assumption σ(M) lies in H− 1 , so zν belongs to the resolvent set of M. Therefore the matrix (zν I − M) is invertible and the previous equality may be written as ykN +w = e

−iν

Θ%(w) (0)(z∈ I − M)

−1

(zνk I

k

− M )Fν b +

e−iν zνk

w X l=1

Θ%(w) (l)eiνl b.

26

Taking norm of both sides, we get kykN +w k ≤ kΘ%(w) (0)(z∈ I − M)−1 zνk Fν bk −1

k

+kΘ%(w) (0)(z∈ I − M) M Fν bk + k

w X

Θ%(w) (l)bk

l=1 −1

≤ kΘ%(w) (0)kk(zν I − M) kkFν bk +kΘ%(w) (0)kk(zν I − M)−1 kkMk Fν bk +

w X

kΘ%(w) (l)bk.

l=1

¯1, λ ¯2, . . . , λ ¯ h } and let PM (λ) ¯ = (λ ¯ −λ ¯ 1 )m1 . . . (λ ¯ −λ ¯ h )mh be the characLet σ(M ) = {λ teristic polynomial of M. Here each mi is a natural number and m1 +m2 +· · ·+mh = m. By spectral Decomposition Theorem [ [69], Theorem 1], we have that ¯ k f1 (k) + λ ¯ k f2 (k) + · · · + λ ¯ k fh (k), Mk F ν b = λ 1 2 h where each fi (k) is Cm -valued polynomial having degree at most (mi − 1) for any i ∈ ¯ i | < 1 for each i ∈ {1, 2, . . . , h}. Thus kMk Fν bk → 0 {1, 2, . . . , h}. By assumption |λ as k → ∞. Therefore the subsequence (YkN +w )k is bounded for any w ∈ {0, 1, 2, . . . , N − 1}, that is, the sequence (Yn )n≥0 is bounded. Now we present our second main result. Theorem 3.1.2. If for each ν ∈ R and each non-zero vector b ∈ Cm the state sequence N P ykN is bounded and the switched transition matrix Θ%(N ) (l)eiνl is invertible then l=1

the switching matrix ΘN (0) is stable. Proof: Suppose on contrary that the spectrum of the operator M is not contained T ¯ r ∈ σ(M) T H1 and in the unite circle. When σ(M) H1 is a non empty set, let λ ¯ r y. Then for each natural number let y ∈ Cm be a non zero vector such that My = λ ¯ k y. Choose ν0 ∈ R such that eiν0 N = λ ¯ r . Given that Mν is k we have that Mk y = λ 0 r

27

invertible, therefore, there exists b0 ∈ Cm such that y = Mν0 b0 . Then   Fν0 b0 ykN = e−iν0 Mk−1 zν00 + Mk−2 zν10 + · · · + M0 zνk−1 0 −iν0

= e =

k−1 X

zνk−s−1 (Ms Fν0 b0 ) 0

−iν0

=e

s=0 −iν0 k−1 zν0 y. ke

k−1 X

zνk−1 (Fν0 b0 ) 0

s=0

Taking norm of both sides, we get kykN (ν0 , b0 )k = kkyk → ∞, as k → ∞, and a contradiction arises. ¯r Now if M is dichotomic and the spectrum of M contains a complex number λ ¯ r | > 1 then the switched matrix (zν I − M) is invertible for any switching such that |λ ¯ r y. path %, for each ν ∈ R and there exists a nonzero vector y such that My = λ Thus ykN (ν, b) = e−iν (zν I − M)−1 (zνk − Mk )Fν b = Sk (ν, b) + Pk (ν, b) where Sk (ν, b) = e−iν (zν I − M)−1 zνk Fν b and Pk (ν, b) = −e−iν (zν I − M)−1 Mk Fν b. Clearly the sequence (Sk (ν, b))k is bounded for any switching path %, for each real number ν and any m-vector b. Now let ν ∈ R be fixed, for any switching path % and let b1 ∈ Cm such that Fν b1 = y. Then ¯ k |k(zν I − M)−1 yk → ∞ as k → ∞, kPk (ν, b1 )k = |λ r

28

which is a contradiction.

3.2

Example

Here we give an example to support our Theorem 3.1.2. Example 3.2.1. Let N = 2. Then Fν = eiν (Θ%(2) (1) + eiν I) and it is invertible for each real number T ν ∈ R if and only if the matrix Θ%(2) (1) is dichotomic, or equivalently if σ(Θ%(2) (1)) H1 is the empty set. Take   1  0 1 0 Θ%(2) (1) = , Θ%(2) (0) = 2 . 0 1 0 −1 T Clearly σ(Θ%(2) (1)) H1 is a non-empty set, thus Fν is not invertible for some real number ν. Moreover Y2k =

k−1 X

Mk−s−1 zνs (Θ%(2) (1) + zν I)b

s=0

. Y2k

k−1 P 1 k−s−1 s zν  (2) s=0  = 0

 0   b. k−1 P s zν s=0

When zν 6= 0, we have that Y2k

  h1 0 = b 0 h2

where h1 =

|( 12 )k − zνk | 2 ≤ 1 ≤4 1 | 2 − zν | | 2 − zν |

h2 =

|(zνk − 1| 2 ≤ . |1 − zν | |1 − zν |

. So for the corresponding values of ν ∈ R and each nonzero vector b ∈ C2 the sequence (Y2k )k , is bounded. If zν = 1 then Y2k =

k−1 X s=0

Mk−s−1 zνs (Θ%(2) (1) + zν I)b

29

Y2k

 2(1 − = 0

1 ) 2k

0 k

   2 0 4(1 − b= 0 0 0

1 ) 2k

 0 b. k

We have again that (Y2k )k is bounded. On the other hand 1 ∈ σ(Θ%(2) (0)), i.e. the matrix Θ%(2) (0) is unstable.

Chapter 4 Kallman-Rota Inequality 4.1

Introduction

In this chapter, we discuss the Kallman-Rota inequality for the system of differential equations, difference equations and fractional difference equations. We use the different approaches for such varieties of systems, system of differential equations solve by using the evolution semigroup approach, system of difference equations solve by using discrete evolution semigroup approach and system of fractional difference equations solve by using (r, q)-resolvent operators families, which must satisfy the CP-condition. In 1970, Kallman and Rota established the Kallman-Rota inequality kfˆ0 k22 ≤ 2kfˆk2 kfˆ00 k2 for any function fˆ on R+ , where fˆ, fˆ0 , fˆ00 ∈ L2 (R+ ). We also present applications of inequality as a corollaries, for each main result.

4.2

Kallman-Rota Inequality by Continuous and Discrete Evolution Family

In this section, we discuss Kallman-Rota inequality for evolution semigroup and for discrete evolution semigroup approach. We also present application of inequality as corollaries in continuous and in discrete sense. 30

31

Lemma 4.2.1. Let T c := {T c (t) : t > 0} be a strongly continuous one-parameter semigroup and A : D(A) ⊂ Ω → Ω its infinitesimal generator. If T c is uniformly stable, supt>0 kT c (t)k ≤ W0 , then kAxk2 ≤ 4W02 kxkkA2 xk, for all x ∈ D(A2 ).

(4.2.1)

The strongly continuous semigroup T c defined in (2.3.3) is called evolution semigroup associated to U c on the space Ω. We will state here inequality for evolution semigroup associated to U c on the space Ω. Theorem 4.2.2. Let we have the following hypothesis: Rt (i) U c (t, s)ˆ g (s)ds on Ω. 0

(ii)

Rt

(t − s)U c (t, s)ˆ g (s)ds on Ω.

0

Then the following inequality hold:

Zt

2

Zt



2 c c g (s)ds ≤ W0 kˆ g kΩ × (t − s)U (t, s)ˆ g (s)ds ,

U (t, s)ˆ Ω

0

(4.2.2)

Ω

0

where W0 is a constant from the estimation of (2.3.2), gˆ on Ω and U c is uniformly stable. Proof: Let (A, D(A)) be infinitesimal generator of evolution semigroup T c associated to U c on the space Ω. Let gˆ is any element of the space Ω. From item (i) we know Rt Rt that U c (t, s)ˆ g (s)ds is also an element of Ω, so let us denoted U c (t, s)ˆ g (s)ds by ϕ 0

0

i.e. ϕ(t) :=

Rt

that y˘(t) :=

Rt

U c (t, s)ˆ g (s)ds, using Lemma 2.3.1 we have Aϕ(t) = −ˆ g . We claimed

0

0

U c (t, r)ϕ(r)dr belongs to Ω, because by using the Fubini Theorem, we

32

have y˘(t) =

Zt h

c

U (t, r)

0

=

=

=

0

=

U (r, s)ˆ g (s)ds dr

i U c (t, s)ˆ g (s)ds dr

i 1[0,r] (s)U c (t, s)ˆ g (s)ds dr

0

Zt h Zt Zt

i

0

Zt h Zt 0

c

0

Z t h Zr 0

Zr

i U c (t, s)ˆ g (s)dr ds

s

(t − s)U c (t, s)ˆ g (s)ds,

0

where 1[0,r] is the indicator function of the interval [0, r]. Using again Lemma 2.3.1 follows that y˘ ∈ D(A2 ) and A2 y˘ = A(A˘ y ) = −Aϕ = gˆ. If we replace x with y˘ in (4.2.1), we get inequality (4.2.2). The same inequality in Lemma 4.2.1 is presented in discrete sense, which facilitate the next theorem. Lemma 4.2.3. If a discrete semigroup T d = {T d (n) : n ∈ Z+ } is uniformly stable i.e. supn∈Z+ kT d (n)k ≤ W, then kAxk2 ≤ 4W 2 kxkkA2 xk, for all x ∈ D(A2 ).

(4.2.3)

For proof see [40]. Now we state the same inequality for discrete evolution semigroups on the space Ω. Theorem 4.2.4. Suppose the evolution family U d is uniformly stable and bounded linear operators acting on Ω, and let fn ∈ Ω, that the following conditions are fulfilled: n P (i) U d (n, s)f (s) belong to Ω. s=0

33

(ii)

n P

(n − s)1{0,1,...,r} (s)U d (n, s)f (s) belong to Ω.

s=0

Then the following inequality hold: n n

X

2

X



d 2 d (n − s)1 (s)U (n, s)f (s) ≤ 4W kf k × U (n, s)f (s)

, (4.2.4)

Ω {0,1,...,r} Ω

s=0

Ω

s=0

where W is a constant from the estimation of (2.3.6) and 1{0,1,...,r} is the indicator function. Proof: Let T d be the evolution semigroup associated to U d on the space Ω and A is its algebraic generator. Let f is any arbitrary element of the space Ω, From n P condition (i) we know that U d (n, s)f (s) is also an element of Ω, so let us denote n P

s=0

U d (n, s)f (s) by ϑn i.e.

s=0

ϑn :=

n X

U d (n, s)f (s).

s=0

Then by using Lemma 4.3.1 we have Aϑn = −fn . Also we claimed that

n P

U d (n, s)ϑ(s) ∈ Ω, because,

s=0 n X

U d (n, s)ϑ(s) =

s=0

n X

U d (n, r)

r=0

=

n X n X r=0 s=0

r X

U d (r, s)f (s) where n ≥ r ≥ s ≥ 0

s=0

1{0,1,...,r} (s)U d (n, s)f (s)

(4.2.5)

34

where 1{0,1,...,r} is the indicator function on the non-empty set Z+ = = +

n X n X

1{0,1,...,r} (s)U d (n, s)f (s)

s=0 r=0 n X

n X

r=0

r=0

1{0,1,...,r} (0)U d (n, 0)f (0) +

n X

1{0,1,...,r} (1)U d (n, 1)f (1) + . . .

1{0,1,...,r} (n)U d (n, n)f (n)

r=0

= nU d (n, 0)f (0) + (n − 1)U d (n, 1)f (1) + · · · + (n − r)U d (n, r)f (r) n X (n − s)1{0,1,...,r} (s)U d (n, s)f (s), = s=0

i.e.

n P

U d (n, s)ϑ(s) =

s=0

By condition (ii) Ω. Let ˆbn =

n P

n P

n P

(n − s)1{0,1,...,r} (s)U d (n, s)f (s).

s=0

(n−s)1{0,1,...,r} (s)U d (n, s)f (s) ∈ Ω implies that

s=0

n P

U d (n, s)ϑ(s) ∈

s=0

U d (n, s)ϑ(s) i.e. ˆbn , ϑn ∈ Ω, then again by Lemma 2.3.3, we get

s=0

Aˆbn = −ϑn . Applying A on both sides, A(Aˆbn ) = A(−ϑn ) = −Aϑn ,

(using (4.2.5))

= −(−fn ), A2ˆbn = fn ∈ Ω.

(4.2.6)

But we took fn from Ω this implies ˆbn belong to Ω, i.e. ˆbn ∈ D(A2 ). If we replace x by ˆbn and Ax, A2 x from (4.2.5) and (4.2.6) respectively, from (4.3.1), we get n n

X

2

X



d 2 d U (n, s)f (s) ≤ 4W kf kΩ × (n − s)1{0,1,...,r} (s)U (n, s)f (s) ,

s=0

Ω

s=0

where n ≥ r ≥ s ≥ 0. Hence this complete the proof.

Ω

35

4.2.1

Applications of Inequality

In this section, we evaluate applications of inequality in continuous and discrete sense. First we present some inequalities in continuous sense. Corollary 4.2.5. Let V˘ : R+ → R be a continuous function such that V˘ (0) = V˘ (∞) := limt→∞ V˘ (t) = 0. If the functions: Z t V˘ (s)ds t 7→ y˘(t) := 0

and ˘ := t 7→ J(t)

Z

t

(t − s)V˘ (s)ds

0

˘ confirms the condition y˘(∞) = J(∞) = 0. Then the following inequality is true: Z t Z t 2 sup | V˘ (s)ds| ≤ 4 sup |V˘ (t)| × | (t − s)V˘ (s)ds|. t≥0

t≥0

0

0

Proof: We apply Theorem 4.2.2 for Ω = C00 (R+ , R) and for U c (t, s)x = x, where t ≥ s ≥ 0 and x ∈ R. ˘ Corollary 4.2.6. Let V˘ , y˘, J˘ as in Corollary 4.2.5 and λ(t) be a positive nondecreasing function on R+ . The below inequality is true:   2   Rt Rt ˘ ˘  λ(s)V˘ (s)ds   (t − s)λ(s)V˘ (s)ds     0  sup  0  ≤ 4 sup |V˘ (t)| sup  . 2 ˘ ˘   t≥0  t≥0 t≥0  λ(t) λ(t) Proof: Follows by Theorem 4.2.2 for Ω = C00 (R+ , R) and U c (t, s) = t ≥ s ≥ 0. ˘ ∈ Lp (R+ , R). If the functions Corollary 4.2.7. Let 1 ≤ p < ∞ and λ Zt t 7→ g˘(t) := 0

˘ λ(s)ds

Zt and

t 7→ y˘(t) := 0

˘ (t − s)λ(s)ds

˘ λ(s) ˘ , λ(t)

where

36

belong to Lp (R+ , R), then the below inequality is true: Zt k

2

˘ p × k˘ g˘(s)dskp ≤ 4kλk y kp .

0

Proof: We apply Theorem 4.2.2 for Ω = Lp (R+ , R) and for U c (t, s)x = x, where t ≥ s ≥ 0 and x ∈ R. Now we prove some applications of inequality (4.2.4) on the spaces K00 (Z+ , Z) and lp (Z+ , Z) as corollaries. Corollary 4.2.8. Let {˘ ν }Z+ or ν˘ : Z+ → Z is a sequence such that it decay to zero at 0 and at ∞, so limn→∞ ν˘n = 0. Suppose that the sequences: n 7→ y(n) :=

n X

ν˘(s)

s=0

and n X n 7→ J(n) := (n − s)˘ ν (s) s=0

confirms the stipulation limn→∞ y(n) = limn→∞ J(n) = 0. Then the below inequality is true: sup | n∈Z+

n X

n X ν˘(s)| ≤ 4 sup |˘ ν (s)| × | (n − s)˘ ν (s)|.

s=0

2

n∈Z+

s=0

Proof: We apply Theorem 4.2.4 for Ω = K00 (Z+ , Z) and for U d (n, m)x = x, where n ≥ m ≥ 0 and x ∈ Z. Corollary 4.2.9. Let ν˘, y, J as in Corollary 4.2.8 and λn be a positive non-decreasing sequence on Z+ . The below inequality is true:   2   n n P P (n − s)λ(s)˘ ν (s) ν (s)   λ(s)˘   s=0   s=0  . |˘ ν (s)| sup ≤ 4 sup sup     λ(n)2 λ(n)  n∈Z n∈Z n∈Z  +

+

+

37

Proof: Follows by Theorem 4.2.4 for Ω = K00 (Z+ , Z) and U d (n, m) =

λ(m) , λ(n)

where

n ≥ m ≥ 0. Corollary 4.2.10. Let 1 ≤ p ≤ ∞ and fˆ ∈ lp (Z+ , Z). If the sequences n 7→ gˆ(n) :=

n X

fˆ(s)

and

ˆ n 7→ h(n) :=

s=0

n X

(n − s)fˆ(s)

s=0

belong to lp (Z+ , Z), then the following inequality is true: k

n X

2

fˆ(s)kp ≤ 4kfˆkp .k

s=0

n X

(n − s)fˆ(s)kp .

s=0

Proof: We apply Theorem 4.2.4 for Ω = lp (Z+ , Z) and for U d (n, m)x = x, where n ≥ m ≥ 0 and x ∈ Z.

4.3

Kallman-Rota Inequality by (r, q)-Resolvent Operators Families

Here we state the Kallman-Rota inequality for (r, q)-resolvent operators families, when both sequences rn ∈ l1 (Z+ , Ω) and qn ∈ K00 (Z+ , Ω) are positive. Theorem 4.3.1. Suppose (r, q) be the pair fulfilling the CP-condition and C ¸ r,q := sup n>0

(r ∗ r ∗ q)n qn < ∞. (q ∗ r)2n

(4.3.1)

Assume that A is the algebraic generator of the (r, q)-resolvent {Kn }n≥0 , such that kKn k ≤ ηqn , n ≥ 0,

(4.3.2)

with η ≥ 1. Then the Kallman-Rota inequality, kAxk2 ≤ 8η 2 C ¸ r,q kxkkA2 xk, is true for all x ∈ D(A2 ).

(4.3.3)

38

Proof: For all x ∈ D(A2 ) and n ∈ Z+ , we have Kn x ∈ D(A) and AKn x ∈ D(A), hence Kn x = (r ∗ AK)n x + qn x = r ∗ A[(r ∗ AK)n x + qn x] + qn x = (r ∗ r ∗ A2 K)n x + (r ∗ q)n Ax + qn x. Therefore, k(r ∗ q)n Axk ≤ kKn xk + k(r ∗ r ∗ A2 K)n xk + kqn xk n X (r ∗ r)(n−s) A2 Ks xk + kqn xk = kKn xk + k s=0

≤ kKn xk +

n X

(r ∗ r)(n−s) kKs A2 xk + kqn xk

s=0

≤ kKn xk +

n X

(r ∗ r)(n−s) kKs kkA2 xk + kqn xk

s=0 n X

≤ ηqn kxk + η

(r ∗ r)(n−s) qn kA2 xk + qn kxk, using (4.3.2)

s=0

k(r ∗ q)n Axk ≤ ηqn kxk + η(r ∗ r ∗ q)n kA2 xk + qn kxk, or, equivalently, kAxk ≤ 2η

qn (r ∗ r ∗ q)n 2 kxk + η kA xk, n > 0. (r ∗ q)n (r ∗ q)n

(4.3.4)

Let d = 2ηkxk and e = ηkA2 xk, we define yn = d

qn (r ∗ r ∗ q)n +e , (r ∗ q)n (r ∗ q)n

√ √ √ p √ p ( e (r ∗ r ∗ q)n − d qn )2 + 2 de (r ∗ r ∗ q)n qn yn = . (r ∗ q)n

(4.3.5)

39 √ √ √ p As ( e (r ∗ r ∗ q)n − d qn )2 ≥ 0, the equation (4.3.5) can be written as s √ (r ∗ r ∗ q)n qn yn ≥ 2 de , for all n > 0, (r ∗ q)2n

(4.3.6)

and s √ (r ∗ r ∗ q)n qn yn = 2 de , (4.3.7) (r ∗ q)2n √ √ √ p for those n > 0 such that e (r ∗ r ∗ q)n − d qn = 0. Since the pair (r, q) satisfies the CP-condition, we conclude that there exists n0 > 0, depending on d and e, such that d q(n0 ) = (r ∗ r ∗ q)(n0 ). e

(4.3.8)

Hence, yn0

s √ (r ∗ r ∗ q)n0 qn0 qn 0 . = 2 de = 2d (r ∗ q)2n0 (r ∗ q)n0

(4.3.9)

From (4.3.4), we deduce that for all x ∈ D(A2 ), kAxk ≤ min y(n) ≤ yn˘ 0 n>0

s p (r ∗ r ∗ q)n qn . ≤ 2η 2kxkkA2 xk (r ∗ q)2n

(4.3.10)

Putting (4.3.1) in (4.3.10), we get kAxk2 ≤ 8η 2 C ¸ r,q kxkkA2 xk. Hence the proof is complete.

In the next main result, we used the sequence hα ,

define by hα (b) :=

Γ(α + b + 2)2 , b > −1, Γ(α + 1)Γ(α + 2b + 3)

(4.3.11)

for α > −1 will participate a important position in several guesses, see Theorem 4.3.3 below. In the below proposition, we show some noticing behaviors of a function hα .

40

Proposition 4.3.2. Let α > −1 and hα defined by (4.3.11). Then hα (−1) = 1, 0 < hα (b) ≤ 1, hα is a decreasing sequence in (−1, +∞) for any α > −1 and limα→∞ hα (b) = 1 for any b ∈ (−1, +∞). Proof: We honestly verify that hα (−1) = 1. To prove that hα is decreasing, we show that ∆hα (b) < 0 for b > −1. Remind that ∆hα (b) :=



 Γ(α + b + 2)2 Γ(α + b + 3)2 − < 0, Γ(α + 1)Γ(α + 2b + 5) Γ(α + 1)Γ(α + 2b + 3)

⇔ Γ(α + 2b + 3) Γ(α + b + 2)2 < , for all b > −1, Γ(α + 2b + 5) Γ(α + b + 3)2 the above inequality is obvious, so we conclude that hα is decreasing for any α > −1. Then 1 = hα (−1) ≥ hα (b) for any b > −1. It is known that Γ(α + b + 2)2 α→∞ Γ(α + 1)Γ(α + 2b + 3)  2 (α + b + 1)! = lim α→∞ α!(α + 2b + 2)!

lim hα (b) =

α→∞

lim

= 1 and we conclude the proof. Now we will find out application to abstract evolution equations creates different types of algebraic generators families, discuss in the next main result. We start for generators of α-times β-resolvents (Ψα,β (n))n∈Z+ families form norm inequalities. From the definition of (r, q)-resolvent family given in chapter 2, that they satisfy: n−1

X (n − s)β nα Ψα,β (n)x = x+ AΨα,β (s)x, Γ(α + 1) Γ(β + 1) s=1

n ∈ Z+ ,

x ∈ Ω,

41

i.e. a (gβ+1 , gα+1 )-resolvent for some α, β > −1. Remind that for α = 0, gives (Ψ0,β (n))n∈Z+ , which is the solution of well-posedness of the abstract fractional difference equation ∆β+1 v(n) = Av(n), n ∈ Z+ , β > −1, c

(4.3.12)

with some initial conditions, where ∆β+1 denotes the Caputo’s fractional difference, c see [45]. In case α > 0, the (gβ+1 , g1 )-resolvent families lead to the idea of α-times solutions of the above equation. Theorem 4.3.3. Suppose the generator of α-times β-resolvent (Ψα,β (n))n∈Z+ is A for some α, β > −1 and suppose that there is η ≥ 1 such that kΨα,β (n)k ≤ η

nα , n ∈ Z+ . Γ(α + 1)

Then Γ(α + β + 2)2 kAxk ≤ 8η kxkkA2 xk, Γ(α + 1)Γ(α + 2β + 3) 2

2

(4.3.13)

for all x ∈ D(A2 ). Proof: The pair (gβ+1 , gα+1 ) satisfies the CP-condition and the following inequality Γ(α + β + 2)2 (r ∗ r ∗ q)n qn ≤ (r ∗ q)2n Γ(α + 1)Γ(α + 2β + 3) holds for any n ∈ Z+ . Hence, the conclusion follows from Theorem 4.3.1.

4.3.1

Applications of Inequality

In this subsection, we present remark, in which we investigate different behavior of Kallman-Rota inequality. Moreover, we also prove applications of inequality as corollaries.

42

Remark 4.3.4. For the study of the Kallman-Rota type inequality (4.3.13), we discuss the case of the well-posed fractional difference equation (4.3.12), for different values of α and the qualitative performance of hα (β) discuss in Proposition 4.3.2. (i) If α goes to −1, then hα (β) goes to zero for some β > −1. So we can select smaller or bigger constant ω ¯ α,β as we need, such that kAxk2 ≤ ω ¯ α,β kxkkA2 xk, x ∈ D(A2 ). (ii) When α = 0 then h0 (β) = Here, we discuss two cases

Γ(β+2)2 Γ(2β+3)

(4.3.14)

for β > −1, which is a decreasing sequence.

• The first order abstract fractional difference equation gives the inequality having constant is 12 . • The second order abstract fractional difference equation also gives the inequality having constant is 16 . From these cases, we conclude that constant obtained by second order abstract fractional difference equation is smaller than first order. (iii) When α goes to ∞, then hα (β) goes to 1. The behavior is different as in case (i) and (ii): If the inequality obtained the above position, then the constant ω ¯ α,β in 2 (4.3.14) gives 8η . Moreover, the second order abstract difference equation obtained constant is smaller than the constant in the abstract difference equation of first order, for the same value of α. The Theorem 4.3.3 gives special estimation in the cases of β = 0 and β = 1, respectively, the following corollaries. Corollary 4.3.5. Suppose the algebraic generator of the α-times semigroup (Ψα (n))n∈Z+ is A for some α ≥ 0 and suppose that there is η ≥ 1 such that k(Ψα (n))n∈Z+ k ≤ η

nα , n ∈ Z+ . Γ(α + 1)

Then kAxk2 ≤ η 2

(α + 1) kxkkA2 xk, (α + 2)

for all x ∈ D(A2 ). As a simple consequence of the next corollary.

(4.3.15)

43

Corollary 4.3.6. Suppose the algebraic generator of the α-times sums cosine sequence (Cα (n))n∈Z+ is A for some α ≥ 0 and suppose that there is η ≥ 1 such that kCα (n)k ≤ η

nα , n ∈ Z+ . Γ(α + 1)

Then kAxk2 ≤ η 2 for all x ∈ D(A2 ).

(α + 1)(α + 2) kxkkA2 xk, (α + 3)(α + 4)

(4.3.16)

Chapter 5 Controllability In this chapter, we have discussed controllability of linear time-varying system or discrete time-varying system, using gramian controllability conditions in both continuous and discrete sense.

5.1

Controllability of Linear Time-Varying System

In section 4.1, we have discussed definition of controllability and also present theorem, in which we show that the linear time-varying system is controllable on interval [t0 , tf ]. The following definition is helpful for the next theorem. Definition 5.1.1. Linear time-varying system (2.4.1) is controllable on time interval [t0 , tf ] if for every ξ0 ∈ Rn , there exists T > 0 and a bounded control input factor o(t) such that Z T ˆ ℵAˆ(t, 0)ξ0 + ℵAˆ(t, s)B(s)o(s)ds = ξ(tf ). t0

Theorem 5.1.2. (Gramian Controllability) The linear system (2.4.1) is controllable on [t0 , tf ] if and only if the n × n gramian controllability matrix given by Z

tf

G(tf , t0 ) = t0

ˆ BˆT (t)ℵTˆ(tf , t)dt, ℵAˆ(tf , t)B(t) A

is invertible. 44

45

Proof: Suppose G(t0 , tf ) is invertible. Then, given ξ0 and ξf , we can choose the input function o(t) as o(t) = −BˆT (t)ℵT (t , t)G −1 (tf , t0 )(ℵAˆ(tf , t0 )ξ0 − ξf ), Aˆ f

t ∈ (t0 , tf ),

and extend o(t) continuously for all other values of t. The corresponding solution of the system at t = tf can be written as Z

tf

ξ(tf ) = ℵAˆ(tf , t0 )ξ0 +

ˆ ℵAˆ(tf , t)B(t)o(t)dt

t0

= ℵAˆ(tf , t0 )ξ0 Z tf ˆ BˆT (t)ℵTˆ(tf , t)G −1 (tf , t0 )(ℵ ˆ(tf , t0 )ξ0 − ξf )dt ℵAˆ(tf , t)B(t) − A A t0

= ℵAˆ(tf , t0 )ξ0 Z tf ˆ BˆT (t)ℵTˆ(tf , t)dtG −1 (tf , t0 )(ℵ ˆ(tf , t0 )ξ0 − ξf ) − ℵAˆ(tf , t)B(t) A A t0

= ℵAˆ(tf , t0 )ξ0 − ℵAˆ(tf , t0 )ξ0 + ξf = ξf . so that the state equation is controllable on [t0 , tf ]. For the converse, suppose that the state equation is controllable, but for the sake of a contradiction, assume that the matrix G(t0 , tf ) is not invertible. If G(t0 , tf ) is not invertible, then there exists a vector ξa 6= 0 such that 0=

ξaT G(t0 , tf )ξa

Z

tf

= t

Z 0tf =

ˆ BˆT (t)ℵTˆ(tf , t)ξa dt ξaT ℵAˆ(tf , t)B(t) A 2 ˆ kξaT ℵAˆ(tf , t)B(t)k dt,

(5.1.1)

t0

and hence ˆ = 0, ξaT ℵAˆ(tf , t)B(t)

t ∈ [t0 , tf ).

(5.1.2)

46

However, the state equation is controllable on [t0 , tf ], and so choosing ξ0 = ℵ−1 (tf , t0 )(ξa + Aˆ ξf ), there exists an input function oa (t) such that Z tf ˆ ℵAˆ(tf , t)B(t)o ξf = ℵAˆ(tf , t0 )ξ0 + a (t)dt, t0

which is equivalent to the equation Z tf ˆ ℵAˆ(tf , t)B(t)o ξa = − a (t)dt. t0

Multiplying through by ξa and using (5.1.1) and (5.1.2) yields ξaT ξa = 0, a contradiction. Thus, the matrix G(t0 , tf ) is invertible.

5.2

Controllability of Discrete Linear Time-Varying System

This section present the idea of controllability of linear time-varying system in discrete sense. We have proved that the linear discrete time-varying system is controllable on Z+ having control input factor Oj , while the control input factor Oj contains gramian controllability matrix which must be invertible. Now we state helpful definition for the next theorem. Definition 5.2.1. Linear discrete time-varying system (2.4.4) is controllable on Z+ if for every ω0 ∈ Z n , there exists n > 0 and a bounded control input factor On such that ℵA¯(n, n0 )ω0 +

n X

ℵA¯(n, j)B¯j Oj = ω(n).

j=n0

Theorem 5.2.2. (Gramian Controllability) The linear system (2.4.4) is controllable on Z+ if and only if the n × n gramian controllability matrix given by G(n, n0 ) =

n X j=n0

is invertible.

ℵA¯(n, j)B¯j B¯jT ℵT A¯(n, j),

47

Proof: Suppose G(n, n0 ) is invertible, then for given ω0 and ωn , we can decide the input factor O(n) as −1 O(j) = −B¯jT ℵT A¯(n, j)G (n, n0 )(ℵA¯(n, n0 )ω0 − ωn ),

j ∈ Z+ , n0 ≤ j ≤ n

and extend O(j) continuously for all other values of j ∈ Z+ . The solution of the system (2.4.4) at j = n can be written as ω(n) = ℵA¯(n, n0 )ω0 +

n X

ℵA¯(n, j)B¯j Oj

j=n0

= ℵA¯(n, n0 )ω0 n X T −1 − ℵA¯(n, j)B¯j B¯jT ℵA ¯(n, j)G (n, n0 )(ℵA¯(n, n0 )ω0 − ωn ) j=n0

= ℵA¯(n, n0 )ω0 −G −1 (n, n0 )

n X

ℵA¯(n, j)B¯j B¯jT ℵT A¯(n, j)(ℵA¯(n, n0 )ω0 − ωn )

j=n0

= ℵA¯(n, n0 )ω0 − ℵA¯(n, n0 )ω0 + ωn = ωn . so that the state equation is controllable on Z+ . For the converse, suppose that the state equation is controllable, but for the sake of a contradiction, assume that the matrix G(n, n0 ) is not invertible. If G(n, n0 ) is not invertible, then there exists a vector ωa 6= 0 such that 0=

ωaT G(n, n0 )ωa

= =

n X j=n0 n X

¯ B¯T (j)ℵT¯(n, j)ωa ωaT ℵA¯(n, j)B(j) A 2 ¯ kωaT ℵA¯(n, j)B(j)k ,

(5.2.1)

j=n0

and hence ¯ = 0, ωaT ℵA¯(n, j)B(j)

n ≥ j ≥ n0 .

(5.2.2)

48

However, the state equation is controllable on Z+ , and so choosing ωa = ℵAˆ(n, n0 )ω0 − ωn ), there exists an input function Oa (j) such that ωn = ℵA¯(n, n0 )ω0 +

n X

¯ ℵA¯(n, j)B(j)O a (j),

j=n0

which is equivalent to the equation ωa = −

n X

¯ ℵA¯(n, j)B(j)O a (j).

j=n0

Multiplying through by ωa and using (5.2.1) and (5.2.2) yields ωaT ωa = 0, a contradiction. Thus, the matrix G(n, n0 ) is invertible.

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