Approximate controllability for semilinear retarded control equations using surjectivity results

Systems & Control Letters 131 (2019) 104496

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Approximate controllability for semilinear retarded control equations using surjectivity results ∗

Daewook Kim a , Jin-Mun Jeong b , , Seong Ho Cho b a b

Department of Mathematics Education, Seowon University, Chungbuk, Korea Department of Applied Mathematics, Pukyong National University, Busan, Korea

article

info

a b s t r a c t Sufficient conditions for approximate controllability of semilinear retarded functional control equations are obtained under general conditions on nonlinear terms. To get the main result, we transform the controllability problem into surjectivity results similar to Fredholm alternative for nonlinear operators under restrictive assumptions. Finally, a simple example to which our main result can be applied is given. © 2019 Elsevier B.V. All rights reserved.

Article history: Received 19 February 2019 Received in revised form 28 May 2019 Accepted 21 June 2019 Available online xxxx Keywords: Approximate controllability Semilinear control equations Retarded control equations Controller Surjectivity result

1. Introduction In this paper, we deal with the approximate controllability for the following semilinear retarded functional differential equation in a Hilbert space H:

∫ t ⎧ d ⎪ ⎨ x(t) = A0 x(t) + A1 x(t − h) + k(t − s)g(s, x(s), u(s))ds dt

⎪ ⎩

+ Bu(t), x(0) = φ 0 , x(s) = φ 1 (s),

0

−h ≤ s < 0. (1.1) ∗

Let V be embedded in H as a dense subspace and V be the dual space of V . Here, let A0 be the operator associated with a sesquilinear form defined on V × V satisfying Gårding’s inequality and A1 be a bounded linear operator from V to V ∗ such that A1 maps from D(A0 ) endow with graphic norm of A0 to H continuously. The nonlinear mapping is given by f (t , x, u) =

∫

t

k(t − s)g(s, x(s), u(s))ds,

0

where k belongs to L2 (0, T ) and g : [0, T ] × V × U → H is a nonlinear mapping such that t → g(t , x, u) satisfies Lipschitz continuity as detailed in Section 2. The controller B is a bounded linear operator from another Hilbert space U to H. ∗ Corresponding author. E-mail addresses: [email protected] (D. Kim), [email protected] (J.-M. Jeong), [email protected] (S.H. Cho). https://doi.org/10.1016/j.sysconle.2019.104496 0167-6911/© 2019 Elsevier B.V. All rights reserved.

There are many literature works which are studied for the structural properties for retarded systems in infinite dimensional spaces (see the bibliographies of [1–4]). The first approach to deal with the approximate controllability of a semilinear control system is related to the range condition argument of controller in order as in [4–7]. It is to reveal the relationship between the reachable trajectory set of semilinear equation and that of its corresponding linear equation. Another approach used to obtain sufficient conditions for approximate solvability of nonlinear equations is a fixed point theorem combined with technique of operator transformations by configuring the resolvent as seen in [8]. The approximate controllability for various nonlinear control equations has been studied by many authors, for example, see [9–13] for local controllability of neutral functional differential systems with unbounded delay, and [14] for impulsive neutral functional evolution integrodifferential systems with infinite delay. Moreover, the approximate controllability for semilinear retarded stochastic systems has been studied by [3,15, 16]. As a sufficient condition for the approximate controllability for the general retarded initial value problem, Sukavanam and Tomar [17] needed a hypothesis that the Lipschitz constant of the nonlinear term is less than one, and Wang [4] used the condition of the growth condition of the nonlinear term and the compactness of the semigroup. In this paper, we propose a different approach than the previous one. Sufficient conditions for approximate controllability of semilinear retarded functional control equations are obtained under complete continuity condition on nonlinear terms. To get the main result, we transform the controllability problem into surjectivity results similar to Fredholm alternative. That is, the

2

D. Kim, J.-M. Jeong and S.H. Cho / Systems & Control Letters 131 (2019) 104496

equation λT (x) − F (x) = y in dependence on the real number λ is solvable, where T and F are nonlinear operators defined a Banach space X with values in a Banach space Y . It is necessary to suppose that T acts as the identity operator while F related to the nonlinear term of (1.1) is completely continuous. In Section 2, we introduce L2 -regularity properties for (1.1). To ensure complete continuity of nonlinear terms, we assume that D(A0 ) is compactly embedded in V . Then by virtue of Aubin [18], we have that the solution mapping of a control space to the terminal state space is completely continuous. This method has been applied in many papers to study the approximate controllability for various semilinear systems by using the homotopy property of topological degree theory, see [2,5,19]. Based on Section 2, Section 3 gives the sufficient conditions on the controller and nonlinear terms for approximate controllability for (1.1) using surjectivity results. Finally, a simple example to which our main result can be applied is given. 2. Semilinear functional equations Let V and H be complex Hilbert spaces forming a Gelfand triple V ↪→ H ≡ H ↪→ V ∗

∗

By virtue of Theorem 3.3 of [1], we have the following result on the corresponding linear equation (2.2) in case where g ≡ 0. Proposition 2.1. Suppose that the assumptions for the principal operator A0 stated above are satisfied. Then the following properties hold: (1) Noting that V = (D(A0 ), H)1/2,2 and g ≡ 0, for (φ 0 , φ 1 ) ∈ V ×L2 (−h, 0; D(A0 )) and j ∈ L2 (0, T ; H), T > 0, there exists a unique solution x of (2.2) belonging to L2 (−h, T ; D(A0 )) ∩ W 1,2 (0, T ; H) ⊂ C ([0, T ]; V ) and satisfying

∥x∥L2 (−h,T ;D(A0 ))∩W 1,2 (0,T ;H) ≤ C1 (∥φ 0 ∥ + ∥φ 1 ∥L2 (−h,0;D(A0 )) + ∥j∥L2 (0,T ;H) ),

(2.3)

where C1 is a constant depending on T . (2) Let (φ 0 , φ 1 ) ∈ H × L2 (−h, 0; V ) and j ∈ L2 (0, T ; V ∗ ), T > 0. Then, noting that (V , V ∗ )1/2,2 = H, there exists a unique solution x of (2.2) belonging to L2 (−h, T ; V ) ∩ W 1,2 (0, T ; V ∗ ) ⊂ C ([0, T ]; H) and satisfying

by identifying the antidual of H with H. Therefore, for the brevity, we may regard that ∥u∥∗ ≤ |u| ≤ ∥u∥ for all u ∈ V , where the notations |·|, ∥ · ∥ and ∥ · ∥∗ denote the norms of H , V and V ∗ , respectively as usual. Let a(u, v ) be a bounded sesquilinear form defined in V × V satisfying Gårding’s inequality Re a(u, u) ≥ c0 ∥u∥ − c1 |u| , 2

c0 > 0,

2

c1 ≥ 0.

∥x∥L2 (−h,T ;V )∩W 1,2 (0,T ;V ∗ ) ≤ C1 (|φ 0 | + ∥φ 1 ∥L2 (−h,0;V ) + ∥j∥L2 (0,T ;V ∗ ) ), (2.4) where C1 is a constant depending on T .

∫t

Let A0 be the operator associated with this sesquilinear form:

Lemma 2.1. Suppose that k ∈ L2 (0, T ; H) and x(t) = 0 S(t − s)k(s)ds for 0 ≤ t ≤ T . Then there exists a constant C2 such that

(A0 u, v ) = −a(u, v ),

∥x∥L2 (0,T ;H) ≤ C2 T ∥k∥L2 (0,T ;H) ,

u, v ∈ V . ∗

Then A0 is a bounded linear operator from V to V . The realization of A0 in H which is the restriction of A0 to D(A0 ) = {u ∈ V : A0 u ∈ H } is also denoted by A0 . Moreover, for each T interpolation theory, we have

> 0, by using

and

√ ∥x∥L2 (0,T ;V ) ≤ C2 T ∥k∥L2 (0,T ;H) . Proof. By a consequence of (2.3), it is immediate that

c0 ∥u∥2 ≤ Re a(u, u)+c1 |u|2 ≤ |A0 u| |u|+c1 |u|2 ≤ (|A0 u|+c1 |u|)|u|,

∫ T ∫ t ∫ t ( |k(s)|ds)2 dt | S(t − s)k(s)ds|2 dt ≤ M 2 0 0 0 0 ∫ T ∫ t ∫ T (MT )2 2 2 t |k(s)| dsdt ≤ ≤M |k(s)|2 ds,

∥x∥2L2 (0,T ;H) =

it follows that there exists a constant C0 > 0 such that

∫

T

1/2

0

∥u∥ ≤ C0 ∥u∥D(A0 ) |u|1/2 .

2

0

0

where M is the constant of (2.1), it follows that

Therefore, in terms of the intermediate theory, we can see that (D(A0 ), H)1/2,2 = V , and (V , V ∗ )1/2,2 = H , where (V , V ∗ )1/2,2 denotes the real interpolation space between V and V ∗ (Section 1.3.3 of [20];[21]). For the sake of simplicity we assume that c1 = 0 and hence the closed half plane {λ : Re λ ≥ 0} is contained in the resolvent set of A0 . It is known that A0 generates an analytic semigroup S(t) in both H and V ∗ . As seen in Lemma 3.6.2 of [22], there exists a constant M > 0 such that

|S(t)x| ≤ M |x|(x ∈ H) and ∥S(t)x∥∗ ≤ M ∥x∥∗ (x ∈ V ∗ ).

(2.1)

The following initial value problem:

√ ∥x∥L2 (0,T ;H) ≤ TM / 2∥k∥L2 (0,T ;H) .

(2.8)

From (2.7) and (2.8) it holds that

√ ∥x∥L2 (0,T ;V ) ≤ C0 C1 T (M /2)1/4 ∥k∥L2 (0,T ;H) . So, if we take a constant C2 > 0 such that

√

C2 = max{M / 2, C0

√

C1 (M /2)1/4 },

Thus, (2.5) and (2.6) are satisfied. □ Consider the following initial value problem for the abstract semilinear parabolic equation

⎧ ∫ t d ⎪ ⎪ k(t − s)g(s, x(s), u(s))ds ⎨ x(t) = A0 x(t) + A1 x(t − h) +

⎧ ∫ t d ⎪ ⎪ k(t − s)g(s, x(s), u(s))ds ⎨ x(t) = A0 x(t) + A1 x(t − h) + + j(t), x(0) = φ 0 , x(s) = φ 1 (s),

(2.7)

Since

From the following inequalities

dt

(2.6)

∥x∥L2 (0,T ;D(A)) ≤ C1 ∥k∥L2 (0,T ;H) ,

L2 (−h, T ; V ) ∩ W 1,2 (0, T ; V ∗ ) ⊂ C ([0, T ]; H).

⎪ ⎪ ⎩

(2.5)

dt

0

⎪ ⎪ ⎩

−h ≤ s < 0 (2.2)

+ Bu(t), x(0) = φ 0 , x(s) = φ 1 (s),

0

−h ≤ s < 0 (2.9)

D. Kim, J.-M. Jeong and S.H. Cho / Systems & Control Letters 131 (2019) 104496

3

Let U be a Hilbert space and the controller operator B be a bounded linear operator from U to H. Let g : R+ × V × U → H be a nonlinear mapping satisfying the following:

Corollary 2.1. Assume that the embedding D(A) ⊂ V is completely continuous. Let Assumption (F) be satisfied, and xu be the solution of Eq. (2.9) associated with u ∈ L2 (0, T ; U). Then the mapping u ↦ → xu is completely continuous from L2 (0, T ; U) to L2 (0, T ; V ).

Assumption (F).

Proof. If u ∈ L2 (0, T ; U), then in view of (2.4) in Proposition 2.1

(i) For any x ∈ V , u ∈ U the mapping g(·, x, u) is strongly measurable; (ii) There exist positive constants L0 , L1 , L2 such that (a) u ↦ → g(t , x, u) is an odd mapping (g(·, x, −u) = −g(·, x, u)); (b) for all t ∈ R+ , x, xˆ ∈ V , and u, uˆ ∈ U,

|g(t , x, u) − g(t , xˆ , uˆ )| ≤ L1 ∥x − xˆ ∥ + L2 ∥u − uˆ ∥U , |g(t , 0, 0)| ≤ L0 . For x ∈ L2 (−h, T ; V ), we set f (t , x, u) =

∫

t

k(t − s)g(s, x(s), u(s))ds

0

where k belongs to L2 (0, T ). Lemma 2.2. Let Assumption (F) be satisfied. Assume that x ∈ L2 (0, T ; V ) for any T > 0. Then f (·, x, u) ∈ L2 (0, T ; H) and

√ ∥f (·, x, u)∥L2 (0,T ;H) ≤ L0 ∥k∥L2 (0,T ) T / 2 √ + ∥k∥L2 (0,T ) T (L1 ∥x∥L2 (0,T ;V ) + L2 ∥u∥L2 (0,T ;U) ).

(2.10)

∥xu ∥L2 (−h,T ;V )∩W 1,2 (0,T ;V ∗ ) ≤ C3 (|φ 0 | + ∥φ 1 ∥L2 (−h,0;V ) + ∥B∥∥u∥L2 (0,T ;U) ).

(2.13)

Since xu ∈ L (0, T ; V ), we have f (·, xu , u) ∈ L (0, T ; H). Consequently 2

2

xu ∈ L2 (−h, T ; D(A)) ∩ W 1,2 (0, T ; H). Hence, with aid of (2.3) of Proposition 2.1, (2.10) and (2.12),

∥xu ∥L2 (−h,T ;D(A))∩W 1,2 (0,T ;H) ≤ C1 (∥φ 0 ∥ + ∥φ 1 ∥L2 (−h,0;D(A0 )) + ∥f (·, xu , u) + Bu∥L2 (0,T ;H) ) √ ≤ C1 {∥φ 0 ∥ + ∥φ 1 ∥L2 (−h,0;D(A0 )) + L0 ∥k∥L2 (0,T ) T / 2 √ + ∥k∥L2 (0,T ) T (L1 ∥x∥L2 (0,T ;V ) + L2 ∥u∥L2 (0,T ;U) ) + ∥Bu∥L2 (0,T ;H) } √ [ ≤ C1 ∥φ 0 ∥ + ∥φ 1 ∥L2 (−h,0;D(A0 )) + L0 ∥k∥L2 (0,T ) T / 2 √ { } + ∥k∥L2 (0,T ) T L1 C3 (1 + |φ 0 | + ∥u∥L2 (0,T ;U) ) + L2 ∥u∥L2 (0,T ;U) ] + ∥Bu∥L2 (0,T ;H) . Thus, if u is bounded in L2 (0, T ; U), then so is xu in L2 (−h, T ; D(A)) ∩ W 1,2 (0, T ; H). Since D(A) is compactly embedded in V by assumption, the embedding

Moreover if x, xˆ ∈ L2 (0, T ; V ), then

L2 (−h, T ; D(A)) ∩ W 1,2 (0, T ; V ) ⊂ L2 (0, T ; V )

∥f (·, x, u) − f (·, xˆ , uˆ )∥L2 (0,T ;H) √ ≤ ∥k∥L2 (0,T ) T (L1 ∥x − xˆ ∥L2 (0,T ;V ) + L2 ∥u − uˆ ∥L2 (0,T ;U) ).

is completely continuous in view of Theorem 2 of [18], the mapping u ↦ → xu is completely continuous from L2 (0, T ; U) to L2 (0, T ; V ). □

(2.11)

Proof. From Assumption (F), and using the Hölder inequality, it is easily seen that

3. Surjectivity and approximate controllability

∥f (·, x, u)∥L2 (0,T ;H) ≤ ∥f (·, 0, 0)∥L2 (0,T ;H) + ∥f (·, x, u) − f (·, 0, 0)∥L2 (0,T ;H) (∫ T ∫ t )1/2 2 ≤ | k(t − s)g(s, 0, 0)ds| dt

In this section, we first introduce surjectivity results similar to Fredholm alternative for nonlinear operators under restrictive assumptions. Throughout this section, we assume that D(A0 ) is compactly embedded in V . Let x(T ; f , u) be a state value of the system (2.9) at time T corresponding to the nonlinear term f and the control u. We define the reachable sets for the system (2.9) as follows:

0

0 T

(∫

t

∫

k(t − s){g(s, x(s), u(s)) − g(s, 0, 0)}ds|2 dt

|

+ 0

)1/2

RT (f ) = {x(T ; f , u) : u ∈ L2 (0, T ; U)},

0

√ √ ≤ L0 ∥k∥L2 (0,T ) T / 2 + ∥k∥L2 (0,T ) T ∥ g(·, x, u) − g(·, 0, 0) ∥L2 (0,T ;H) √ √ ≤ L0 ∥k∥L2 (0,T ) T / 2 + ∥k∥L2 (0,T ) T (L1 ∥x∥L2 (0,T ;V ) + L2 ∥u∥L2 (0,T ;U) ).

RT (0) = {x(T ; 0, u) : u ∈ L2 (0, T ; U)}. Definition 3.1. The system (2.9) is said to be approximately controllable in the time interval [0, T ] if for every desired final state x1 ∈ H and ϵ > 0 there exists a control function u ∈ L2 (0, T ; U) such that the solution x(T ; f , u) of (2.9) satisfies |x(T ; f , u) − x1 | < ϵ.

The proof of (2.11) is similar. □ By virtue of Theorem 2.1 of [23], we have the following result on (2.9). Proposition 2.2. Let Assumption (F) be satisfied and (φ 0 , φ 1 , u) ∈ H × L2 (−h, 0; V ) × L2 (0, T ; U). Then there exists a unique solution x of (2.9) such that x ∈ L2 (−h, T ; V ) ∩ W 1,2 (0, T ; V ∗ ) ⊂ C ([0, T ]; H) for any x0 ∈ H. Moreover, there exists a constant C3 such that

∥x∥L2 (−h,T ;V )∩W 1,2 (0,T ;V ∗ ) ≤ C3 (1+|φ 0 |+∥φ 1 ∥L2 (−h,0;V ) +∥u∥L2 (0,T ;U) ). (2.12)

Let us introduce the theory of the degree for completely continuous perturbations of the identity operator, which is the infinite dimensional version of Borsuk’s theorem. Let 0 ∈ D be a bounded open set in a Banach space X , D its closure and ∂ D its boundary. The number d[I − T ; D, 0] is the degree of the mapping I − T with respect to the set D and the point 0 (see Fučik et al. [24] or Lloid [25]). Theorem 3.1 (Borsuk’s Theorem). Let D be a bounded open symmetric set in a Banach space X , 0 ∈ D. Suppose that T : D → X be odd completely continuous operator satisfying T (x) ̸ = x for x ∈ ∂ D. Then d[I − T ; D, 0] is odd integer. That is, there exists at least one point x0 ∈ D such that (I − T )(x0 ) = 0.

4

D. Kim, J.-M. Jeong and S.H. Cho / Systems & Control Letters 131 (2019) 104496

Definition 3.2. Let T be a mapping defined on a Banach space X with value in a real Banach space Y . The mapping T is said to be a (K , L, α )-homeomorphism of X onto Y if

We can choose x0 ∈ X satisfying λT (x0 ) = y1 , and so, λT (x0 ) − F (x0 ) = y0 . Thus, it implies that λT − F is a mapping of X onto Y. □

(i) T is a homeomorphism of X onto Y ; (ii) there exist real numbers K > 0, L > 0, and α > 0 such that

Combining Lemma 3.1. and Proposition 3.1, we have the following results.

L∥x∥αX ≤ ∥T (x)∥Y ≤ K ∥x∥αX ,

∀x ∈ X .

Lemma 3.1. Let T be an odd (K , L, α )-homeomorphism of X onto Y and F : X → Y a continuous operator satisfying lim sup ∥x∥X →∞

lim

∥x∥X →∞

N L

lim sup ∥x∥X →∞

∥F (x)∥Y = N ∈ R+ . ∥x∥αX

If |λ| ∈ / [ NK ,

Corollary 3.1. Let T be an odd (K , L, α )-homeomorphism of X onto Y and F : X → Y an odd completely continuous operator satisfying

∥F (x)∥Y = N ∈ R+ . ∥x∥αX

Then if |λ| ∈ / [ NK , NL ] ∪ {0} then λT − F maps X onto Y . Therefore, if N = 0, then for all λ ̸ = 0 the operator λT − F maps X onto Y .

] ∪ {0} then

∥λT (x) − F (x)∥Y = ∞.

Proof. Suppose that there exist a constant M > 0 and a sequence

{xn } ⊂ X such that

First we consider the approximate controllability of the system (2.9) in case where the controller B is the identity operator on H under Assumption (F) on the nonlinear operator f in Section 2. Hence, noting that H = U, we consider the linear system given by

∥λT (xn ) − F (xn )∥Y ≤ M

⎧ ⎨d

as xn → ∞. From this it follows that

⎩y(0)

F (xn ) λT (xn ) − → 0. ∥xn ∥αX ∥xn ∥αX

and the following semilinear control system

dt

⎧ ⎨d

dt

Hence, we have lim sup n→∞

|λ|∥T (xn )∥Y = N, ∥xn ∥αX

Proposition 3.1. Let T be an odd (K , L, α )-homeomorphism of X onto Y and F : X → Y an odd completely continuous operator. Suppose that for λ ̸ = 0, lim

∥λT (x) − F (x)∥Y = ∞.

(3.1)

Then λT − F maps X onto Y . Proof. We follow the proof of Theorem 1.1 in Chapter II of Fučik et al. [24]. Suppose that there exists y ∈ Y such that λT (x) = y. Then from (3.1) it follows that FT −1 : Y → Y is an odd completely continuous operator and y

∥y − FT −1 ( )∥Y = ∞. ∥y∥Y →∞ λ lim

y

∥y − FT −1 ( )∥Y > ∥y0 ∥Y ≥ 0 λ for each y ∈ Y satisfying ∥y∥Y = r. Let Yr = {y ∈ Y : ∥y∥Y < r } be a open ball. Then by view of Theorem 3.1, we have y d[y − FT −1 ( λ ); Yr , 0] is an odd number. For each y ∈ Y satisfying ∥y∥Y = r and t ∈ [0, 1], there is y

∥y − FT −1 ( ) − ty0 ∥Y ≥ ∥y − FT −1 ( )∥Y − ∥y0 ∥Y > 0 λ λ and hence, by the homotopic property of degree, we have y y d[y − FT −1 ( ); Yr , y0 ] = d[y − FT −1 ( ); Yr , 0] ̸ = 0.

λ

λ

Hence, by the existence theory of the Leray–Schauder degree, there exists a y1 ∈ Yr such that y1 − FT

−1

(

y1

λ

) = y0 .

y(s) = φ 1 (s)

− h ≤ s < 0,

= A0 x(t) + A1 x(t − h) + f (t , x(t), v (t)) + v (t), = φ0,

x(s) = φ 1 (s)

− h ≤ s < 0.

Theorem 3.2. Assume that ∥f (·, xu , u)∥L2 (0,T ;H) ̸= 1. lim sup ∥u∥L2 (0,T ;H) ∥u∥→∞

(3.2)

(3.3)

(3.4)

Under Assumption (F) we have RT (0) ⊂ RT (f ). Therefore, if the linear system (3.2) with f = 0 is approximately controllable, then so is the semilinear system (3.3). Proof. Let y(t) be solution of (3.2) corresponding to a control u. First, we show that there exists a v ∈ L2 (0, T ; H) such that

{

v (t) = u(t) − f (t , x(t), v (t)), v (0) = u(0).

0 < t ≤ T,

Let us define an operator F : L2 (0, T ; H) → L2 (0, T ; H) as F v = −f (·, xv , v ).

Let y0 ∈ Y . There exists r > 0 such that

y

= φ0,

x(t)

⎩x(0)

and so, |λ|K ≥ N ≥ |λ|L. It is a contradiction with |λ| ∈ / [ NK , NL ]. □

∥x∥X →∞

y(t) = A0 y(t) + A1 y(t − h) + u(t),

Then by Corollary 2.1, F is a compact mapping from L2 (0, T ; H) to itself, and we have lim ∥λI(v ) − F (v )∥L2 (0,T ;H) = ∞,

∥v∥→∞

where the identity operator I on L2 (0, T ; H) is an odd (1, 1, 1)homeomorphism. Thus, from (3.4) and Corollary 3.1, it follows that λI − F maps L2 (0, T ; H) onto itself. Hence, we have showed that there exists a v ∈ L2 (0, T ; H) such that v (t) = u(t) − f (t , y(t), v (t)). Let y and x be solutions of (3.2) and (3.3) corresponding to controls u and v , respectively. Then, Eq. (3.3) is rewritten as d x(t) = A0 x(t) + A1 x(t − h) + f (t , x(t), v (t)) + v (t), 0 < t ≤ T dt = A0 x(t) + A1 x(t − h) + f (t , x(t), v (t)) + u(t)

− f (t , x(t), v (t)) = A0 x(t) + A1 x(t − h) + u(t)

D. Kim, J.-M. Jeong and S.H. Cho / Systems & Control Letters 131 (2019) 104496

with x(0) = φ 0 , x(t) =S(t)φ + 0

=S(t)φ 0 +

x(s) = φ 1 (s)

with x(0) = φ 0 , x(s) = φ 1 (s), −h ≤ s < 0. Hence, we have

− h ≤ s < 0, which means

t

∫

S(t − s){A1 x(s − h) + f (s, x(s), v (s)) + v (s)}ds

∫0 t

5

x(t) =S(t)φ +

S(t − s)A1 x(s − h) + u(s)ds = y(t),

t

∫

0

S(t − s){A1 x(s − h)f (s, x(s), v (s)) + Bv (s)}ds

∫0 t

=S(t)φ 0 +

0

S(t − s){A1 x(s − h) + u(s)}ds = y(t).

0

where y be solution of (3.2) corresponding to a control u. Therefore, we have proved that RT (0) ⊂ RT (f ). □

Thus, we obtain that RT (0) ⊂ RT (f ).

□

Corollary 3.2. Let us assume that

Example. We consider the semilinear heat equation dealt with by Zhou [7], and Naito [5]. Let

√ ( ) ∥k∥L2 (0,T ) T L1 C3 + L2 ̸= 1,

H = L2 (0, π ), V = H01 (0, π ), V ∗ = H −1 (0, π ),

where C3 is the constant in Proposition 2.2. Under Assumption (F), we have

a(u, v ) =

RT (0) ⊂ RT (f )

and

in case where B ≡ I.

A = d2 /dx2

Proof. By Lemma 2.2 and Proposition 2.2, we have

We consider the following retarded functional differential equation

∥Fu∥L2 (0,T ;H) = ∥f (·, xu , u)∥L2 (0,T ;H) √ √ ≤ L0 ∥k∥L2 (0,T ) T / 2 + ∥k∥L2 (0,T ) T (L1 ∥x∥L2 (0,T ;V ) + L2 ∥u∥L2 (0,T ;U) ) √ √ { ( ) ≤ L0 ∥k∥L2 (0,T ) T / 2 + ∥k∥L2 (0,T ) T L1 C3 1 + |φ 0 | + ∥u∥L2 (0,T ;U) } + L2 ∥u∥L2 (0,T ;U) .

π

∫

du(x) dv (x) dx

0

dx

dx

D(A) = {y ∈ H 2 (0, π ) : y(0) = y(π ) = 0}.

with

⎧ ∫ t d ⎪ ⎪ k(t − s)g(s, x(s), u(s))ds + Bu(t), ⎨ y(x, t) = Ay(x, t) + dt

0

y(t , 0) ⎪ ⎪ ⎩ y(0, x)

= y(t , π ) = 0, t > 0 = φ 0 (x), y(x, s) = φ 1 (x, s)

− h ≤ s < 0, (3.6)

Hence, we have lim sup ∥u∥→∞

where k belongs to L (0, T ). The eigenvalue and the eigenfunction of A are λn = −n2 and φn (x) = sin nx, respectively. Let 2

√ ( ) ∥F (u)∥L2 (0,T ;H) ≤ ∥k∥L2 (0,T ) T L1 C3 + L2 . ∥u∥L2 (0,T ;U)

Thus, from Corollary 3.1, it follows that I − F maps L2 (0, T ; H) onto itself, and so, by the same argument as in the proof of theorem it holds that RT (0) ⊂ RT (f ). □ From now on, we consider the initial value problem for the semilinear parabolic equation (2.9). Let U be some Hilbert space and the controller operator B be a bounded linear operator from U to H. Assumption (B). There exists a constant β > 0 such that R(f ) ⊂ R(B) and

∥Bu∥ ≥ β∥u∥,

∀u ∈ L2 (0, T ; U).

dt

un φn :

n=2

∞ ∑

un φn ,

for

u=

n=2

y(t)

= A0 y(t) + A1 y(t − h) + Bu(t), =φ , 0

y(s) = φ (s) 1

− h ≤ s < 0.

(3.5)

RT (0) ⊂ RT (f ).

Then

∞ ∑

(sin xn )φn (x) +

√ n ∥u∥φ2 (x)

= A0 x(t) + A1 x(t − h) + f (t , x(t), v (t)) + Bu(t) − f (t , x(t), v (t))

u

, n > 2.

t

∫

k(t − s)g(s, x(s), u(s))ds. 0

∥f (·, x, u)∥L2 (0,T ;H) =

Proof. Let y be a solution of the linear system (3.5) with f = 0 corresponding to a control u, and let x be a solution of the semilinear system (3.3) corresponding to a control v . Set v (t) = u(t) − B−1 f (t , x(t), v (t)). Then, system (2.9) is rewritten as x(t) = A0 x(t) + A1 x(t − h) + f (t , x(t), v (t)) + Bv (t)

un ∈ U .

n=2

It is easily seen that Assumption (F) is satisfied. For x ∈ L2 (0, T ; V ) and k ∈ L2 (0, π ), we set

Theorem 3.3. Under the Assumptions (3.4), (B) and (F), we have

Therefore, if the linear system (3.5) with f = 0 is approximately controllable, then so is the semilinear system (2.9).

∞ ∑

It is easily seen that the operator B is one to one and R(B) is closed. It follows that the operator B satisfies hypothesis as in Theorem 3.3. We can see many examples which satisfy Assumption (B) as seen in [7,26]. ∑∞ 2 For any x = n=1 xn φn ∈ L (0, π ), consider the nonlinear term g given by

f (t , x, u) =

d

u2n < ∞},

n=1

⎩y(0)

dt

∞ ∑ n=2

Bu = 2u2 φ1 +

g(t , x, u) =

Consider the linear system given by

⎧ ⎨d

∞ ∑

U ={

{∫

T 0

∫ t ∞ (∑ | k(t − s) (sin xn )φn (x) 0

n=1

√ ) + n ∥u∥φ2 (x) ds|dt √ ≤

T ∥k∥L2 (0,π )

∞ √ (∑ ) |sin xn |2 + n ∥u∥2 . n=1

Hence, we have lim sup ∥u∥→∞

∥f (·, x, u)∥L2 (0,T ;H) = 0, ∥u∥L2 (0,T ;H)

}1/2

6

D. Kim, J.-M. Jeong and S.H. Cho / Systems & Control Letters 131 (2019) 104496

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