This paper is divided in two parts. In the first part we study a second order neutral partial differential equation with state dependent delay and noninstantaneous impulses. The conditions for existence and uniqueness of the mild solution are investigated via Hausdorff measure of noncompactness and Darbo Sadovskii fixed point theorem. Thus we remove the need to assume the compactness assumption on the associated family of operators. The conditions for approximate controllability are investigated for the neutral second order system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. A simple range condition is used to prove approximate controllability. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in (Balachandran and Park, 2003), which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in (Dauer and Mahmudov, 2002), which are practically difficult to verify and apply. Examples are provided to illustrate the presented theory.

1. Introduction

Neutral differential equations appear as mathematical models in electrical networks involving lossless transmission, mechanics, electrical engineering, medicine, biology, ecology, and so forth. Neutral differential equations are functional differential equations in which the highest order derivative of the unknown function appears both with and without derivatives. Second order neutral differential equations model variational problems in calculus of variation and appear in the study of vibrating masses are attached to an electric bar.

Impulsive differential equations are known for their utility in simulating processes and phenomena subject to short term perturbations during their evolution. Discrete perturbations are negligible to the total duration of the process which have been studied in [1–6].

However noninstantaneous impulses are recently studied by Ahmad [7]. Stimulated by their numerous applications in mechanics, electrical engineering, medicine, ecology, and so forth, noninstantaneous impulsive differential equations are recently investigated.

Recently, much attention is paid to partial functional differential equation with state dependent delay. For details see [7–12]. As a matter of fact, in these papers their authors assume severe conditions on the operator family generated by A, which imply that the underlying space X has finite dimension. Thus the equations treated in these works are really ordinary and not partial equations. The literature related to state dependent delay mostly deals with functional differential equations in which the state belongs to a finite dimensional space. As a consequence, the study of partial functional differential equations with state dependent delay is neglected. This is one of the motivations of our paper.

The papers [13, 14] study existence of differential equation via measure of noncompactness. Measure of noncompactness significantly removes the need to assume Lipschitz continuity of nonlinear functions and operators.

In recent years, controllability of infinite dimensional systems has been extensively studied for various applications. In the papers [15, 16] the authors discuss the exact controllability results by assuming that the semigroup associated with the linear part is compact. However, if the operator B is compact or C₀-semigroup T(t) is compact then the controllability operator is also compact. Hence the inverse of it does not exist if the state space X is infinite dimensional [17].

Another available method in the literature involves the invertibility of operator , where is the controllability Gramian and a limit condition which is difficult to check and apply in practical real world problems. See for details [18]. Also it is practically difficult to verify their condition directly. This is one of the motivations of our paper.

However our work is a continuation of coauthor Sukavanam′s novel approach in article [19]. We extend our work [20–22] in this paper.

Controllability results are available in overwhelming majority for abstract differential delay systems (see [1, 3–6, 9–12, 14–17, 19–34]), rather than for neutral differential with state dependent delay.

The organization of the paper is as follows. In Section 3 we study the existence and uniqueness of mild solution of the second order equation modelled in the form

()

where A is the infinitesimal generator of a strongly continuous cosine family

of bounded linear operators on a Banach space X. The history valued function x_t : (−∞, 0] → X, x_t(θ) = x(t + θ) belongs to some abstract phase space

defined axiomatically;

are appropriate functions. 0 = t₀ = s₀ < t₁ ≤ s₁ ≤ t₂, <⋯, <t_n ≤ s_n ≤ t_n+1 = a are prefixed numbers. In Section 5 we study the approximate controllability of

()

where A is the infinitesimal generator of a strongly continuous cosine family

of bounded linear operators on a Hilbert space X. The history valued function x_t : (−∞, 0] → X, x_t(θ) = x(t + θ) belongs to some abstract phase space

defined axiomatically; g, f are appropriate functions. B is a bounded linear operator on a Hilbert space U.

2. Preliminaries

In this section some definitions, notations, and lemmas that are used throughout this paper are stated. The family

of operators in B(X) is a strongly continuous cosine family if the following are satisfied:

(a)
C(0) = I (I is the identity operator in X);
(b)
C(t + s) + C(t − s) = 2C(t)C(s) for all ;
(c)
the map t → C(t)x is strongly continuous for each x ∈ X.

is the strongly continuous sine family associated to the strongly continuous cosine family

. It is defined as

, x ∈ X,

The operator A is the infinitesimal generator of a strongly continuous cosine function of bounded linear operators and S(t) is the associated sine function. Let N, be certain constants such that ∥C(t)∥ ≤ N and for every t ∈ J = [0, a]. For more details see book by Fattorini [28] and articles [35–37]. In this work we use the axiomatic definition of phase space , introduced by Hale and Kato [30].

Definition 1 (see [30].)Let be a linear space of functions mapping (−∞, 0] into X endowed with seminorm and satisfy the following conditions:

(A)
If x : (−∞, σ + b] → X, b > 0, such that and x|_[σ,σ+b] ∈ C([σ, σ + b] : X), then for every t ∈ [σ, σ + b) the following conditions:
- (i)
  x_t is in ,
- (ii)
  ,
- (iii)
  ,
where H > 0 is a constant K, M : [0, ∞)→[1, ∞), K is continuous, M is locally bounded and H, K, M are independent of x(·).
(B)
The space is complete.

Definition 2 (see [31].)Hausdorff’s measure of noncompactness χ_Y for a bounded set B in any Banach space Y is defined by χ_Y(B) = inf⁡{r > 0, B can be covered by finite number of balls with radii r.}

Lemma 3 (see [31].)Let Y be a Banach space and B, C ⊂ Y be bounded, then the following properties hold:

(1)
B is precompact if and only if χ_Y(B) = 0;
(2)
, where and conv B are closure and convex hull of B, respectively;
(3)
χ_Y(B) ≤ χ_Y(C) when B ⊂ C;
(4)
χ_Y(B + C) ≤ χ_Y(B) + χ_Y(C) where B + C = {x + y; x ∈ B, y ∈ C};
(5)
χ_Y(B ∪ C) = max⁡{χ_Y(B), χ_Y(C)};
(6)
χ_Y(λB) = ∥λ∥χ_Y(B) for any ;
(7)
if the map Q : D(Q) ⊂ Y → Z is Lipschitz continuous with constant k then χ_Z(QB) ≤ kχ_Y(B) for any bounded subset B ⊂ D(Q), where Z is a Banach space;
(8)
if is a decreasing sequence of bounded closed nonempty subset of Y and lim⁡_n→∞χ_Y(W_n) = 0, then is nonempty and compact in Y.

Definition 4 (see [31].)The map Q : W ⊂ Y → Y is said to be a χ-contraction if there exists a positive constant k < 1 such that χ_YQ(C) ≤ kχ_Y(C) for any bounded close subset C ⊂ W where Y is a Banach space.

Lemma 5 (Darbo-Sadovskii [31]). If W ⊂ Y is closed and convex and 0 ∈ W, the continuous map Q : W → W is χ-contraction, then the map Q has at least one fixed point.

PC([0, a], X) is the space formed by normalized piecewise continuous function from [0, b] into X. In particular it is the space PC formed by all functions u : [0, b] → X such that u is continuous at t ≠ t_i,

and

exists for all i = 1,2, …, n. It is clear that PC endowed with the norm ∥x∥_PC = sup_t∈J∥x(t)∥ is a Banach space. For any x ∈ PC

()

So,

Lemma 6 (see [31].) (1) If W ⊂ PC([a, b]; X) is bounded, then χ(W(t)) ≤ χ_PC(W) for any t ∈ [a, b] where W(t) = {u(t) : u ∈ W} ⊂ X.

(2) If W is piecewise equicontinuous on [a, b], then χ(W(t)) is piecewise continuous for t ∈ [a, b], and

()

(3) If W ⊂ PC([a, b]; X) is bounded and piecewise equicontinuous, then χ(W(t)) is piecewise continuous for t ∈ [a, b] and

()

Lemma 7 (see [35].)If the semigroup S(t) is equicontinuous and η ∈ L([0, a]; R⁺), then the set is equicontinuous for t ∈ [0, b].

3. Existence and Uniqueness of Mild Solution

We define mild solution of problem (1) as follows.

Definition 8. A function x : (−∞, a] → X is a mild solution of the problem (1) if x₀ = ϕ; x(·)|_[0,a] ∈ PC(X), , and

()

To prove our result we always assume

is a continuous function. The following hypotheses are used.

H_ϕ The function t → ϕ_t is continuous from into and there exists a continuous bounded function such that for every .
(Hf)
satisfies the following.
- (1)
  For every and x|_J ∈ PC, the function f(·, ψ) : J → X is strongly measurable for every and f(·, t) is continuous for a.e. t ∈ J.
- (2)
  There exists an integrable function α : J → [0, +∞) and a monotone continuous nondecreasing function Ω : [0, +∞)→(0, +∞) such that and .
- (3)
  There exists an integrable function η : J → [0, ∞) such that χ(S(s)f(t, D)) ≤ η(t)sup⁡_−∞<θ<0χ(D(θ)) for a.e. s, t, ∈J, where D(θ) = {v(θ) : v ∈ D}.
(Hg)
The function g(·) is continuous and g(t, ·) is Lipschitz continuous such that there exists positive constant L_g such that
()
(HJ)
- (1) There exist positive constants such that and .
- (2) for all .
- (1) ;
- (2) .
(H1)
- (1) ;
- (2) .

Lemma 9 (see [9].)If y : (−∞, a] → X is a function such that y₀ = ϕ and y|_J ∈ PC(X) then

()

where

In this section y : (−∞, a] → X is the function defined by y₀ = ϕ and y(t) = C(t)ϕ(0) + S(t)(z + g(0, ϕ)) on J₁ = [0, t₁]. Clearly where ∥y∥_b = sup_0≤t≤b∥y(t)∥.

Theorem 10. If the hypotheses (Hf), (Hg), (HI), (H1) are satisfied, then the initial value problem (1) has at least one mild solution.

Proof. Let S(a) be the space S(a) = {x : (−∞, a] → X∣x₀ = 0, x|_J ∈ PC} endowed with supremum norm ∥·∥_a.

Let Γ : S(a) → S(a) be the map defined by (Γx) ₀ = 0 and :

()

where

and

on J. It is easy to see that

()

where ∥x∥_t = sup_0≤s≤t∥x(s)∥

()

Thus Γ is well defined and has values in S(a). Also by axioms of phase space, the Lebesgue dominated convergence theorem, and the conditions (Hf), (Hg) it can be shown that Γ is continuous.

Step 1. There exists k > 0 such that Γ(B_k) ⊂ B_k, where B_k = {x ∈ S(a) : ∥x∥_a ≤ k}. In fact, if we assume that the assertion is false, then for k > 0 there exist x_k ∈ B_k and t_k ∈ (s_i, t_i+1] such that k < ∥Γx_k(t_k)∥:

()

Hence

()

which is a contradiction to the hypothesis (H1). Similarly (Γx)(t) < k, for t_k ∈ (t_i, s_i] ∀ i = 1,2, …, n. Suppose on the contrary,

()

Hence,

()

which is a contradiction.

Step 2. To prove that Γ is a χ-contraction. Let be split into for t > 0

()

For arbitrary x₁, x₂ ∈ B_k, and t ∈ (s_i, t_i+1]

()

So, is Lipschitz continuous with Lipschitz constant NL_gaK_a.

For any , W is piecewise equicontinuous since S(t) is equicontinuous. Hence from the fact that , s ∈ [0, a] and Lemma 6 and χ_PC(W) = sup⁡{χ(W(t)), t ∈ J} we have

()

For arbitrary x₁, x₂ ∈ B_k and t ∈ (s_i, t_i+1]

()

So,

∀i = 1, …, n is Lipschitz continuous with Lipschitz constant

For arbitrary x₁, x₂ ∈ B_k and t ∈ (t_i, s_i],

()

For each bounded set W ∈ PC(J; X) and t ∈ (s_i, t_i+1], ∀i = 0, …, n we have

()

For each bounded set W ∈ PC(J; X) and t ∈ (t_i, s_i] ∀ i = 1,2, …, n we have

()

Therefore, Γ is a χ-contraction. So, by Darbo-Sadovskii fixed point theorem we conclude that Γ has a fixed point in S(a). hence, z = x + y is a mild solution of (1).

4. Approximate Controllability

In this section the approximate controllability of the control system (1) without the impulsive conditions is studied. We consider

()

where A is the infinitesimal generator of a strongly continuous cosine family

of bounded linear operators on a Hilbert space X. The history valued function x_t : (−∞, 0] → X, x_t(θ) = x(t + θ) belongs to some abstract phase space

defined axiomatically; g, f are appropriate functions. B is a bounded linear operator on a Hilbert space U. We define mild solution of problem (24) as follows.

Definition 11. A function x : (−∞, a] → X is a mild solution of the problem (24) if x₀ = ϕ; x(·)|_[0,a] ∈ C(J, X), the functions and g(s, x_s) are integrable and the integral equation is satisfied:

()

Lemma 12 (see [11].)Under the assumption that h : [0, a] → X is an integrable function, such that

()

and h is a function continuously differentiable, then

()

Set a∶ = T.

Definition 13. The set given by is the mild solution of (24)} is called reachable set of the system (24). is the reachable set of the corresponding linear control system (31).

Definition 14. The system (24) is said to be approximately controllable on [0, T] if is dense in X. The corresponding linear system is approximately controllable if is dense in X.

Lemma 15. Let X be Hilbert space and X₁, X₂ closed subspaces such that X = X₁ + X₂. Then there exists a bounded linear operator P : X → X₂ such that for each x ∈ X, x = x − Px ∈ X₁ and ∥x₁∥ = min⁡{∥y∥ : y ∈ X₁, (1 − Q)(y) = (1 − Q)(x)} where Q denotes the orthogonal projection on X₂.

Let us define a continuous linear operator

()

Let us denote the kernel of the operator

by N which is a closed subspace of

. Let

denote the corresponding orthogonal subspace of

. Let

be a projection on

with range

. Let

denote the closure of the range of operator B. The following hypothesis is required to prove the approximate controllability

(HR)
∀ϵ > 0 and , ∃u(·) ∈ U such that .

It is easily seen that hypothesis (HR) is equivalent to the

. Theorem 16 shows that (HR) implies approximate controllability of the system (29). It is also known that approximate controllability of (31) implies

. Hence the closeness of the product space implies that (HR) is equivalent to approximate controllability of (29).

Theorem 16. If the assumptions (Hg) and (HR) hold then the corresponding neutral system

()

with f ≡ 0 is approximately controllable.

Proof. It is sufficient to prove that since D(A) is dense in X. Let for any chosen ξ ∈ D(A), then ξ − h(T, ϕ) ∈ D(A). It can be easily seen from Lemma 12 and [28] that there exists some p ∈ C¹([0, T]; X) such that

()

By hypothesis (HR) there exists a control function

such that

. As ϵ is arbitrary it implies that K_T(0) ⊂ D(A). Since the D(A) is dense in X, K_T(0) is dense in X. Hence the neutral system with f ≡ 0 is approximately controllable.

We state the corresponding linear control system

()

Both exact and approximate controllability of the above system are studied extensively in [33, 38] and so forth.

Assume that f, g satisfy the following conditions with μ_f, μ_g, ν_f, ν_g ∈ L²(J). For a fixed

and x ∈ C(J, X) such that x(0) = ϕ(0), we define maps F, G : C₀(J, X) → L²(J, X) by F(z)(t) = f(t, z_t + x_t) and G(z)(t) = g(t, z_t + x_t). Here x_t(θ) = x(t + θ), for t + θ ≥ 0 and x_t(θ) = ϕ(t + θ) for t + θ ≤ 0 and z_t(θ) = z(t + θ) for t + θ ≥ 0 and z_t(θ) = 0 for t + θ ≤ 0. Clearly, F, G are continuous maps.

(C1)
The function is continuous for almost all t ∈ I and F(·, z) : J → X is strongly measurable, .
(C2)
There exists integrable functions μ_F, ν_F : I → [0, ∞) and a continuous nondecreasing function W_F : [0, ∞)→(0, ∞) such that ∥F(t,z)∥₂ ≤ μ_F(t)W_F(∥z∥₁) + ν_F(t), .
(C3)
The function f(·) is continuous and f(t, ·) is Lipschitz continuous such that there exists positive constant L_f such that
()

The above same conditions also hold for G.

Also, y : (−∞, a] → X is the function defined by y₀ = ϕ and y(t) = C(t)ϕ(0) + S(t)(z + g(0, ϕ)) on J. Clearly where ∥y∥_b = sup_0≤t≤b∥y(t)∥.

The operators Λ_i : L²(J, X) → X i = 1,2 are defined as

()

Clearly Λ_i are bounded linear operators. We set

and

. Let C₀(J, X) denote the space consisting of continuous functions x : J → X such that x(0) = 0, endowed with the norm of uniform convergence. Let J_i : L²(J, X) → C₀(J, X), i = 1,2 be maps defined as follows:

()

So, J_ix(a) = Λ_i(x), i = 1,2.

As a continuation of coauthor Sukavanam’s work [19] and from hypothesis (B1) in [39] we assume that .

By using Lemma 15 we denote P_i the map associated to this decomposition and construct and . Also set c_i = ∥P_i∥.

We introduce the space

()

and we define the map

()

Lemma 17. If the hypothesis (H_ϕ)–(Hg) and conditions (C1)-(C2) hold for f, g and then Γ has a fixed point.

Proof. For let and . ∀0 ≤ t ≤ a

()

Now

()

Similarly we find for g. So,

()

where

. Repeating this we get

()

and

as n → ∞, the map Γⁿ is a contraction for n sufficiently large and therefore Γ has a fixed point.

Theorem 18. If the associated linear control system (31) is approximately controllable on J, the space and condition of the preceding Lemma 17 hold then the semilinear control system (24) with state dependent delay is approximately controllable on J.

Proof. Assume x(·) to be the mild solution and u(·) to be an admissible control function of system (31) with initial conditions x(0) = ϕ(0) and x^′(0) = w + g(0, ϕ). Let z be the fixed point of Γ. So, z(0) = 0 and z(a) = Λ₁(P₁(F(z))) − Λ₂(P₂(G(z))) = 0. By Lemma 12 we can split the functions F(z), G(z) with respect to the decomposition , respectively, by setting q₁ = F(z) − P₁(F(z)) and q₂ = G(z) − P₂(G(z)). We define the function y(t) = z(t) + x(t) for t ∈ J and y₀ = ϕ. So, x(a) = y(a). Thus by the properties of x and z

()

is dense in L²(J, U) we can choose a sequence

and a sequence

such that

and

as n → ∞. By Lemma 15 we get

()

Hence by Definition 11 and the last expression we conclude that yⁿ is the mild solution of the following equation:

()

Hence

. Since the solution map is generally continuous, yⁿ → y as n → ∞. Thus

. Therefore

, which means

is dense in X. Thus the system (1) is controllable.

5. Examples

Example 1. In this section we discuss a partial differential equation applying the abstract results of this paper. In this application, is the phase space C₀ × L²(h, X) (see [10]).

Consider the second order neutral differential equation:

()

where ϕ ∈ C₀ × L²(h, X), 0 < t₁ < , …, t_n < a. For y ∈ D(A),

and

, where

is the eigenfunction corresponding to the eigenvalue λ_n = −n² of the operator A. ϕ_n is an orthonormal base. A will generate the operators S(t), C(t) such that

, and the operator

. To find a solution to this problem we will assume that h(·) satisfies the conditions (g-5)–(g-7) in [34]. From Theorems 1.37 and 7.1.1 in [34] we conclude that C_a((−∞, 0], X) is continuously included in

. Let us suppose that the functions

are piecewise continuous. By defining maps

()

the system (51) can be transformed into system (1). Assume that the following conditions hold:

(a)
the functions b(s, η, ξ), ∂b(s, η, ξ)/∂ξ are measurable, b(s, η, π) = b(s, η, 0) = 0 and
()
such that .
(b)
The function is continuous and there is continuous function and .
(c)
The functions and for all i = 1,2, …, n j = 1,2.

Moreover

are bounded linear operators.

Hence by assumptions (a)–(c) and Theorem 10 it is ensured that mild solution to the problem (51) exists.

Now let us consider a particular example from the point of view of an application:

()

where

. The functions

are piecewise continuous. We assume the existence of positive constants b₁, b₂ such that

()

If we define maps

()

and g(t, ψ)(ξ) as in the problem (51) we can transform (47) into (1). Also a simple estimate shows that

Also if we define

and

for all i = 1, …, n then the hypotheses (HJ) can be easily proved. For instance,

()

Similarly it is easily seen for

. Now, if ϕ satisfies the hypothesis (H_ϕ) then ∃ a mild solution of (47).

Example 2. Consider the second order neutral differential equation:

()

where ϕ ∈ C₀ × L²(h, X), 0 < t₁ < , …, t_n < a. For y ∈ D(A),

, and

, where

is the eigenfunction corresponding to the eigenvalue λ_n = −n² of the operator A. ϕ_n is an orthonormal base. A will generate the operators S(t), C(t) such that

, and the operator

. Let the infinite dimensional control space be defined as

with norm

. Thus U is a Hilbert space. By defining maps

()

the system (51) can be transformed into system (1). Assume that the functions

are continuous and satisfy the following conditions.

(a)
The functions b(s, η, ξ), ∂b(s, η, ξ)/∂ξ are measurable, b(s, η, π) = b(s, η, 0) = 0 and
()
such that .
(b)
The function is continuous and there is continuous function and .
(c)
The functions and for all i = 1,2, …, n j = 1,2.

Moreover g(t, ·) is a bounded linear operator.

Here we examine the conditions (HR) for this control system. Then by using Theorem 18 we show its approximate controllability. Let for . The bounded linear operator B : L₂([0, T]; U) → L₂([0, T]; X) is defined by .

Let α ∈ N ⊂ L₂(0, T : X), N is the null space of Γ. Also

. Therefore

()

This implies that

()

The Hilbert space L₂(0, T) can be written as

()

Thus for h₁, h₂ ∈ L₂(0, T) there exists α₁ ∈ {sins}⊥, α₂ ∈ {sin4s} ^⊥ such that h₁ − 2h₂ = α₁ − 2α₂. So let u₂ = h₂ − α₂. Then h₁ = α₁ + 2u₂, h₂ = α₂ + u₂ also let u_n = h_n, n = 3,4, … and α_n = 0, n = 3,4, ….. Thus we see that hypothesis (HR) is satisfied as

and

Hence by assumptions (a)–(c) and Theorem 18 it is ensured that the problem (51) is approximately controllable.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express sincere gratitude to the reviewer for his valuable suggestions. The first author would like to thank Ministry of Human Resource and Development with Grant no. MHR-02-23-200-429/304 for their funding.

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Existence of Solution and Approximate Controllability for Neutral Differential Equation with State Dependent Delay

Abstract

1. Introduction

2. Preliminaries

3. Existence and Uniqueness of Mild Solution

4. Approximate Controllability

5. Examples

Conflict of Interests

Acknowledgments

References

Citing Literature

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Existence of Solution and Approximate Controllability for Neutral Differential Equation with State Dependent Delay

Abstract

1. Introduction

2. Preliminaries

3. Existence and Uniqueness of Mild Solution

4. Approximate Controllability

5. Examples

Conflict of Interests

Acknowledgments

References

Citing Literature

References

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