Analysis and control of integro-differential Volterra equations with delays
AANALYSIS AND CONTROL OF INTEGRO-DIFFERENTIALVOLTERRA EQUATIONS WITH DELAYS
Y. EL KADIRI, S. HADD AND H. BOUNIT
Abstract.
The purpose of this paper is to introduce a semigroup approach to linearintegro-differential systems with delays in state, control and observation parts. On theone hand, we use product spaces to reformulate state-delay integro-differential equationsto a standard Cauchy problem and then use a perturbation technique (feedback) to provethe well-posendess of the problem, a new variation of constants formula for the solutionas well as some spectral properties. On the other hand, we use the obtained resultsto prove that integro-differential systems with delays in state, control and observationparts form a subclass of distributed infinite-dimensional regular linear systems in theSalamon-Weiss sense. Introduction
Integro-gdifferential equations have attracted the attention of many researchers formany years and have been the subject of much interesting work, see e.g. [18],[19],[21], andthe references therein. In such a class of evolution equations, the differential and integraloperators can appear together at the same time.In the first part of this work, we are concerned with introducing a unified semigroupapproach to the following class of delay-integro-differential equations ˙ x ( t ) = Ax ( t ) + (cid:90) t a ( t − s ) Ax ( s ) ds + Lx t + f ( t ) , t ≥ x (0) = x, x = ϕ (1.1)where A is a generator of a strongly continuous semigroup ( T ( t )) t ≥ on a Banach space X , x ∈ X, ϕ ∈ L p ([ − r, , X ) , a ∈ W ,p ( R + , C ) with p ∈ (1 , ∞ ) and L is a Riemann-Stieljesintegral of the form Lϕ := (cid:90) − r dµ ( θ ) ϕ ( θ ) , ϕ ∈ W ,p ([ − r, , X ) , (1.2)with µ : [ − r, → L ( X ) is a function of bounded variation continuous in 0 and with totalvariation | µ | (which is a positive Borel measure) satisfying | µ | ([ − ε, → ε → t ≥
0, the history function x t : [ − r, → X is defined by x t ( θ ) = x ( t + θ )for θ ∈ [ − r, Mathematics Subject Classification.
Key words and phrases.
Integro-differential equations, Delay equations, Boundary systems, admissi-bility, infinite-dimensional systems. a r X i v : . [ m a t h . A P ] F e b ote that a complete theory already exists for the problem (1.1) in the case a ≡ L ≡ X = X × L p ( R + , X ) × L p ([ − r, , X ) . Using this space, we embed the problem (1.1) in a large free-delay Cauchy problem in X of the form ˙ z ( t ) = A L z ( t ) , z (0) = (cid:16) xfϕ (cid:17) , t ≥ , where A L : D ( A L ) ⊂ X → X is defined in (3.9). This technique is originating in [19] hasbeen widely investigated by many authors, in order to study the well-posedness of free-delay integro-differential equation. More references can be found in [21]. According tothis transformation, we say that the delay-integro-differential equation (1.1) is well-posedif and only if the operator A L generates a strongly continuous semigroup ( T L ( t )) t ≥ on X . To prove that A L is a generator on X , we first adopt the decomposition A L = A + L , where the operators A and L are defined in (3.11) and (3.10), respectively. Second, weuse a perturbation theorem in [14] to prove that the operator A generates a stronglycontinuous semigroup on X (see Theorem 3.2). Third, based on regular linear systemswe show that the operator L is a Miyadera-Voigt perturbation for A . Finally, we apply[8, Chap.III] to deduce that A L is a generator on X .If we denote by L Λ the Yosida extension of L with respect to the left shift semigroup on L p ([ − r, , X ) (see (2.6) for the definition), we prove that (see Theorem 3.4) the solutionof the above Cauchy problem (hence of the delay-integro-differential equation (1.1)) isgiven by z ( t ) = x ( t ) v ( t, · ) x t , t ≥ , where the function t (cid:55)→ x ( t ) is given by the following variation of parameters formula x ( t ) = R ( t ) x + (cid:90) t R ( t − s )( L Λ x s + f ( s )) ds, and v ( t, · ) is the solution of boundary system (2.9). Here ( R ( t )) t ≥ is the resolvent familyassociated the free-delay integro-differential equation problem (see. Definition (3.1)). oncerning the spectral theory for the generator A L , we have proved in Theorem 4.1that for λ ∈ C ∩ ρ ( A δ ) ∩ ρ ( A ) λ ∈ ρ ( A L ) ⇐⇒ λ ∈ ρ (cid:16) a ( λ )) A + Le λ (cid:17) , where ˆ a is the Laplace transform of a and A δ and A are the operators defined in (3.2).Remark that for a ( · ) ≡ L ≡ , we identify the operators A L and A δ , and then we retrieve the well-knownresults on the spectrum of A δ , see e.g. [8, Chap.7] and [21].In the second part of this paper (see Section 5), we prove that integrodifferential equa-tions with state, input and output delays (see equation (5.1)) form a subcalss of infinite-dimensional linear systems in the Salamon-Weiss sense [23]. This extends some existingresults for standard delay and neutral systems [11], [13] and [4] respectively.The organization of the paper is as follow: Section 2 is devoted to a brief background onfeedback theory of regular linear systems in Salamon-Weiss sense. In Section 3 we provethe well-posedness of the problem (1.1). In Section 4 we propose a spectral study forstate-delay integrodifferential equations. The last section is concerned with reformulatingintegrodifferential equations as distributed systems. Notation
Throughout the paper we shall frequently use the following symbols: Let ( X, || || ) be aBanach space and γ be a real constant. For p ∈ [1 , ∞ ), we denote by L pγ ( R + , X ) standsfor the space of functions f : R + → X , such that e − γt f ( t ) is Bochner integrable on R + . W ,p ( R + , X ) is the Sobolev space associated to L p ( R + , X ) := L p ( R + , X ). Giventwo Banach spaces X and Y , we denote the space of bounded linear operators X → Y by L ( X, Y ), and for G ∈ L ( X, Y ) , (cid:107) G (cid:107) L ( X,Y ) means its operator norm (some time weonly write (cid:107) G (cid:107) ). We shall omit the subscripts if no confusion is possible. The identityoperator will be denoted by I . The Laplace transform of a function f ∈ L ω ( R + , X ) isdefined by ˆ f ( λ ) = (cid:82) ∞ e − λt f ( t ) dt for λ > γ . If G : R + → L ( X, Y ) is an operator valuedfunction such that for each x ∈ X the function G ( t ) x is in L ω ( R + , X ), then we define thelinear operator ˆ G ( λ ) by ˆ G ( λ ) x = (cid:82) ∞ e − λt G ( t ) xdt . Moreover, we shall use convolution ofoperator valued functions: Let X, Y, Z be Banach spaces, and G : R + → L ( X, Y ) and F : R + → L ( Y, Z ) functions such that G ( t ) x and F ( t ) x are measurable for all x ∈ X or Y ,respectively, and || G ( t ) || , || F ( t ) || are in L γ ( R + , R ). Then F ∗ G : R + → L ( X, Z ) is definedby ( F ∗ G )( t ) x = (cid:82) t F ( t − s ) G ( s ) xdt for x ∈ X . One can prove that (cid:92) ( F ∗ G ) = ˆ F · ˆ G .For a closed, linear operator A : D ( A ) ⊂ X → X , the resolvent set of A is given by ρ ( A ) := { λ ∈ C : λ − A ; D ( A ) ⊂ X → X is bijective } .Let A : D ( A ) ⊂ X → X be a generator of a strongly continuous semigroup T :=( T ( t )) t ≥ on X . We introduce a new norm (cid:107) x (cid:107) − := (cid:107) R ( α, A ) x (cid:107) for x ∈ X and some(hence all) α ∈ ρ ( A ). The completion of X with respect to the norm (cid:107) · (cid:107) − is a Banachspace denoted by X − . Moreover, the semigroup T can be extended to another strongly ontinuous semigroup T − := ( T − ( t )) t ≥ on X , whose generator A − : X → X − is theextension of A to X . For more details on extrapolation theory we refer to [8, ChapII].2. Some background on regular linear systems
In this section, we briefly recall the concept of regular linear systems (for more detailson this theory we refer to [23]. Let
X, U and Z be Banach spaces such that Z ⊂ X densely and continuously. Let A m : Z → X be a closed operator on X, G : Z → U alinear operator (trace operator), and consider the boundary control problem (cid:40) ˙ z ( t ) = A m z ( t ) , z (0) = x, t > ,Gz ( t ) = u ( t ) , t ≥ , (2.1)for x ∈ X and u ∈ L p ([0 , + ∞ ) , U ). Under the following conditions (H1) A := A m with domain D ( A ) = ker( G ) generates a strongly continuous semigroup T = ( T ( t )) t ≥ , (H2) G is surjective,the inverse D λ := (cid:0) G | ker( λ − A m ) (cid:1) − ∈ L ( U, X ) , λ ∈ ρ ( A ) , exists, see [9]. In addition ifwe seclect B := ( λ − A − ) D λ , λ ∈ ρ ( A ), then we have A m = ( A − + BG ) | Z . This impliesthat the boundary system (2.1) can be reformulated as a distributed system of the form (cid:40) ˙ z ( t ) = A − z ( t ) + Bu ( t ) , t ≥ ,z (0) = z. (2.2)The mild solution of (2.2) is given by z ( t ) = T ( t ) x + (cid:90) t T − ( t − s ) Bu ( s ) ds = T ( t ) x + Φ A,Bt u (2.3)for any t ≥ , x ∈ X and u ∈ L p ([0 , + ∞ ) , U ). This solution takes its values in X − . It is then more practice to consider control operators B ∈ L ( U, X − ) for whichthe solution z ( t ) ∈ X for any t ≥ , initial condition z ∈ X and control func-tion u ∈ L p ([0 , + ∞ ) , U ). This situation happens if there exists τ > A,Bτ u ∈ X for any u ∈ L p ([0 , + ∞ ) , U ), see [22]. In fact, this condition also impliesthat Φ A,Bt ∈ L ( L p ([0 , + ∞ ) , U ) , X ) for any t ≥
0, due to the closed graph theorem. In thiscase, we say that B is an admissible control operator for A .Let us now consider the observation function y ( t ) = C z ( t ) ,t ≥ , (2.4)where t (cid:55)→ z ( t ) is the solution of (2.2) and C : Z → U is a linear operator.The system (2.2)-(2.4) is called well-posed if t (cid:55)→ y ( t ) can be extended to a function y ∈ L ploc ([0 , + ∞ ) , U ) that satisfies (cid:107) y (cid:107) L p ([0 ,α ] ,U ) ≤ c (cid:0) (cid:107) x (cid:107) + (cid:107) u (cid:107) L p ([0 ,α ] ,U ) (cid:1) (2.5) or any x ∈ X, u ∈ L loc ([0 , + ∞ ) , U ) , and some constants α > c := c ( α ) >
0. Let usnow introduce conditions on B and C that guaranties the well-posedness of the system(2.2)-(2.4). To this end, first consider the operator C := C | D ( A ) ∈ L ( D ( A ) , U ) . We say that C is an admissible observation operator for A if for some (hence all) α > κ such that (cid:90) α (cid:107) CT ( t ) x (cid:107) p dt ≤ κ p (cid:107) x (cid:107) p for all x ∈ D ( A ) . In this case, we have T ( t ) x ∈ D ( C Λ ) for all x ∈ X and a.e. t > (cid:90) α (cid:107) C Λ T ( t ) x (cid:107) p dt ≤ γ p (cid:107) x (cid:107) p , ∀ x ∈ X, where C Λ is the Yosida extension of C with respect to A defined as follow D ( C Λ ) := { x ∈ X : lim s → + ∞ sCR ( s, A ) x exists in U } C Λ x := lim s → + ∞ sCR ( s, A ) x. (2.6)We need the following space W ,p ,t ( U ) := (cid:8) u ∈ W ,p ([0 , t ] , U ) : u (0) = 0 (cid:9) , t > . which is dense in L p ([0 , t ] , U ). Now a simple integration by parts yields Φ A,Bt u ∈ Z forany t ≥ u ∈ W ,p ,t ( U ). This allows us to define the so-called input-output map( F u )( t ) = C Φ A,Bt u, t ≥ , u ∈ W ,p ,t ( U ) . Using (2.3), for x ∈ D ( A ) and u ∈ W ,p ,τ ( U ) ( τ > y defined in(2.4) satisfies y ( t ) = CT ( t ) x + ( F u )( t )for any t ∈ [0 , τ ] and u ∈ L p ([0 , τ ] , U ). Definition 2.1.
Let
A, B, C and F as above. We say that the operator triple ( A, B, C ) iswell-posed if the following assertions hold: (i) B is an admissible control operator for A , (ii) C is an admissible observation operator for A , and (iii) There exist τ > and κ > such that (cid:107) F u (cid:107) L p ([0 ,τ ] ,U ) ≤ κ (cid:107) u (cid:107) L p ([0 ,τ ] ,U ) for all u ∈ W ,p ,τ ( U ) . If the triple (
A, B, C ) is well-posed, by density of W ,p ,τ ( U ) in L p ([0 , τ ] , U ), we can extend F to a bounded operator in L ( L p ([0 , τ ] , U )), for any τ >
0. In particular, the estimationof the observation (2.5) holds, so that the system (2.2)-(2.4) is well-posed. efinition 2.2. A well-posed triple ( A, B, C ) is called regular (with feedthrough zero) ifthe following limit exists lim τ → τ (cid:90) τ ( F ( R + · v )) ( σ ) dσ = 0 , for any v ∈ U .Remark . (i) According to [23], if ( A, B, C ) is a regular triple, then Φ
A,Bt u ∈ D ( C Λ )and ( F u )( t ) = C Λ Φ A,Bt u for a.e. t ≥ u ∈ L ploc ([0 , + ∞ ) , U ). This showsalso that the state trajectory and the output function of the system (2.2)-(2.4)satisfy x ( t ) ∈ D ( C Λ ) and y ( t ) = C Λ z ( t ) for a.e. t >
0, and all initial state z (0) = x ∈ X and all input u ∈ L ploc ([0 , + ∞ ) , U ).(ii) Clearly, if one of the operators B or C is bounded and the other is admissible for A , then the triple ( A, B, C ) is regular.We end this section with an example of regular linear system which will be frequentlyused in some proofs in the next sections.
Example . Let X be a Banach space and p > S ( t )) t ≥ defined by( S ( t ) ϕ )( θ ) := (cid:40) , t + θ ≥ ,ϕ ( t + θ ) , t + θ ≤ f ∈ L p ([ − r, , X ), t ≥ θ ∈ [ − r, L p ([ − r, , X ) , (called the left shift semigroup). The generator of this semigroup is Qϕ = ϕ (cid:48) , D ( Q ) = (cid:8) ϕ ∈ W ,p ([ − r, , X ) : ϕ (0) = 0 (cid:9) . (2.8)Now, consider the boundary system ∂v ( t, θ ) ∂t = ∂v ( t, θ ) ∂θ , t ≥ , θ ∈ [ − r, ,v (0 , θ ) = ϕ ( θ ) , θ ∈ [ − r, ,v ( t,
0) = x ( t ) , t ≥ , (2.9)We select Q m := ∂∂θ with maximal domain D ( Q m ) = W ,p ([ − r, , X ), so Q = Q m and D ( Q ) = ker G with G = δ where δ f = f (0) is the Dirac operator. The Dirichlet operator d λ associated to (2.9) is given by d λ x = e λ x := e λ · x, λ ∈ ρ ( Q ) = C , x ∈ X. We put β := ( λ − Q − ) d λ , λ ∈ C . The control maps associated with the control operator β are given by (cid:0) Φ Xt u (cid:1) ( θ ) = (cid:40) u ( t + θ ) , − t ≤ θ ≤ , , − r ≤ θ < − t, or any t ≥ u ∈ L p ([0 , + ∞ ) , X ), see [13]. Thus the operator β ∈L ( X, ( L p ([ − r, , X )) − ) is an admissible control operator for Q . Now consider the oper-ator L : W ,p ([ − r, , X ) → X defined by (1.2) and define L := L | D ( Q ) ∈ L ( D ( Q ) , X ) . Then L is an admissible observation operator for Q , see [10, Lemma 6.2.]. We select( F X u )( t ) = L Φ t u, u ∈ W ,p ,t ( X ) . As in [13], we show that for any τ > u ∈ W ,p ,τ ( X ), (cid:90) τ (cid:107) ( F X u )( t ) (cid:107) p dt ≤ (cid:90) τ (cid:18)(cid:90) − t (cid:107) u ( t + θ ) (cid:107) d | µ | ( θ ) (cid:19) p dt ≤ ( | µ | ([ − τ, pq (cid:90) τ (cid:90) − t (cid:107) u ( t + θ ) (cid:107) p d | µ | ( θ ) dt ≤ ( | µ | ([ − τ, pq (cid:90) − τ (cid:90) τ − θ (cid:107) u ( t + θ ) (cid:107) p dtd | µ | ( θ ) ≤ ( | µ | ([ − τ, p (cid:90) τ (cid:107) u ( σ ) (cid:107) p dσ, due to H¨older inequality and Fibini’s theorem. This shows that the triple ( Q, β, L ) iswell-posed. On the other hand, it is shown in [13, Theorem 3] that the triple ( Q, β, L )is a regular linear system.3. Well-posedness of the state-delay integro-differential equation
In this section, we will study the well-posedness of the state-delay integro-differentialequation (1.1). But first we recall some facts about the following free-delay integro-differential equation˙ x ( t ) = Ax ( t ) + (cid:90) t a ( t − s ) Ax ( s ) ds + f ( t ) , x (0) = x, t > . (3.1)It is known that the solvability of (3.1) is related to the concept of strongly continuousfamily defined as follows: Definition 3.1.
Let a ∈ L loc ( R + ) . A strongly continuous family ( R ( t )) t ≥ ⊂ L ( X ) iscalled resolvent family for the homogeneous free-delay integro-differential equation, if thefollowing three conditions are satisfied: • R (0) = I . • R ( t ) commutes with A, which means R ( t ) D ( A ) ⊂ D ( A ) for all t ≥ , and AR ( t ) x = R ( t ) Ax for all x ∈ D ( A ) and t ≥ . • For each x ∈ D ( A ) and all t ≥ the resolvent equations holds: R ( t ) x = x + (cid:90) t a ( t − s ) AR ( s ) xds. enerally, in the definition above, it is not necessary that A be the generator of asemigroup. It is well-known (see, e.g., [7, Section 1]) that the homogeneous free-delayintegro-differential system is well-posed if and only if it has a resolvent R ( t ). In thissituation, x ( t ) = R ( t ) x, t ≥ x ∈ X is the mild solution, which gives the uniqueclassical solution if x ∈ D ( A ). For more details on resolvent families we refer to themonograph by Pruss [21].It is well-known that the assumptions a ( · ) ∈ W ,p ( R + , C ) and A generates a semigroupon X imply that the free-delay integro-differential equation has a unique resolvent family( R ( t )) t ≥ , see e.g. [7], [21].In the sequel, we will use matrices operators to solve the equation (3.1). We then select X := X × L p ( R + , X ) with norm (cid:107) ( xf ) (cid:107) := (cid:107) x (cid:107) + (cid:107) f (cid:107) p . In this space, we consider the following unbounded operator matrices A := A δ dds , A δ := A δ a ( · ) A dds D ( A δ ) = D ( A ) := D ( A ) × W ,p ( R + , X ) , (3.2)where dds is the first derivative with D ( dds ) = W ,p ( R + , X ).The following result is well known, see e.g. [8], [21, p.339] and [2]. Theorem 3.1.
The operator A δ generates a strongly continuous semigroup ( T δ ( t )) t ≥ on X . Moreover, if we assume that a ( · ) ∈ W ,p ( R + , C ) , then T δ ( t ) = (cid:18) R ( t ) Υ( t ) ∗ ∗ (cid:19) , t ≥ , with Υ( t ) f := (cid:90) t R ( t − s ) f ( s ) ds, f ∈ L p ( R + , X ) . (3.3) In particular R ( t ) is exponentially bounded since ( T δ ( t )) t ≥ is so. Let us now solve the state-delay integro-differential equation (1.1). The latter can bereformulated in X as the following problem ˙ (cid:37) ( t ) = A δ (cid:37) ( t ) + (cid:16) Lx t (cid:17) , t ≥ ,(cid:37) (0) = ( xf ) , x = ϕ. (3.4)This equation is not yet a Cauchy problem because we still have a delay term. In orderto reformulate it as Cauchy problem, we need the following larger product state space X := X × L p ([ − r, , X ) , (3.5) nd the new state z ( t ) := (cid:18) (cid:37) ( t ) x ( t + · ) (cid:19) , t ≥ . (3.6)By combining (2.9) and (3.4), we rewrite the problem (1.1) as the following boundaryvalue problem ˙ z ( t ) = A m,L z ( t ) , t ≥ z (0) = z ,Gz ( t ) = M z ( t ) , t ≥ , (3.7)where the operator A m,L : Z → X is given by A m,L := A δ L
00 0 ddθ , Z := D ( A δ ) × W ,p ([ − r, , X ) , and G : Z → X and M : X → X are the linear operators G := (cid:2) δ (cid:3) , M := (cid:2) I (cid:3) , and the initial state z = (cid:16) xfϕ (cid:17) . To solve the problem (1.1), it suffices to solve the following Cauchy problem (cid:40) ˙ z ( t ) = A L z ( t ) , t ≥ ,z (0) = z , (3.8)where A L := A m,L , D ( A L ) := { z ∈ Z : Gz = M z } . (3.9)This means that it suffices to show that the operator A L is a generator of a stronglycontinuous semigroup on X . Thus, we define A m, := A δ
000 0 ddθ and L := L
00 0 0 (3.10)on D ( A m,L ) = Z , and A := A m, ,D ( A ) := { z ∈ Z , Gz = M z } . (3.11)Observe that we have A L = A + L . Next we follow the following strategy: We first prove that A is a generator on X andsecond show that L is a Miyadera-Voigt perturbation for A . et us start by proving that A is a generator. In fact, define the operator A := A m, , with domain D ( A ) = ker( G ) . It is clear that the operator A generates a strongly continuous semigroup ( T ( t )) t (cid:62) on X given by T ( t ) xfϕ = T δ ( t ) (cid:18) xf (cid:19) S ( t ) ϕ , t ≥ , xfϕ ∈ X . (3.12)Now let us compute D λ , the Dirichlet operator associated with G and A m, . To determine D λ , it suffices to determine Ker( λ − A m, ), for λ ∈ ρ ( A δ ). For λ ∈ ρ ( A ) = ρ ( A δ ),Ker( λ − A m, ) = Ker( λ − A δ ) × Ker( λ − Q m )= (cid:110)(cid:16) (cid:17)(cid:111) × (cid:110) e λ v : v ∈ X (cid:111) . This implies that D λ v = e λ v , λ ∈ ρ ( A δ ) , v ∈ X. Now, let us consider the operator B := ( λ − A − ) D λ ∈ L ( X, X − ) , λ ∈ ρ ( A ) . (3.13)The facts mentioned above give the following variation of the formula of the parameterswhich will be useful later. Theorem 3.2.
The operator A generates a strongly continuous semigroup ( T ( t )) t ≥ on X satisfying T ( t ) z = T ( t ) z + (cid:90) t T − ( t − s ) B M T ( s ) z ds (3.14) for any z ∈ X and t ≥ .Proof. For u ∈ L p ([0 , ∞ ) , X ) and λ > (cid:92) ( T − ∗ B u )( λ ) = R ( λ, A − ) B (cid:98) u ( λ ) = D λ ˆ u ( λ ) = e λ ˆ u ( λ ) . Thus (cid:92) ( T − ∗ B u )( λ ) = (cid:92) Φ Q,β · u ( λ ) , here (Φ Q,βt ) t ≥ is the control maps associated to the regular system ( Q, β, L ) definedbefore in Example 2.1. By injectivity of Laplace-transform, we deduce that (cid:90) t T − ( t − s ) B u ( s ) ds = Q,βt u ∈ X . (3.15)It follows that B is an admissible control operator for A . The fact that M ∈ L ( X , X ),( A , B , M ) is a regular system with I : X → X as an admissible feedback (see. [23]).Therefore, the operator A cl := A − + B MD ( A cl ) := (cid:8) z ∈ X , A cl z ∈ X (cid:9) (3.16)generates a strongly continuous semigroup ( T cl ( t )) t ≥ on X satisfying T cl ( t ) z = T ( t ) z + (cid:90) t T − ( t − s ) B M T cl ( s ) z ds for any z ∈ X and t ≥
0, see [23]. Finally, we mention that the operator A cl coincideswith the operator A , due to [14, Theorem 4.1]. (cid:3) The following theorem is the main results of this section.
Theorem 3.3.
The operator A L := A + L , D ( A L ) := D ( A ) generates a strongly continuous semigroup ( T L ( t )) t ≥ on X .Proof. To prove that A L is a generator it suffices to show that L is an admissible obser-vation operator for A (see. [10, Theorem 2.1]). To this end, let us define the operator L = L | D ( A ) ∈ L ( D ( A ) , X ) . Let us first show that L is an admissible observation operator for A . For α > z = ( x, f, ϕ ) (cid:62) ∈ D ( A ) = D ( A δ ) × D ( Q ) , we have (cid:90) α (cid:107)L T ( t ) z (cid:107) p dt = (cid:90) α (cid:107) L S ( t ) ϕ (cid:107) p dt, where L is the operator defined in Example 2.1, which is an admissible observationoperator for Q . It follows from the above estimate that L is admissible for A . On theother hand, by using a similar argument as in [10, Lemma 6.3], one can see that theYosida extension of L with respect to A is explicitly given by D ( L , Λ ) = X × D ( L , Λ ) , L , Λ = L , Λ
00 0 0 . (3.17) n addition for any z ∈ D ( A ), we have (cid:90) α (cid:107)L , Λ T ( t ) z (cid:107) p dt ≤ c p (cid:107) z (cid:107) p (3.18)for a constant c >
0. According to [14, Lemma 3.6], we have
Z ⊂ D ( L , Λ ) , and ( L , Λ ) |Z = L . Thus for z ∈ D ( A ), (cid:90) α (cid:107)L T ( t ) z (cid:107) p dt = (cid:90) α (cid:107)L , Λ T ( t ) z (cid:107) p dt. (3.19)If we consider u ( t ) := M T ( t ) z , t ≥ , then by using the proof of Theorem 3.2, we obtain (cid:90) t T − ( t − s ) B M T cl ( s ) z ds = Q,βt u . (3.20)Now from Example 2.1, ( Q, β, L ) is regular and that Φ Q,βt u ∈ D ( L , Λ ) for a.e. t ≥ (cid:90) t T − ( t − s ) B M T cl ( s ) z ds ∈ D ( L , Λ ) , and (cid:90) α (cid:13)(cid:13)(cid:13)(cid:13) L , Λ (cid:90) t T − ( t − s ) B M T cl ( s ) z ds (cid:13)(cid:13)(cid:13)(cid:13) p dt = (cid:90) α (cid:107) L , Λ Φ Q,βt u (cid:107) p dt ≤ κ p (cid:107) u (cid:107) pL p ([0 ,α ] ,X ) ≤ ˜ κ p (cid:107) z (cid:107) p , (3.21)for a constant ˜ κ >
0; due to the facts M is bounded and that T is exponentially bounded.Finally, using on the one side, (3.14) and (3.19), on the other side, the estimates (3.18)and (3.21), we obtain (cid:90) α (cid:107)L T ( t ) z (cid:107) p dt ≤ γ p (cid:107) z (cid:107) p dt, for any z ∈ D ( A ) and some constant γ := γ ( α ) >
0. This ends the proof. (cid:3)
Remark . Observe that M D λ = 0 for any λ ∈ ρ ( A ). It is shown in [14, Theorem 4.1]that for λ ∈ ρ ( A ) we have λ ∈ ρ ( A ) ⇐⇒ ∈ ρ ( D λ M ) ⇐⇒ ∈ ρ ( M D λ ) = C ∗ . Thus ρ ( A ) = ρ ( A δ ) ⊂ ρ ( A ) . (3.22) gain by [14, Theorem 4.1], for λ ∈ ρ ( A ), we have R ( λ, A ) = ( I − D λ M ) − R ( λ, A ) . In the following result, we give a new expression of the solution of the Cauchy problem(3.8) by appealing only to semigroup ( T ( t )) t ≥ . Proposition 3.1.
The solution z ( · ) of the Cauchy problem (3.8) satisfies z ( s ) ∈ D ( L Λ ) , a.e. s ≥ ,z ( t ) = T ( t ) z (0) + (cid:90) t T − ( t − s ) B M z ( s ) ds + (cid:90) t T ( t − s ) L Λ z ( s ) ds (3.23) for any t ≥ , z (0) ∈ X, where L Λ is the Yosida extension of L with respect to A .Proof. Using [10, Theorem 5.1] and the fact that L is an admissible observation operatorfor A (see the proof of the previous result), the solution of the Cauchy problem (3.8)satisfies z ( s ) ∈ D ( L Λ ) for a.e. s ≥
0, and z ( t ) = T ( t ) z (0) + (cid:90) t T ( t − s ) L Λ z ( s ) ds, for any t ≥ , z (0) ∈ X .For a sufficiently large λ > , the Laplace transform of z yields (cid:98) z ( λ ) = R ( λ, A ) z (0) + R ( λ, A ) L Λ (cid:98) z ( λ )]= ( I − D λ M ) − [ R ( λ, A ) z (0) + R ( λ, A ) L Λ (cid:98) z ( λ )] . This in turn implies that (cid:98) z ( λ ) = R ( λ, A ) z (0) + D λ M (cid:98) z ( λ ) + R ( λ, A ) L Λ (cid:98) z ( λ ) . (3.24)Let (cid:36) ( t ) := T ( t ) z (0) + (cid:90) t T − ( t − s ) B M z ( s ) ds + (cid:90) t T ( t − s ) L Λ z ( s ) ds, t ≥ . According to (3.13) and (3.24), the Laplace transform of (cid:36) satisfies (cid:98) (cid:36) ( λ ) = R ( λ, A ) z (0) + R ( λ, A − ) B M (cid:98) z ( λ ) + R ( λ, A ) L Λ (cid:98) z ( λ )= (cid:98) z ( λ ) , and the result follows by virtue of the injectivity the Laplace transform. (cid:3) Note that for a ( · ) ≡
0, the resolvent familly correspond to the semigroup generatedby A . Therefore the following result which establishes the relation between the solutionsof (1.1) and (3.8), extends [13, Proposition 2] from delay differential equations to delayintegro-differential one. Theorem 3.4.
For any initial condition ( x, f, ϕ ) (cid:62) ∈ X , there exists a unique solution x ( · ) of (1.1) satisfying x t ∈ D ( L Λ ) for a.e. t ≥ and x ( t ) = R ( t ) x + (cid:90) t R ( t − s )( L Λ x s + f ( s )) ds, t ≥ . roof. Let z (0) = ( x, f, ϕ ) (cid:62) ∈ X and z ( t ) = ( x ( t ) , v ( , · ) , w ( t )) (cid:62) be the solution of theCauchy problem (3.8) associated to z (0). Appealing to Proposition (3.1) and by combining(3.23) together with (3.12), (3.20), (3.3) we deduce that w ( t ) = x t and (3.8) becomes x ( t ) v ( t, · ) x t = T δ ( t ) (cid:18) xf (cid:19) S ( t ) ϕ + Q,β x + (cid:90) t T δ ( t − s ) (cid:18) L Λ x s (cid:19) ds = (cid:18) R ( t ) Υ( t ) ∗ ∗ (cid:19) (cid:18) xf (cid:19) S ( t ) ϕ + Q,β x + (cid:90) t (cid:18) R ( t − s ) Υ( t − s ) ∗ ∗ (cid:19) (cid:18) L Λ x s (cid:19) ds. Thus by equalizing the two first components of the above equality we get the requiredresult. (cid:3) Spectral theory
In this section, we attempt to study the spectral theory for the generator A L and we getsome results extending those obtained in [1] and [8]. First of all, we require the followingstraightforward lemma. Lemma 4.1.
For λ ∈ ρ ( A δ ) , we have λ ∈ ρ ( A L ) ⇐⇒ ∈ ρ ( L R ( λ, A )) . (4.1) Proof.
For λ ∈ ρ ( A δ ), we write λ − A L = λ − A − L . By Remark (3.1), we have λ ∈ ρ ( A ) λ − A L = (1 − L R ( λ, A ))( λ − A ) . This implies that λ ∈ ρ ( A L ) if and only if (1 − L R ( λ, A )) − exists. (cid:3) From the previous results we immediately obtain the following result which characterisesthe spectrum of A L and then extends [1, Lemma 4.1] from delay differential or integro-differential equations to delay integro-differential equations (1.1) . Theorem 4.1.
For λ ∈ C ∩ ρ ( A δ ) ∩ ρ ( A ) , we have λ ∈ ρ ( A L ) ⇐⇒ λ ∈ ρ (cid:16) (1 + ˆ a ( λ )) A + Le λ (cid:17) . Proof.
By virtue of Remark (3.1), we have λ ∈ ρ ( A ) and therefore R ( λ, A ) = ( I − D λ M ) − R ( λ, A )= e λ R ( λ, A δ ) 000 0 R ( λ, Q ) = (cid:18) R ( λ, A δ ) 0 (cid:0) e λ (cid:1) R ( λ, A δ ) R ( λ, Q ) (cid:19) . ppealing to [8, Proposition. 7.25], (see. [2]) leads to λ a ( λ ) ∈ ρ ( A ) and by virtue of[2, Lemma 7 (ii)], we obtain R ( λ, A δ ) = H ( λ ) H ( λ ) δ R ( λ, dds ) R ( λ, dds ) V H ( λ ) R ( λ, dds ) V H ( λ ) δ R ( λ, dds ) + R ( λ, dds ) , where H ( λ ) = ( λ − (1+ ˆ a ( λ ) A )) − and V ∈ L ( D ( A ) , W ,p ( R + , X )) is given by ( V x )( t ) := a ( t ) Ax for x ∈ D ( A ) and a.e. t > R ( λ, A ) = H ( λ ) H ( λ ) δ R ( λ, dds ) 0 R ( λ, dds ) V H ( λ ) R ( λ, dds ) V H ( λ ) δ R ( λ, dds ) + R ( λ, dds ) 0 e λ H ( λ ) e λ H ( λ ) δ R ( λ, dds ) R ( λ, Q ) and L R ( λ, A ) = Le λ H ( λ ) Le λ H ( λ ) δ R ( λ, dds ) LR ( λ, Q )0 0 00 0 0 (4.2)By combining (4.1) and (4.2), we deduce that λ ∈ ρ ( A L ) if and only if 1 ∈ ρ ( Le λ H ( λ )).Rewriting ( λ − (1 + ˆ a ( λ )) A ) − Le λ = (1 − Le λ H ( λ )) ( λ − (1 + ˆ a ( λ )) A )we conclude that λ ∈ ρ ( A L ) ⇐⇒ λ ∈ ρ ((1 + ˆ a ( λ )) A + Le λ ) . (cid:3) Remark . In the particular case of a ( · ) ≡
0, we obtain the known result which for λ ∈ ρ ( A ) λ ∈ ρ ( A L ) ⇐⇒ λ ∈ ρ ( A + Le λ )which coincides perfectly with the classic problem with delay (see [1, Lemma 4.1]). Onthe other hand, for the free-delay integro-differential equation we can identify ρ ( A L ) with ρ ( A δ ). Thus for λ ∈ C ∩ ρ ( A ), we retrieve [8, Proposition 7.25] saying that λ ∈ ρ ( A δ ) ⇐⇒ λ ∈ ρ ((1 + ˆ a ( λ )) A ) . . Intego-differential systems with delays as regular Salamon-Weisssystems
In this section, we keep the same notation as in the previous sections. In addition, setthe following Riemann Stieltjes integrals Kψ = (cid:90) − r dµ K ( θ ) ψ ( θ ) , µ K : [ − r, → L ( U, X ) Cϕ = (cid:90) − r dµ C ( θ ) ϕ ( θ ) , µ C : [ − r, → L ( X, Y ) ,Dψ = (cid:90) − r dµ D ( θ ) ψ ( θ ) , µ D : [ − r, → L ( U, Y ) , for ϕ ∈ W ,p ([ − r, , X ) and ψ ∈ W ,p ([ − r, , U ), where µ K , µ C , and µ D are functions ofbounded variations assumed continuous on [ − r,
0] and vanishing at zero.Now consider the state, input and output delay system ˙ x ( t ) = Ax ( t ) + (cid:90) t a ( t − s ) Ax ( s ) ds + Lx t + Ku t + f ( t ) , t ≥ x (0) = x, x = ϕ, u = ψy ( t ) = Cx t + Du t , t ≥ , (5.1)for initial data x ∈ X, ϕ ∈ L p ([ − r, , X ) and ψ ∈ L p ([ − r, , U ) , where A, L, a and thehistory function ( t (cid:55)→ x t ) are defined in the previous sections, u ∈ L p ([ − r, + ∞ ) , U is thecontrol function, and t (cid:55)→ u t is the control history function defined by u t ( θ ) = u ( t + θ ) forany t ≥ θ ∈ [ − r, L p ([ − r, , U ) , we define the linear operator Q U ψ = ψ (cid:48) , D ( Q U ) = { ψ ∈ W ,p ([ − r, , U ) : ψ (0) = 0 } . The operator Q U generates the left shift semigroup ( S U ( t )) t ≥ on L p ([ − r, , U ) definedby ( S U ( t ) ψ )( θ ) = ψ ( t + θ ) if t + θ ≤ t ≥ θ ∈ [ − r, u, we have the following results on the existence of thesolution of the problem (5.1). Theorem 5.1.
Let the initial conditions x ∈ X, ϕ ∈ L p ([ − r, , U ) , ψ ∈ D ( Q U ) , f ∈ L p ( R + , X ) and a smooth control function u ∈ W ,p ([0 , + ∞ ) , U ) with u (0) = 0 , . Thenthere exists a unique solution x ( · ) of (5.1) satisfying x t ∈ D ( L Λ ) for a.e. t ≥ and x ( t ) = R ( t ) x + (cid:90) t R ( t − s )( L Λ x s + Ku s + f ( s )) ds, t ≥ . (5.2) Proof.
As in Example 2.1, the function t (cid:55)→ u t is the state of a regular linear system( Q U , β U , K ) with control operator β U := ( λ − Q U − ) e λ for λ ∈ C . Thus u t = S U ( t ) ψ + (cid:90) t S U − ( t − s )( − Q U − e ) u ( s ) ds, t ≥ . s ψ ∈ D ( Q U ) and u ∈ W ,p ([0 , + ∞ ) , U ), an integration by parts shows that u t ∈ W ,p ([ − r, , U ). Thus the function g ( t ) = Ku t is well-defined for t ≥
0. On the otherhand, by [14, Lemma 3.6], we have W ,p ([ − r, , U ) ⊂ D ( K Λ ) and g ( t ) = K Λ u t for a.e. t ≥ , where K Λ is the Yosida extension of K with respect to Q U . As t (cid:55)→ K Λ u t is the extendedoutput function of the regular linear system ( Q U , β U , K ), we have g ∈ L ploc ( R + , U ). If weset ζ ( t ) = (cid:16) g ( t )00 (cid:17) , t ≥ , then the problem (5.1) can be reformulated as˙ z ( t ) = A L z ( t ) + ζ ( t ) , z (0) = (cid:16) xfϕ (cid:17) , t ≥ , (5.3)where z ( · ) and A L are given by (3.6) and (3.9), respectively. Now by using the sameargument as in the proof of Proposition 3.1, one can see that the mild solution of theinhomogeneous Cauchy problem (5.3) is given by z ( t ) = T ( t ) z (0) + (cid:90) t T − ( t − s ) B M z ( s ) ds + (cid:90) t T ( t − s ) ( L Λ z ( s ) + ζ ( s )) ds, for t ≥ , where ( T ( t )) t ≥ is the strongly continuous semigroup given by (3.12). The restof the proof follows exactly as in the proof of Theorem 3.4. (cid:3) From our discussion in the proof of Theorem 5.1, we adopt the following definition:
Definition 5.1.
Let the initial conditions x ∈ X, ϕ ∈ L p ([ − r, , X ) and ψ ∈ L p ([ − r, , U ) , and let f ∈ L p ( R + , X ) . A mild solution of the integro-differential equationin (5.1) is a function x : [ − r, + ∞ ) → X such that x ( t ) = R ( t ) x + (cid:90) t R ( t − s ) ( L Λ x s + K Λ u s + f ( s )) ds, t ≥ ϕ ( t ) , a.e. t ∈ [ − r, . (5.4)The following result proves the relationship between smooth input solution and mildsolution of the problem (5.1). Proposition 5.1.
The mild solution of (5.1) is limit of a sequence of smooth inputsolutions of (5.1) .Proof.
Let x ∈ X, ϕ ∈ L p ([ − r, , X ) , ψ ∈ L p ([ − r, , U ) and u ∈ L ploc ( R + , U ) such that u = ψ . Let x ( · ) the corresponding mild solution of the problem (5.1). We approximate x, ϕ , ψ and u by sequences ( x n ) n ⊂ D ( A ) , ( ϕ n ) n ⊂ D ( Q X ) , ( ψ n ) n ⊂ D ( Q U ) and u n ∈ W ,p ,loc ( R + , U ), respectively. For any t ≥ , define the function u nt ( θ ) = (cid:40) u n ( t + θ ) , − t ≤ θ ≤ ,ψ n ( t + θ ) , − r ≤ θ < − t. ccording to Theorem 5.1, the following functions x n ( t ) = R ( t ) x + (cid:90) t R ( t − s ) ( L Λ x ns + Ku ns + f ( s )) ds, t ≥ , n ∈ N . (5.5)define smooth input solutions of the problem (5.1) with initial conditions x n , ϕ n and ψ n .Let Σ U := ( S U , Ψ U , Φ U , F U ) the regular linear system generated by the triple ( Q U , β U , K ).As both functions ( t (cid:55)→ K Λ u t ) and ( t (cid:55)→ Ku nt ) are output functions of Σ U , then K Λ u t = (cid:0) Ψ U ψ (cid:1) ( t ) + (cid:0) F U u (cid:1) ( t ) ,Ku nt = (cid:0) Ψ U ψ n (cid:1) ( t ) + (cid:0) F U u n (cid:1) ( t ) . (5.6)Let α > t ∈ [0 , α ], and set γ := max (cid:0) | µ K | ([ − r, , | µ L | ([ − r, (cid:1) . ByH¨older inequality, there exists a constant c α > (cid:107) K Λ u · − Ku n · (cid:107) L p ([0 ,α ] ,X ) ≤ γc t (cid:0) (cid:107) ψ − ψ n (cid:107) L p ([ − r, ,U ) + (cid:107) u − u n (cid:107) L p ([0 ,α ] ,U ) (cid:1) Thus (cid:107) K Λ u · − Ku n · (cid:107) L p ([0 ,α ] ,X ) → n → ∞ . Similarly, for any m, n ∈ N , (cid:107) L Λ x n · − L Λ u m · (cid:107) L p ([0 ,α ] ,X ) ≤ γc α (cid:0) (cid:107) ϕ n − ϕ m (cid:107) L p ([ − r, ,X ) + (cid:107) x n ( · ) − x m ( · ) (cid:107) L p ([0 ,α ] ,X ) (cid:1) . On the other hand, by H¨older’s inequality, for m, n ∈ N , (cid:107) R ∗ (( L Λ x n · − L Λ x m · ) + ( K Λ u n · − K Λ u m · )) (cid:107) L p ([0 ,α ] ,X ) ≤ αc ( α ) (cid:0) (cid:107) ϕ n − ϕ m (cid:107) p + (cid:107) x n ( · ) − x m ( · ) (cid:107) L p ([ − r, ,X ) + (cid:107) K Λ u n · − K Λ u m · (cid:107) L p ([0 ,α ] ,X ) (cid:1) for a constant c ( α ) >
0. We choose α > αc ( α ) < . For a such α, we have (cid:107) x n ( · ) − x m ( · ) (cid:107) L p ([ − r, ,X ) ≤ κ ( α ) (cid:0) (cid:107) x n − x m (cid:107) + (cid:107) ϕ n − ϕ m (cid:107) p + (cid:107) K Λ u n · − K Λ u m · (cid:107) L p ([0 ,α ] ,X ) (cid:1) . We deduce that ( x n ( · )) n is a Cauchy sequence in L p ([0 , α ] , X ), so it converges to function x ( · ) ∈ L p ([0 , α ] , X ). We select x t ( θ ) = (cid:40) x ( t + θ ) , − t ≤ θ ≤ ,ϕ ( t + θ ) , − r ≤ θ < − t. As above, L Λ x n · converges to L Λ x · in L p ([0 , α ] , X ). By passing to limit in (5.5), we obtain x ( t ) = R ( t ) x + (cid:90) t R ( t − s ) ( L Λ x s + K Λ u s + f ( s )) ds, t ≥ . this ends the proof. (cid:3) In the rest of this section we will show that the delay system (5.1) is equivalent toa regular distributed linear system in the Salamon-Weiss sense (see Section 2 for thedefinition of such systems). In fact, as shown in the proof of Theorem 5.1, the problem(5.1) is reformulated as the following system˙ z ( t ) = A L z ( t ) + (cid:16) K Λ u t (cid:17) , z (0) = (cid:16) xfϕ (cid:17) , t ≥ . (5.7) n particular, the mild solution of the equation (5.7) is given by z ( t ) = T L ( t ) (cid:16) xfϕ (cid:17) + (cid:90) t T L ( t − s ) (cid:16) K Λ u s (cid:17) ds. (5.8)Let us introduce the a new product state space˜ X = X × L p ([ − r, , U ) , where X is the product space defined in (3.5). Moreover, we select a new state function w : t ∈ [0 , + ∞ ) (cid:55)→ w ( t ) = (cid:18) z ( t ) u t (cid:19) ∈ ˜ X . The main result of this section is the following:
Theorem 5.2.
There exist a generator ˜ A : D ( ˜ A ) ⊂ ˜ X → ˜ X of a strongly continuoussemigroup on ˜ X , an admissible control operator ˜ B ∈ L ( U, ˜ X − ) for ˜ A and an admissibleobservation operator ˜ C ∈ L ( D ( ˜ A ) , Y ) for ˜ A such that the triple ( ˜ A , ˜ B , ˜ C ) is regular on ˜ X , U, Y and the integro-differential equation with state and input delays (5.1) is reformu-lated as the following input-output distributed linear system ˙ w ( t ) = ˜ A w ( t ) + ˜ B u ( t ) , w (0) = (cid:18) xfϕψ (cid:19) , t ≥ ,y ( t ) = ˜ C w ( t ) , t ≥ . Proof.
By using (5.6), (5.8) and the notation M ( t ) ψ := (cid:90) t T L ( t − s ) (cid:16) (Ψ U ψ )( s )00 (cid:17) ds, N t u := (cid:90) t T L ( t − s ) (cid:16) ( F U u )( s )00 (cid:17) ds for ψ ∈ L p ([ − r, , U ) and u ∈ L p ( R + , U ) with u = ψ, we can also write z ( t ) = T L ( t ) (cid:16) xfϕ (cid:17) + M ( t ) ψ + N t u. Now, the fact that u t = S U ( t ) ψ + Φ Ut u, where (Φ Ut ) t ≥ is the family of control mapsassociated with the regular triple ( Q U , β U , K ), implies that w ( t ) = ˜ T ( t ) (cid:18) xfϕψ (cid:19) + ˜Φ t u, where ˜ T ( t ) := (cid:18) T L ( t ) M ( t )0 0 0 S U ( t ) (cid:19) , and ˜Φ t u := (cid:18) N t u Φ Ut u (cid:19) . similar argument as in the proof of [11, Theorem 3.1] shows that the operators family( ˜ T ( t )) t ≥ define a strongly continuous semigroup on ˜ X with generator˜ A = A L K
000 0 0 Q U , D ( ˜ A ) = D ( A L ) × D ( Q U ) . We know that (see Example 2.1) (cid:107) Φ U u (cid:107) L p ([ − r, ,U ) ≤ (cid:107) u (cid:107) L p ([0 ,t ] ,U ) and (cid:107) F U u (cid:107) L p ([0 ,t ] ,U ) ≤ | µ K | ([ − r, (cid:107) u (cid:107) L p ([0 ,t ] ,U ) for any t ≥ u ∈ L p ( R + , U ). Then by H¨older inequality, we obtain (cid:107) ˜Φ t u (cid:107) ˜ X = (cid:107) N t u (cid:107) X + (cid:107) Φ U u (cid:107) L p ([ − r, ,U ) ≤ ( c | µ K | ([ − r, (cid:107) u (cid:107) L p ([0 ,t ] ,U ) for some constant c := c ( t ) >
0. In addition, the fact thatΦ Ut + s u = S U ( t )Φ Us ( u | [0 ,s ] ) + Φ Ut u ( · + s ) , ( F U u )( t + s ) = (Ψ U Φ s u )( t ) + ( F U u ( · + s ))( t ) , implies that ˜Φ t + s u = ˜ T ( t ) ˜Φ s u + ˜Φ t u ( · + s ) . According to [22], this functional equation shows that there exists an admissible controloperator ˜
B ∈ L ( U, ˜ X ) for ˜ A such that˜Φ t u = (cid:90) t ˜ T − ( t − s ) ˜ B u ( s ) ds for any t ≥ u ∈ L p ( R + , U ). Thus the function w ( · ) is the solution of the controlproblem ˙ w ( t ) = ˜ A w ( t ) + ˜ B u ( t ) , w (0) = (cid:18) xfϕψ (cid:19) , t ≥ C := (cid:0) C D (cid:1) : D ( ˜ A ) → Y. Then the out-put of the system (5.1) satisfies y ( t ) = ˜ C w ( t ) , t ≥ . First, we prove that ˜ C is an admissible observation operator for ˜ A . According to theexpression the semigroup ( ˜ T ( t )) t ≥ , the fact that the operator D ∈ L ( D ( Q U ) , Y ) is ad-missible observation for Q U , and [10, Proposition 3.3] it suffices to show that the operator C = (0 0 C ) ∈ L ( D ( A ) , Y ) is an admissible observation operator for A . To this end, as A L = A + L and L is an admissible observation operator for A , (see the proof of Theorem3.3), by [12] it suffices to prove that C is an admissible observation operator for A . Thisis obvious due to the expression of the semigroup ( T ( t )) t ≥ given in (3.12) and the fact hat the operator C ∈ L ( D ( Q ) , X ) is an admissible observation operator for the left shiftsemigroup ( S ( t )) t ≥ on L p ([ − r, , X ).Second, we prove that the triple ( ˜ A , ˜ B , ˜ C ) is regular. To that purpose, we need first tocompute the Yosida extension of ˜ C with respect to ˜ A . In fact, by taking the Laplacetransform of ˜ T ( t ) we obtain R ( λ, ˜ A ) = (cid:32) R ( λ, A L ) R ( λ, A L ) (cid:16) KR ( λ,Q U )00 (cid:17) R ( λ, Q U ) (cid:33) , λ ∈ ρ ( A L ) . Let ( x f ϕ ψ ) (cid:62) ∈ ˜ X . For λ > C λR ( λ, ˜ A ) (cid:18) xfϕψ (cid:19) = C λR ( λ, A L ) (cid:16) xfϕ (cid:17) + C R ( λ, A L ) (cid:16) KλR ( λ,Q U ) ψ (cid:17) + DλR ( λ, Q U ) ψ. Observe that if ψ ∈ D ( K Λ ), then (cid:13)(cid:13)(cid:13) C R ( λ, A L ) (cid:16) KλR ( λ,Q U ) ψ (cid:17)(cid:13)(cid:13)(cid:13) ≤ (cid:107)C R ( λ, A L ) (cid:107) (cid:0) (cid:107) KλR ( λ, Q U ) ψ − K Λ ψ (cid:107) + (cid:107) K Λ ψ (cid:107) (cid:1) . As C is an admissible observation operator for A L , it follows that (cid:107)C R ( λ, A L ) (cid:107) goes to 0when λ → + ∞ . Thenlim λ → + ∞ (cid:13)(cid:13)(cid:13) C R ( λ, A L ) (cid:16) KλR ( λ,Q U ) ψ (cid:17)(cid:13)(cid:13)(cid:13) = 0 , ∀ ψ ∈ D ( K Λ ) . Then we haveΩ := D ( C Λ ) × ( D ( K Λ ) ∩ D ( D Λ )) ⊂ D ( ˜ C Λ ) and ( ˜ C Λ ) | Ω = (cid:0) C Λ D Λ (cid:1) . (5.9)By [10, Proposition 3.3], for u ∈ L ploc ( R + , U ) we have N t u ∈ D ( C Λ ) and (cid:107)C Λ N · u (cid:107) L p ([0 ,τ ] ,Y ) ≤ cτ q (cid:107) F U u (cid:107) L p ([0 ,τ ] ,X ) ≤ cτ q | µ K | ([ − τ, (cid:107) u (cid:107) L p ([0 ,τ ] ,U ) , for a.e. t ≥ c > p + q = 1. On the other hand, thetriple ( Q U , β U , K ) and ( Q U , β U , D ) are regular and having the same control maps. ThenΦ Ut u ∈ D ( K Λ ) ∩ D ( D Λ ) for a.e. t ≥
0. Moreover, the extended input-output operator F D of ( Q U , β U , D ) is given by ( F D u )( t ) = D Λ Φ Ut u for almost every t ≥ , and (cid:107) F D u (cid:107) L p ([0 ,τ ] ,Y ) ≤ | µ D | ([ − τ, (cid:107) u (cid:107) L p ([0 ,τ ] ,U ) . Now according to (5.9), we have˜Φ t u ∈ Ω ⊂ D ( ˜ C Λ ) and (˜ F u ) := ˜ C Λ ˜Φ t u = C Λ N t u + ( F D u )( t )for a.e. t ≥ u ∈ L ploc ( R + , U ). Moreover (cid:107) ˜ F u (cid:107) L p ([0 ,τ ] ,Y ) ≤ c ( τ ) (cid:107) u (cid:107) L p ([0 ,τ ] ,U ) , ∀ u ∈ L p ([0 , τ ] , U ) , (5.10)where c ( τ ) := 2 p − (cid:16) cτ p | µ K | ([ − τ, | µ D | ([ − τ, (cid:17) −→ τ → . ence the triple ( ˜ A , ˜ B , ˜ C ) is well-posed in ˜ X , U, Y . Let b ∈ U be a fixed control and set u b ( t ) = b for any t ≥
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