International Journal of Control ISSN: 0020-7179 (Print) 1366-5820 (Online) Journal homepage: http://www.tandfonline.com/loi/tcon20 Control problems for semilinear retarded integrodifferential equations by the Fredholm theory Yong Han Kang & Jin-Mun Jeong To cite this article: Yong Han Kang & Jin-Mun Jeong (2017): Control problems for semilinear retarded integrodifferential equations by the Fredholm theory, International Journal of Control, DOI: 10.1080/00207179.2017.1390260 To link to this article: http://dx.doi.org/10.1080/00207179.2017.1390260 Accepted author version posted online: 27 Oct 2017. Submit your article to this journal View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tcon20 Download by: [Chalmers University of Technology] Date: 28 October 2017, At: 22:20 Publisher: Taylor & Francis Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017 Journal: International Journal of Control DOI: https://doi.org/10.1080/00207179.2017.1390260 Control problems for semilinear retarded integrodifferential equations by the Fredholm theory Yong Han Kang1 and Jin-Mun Jeong2 1 Institute of Liberal Education, Catholic University of Daegu Kyeongsan 680-749, Korea 2 Department of Applied Mathematics, Pukyong National University Busan 608-737, Korea Abstract In this paper, we deal with the approximate controllability for semilinear retarded functional integrodifferential equations by using the Fredholm theory in Hilbert spaces. We no longer require the compactness of structural operators to obtain the approximate controllability for the nonlinear differential system, but instead we use the theory of interpolation spaces and the regularity of solutions of semilinear given equations with unbounded principal operators. Finally, based on the properties of general degree theory in infinite dimensional spaces, we investigate the relation between the reachable set of trajectories of the semilinear retarded functional integrodifferential system and that of its corresponding linear system excluded by the nonlinear term. Email: 1 yhkang2642@gmail.com, 2 jmjeong@pknu.ac.kr( Corresponding author) 1 Keywords: retarded differential equation; semilinear equation; Fredholm theory; approximate controllability; reachable set. AMS Classification: Primary 35B37; Secondary 93C20 Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017 1 Introduction This paper is concerned with the approximate controllability for the following semilinear retarded functional integrodifferential equation: ∂ x) + A(x, Dx )u(t, x) + A1 (x, Dx )u(t − h, x) ∂t u(t, R0 (1.1) + −h a(s)A2 (x, Dx )u(t + s, x)ds Rt = 0 k(t − s)g(s, u(s, x(s))ds + (Bw)(t), (t, x) ∈ [0, T ] × Ω. Here, Ω ⊂ Rn is a bounded domain with smooth boundary ∂Ω, A(x, Dx ) and Aι (x, Dx )(ι = 1, 2) are second order linear differential operators with real coefficients, and A(x, Dx ) is an elliptic operator in Ω. The function a(s) is a real scalar function on [−h, 0], where h > 0 is a delay time. The controller B is a bounded linear operator from L2 (0, T ; U) to L2 (0, T ; H), where U is some Banach space. The boundary condition attached to (1.1) is given by Dirichlet boundary condition u|∂Ω = 0, 0 < t ≤ T, (1.2) and the initial condition is given by u(0, x) = g 0 (x), u(s, x) = g 1 (s, x) − h ≤ s ≤ 0. (1.3) Precise assumptions on the nonlinear function g are given in the next section. This kind of abstract formulations of many partial integrodifferential equations arise in the mathematical description of the dynamical processes with heat flow in material with memory, viscoelasticity, and many physical phenomena (See Fitzgibbon, 1980;Heard, 1981). Many authors have discussed the structural properties for retarded systems(see Di Blasio, Kunisch, Sinestrari, 1984;Jeong, 1993;Manitius, 1980;Nakagiri, 1988;Tanabe, 1988), and references therein. The numerical solution of FredholmVolterra integro-differential equations for periodic boundary value problems has discussed in (Abu Arqub and M. Al-Smadi, 2004;Abu Arqub and Rashaideh, 2017; Momani, Abu Arqub, Hayat, and Al-Sulami; 2014). To obtain sufficient conditions for the approximate controllability, many authors frequently used either the range condition of controller with a compact 2 Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017 semigroup introduced in Naito (1987) or the so called resolvent condition for using a fixed point theorem introduced in (Bashirov and Mahmudov, 1999;Fu, 2004). Recently, approximate controllability for semilinear control systems with a range condition of the control action operator can be founded in (Jeong, Kwun and Park, 1999;Mahmudov, 2006;Radhakrishnan and Balachandran, 2012;Sukavanam and Tomar, 2007;Wang, 2009). In particular, with conditions the range condition of controller with a compact semigroup, Wang (2009) established the approximate controllability for System (1,1) assuming ||f (·, u)||L2(0,T ;H) := γ ||u||L2(0,T ;H) |||u||L2 (0,T ;H) →∞ lim sup (1.4) is sufficiently small, where f (t, x) = Z 0 t k(t − s)g(s, x(s))ds for a k belonging to L2 (0, T ). In this paper, we will show the approximate controllability for the system (1,1) even if γ 6= 1 of (1.4) by using so called Fredholm theory: (λT − S)(u) = f is solvable, where T and S are nonlinear operators defined on a Banach space X with value in a Banach space Y . In our case, the operator T acts as the identity operator while S is completely continuous. Instead of the uniform boundedness of the nonlinear terms and the compactness of the principal operators studied by most parts of previous results, we make the natural assumption that the embedding D(A0 ) := −D(A(x, Dx ) ⊂ V is compact. Then the embedding L2 (0, T ; D(A0 )) ∩ W 1,2 (0, T ; H) ⊂ L2 (0, T ; V ) is compact in view of Aubin (1963), and we show that the mapping which maps w to the mild solution of (1.1) is a compact operator from L2 (0, T ; U) to L2 (0, T ; V ). The paper is organized as follows. In Section 2, we present the regularity and variations of constant formula of solutions for the system (1.1), and introduce basic properties of the Fredholm theory by using general degree theory in infinite dimensional spaces in Section 3. In Section 4, we will obtain the equivalent relations between the reachable set of the semilinear retarded integrodifferential equation and that of its corresponding linear system. Finally we give a simple example to which our main result can be applied. 2 Retarded control equations Let H and V be two complex Hilbert spaces such that V is a dense subspace of H. The norm of H(resp. V ) is denoted by | · | ( resp. || · ||). Assume that the injection 3 Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017 of V into H is continuous. The antidual of V is denoted by V ∗ , and the norm of V ∗ by || · ||∗. Identifying H with its antidual, we may consider that H is embedded in V ∗ . Hence we have V ⊂ H ⊂ V ∗ densely and continuously. The duality pairing between the element v1 of V ∗ and the element v2 of V is denoted by (v1 , v2 ), which is the ordinary inner product in H if v1 , v2 ∈ H. We realize the operator A(x, Dx ), Aι (x, Dx ), ι = 1, 2, in Hilbert spaces by A0 v = −A(x, Dx )v, Aι v = −Aι (x, Dx )v, ι = 1, 2, v∈V (2.1) in the distribution sense. Let b(·, ·) be a sesquilinear form defined in V × V , that is, for each u, v ∈ V there corresponds a complex number a(u, v) which is linear in u and antilinear in v: b(u1 + u2 , v) = b(u1 , v) + b(u2 , v), b(u, v1 + v2 ) = b(u, v1 ) + b(u, v2 ), b(λu, v) = λb(u, v), b(u, λv) = λb(u, v). We assume that b(·, ·) is bounded and satisfies Gårding’s inequality Re b(v, v) ≥ c0 ||v||2 − c1 |v|2 , c0 > 0, c1 ≥ 0. (2.2) The operator A0 defined as (2.1) may be associated with the sesquilinear form −b(·, ·) as follows: (A0 v1 , v2 ) = −b(v1 , v2 ), v1 , v2 ∈ V. The following result is known as the Lax-Milgram theorem. Lemma 2.1. Let X be a Hilbert space. Assume that b(u, v) is sesquilnear form defined on X × X satisfying (2.2). Then if f ∈ X ∗ , then there uniquely exists an element u ∈ X such that f (v) = b(u, v) for every v ∈ X. We can apply Lemma 2.1 to show that if f ∈ V ∗ then there exists a u ∈ V such that (f, u) = b(u, v) for all v ∈ V , i.e., f = A0 u and hence, A0 is a bounded linear operator from V to V ∗ . Moreover, it is well known that A0 and generates an analytic semigroup S(t)(t ≥ 0) in both of H and V ∗ (see Tanabe, 1979), and its realization in H which is the restriction of A0 to D(A0 ) = {v ∈ V ; A0 v ∈ H} is also denoted by A0 . From the following inequalities c0 ||v||2 ≤ Re b(v, v) + ω2 |v|2 ≤ |A0 u| |u| + c2 |u|2 ≤ max{1, c2 }||u||D(A0) |u|, 4 where ||v||D(A0) = (|Av|2 + |v|2)1/2 is the graph norm of D(A0 ), it follows that there exists a constant C0 > 0 such that 1/2 ||v|| ≤ C0 ||v||D(A0) |v|1/2 . (2.3) Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017 Thus, we have the following sequence D(A0 ) ⊂ V ⊂ H ⊂ V ∗ ⊂ D(A0 )∗ , (2.4) where each space is dense in the next one, which is continuous injection. Lemma 2.2. With the notations (2.3), (2.4), we have (V, V ∗ )1/2,2 = H and (D(A0 ), H)1/2,2 = V, where (V, V ∗ )1/2,2 denotes the real interpolation space between V and V ∗ (Section 1.3.3 of Butzer and Berens, 1967;Triebel, 1977). The mixed problem (1.1) can be formulated abstractly as R0 d dt u(t) = A0Ru(t) + A1 u(t − h) + −h a(s)A2 u(t + s)ds t + 0 k(t − s)g(s, u(t, x(s))ds + (Bw)(t), 0 ≤ t ≤ T, u(0) = g 0, u(s) = g 1 (s), −h ≤ s ≤ 0. (2.5) The operators A1 and A2 are bounded linear operators from V to V ∗ such that their restrictions to D(A0 ) are bounded linear operators from D(A0 ) equipped with the graph norm of A0 to H. The function a(·) is assumed to be real valued and belongs to L2 (−h, 0). We assume the following hypotheses on the nonlinear mappings g in (2.5): Assumption (F). g : [0, T ] × V −→ H is a nonlinear mapping such that (i) t 7→ g(t, x) is measurable; (iii) x 7→ g(t, x) is odd mapping (g(·, x) = −g(·, −x)); (iii) there exists a constant L such that |g(t, x) − g(t, y)| ≤ L||x − y||, x, y ∈ V. We assume that g(t, 0) = 0 for the sake of simplicity. 5 For x ∈ L2 (0, T ; V ) we set f (t, x) = Z 0 t k(t − s)g(s, x(s))ds, Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017 where k belongs to L2 (0, T ). Lemma 2.3. Let u ∈ L2 (0, T ; V ), T > 0. Then f (·, u) ∈ L2 (0, T ; H), √ |f (t, u1(t)) − f (t, u2 (t))| ≤ L t||k||L2(0,t) ||u1 − u2 ||C([0,t];V ) and √ ||f (·, u)||L2(0,T ;H) ≤ L||k||L2(0,T ) T ||u||L2(0,T ;V ) . Moreover if u1 , u2 ∈ L2 (0, T ; V ), then √ ||f (·, u1) − f (·, u2)||L2 (0,T ;H) ≤ L||k||L2(0,T ) T ||u1 − u2 ||L2 (0,T ;V ) . Proof. By using Hölder inequality, we obtain Z t |f (t, u1 (t)) − f (t, u2 (t))| ≤ |k(t − s){g(s, u1(s)) − g(s, u2(s))}|ds 0 ≤ Z t 2 0 |k(t − s)| ds 1/2 Z t 2 0 2 L ||u1(s) − u2 (s)|| ds √ ≤ L t||k||L2(0,t) ||u1 − u2 ||C([0,t];V ) . 1/2 From Assumption (F) and using the Hölder inequality, it is easily seen that Z T Z t 2 ||f (·, u)||L2(0,T ;H) ≤ | k(t − s)g(s, u(s))ds|2dt 0 0 Z TZ t 2 L2 ||u(s)||2dsdt ≤ ||k||L2 (0,T ) 0 0 ≤ L2 ||k||2L2 (0,T ) T ||u||2L2(0,T ;V ) . Finally, if u1 , u2 ∈ L2 (0, T ; V ), then Z T Z t ||f (·, u1) − f (·, u2)||L2 (0,T ;H) = | k(t − s) g(s, u1(s)) − g(s, u2(s)) ds|2 dt 0 0 √ ≤ L||k||L2 (0,T ) T ||u1 − u2 ||L2 (0,T ;V ) , which shows the second paragraph. 6 Let the solution spaces W0 (T ) and W1 (T ) of strong solutions be defined by W0 (T ) = L2 (0, T ; D(A)) ∩ W 1,2 (0, T ; H), W1 (T ) = L2 (0, T ; V ) ∩ W 1,2 (0, T ; V ∗ ). Here, we note that by using interpolation theory, we have Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017 W0 (T ) ⊂ C([0, T ]; V ), W1 (T ) ⊂ C([0, T ]; H). By virtue of Theorem 3.1 of Jeong, Kwun and Park (1999), we have the following result on the corresponding linear equation of (2.5). Proposition 2.1. Suppose that the assumptions stated above are satisfied. Then the following properties hold: 1)(linear case) Let f (t, x) ≡ f (t) ∈ L2 (0, T ; H) as considered a forcing term, and let the controller B be a bounded linear operator from L2 (0, T ; U) to L2 (0, T ; H). Then for (g 0, g 1 ) ∈ V × L2 (−h, 0; D(A0 )) and w ∈ L2 (0, T ; U), T > 0, there exists a unique solution u of the linear equation (2.5) belonging to W0 (T ) ⊂ C([0, T ]; V ), and satisfying ||u||W0(T ) ≤ C1 (||g 0|| + ||g 1||L2 (−h,0;D(A0 )) + ||f ||L2(0,T ;H) + ||Bw||L2(0,T ;H) ), (2.6) where C1 is a constant depending on T . 2) Let the controller B be a bounded linear operator from L2 (0, T ; U) to L2 (0, T ; H), and let Assumption (F) be satisfied. Then for (g 0 , g 1) ∈ V × L2 (−h, 0; D(A0 )) and w ∈ L2 (0, T ; U), there exists a unique solution u of the semilinear equation (2.5) belonging to W0 (T ) ⊂ C([0, T ]; V ), and satisfying ||u||W0(T ) ≤ C1 (1 + ||g 0|| + ||g 1||L2 (−h,0;D(A0 )) + ||Bw||L2(0,T ;H) ), (2.7) where C1 is a constant depending on T . 3) Let the controller B be a bounded linear operator from L2 (0, T ; U) to L2 (0, T ; V ∗ ) and let Assumption (F) be satisfied. Then for (g 0, g 1 ) ∈ H × L2 (−h, 0; V ) and w ∈ L2 (0, T ; U), there exists a unique solution u of the semilinear equation (2.5) belonging to W1 (T ) ⊂ C([0, T ]; H), and satisfying ||u||W1(T ) ≤ C1 (1 + |g 0| + ||g 1||L2 (−h,0;V ) + ||Bw||L2(0,T ;V ∗ ) ), where C1 is a constant depending on T . 7 (2.8) Given u ∈ L2 (0, T ; V ), we extend it to the space L2 (−h, T ; V ) by setting u(s) = g (s) for s ∈ (−h, 0). Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017 1 Theorem 2.1. Suppose that Assumption (F) is satisfied and the controller B is bounded from L2 (0, T ; U) to L2 (0, T ; H). If (g 0, g 1 ) ∈ V × L2 (−h, 0; D(A0 )) and w ∈ L2 (0, T ; U), then u ∈ W0 (T ), and the mapping (g 0 , g 1 , w) 7→ u ∈ W0 (T ) is continuous. Proof. For (g 0, g 1 ) ∈ V × L2 (−h, 0; D(A0 )) and w ∈ L2 (0, T ; U), by Proposition 2.1, u belongs to W0 (T ). Let (gi0 , gi1, wi )∈V ×L2 (−h, 0; D(A0 )) ×L2 (0, T ; H), and ui be the solution of (2.5) with (gi0 , gi1 , wi ) in place of (g 0 , g 1 , w) for i = 1, 2. Then in view of (2.6) and Lemma 2.3 we have ||u1 − u2 ||W0 (T )) ≤ C1 {||g10 − g20 || (2.9) 1 1 + ||g1 − g2 ||L2 (−h,0:D(A0 )) + ||f (·, u1) − f (·, u2)||L2 (0,T ;H) + ||Bw1 − Bw2 ||L2 (0,T ;H) } ≤ C1 {||g10 − g20 || + ||g11 − g21 ||L2 (−h,0:D(A0 )) + ||Bw1 − Bw2 ||L2 (0,T ;H) √ + L||k||L2(0,T ) T ||u1 − u2 ||L2 (0,T :V ) }. Since u1 (t) − u2 (t) = we get ||u1 − u2 ||L2 (0,T ;H) ≤ g10 √ − g20 + Z 0 t (u̇1 (s) − u̇2(s))ds, T T |g01 − g20 | + √ ||u1 − u2 ||W 1,2(0,T ;H) . 2 Hence, arguing as in (2.9) we get 1/2 1/2 ||u1 − u2 ||L2 (0,T ;V ) ≤ C0 ||u1 − u2 ||L2 (0,T ;D(A0 )) ||u1 − u2 ||L2 (0,T ;H) (2.10) 1/2 ≤ C0 ||u1 − u2||L2 (0,T ;D(A0 )) T 1/2 × {T 1/4 |g10 − g20 |1/2 + ( √ )1/2 ||u1 − u2 ||W 1,2 (0,T ;H) } 2 T 1/2 ≤ C0 T 1/4 |g10 − g20 |1/2 ||u1 − u2 ||L2 (0,T ;D(A0 )) + C0 ( √ )1/2 ||u1 − u2 ||W0 (T ) 2 T ≤ 2−7/4 C0 |g10 − g20 | + 2C0 ( √ )1/2 ||u1 − u2 ||W0 (T ) . 2 8 Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017 Combining (2.9) and (2.10) we obtain ||u1 − u2 ||W0 (T ) ≤C1 {||g10 − g20 ||X + ||g11 − g21 ||L2 (−h,0:D(A0 )) + ||Bw1 − Bw2 ||L2 (0,T ;H) } √ + 2−7/4 C0 C1 L||k||L2 (0,T ) T |g10 − g20| √ T + 2C0 C1 ( √ )1/2 L||k||L2 (0,T ) T ||u1 − u2 ||W0 (T ) . 2 (2.11) Suppose that (gn0 , gn1 , wn ) → (g 0 , g 1 , w) in V ×L2 (−h, 0; D(A0 ))×L2 (0, T ; H), and let un and u be the solutions (2.5) with (gn0 , gn1 , wn ) and (g 0, g 1 , w) respectively. Let 0 < T1 ≤ T be such that p √ 2C0 C1 (T1 / 2)1/2 L||k||L2 (0,T ) T1 < 1. Then by virtue of (2.11) with T replaced by T1 , we see that un → u in W0 (T1 ). This implies that (un (T1 ), (un )T1 ) 7→ (u(T1 ), uT1 ) in V ×L2 (−h, 0; D(A0 )). Hence the same argument shows that un → u in L2 (T1 , min{2T1 , T }; D(A0 )) ∩ W 1,2 (T1 , min{2T1 , T }; H). Repeating this process we conclude that un → u in W0 (T ). 3 Basic Fredholm theory 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, Nečas, Souček and Souček, 1973;Lloid, 1978). 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) 6= 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. Definition 3.1. Let T be a mapping defined by 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 (i) T is a homeomorphism of X onto Y ; 9 (ii) there exist real numbers K > 0, L > 0, and α > 0 such that L||x||αX ≤ ||T (x)||Y ≤ K||x||αX , ∀x ∈ X. Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017 Lemma 3.1. Let T be an odd (K, L, α)-homeomorphism of X onto Y and S : X → Y a continuous operator satisfying ||S(x)||Y = N ∈ R+ . α ||x|| ||x||X →∞ X lim sup N N , L ] ∪ {0}, Then for |λ| ∈ / [K lim ||x||X →∞ ||λT (x) − S(x)||Y = ∞. Proof. Suppose that there exist a constant M > 0 and a sequence {xn } ⊂ X such that ||λT (xn ) − S(xn )||Y ≤ M as xn → ∞. From this it follows that λT (xn ) S(xn ) − → 0. α ||xn ||X ||xn ||αX Hence, we have lim sup n→∞ |λ|||T (xn )||Y = N, ||xn ||αX N N , L ]. and so, |λ|K ≥ N ≥ |λ|L. It is a contradiction with |λ| ∈ / [K Proposition 3.1. Let T be an odd (K, L, α)-homeomorphism of X onto Y and S : X → Y an odd completely continuous operator. Suppose that for λ 6= 0, lim ||x||X →∞ ||λT (x) − S(x)||Y = ∞. (3.1) Then λT − S maps X onto Y . Proof. We follow the proof Theorem 1.1 in Chapter II of Fučik, Nečas, Souček and Souček (1973). Suppose that there exists y ∈ Y such that λT (x) = y. Then from (3.1) it follows that ST −1 : Y → Y is an odd completely continuous operator and y ||y − ST −1 ( )||Y = ∞. ||y||Y →∞ λ lim 10 Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017 Let y0 ∈ Y . There exists r > 0 such that y ||y − ST −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 d[y − ST −1 ( λy ); Yr , 0] is an odd number. For each y ∈ Y satisfying ||y||Y = r and t ∈ [0, 1], there is y y ||y − ST −1( ) − ty0 ||Y ≥ ||y − ST −1 ( )||Y − ||y0||Y > 0 λ λ and hence, by the homotopic property of degree, we have y y d[y − ST −1 ( ); Yr , y0 ] = d[y − ST −1( ); Yr , 0] 6= 0. λ λ Hence, by the existence theory of the Leray-Schauder degree, there exists a y1 ∈ Yr such that y1 y1 − ST −1( ) = y0 . λ We can choose x0 ∈ X satisfying λT (x0 ) = y1 , and so, λT (x0 ) − S(x0 ) = y0 . Thus, it implies that λT − S is a mapping of X onto Y . Combining Lemma 3.1. and Proposition 3,1, we have the following results. Corollary 3.1. Let T be an odd (K, L, α)-homeomorphism of X onto Y and S : X → Y an odd completely continuous operator satisfying ||S(x)||Y = N ∈ R+ . α ||x|| ||x||X →∞ X lim sup N N , L ] ∪ {0}, the operator λT − S maps X onto Y . Then for |λ| ∈ / [K 4 Controllability of semilinear retarded control equations Let U is a Banach space of control variables and let the controller B be a linear bounded operator from L2 (0, T ; U) to L2 (0, T ; H). In this section, we concern with the approximate controllability of the following retarded semilinear control system (2.5) with initial data (g 0 , g 1) = (0, 0): ( R0 d u(t) = A0 u(t) + A1 u(t − h) + −h a(s)A2 u(t + s)ds + f (t, u) + (Bw)(t), dt u(0) = 0, u(s) = 0 − h ≤ s < 0. (4.1) 11 Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017 The solution of (4.1) is denoted by u(t; g, w). Let S(·) be the analytic semigroup generated by A0 , and let W (·) be the fundamental solution of the linear equation associated with (4.1) in the sense of Nakagiri (1988);Tanabe (1992), which is the operator valued function satisfying ( Rt R0 W (t) = S(t) + 0 S(t − s){A1 W (s − h) + −h a(τ )A2 W (s + τ )dτ }ds, t > 0 W (0) = I, W (t) = 0 − h ≤ t < 0. Then u(t; g, w) = Z t 0 W (t − s){f (s, u(·, g, w)) + (Bw)(s)}ds, and in view of (2.8) ||u(·; g, w)||L2(0,T ;V )∩W 1,2 (0,T ;V ∗ ) ≤ C1 (1 + ||B||||w||L2(0,T ;U ) ). (4.2) In order to obtain approximate controllability for the system (4.1), we need to impose the following conditions. Assumption (A). The embedding D(A0 ) ⊂ V is completely continuous. Assumption (F1). g : [0, T ] × H −→ H is a nonlinear mapping such that (i) t 7→ g(t, x) is measurable; (iii) x 7→ g(t, x) is odd mapping g(·, x) = −g(·, −x)); (iii) there exists a constant L such that |g(t, x) − g(t, y)| ≤ L|x − y|, x, y ∈ H. We assume that g(t, 0) = 0 for the sake of simplicity. Here, we remark that since Assumption(F) is more general than (F1), the regular property of solutions of (4.1) mentioned in Section 2 can be available. Theorem 4.1. For w ∈ L2 (0, T ; U), let uw be the solution of equation (4.1), and let Assumptions (A) and (F1) be satisfied. Then the mapping w 7→ uw is compact from L2 (0, T ; U) to L2 (0, T ; V ). Proof. If w ∈ L2 (0, T ; U), then in view of (2.8) in Proposition 2.1 ||uw ||W1 (T ) ≤ C1 (1 + ||B|| ||w||L2(0,T ;U ) ). 12 (4.3) Since uw ∈ L2 (0, T ; V ), we have f (·, uw ) ∈ L2 (0, T ; H). Consequently uw ∈ L2 (0, T ; D(A0 ) ∩ W 1,2 (0, T ; H). Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017 Hence, with aid of (2.6) of Proposition 2.1, Lemma 2.3, and (4.3), ||uw ||L2 (0,T ;D(A0 )∩W 1,2 (0,T ;H) ≤ C1 (||f (·, uw ) + Bw||L2(0,T ;H) ) √ ≤ C1 {L||k||L2 (0,T ) T ||u||L2(−h,T −h;V ) + ||Bw||L2(0,T ;H) } √ ≤ C1 [C1 L||k||L2 (0,T ) T {(1 + ||B|| ||u||L2(0,T ;U ) )} + ||B|| ||w||L2(0,T ;U ) ]. (4.4) Hence, if w is bounded in L2 (0, T ; H), then so is uw in L2 (0, T ; D(A0))∩W 1,2 (0, T ; H). Since D(A0 ) is compactly embedded in V by assumption, the embedding L2 (0, T ; D(A0) ∩ W 1,2 (0, T ; V ) ⊂ L2 (0, T ; V ) is completely continuous in view of Theorem 2 of Aubin (1963), and by Theorem 2.1, the mapping w 7→ uw is completely continuous from L2 (0, T ; U) to L2 (0, T ; V ). We define the operator F by F (w) = f (·, uw ), ∀w ∈ L2 (0, T ; U). (4.5) By virtue of Assumption(F) in Section 2, we obtain the following results. Corollary 4.1. Let Assumptions (A) and (F1) be satisfied. Then F defined by (4.5) is a completely continuous mapping from L2 (0, T ; U) to L2 (0, T ; H). Let us define the reachable sets for the system (4.1) as follows: RT (g) = {u(T ; g, w) : w ∈ L2 (0, T ; U)}, RT (0) = {u(T ; 0, w) : w ∈ L2 (0, T ; U)}. Definition 4.1. If RT (g) = H, where RT (g) is the closure of RT (g) in H, then the system (4.1) is called approximately controllable at time T . By the same argument as in the proof of Lemma 2.3 in the sense of Assumption (F1), we know that √ ||f (·, u)||L2(0,T ;H) ≤ L||k||L2 (0,T ) T ||u||L2(0,T ;H) . Hence, we have √ ||f (·, u)||L2(0,T ;H) := γ < L||k||L2 (0,T ) T . ||u||L2(0,T ;H) |||u||L2 (0,T ;H) →∞ lim sup 13 (4.6) Theorem 4.2. Assume that (A) and (F1) are satisfied and γ ∈ R+ − {1}, where γ is the constant in (4.6). Then we have RT (0) ⊂ RT (g). Therefore, if the linear system (4.1) with g = 0 is approximately controllable, then so is the nonlinear system (4.1). Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017 Proof. Let η= Z 0 T W (T − s)Bv(s)ds ∈ RT (0). We are going to show that there exists w such that η = x(T ; g, w). Let 2 N = {p ∈ L (0, T ; H) : Z T 0 W (T − s)p(s)ds = 0} and denote by N ⊥ be the orthogonal complement of N in L2 (0, T ; H). We denote the range of the operator B by HB . For y ∈ N ⊥ , let P y be the unique minimum norm element of {y + N} ∩ H B . Then the proof of Lemma 1 of Naito (1987) can be applied to show that P is a linear and continuous operator from N ⊥ to H B . Let Ỹ = L2 (0, T ; H)/N be the quotient space and the norm of a coset ỹ = y + N ∈ Ỹ is defined of ||ỹ|| = inf{|y + g| : g ∈ N}. We define by Q the isometric isomorphism from Ỹ onto N ⊥ , that is, Qỹ is the minimum norm element in ỹ = {y + g : g ∈ N}. Let F ỹ = F (P Qỹ) + N for ỹ ∈ Ỹ . Then by Lemma 2.3 and Theorem 4.1, F is a completely continuous mapping from Ỹ to itself. Moreover, by (4.6), it holds ||F (y)||Ỹ = γ < ∞. ||y||Ỹ ||y||Ỹ →∞ lim sup (4.7) Since the identity operator IỸ on Ỹ is an odd (1, 1, 1)-homeomorphism and, by Assumption (F1) and (4.7), −F : Ỹ → Ỹ an odd completely continuous operator satisfying lim ||λI(y) + F (y)||Y = ∞, ||y||Ỹ →∞ where λ 6= γ , noting that γ ∈ R+ − {1} and by Corollary 3.1, we have that λI + F maps Ỹ onto Ỹ . Hence, if we set z = Bv, there exists w̃ ∈ Ỹ such that z̃ = w̃ + F w̃. 14 (4.8) Put w = Qw̃ and wB = P Qw̃. Then we have that wB = P w and w−wB = w−P w ∈ N. Hence, z̃ = F (wB ) + w + N = F (wB ) + wB + N. (4.9) Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017 For g ∈ L2 (0, T ; H) let ûg be the solution of equation with B = I Then d u(t) dt = A0 u(t) + A1 u(t − h) R0 + −h a(s)A2 u(t + s)ds + f (t, x) + g(t), u(0) = 0, u(s) = 0 − h ≤ s < 0. ûg (t) = Z 0 (4.10) t W (t − s){f (s, ûg (·)) + g(s)}ds, and by (2.8), ||ûg ||L2 (0,T ;V )∩W 1,2 (0,T ;V ∗ ) ≤ C2 (1 + ||g||L2(0,T ;H) ). Since ûg ∈ L2 (0, T ; V ), f (·, ûg ) ∈ L2 (0, T ; H). Consequently, ûg ∈ L2 (0, T ; D(A0)) ∩ W 1,2 (0, T ; H). Thus, from (4.9) it follows that η= Z T W (T − s)(F (wB )(s) + wB (s))ds 0 = Z 0 T W (T − s)(f (s, ûwB ) + wB (s))ds. Since wB ∈ H B , there exists a sequence {vn } ∈ L2 (0, T ; U) such that Bvn 7→ wB in L2 (0, T ; H). Then by the last part of Theorem 2.1 we have that u(·; g, vn) 7→ ûuB in L2 (0, T ; D(A0)) ∩ W 1,2 (0, T ; H), and hence u(T ; g, vn) 7→ ûuB (T ) = η in V . Thus V V we conclude η ∈ RT (g) , where RT (g) is the closure of RT (g) in V , which says η ∈ RT (g). Remark 4.1. In general, to obtain sufficient conditions for approximate controllability of semilinear control systems, many authors frequently used the following range condition of controller with a compact semigroup introduced in Naito (1987): Assumption (B). For each p ∈ L2 (0, T ; X) there exists a function q ∈ X B : S̃p = S̃q, where Z T S̃p = S(T − s)p(s)ds, 0 where S(t)(t ≥ 0) is an analytic semigroup generated by A0 . 15 In particular, in Wang (2009) established the approximate controllability with local Lipschitz continuity for nonlinear term of System (4,1) assuming Assumption (B) with the compactness of S(t)(t ≥ 0) and ||f (·, u)||L2(0,T ;H) := γ ||u||L2(0,T ;H) |||u||L2 (0,T ;H) →∞ Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017 lim sup is sufficiently small. In this paper, to prove approximate controllability of (4.1) for γ ∈ R+ − {1} more generally, we no longer require the compactness of S(t)(t ≥ 0), Assumption (B) and the uniform bundedness of the nonlinear term, but instead we need the natural assumption that the embedding D(A0 ) ⊂ V is compact. 5 Examples Let Ω be a bounded domain in Rn with smooth boundary ∂Ω. Let aij be a realvalued and smooth function for each i, j = 1, · · · , n. Assume that aij (x) = aji (x) for each x ∈ Ω̄ and {aij (x)} is positive definite uniformly in Ω, i.e., there exists a positive number c0 such that n X i,j=1 aij (x)ξi ξj ≥ c0 |ξ|2 (5.1) for ξ. Let bi ∈ L∞ (Ω) and c ∈ L∞ (Ω). Put βi = Pn all x ∈ Ω̄ and all real vectors ∞ i=1 ∂aij /∂xj + bi , then βi ∈ L (Ω). Consider an elliptic differential operator of second order in (1.1) as n n X X ∂ ∂ ∂ (ai,j (x) )+ bi (x) + c(x). A(x, Dx ) = − ∂xj ∂xi ∂xi i=1 i,j=1 For each u, v ∈ H1 (Ω), we put ) Z (X n n X ∂u ∂v ∂u b(u, v) = aij + βi v̄ + cuv̄ dx. ∂xi ∂xj ∂x i Ω i,j=1 i=1 Since {aij } is real symmetric, by (5.1) the inequality n X i,j=1 aij (x)ζi ζ̄j ≥ c0 |ζ|2 16 Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017 holds for all complex vectors ζ = (ζ1 , · · · , ζn ). On the other hand, by this hypothesis, there exists a certain K such that |βi (x)| ≤ K and |c(x)| ≤ K hold almost everywhere. Hence, Z Z X Z n n X ∂u 2 ∂u Re b(u, u) ≥ c0 |u|2dx ∂xi dx − K ∂xi |u|dx − K Ω Ω i=1 Ω i=1 ! 2 2 Z X Z X n n ∂u ∂u 1 ε 2 ≥c0 ∂xi + 2ǫ |u| dx ∂xi dx − K 2 Ω i=1 Ω i=1 Z −K |u|2dx Ω 2 Z n Z ǫ X ∂u nK = c0 − K |u|2dx. ∂xi dx − 2ε + K 2 Ω i=1 Ω By choosing ǫ = c0 K −1 , we obtain 2 Z n Z c0 X ∂u nK 2 Re b(u, u) ≥ dx − +K |u|2dx 2 i=1 Ω ∂xi 2c0 Ω c0 c0 nK 2 kuk2 . = kuk21 − +K + 2 2c0 2 Therefore, it follows that, for any closed subspace V of H1 (Ω) containing H01 (Ω), the quadratic form b(u, v) satisfies (2.2), where H = L2 (Ω). Let A0 be the operator associated with this sesquilinear form: (A0 u, v) = b(u, v), u, v ∈ V. Then −A0 is a bounded linear operator from V to V ∗ by the Lax-Milgram Theorem. The realization of A0 in H which is the restriction of A0 to D(A0 ) = {u : u ∈ H 2 (Ω) ∩ H01 (Ω)} = {u : u ∈ H 2 (Ω), u|∂Ω = 0}, A0 u = −A(x, Dx )u for any u ∈ D(A) is also denoted by A0 . as the result known as Sobolev’s imbedding theorem, the embedding D(A0 ) ⊂ V is compact. Let Aι (x, Dx )(ι = 1, 2) be second order linear differential operators with real coefficients. We realize the operator Aι (x, Dx ), ι = 1, 2, in Hilbert spaces by Aι v = −Aι (x, Dx )v, ι = 1, 2, 17 v∈V in the distribution sense. We set ′ g(x(t, y)) = g1 (|x(t, y)|2)x(t, y), Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017 where the function g1 given is real valued belong to C 1 ([0, ∞)) which satisfies the conditions: (i) g1 (0) = 0, g1 (r) ≥ 0 for r > 0, ′ (ii) |g1(r) ≤ c for c > 0. If we present f (t, u(t, x)) = Z 0 and t k(t − s)g(s, u(s, x(s))ds, ||f (·, u)||L2(0,T ;H) 6= 1. ||u||L2(0,T ;H) |||u||L2(0,T ;H) →∞ lim sup then Assumption (F1) is satisfied. Hence, from Theorem 4.2, we have RT (0) ⊂ RT (g), that is, if the linear system (1.1) with g = 0 is approximately controllable, then so is the nonlinear system (1.1). 6 Conclusion In this paper, authors establish the approximate controllability for a class of semilinear integro-differential functional control equations, which is provided under general sufficient conditions on the system operator, controller and nonlinear terms. To obtain sufficient conditions for the approximate controllability, many authors frequently used either the range condition of controller with a compact semigroup or the so called resolvent condition for using a fixed point theorem. This paper wants to use a different method than the previous one. The approach followed here is the results similar to Fredholm alternative for nonlinear operators under restrictive assumption, which is on the solution of nonlinear operator equations λT (x) − S(x) = y in dependence on the real number λ, where T and S are nonlinear operators defined a Banach space X with values in a Banach space Y . In order to obtain the approximate controllability for a class of semilinear integrodifferential functional control equations, it is necessary to suppose that T acts as the 18 Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017 identity operator while S related to the nonlinear term of control equations is completely continuous. The approach developed in this paper can be extended to more general sufficient assumptions such as the growth condition on nonlinear terms. Future works will concentrate on extending the developed our results to both nonlinear neutral evolution systems with nonlocal conditions and impulsive control with delays, as well as semilinear retarded stochastic systems Acknowledgement This research was supported by Basic Science Research Program through the National research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2015R1D1A1A09059030). References [1] Abu Arqub, O., & Al-Smadi M. (2014), Numerical algorithm for solving two-point, second-order periodic boundary value problems for mixed integrodifferential equations, Applied Mathematics and Computation, 243, 911–922. [2] Abu Arqub, O., & Rashaideh H.(2017), The RKHS method for numerical treatment for integrodifferential algebraic systems of temporal two-point BVPs, Neural Computing and Applications, doi:10.1007/s00521-017-2845-7. [3] Aubin, J. 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