On the Controllability of a Differential Equation with Delayed and Advanced Arguments

and Applied Analysis 3 LetM be the following nonempty, closed subspace of the topological spaceC∞ −1, 1 , C n : M { φ ∈ C∞ −1, 1 ,C : φ n 0 φ n−1 −1 φ n−1 1 , n 1, 2, 3, . . . } . 2.8 The space C∞ −1, 1 ,C is endowed with the topology induced by the following countable system of seminorms: Pk ( f ) max t∈ −1,1 ‖f k x ‖ Cn , for k 0, 1, 2, . . . . 2.9 The convergence in this topology means the uniform convergence of the function and each of its derivatives of any order. We denote


Introduction
In this paper, we will study the exact controllability of a functional differential equation with both delayed and advanced arguments.Such equations are often referred to in the literature as mixed-type functional differential equations MTFDE or forward-backward equations.The study of this type of equations is less developed compared with other classes of functional equations.Interest in MTDFEs is motivated by problems in optimal control 1 and applications, for example, in economic dynamics 2 and travelling waves in a spatial lattice 3 .See also 4 .In all these references, the reader can find interesting examples and applications.
In order to achieve our goal, first, we rewrite the equation as a classical Cauchy problem in a certain Banach space.Then we introduce the associated semigroup and its infinitesimal generator and prove some important properties of these operators including some spectral properties .This will allow us to characterize the exact controllability, by applying a result of Bárcenas and Diestel see 5 .

Preliminary Results
In 6 , the following differential-difference equation is considered where t ≥ 0 and x : −1, ∞ → C n is differentiable in 0, ∞ .Equation 2.1 may be written as or equivalently x t x t − 1 − x t − 2 .

2.3
From this we have that in order to find the solution x t on the interval m, m 1 , it is necessary to know its value on the interval m − 2, m , with m being a positive integer.In particular, to determine the solution on the interval 1, 2 , it is necessary to know it on the interval −1, 1 .Accordingly, x t is defined for t ∈ −1, 1 as where the function ϕ belongs to the space C ∞ −1, 1 , C n .The solution of the initial value problem 2.1 , 2.4 is constructed via an iterative process using the step derivation method.It is where c 1 i and c 2 i are constants not all necessarily different from zero.This solution may be extended to the left by rewriting 2.1 as which allows to yield an expression for x t analogous to 2.5 .
In order to assure the existence, differentiability, and uniqueness of the solution x t , it is demanded that x t must satisfy the relationship for n 1, 2, 3, . . . .Further, if a differentiable solution x t exists, then it belongs to the space C ∞ −1, ∞ , C n see 6, Theorems 3.1 and 3.2 .
Let M be the following nonempty, closed subspace of the topological space The space C ∞ −1, 1 , C n is endowed with the topology induced by the following countable system of seminorms: The convergence in this topology means the uniform convergence of the function and each of its derivatives of any order.We denote

2.10
A sequence {f n } ∞ n 1 converges to f if and only if f n − f k tends to 0, as n tends to infinity, for each k.
For each t ≥ 0, the operator T t is defined on the solutions x t of 2.1 as follows:

2.11
The following result originally appears on 7 .
Theorem 2.1.The family {T t } t≥0 defines a strongly continuous semigroup on L M .
Proof.That T 0 I and T t s T t T s for each t, s ≥ 0 are straightforward from the definition of T t .Since x θ | θ∈ −1,1 ϕ θ ∈ M, the domain of T t is M. On the other hand, the function
In order to prove that T τ is continuous for each fixed τ ≥ 0, we will prove that there exists n k for each k and some constant c ≥ 0 such that 2.14 In view that x k t 2.15 and using formula 2.5 with m τ , one obtains

2.16
It only remains to prove that T t x → T t 0 x, as t → t o , for each x ∈ M. In fact, let t 0 ≥ 0. We have that lim Assuming that 0 ≤ τ ≤ 1 and taking into account the uniform convergence of x k t in the closed interval a, b : t 0 − 2, t 0 2 t, t ∈ a, b , it follows that max θ∈ −1,1 Abstract and Applied Analysis 5 Some basic definitions and concepts on controllability are recalled below.Let U and X be Banach spaces.We consider the inhomogeneous differential linear system: where A : X → X is the infinitesimal generator operator of a strongly continuous semigroup S t t≥0 ; B : U → U is a bounded linear operator and u : 0, ∞ → U is a strongly measurable essentially bounded function.
Let Ω be a nonempty separable weakly compact convex subset of U.
We recall see 8 that x 0 ∈ X is controllable with respect to x 1 if there exist t≥ 0 and a control u ∈ L ∞ 0, t ; U such that x t x 1 in 2.18 .The controllability map on 0, t for some t≥ 0 is the linear map defined by Now, one says that 2.18 is exactly controllable on 0, t if every point in X can be reached from the origin at time t, that is, if ran B t X, which is equivalent to

2.21
In other words,

2.22
The set Ω t {u ∈ L ∞ 0, t ; U : u ∈ Ω a.e.} is called the set of admissible controls of 2.18 , while the set A t x 0 {S t x 0 t 0 S t −s Bu s ds : u ∈ Ω t } is called the set of accessible points of 2.18 .Therefore, the system 2.18 is controllable if 0 ∈ A t x 0 , for each t> 0.
We will make use of the following theorem, which will be applied to problem 2.1 , 2.4 .Theorem 2.2 Bárcenas and Diestel 5 .Let X and U be Banach spaces.Let B : U → X be a bounded linear operator and A : X → X the infinitesimal generator of a C 0 -semigroup S t t≥0 on X whose dual semigroup is strongly continuous on 0, ∞ .Suppose Ω is nonempty separable weakly compact convex subset of U containing 0. Then, for each t> 0, 0 ∈ A t x 0 if and only if for each The Bárcenas-Diestel Theorem is an important and recent achievement on exact controllability.Throughout the literature on optimal control in Banach Spaces, hypotheses like "separable and reflexive" are frequently encountered.Using techniques from Banach space theory and the theory of vector measures, the authors show how to remove the hypothesis of reflexivity thus giving considerably greater generality to the resulting conclusions and translate the question of accessibility of controls to a problem in semigroups of operators, namely, given a c 0 -semigroup S t t≥0 of operators on a Banach space X, under what conditions is the dual semigroup strongly continuous on 0, ∞ ?
It should be noted that, for each fixed t ≥ 0 and each x * ∈ X * , a bounded linear functional u * ∈ U * is defined by means of u * v : x * S t Bv .The maximum in Theorem 2.2 exists as a consequence of a now classical result of James 9 , stating that a weakly closed subset C of a Banach space Z is weakly compact if and only if each continuous linear functional on Z attains a maximum on C.
On the other hand, the following result is proven in 10 : for each x * ∈ X * , the mapping of 0, τ to 0, ∞ that takes t ∈ 0, τ to max v∈Ω x * S t Bv is continuous see also 5 , and so the integral in Theorem 2.2 exists in the common Riemann sense.

The Cauchy Problem
We will formulate the problem 2.1 , 2.4 in the form 2.18 .One should observe that we are working on a topological space which is not a Banach space.Let us consider the space C n × M, endowed with the product topology, where M is defined in 2.8 , and let N be the closed subspace of all pairs r, f in C n × M such that f 0 r.On N we define the following map: for each t ≥ 0 and θ ∈ −1, 1 , where x t is the solution 2.5 of the initial value problem 2.1 , 2.4 .Now, one has the following result.
Theorem 3.1.The operator S satisfies Proof.i The linearity of S t follows from the linearity of its components.In order to see the continuity for each t ≥ 0, let us bear in mind that we are working with the product topology.The continuity and convergence for this topology are coordinatewise.We say that f 0 , f is close or tends to g 0 , g if f 0 − g 0 C n is close to 0 and f − g k is close to 0, for each k, k 0, 1, 2, . . ., or, equivalently, if for each k 2 is close to 0, where • k is defined as above.In the case of the linear map S t , the second coordinate is the continuous semigroup 2.11 , and so, if

3.3
If ϕ 0 , ϕ • is close to 0, 0 , then, in particular, ϕ • k is close to 0, for each k, and thus, by the previous estimate, S t ϕ 0 ϕ • is close to 0, 0 .Being S t linear, this is enough to prove that it is continuous.
ii Now we will check the semigroup properties.Obviously, S 0 I.To prove S t s S t S s , one defines the function h t x t s , where x • is the solution of 2.1 , 2.4 .Therefore, h t satisfies

3.4
By the definition of S, one has On the other hand, According to Theorem 2.1, we have that lim t → 0 x t s x s ; since ϕ is continuous, then lim t → 0 x t ϕ 0 .Consequently, lim t → 0 S t I and S t is a C 0 -semigroup in N.

Lemma 3.2.
Let A be the infinitesimal generator associated to the semigroup S t .For α ∈ R sufficiently large, the resolvent is given by Further, g satisfies the following relation: Proof.i From 8, Lemma 2.1.11, for α ≥ ω 0 ( ω 0 is the growth bound of the semigroup , one has It is observed that g θ is a solution of the differential equation dg dθ θ αg θ − x θ .

3.10
The variation of constants formula for 3.10 shows that g θ is equal to i .
iii Since, as it was defined above, one has after an integration by parts, 12 which is iii ii On the other hand,

. Then its infinitesimal generator is given by
where the domain of A is ∈ N : ϕ is absolutely continuous, and dϕ dθ ∈ M .

3.16
Proof.Consider the operator A defined by

3.18
We will prove that A A. Suppose α is a sufficiently large number such that Lemma 3.2 is true.In this case, for by differentiating i from Lemma 3.2.Now, let us see that To do this, one takes We will show that αI − A is injective.Suppose that there exists

Abstract and Applied Analysis 11
In other words, for every

3.27
Then A A, as we wanted to see.
For each m ∈ N, let M m be the space M defined in 2.8 provided with the topology of

3.28
Now, let N m be the closed subspace of C n × M m of all pairs r, f such that r f 0 .The semigroup S m t : N m → N m , defined in the same form as S t , is now a semigroup defined on a Banach space for all m ∈ N.

Lemma 3.4. If A m denote the infinitesimal generator of S m t , then
A m A for all m ∈ N, where A is the infinitesimal generator of S t . Proof.

3.29
Theorem 3.5.Let A be the infinitesimal generator of S t .The spectrum of A is discrete and it is defined by where Δλ λ − e λ − e −λ I for all λ ∈ C, and the multiplicity of each eigenvalue is finite.For every δ ∈ R, there exist only a finite number of eigenvalues in C δ {s ∈ C : Re s > δ}.If λ ∈ σ A , then r e λ• r , where r / 0 satisfies Δλr 0, is an eigenvector of A with eigenvalue λ.On the other hand, if Φ is an eigenvector of A with eigenvalue λ, then Φ r e λ• r with Δλr 0.
Proof.According to the previous Lemma 3.2, for α ∈ R sufficiently large, one has where g θ and g 0 are as i and ii , respectively in Lemma 3.2.Denote by Q λ the extension of 3.31 to C, that is, A simple calculation shows that if λ ∈ C satisfies det Δλ / 0, then Q λ is a bonded linear operator from N to N. Furthermore, for these λ we have λI Finally, we will show that the multiplicity of each eigenvalue is finite.From Lemma 3.2, one has Abstract and Applied Analysis 13 where g 0 is given in ii of Lemma 3.2.We deduce from this expression that the resolvent operator, as an operator from N m to N m , is the sum of an operator with finite range and an integral operator.The first operator is compact see 8, Lemma A.3.22a , and so is the second one, as we will see.Therefore, αI − A −1 is compact as an operator of L N m , for all m ∈ N. Therefore, it is enough to prove that T B is compact in C m −1, 1 .Let B be the closed unit ball of C 0, a ; then B ⊆ B which implies

Controllability
Let U be a Banach space.One will consider the following linear system: where A : D A ⊆ N → N is the infinitesimal generator of the semigroup S t t≥0 , B : U → N is a bonded linear operator, u : 0, ∞ → U is a strongly measurable, essentially bounded function.
In this section, we will study the controllability of the system 4.1 .The mild solution of 4.1 is given by Further, we will suppose that Ω is a separable weakly compact convex subset of U.
For τ > 0, the set Before stating the main result on controllability, we need to prove the following lemma.
Lemma 4.2.The map T t : M m → M m , defined by T t u θ u t θ , is compact.
Proof.To prove this lemma, we will follow the next five steps.
1 The Kondrasov's Theorem 12, 13 gives that the following canonical injection is compact where with f k being the derivative of order k in the sense of distributions.H m −1, 1 is endowed with the norm This implies that H m −1, 1 is a Banach space.
2 Using 1 , for m 1, 4 To prove that T t is compact, it must be proven that for every bounded sequence {u i } i∈N in M m , a subsequence {u i j } j∈N can be found such that the sequence {T t u i j } j∈N converges in M m .As T t u i θ u i t θ , then the previous fact is equivalent to obtain a convergent subsequence {u i j } j∈N from {u i } i∈N in M m .
5 Using 3 and the conclusion of 4 , it is obtained that for all t ∈ R, T t is compact as a bounded linear operator from M m to M m .Proof.First, we will prove that S m t , for all t ≥ 0, is compact, for all m ∈ N; in fact, S m t r ϕ • x t x t θ x t 0 0 x t θ .

4.13
The first term has finite rank, therefore, it is compact.According to Lemma 4.2, the second term is compact in M m .Consequently, S m t is compact, as an operator in L C n × M m .In particular, it is compact as an operator in L N m .Hence, the adjoint S m t * is strongly continuous in 0, ∞ see, e.g., 14 .On the other hand, in view of Lemma 4.1, 0 ∈ A τ x 0 if and only if 0 ∈ A m τ x 0 , for all m ∈ N and this is true, after Theorem 1.1 of Bárcenas-Diestel 5 , if and only if x * S m τ x 0 τ 0 max u∈Ω x * S m t Bu dt ≥ 0 for all m ∈ N and each x * ∈ N m * .
then there exists a ξ ∈ C n such that λI − e −λ −e λ ξ 0. The following element of N, z 0 ξ e λ• ξ belongs to D A and λI − A z 0 ⎛ ⎝ λξ − e −λ ξ − e λ {λ ∈ C : det Δλ 0} ⊂ σ A .Let λ be an element in C δ with e δ e −δ < |λ|.For this λ, one has |e λ e −λ | ≤ e δ e −δ < |λ|, and from 8, Corollary A.4.10 , one concludes that λI−e λ −e −λ is invertible in N m , where N m is defined as before.Thus det Δλ / 0 and λ ∈ ρ A .Since det λI − e λ − e −λ is an entire function, it has finitely many zeros in the compact set C δ ∩ {λ ∈ C : |λ| ≤ e δ e −δ } see in 8, Theorem A.1.4.6b and we have showed that in the rest of C δ there are none.Therefore, there are only a finite number of eigenvalues in C δ .Let Φ r ϕ • be an eigenvector of A with eigenvalue λ.From the definition of A, one obtains dϕ/dθ θ λϕ θ for θ ∈ −1, 1 which gives ϕ θ e λθ ϕ 0 .Since Φ ∈ D A , one has ϕ 0 r.Using the first equation of the definition of A, e λ e −λ r λr which shows that Δλr 0. The other implication is obvious.

From 8 ,
Theorem A.4.18 and Lemma A.4.19 , one obtains that the eigenvalue's multiplicity is finite for αI − A and A. In order to prove the compactness of the second operator, let us observe that in 11, Example 1, page 277 , it is seen that the operator Tϕ a a 0 e α a−s ϕ s ds 3.35 is compact in C 0, a .Let us suppose that a ≤ 1 and let B be the closed unit ball in C m −1, 1 ; then B ∩ M is the unit ball of M m and it is valid that T B ∩ M ⊆ T B in the topology of C m −1, 1 .

Therefore, H m 1 − 1 , 1 c→ 1 c→
C m −1, 1 .3By 2 and the definition of M m 1 , it follows thatM m 1 ⊆ C m 1 −1, 1 ⊆ H m 1 −1, C m −1, 1 .4.11Hence, for every bounded sequence {f k } k∈N in M m 1 with the topology of C m −1, 1 , it can be found a convergent subsequence{f k l } l∈N ⊆ {f k } k∈N that converges to f in C m −1, 1 ; but f m is continuous and f m 1 ∈ L 2 −1, 1 .Thus {f m 1 k l } l∈Nconverges to f m 1 a.e, and so this subsequence converges to f in M m 1 .