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Abstract
In this paper, we deal with the null controllability of semilinear functional integrodifferential control systems under the Lipschitz continuity of nonlinear terms. Moreover, we establish the regularity and a variation of constant formula for solutions of the given control systems in Hilbert spaces.
Keywords: controllability, semilinear control system, regularity for solution, analytic semigroup, integrodiﬀerential control system.
1. Introduction
Let H and V be two complex Hilbert spaces. Assume that V is a dense subspace in H and the injection of V into H is continuous. The norm on V (resp. H) will be denoted by ||·|| (resp. |·|) respectively. Let A be a continuous linear operator from V into V^{*} which is assumed to satisfy Gårding’s inequality, and generate an analytic semigroup (S(t))_{t}_{≥0}. We study the following the semilinear functional integrodifferential control systems:
Here, a forcing term k ∈ L^{2}(0, T; V^{*}), x_{0} ∈ H, and g: R^{+} × V → H is a nonlinear mapping as detailed in Section 2.. The controller B is a bounded linear operator from L^{2}(0, T;U) to L^{2}(0, T;H), where U is some Banach space of control variables.
The existence of solutions for a class of semilinear functional integrodifferential control systems has been studied by many authors. For example, one finds parabolic type problems in [3, 17, 18], hyperbolic type problems in [11, 18], and linear cases in [4, 5, 7, 13, 14]. The background of these problems is to serve as an initial value problem for many partial integrodifferential equations which arise in problems connected with heat flow in materials, random dynamical systems, and other physical phenomena. For more details on applications of the theory we refer to the survey of Balachandran and Dauer [2] and the book by Curtain and Zwart [7].
In recent years, as for the controllability of semilinear differential equations, Carrasco and Leiva [6] discussed sufficient conditions for approximate controllability of parabolic equations with delay, Mahmudov [15] in the case that the semilinear equations with nonlocal conditions with condition on the uniform boundedness of the Frechet derivative of nonlinear term, and Sakthivel et al. [19] on impulsive and neutral functional differential equations. As for some considerations on the trajectory set of (1.1) and that of its corresponding linear system (in case g ≡ 0) as matters related to (1.1), we refer the reader to Naito [16], Sukavanam and Tomar [22] and references therein.
In [2, 10] the authors dealt with the approximate controllability of a semilinear control system as a particular case of sufficient conditions for the approximate solvability of semilinear equations by assuming that
S(t) is compact operator, and
the linear operator ${S}^{T}u:={\int}_{0}^{T}S(t-s)u(s)ds$ has a bounded inverse operator.
The paper [15] replaces the above condition (2) with
(2-1) The Frechet derivative of nonlinear term is uniformly bounded, and
(2-2) the corresponding linear system (1.1) in case g ≡ 0 and x_{0} ≡ 0 is approximately controllable.
In [22] and [23] they studied the control problems of the semilinear equations by assuming conditions (1), (2-2), a Lipschitz continuity of the nonlinear term, and a range condition of the controller B with an inequality constraint.
In this paper we replace the condition (1) by the compactness of the embedding D(A) ⊂ V, and instead of (2.2) and the uniform boundedness f the nonlinear term, we require the following inequality constraint on the range condition of the controller B: for any p ∈ L^{2}(0, T;H) there exists a u ∈ L^{2}(0, T;U) such that
$${S}^{T}p={S}^{T}Bu.$$
In Section 2, we will obtain that most parts of the regularity for parabolic linear equations can also be applicable to (1.1) with nonlinear perturbations. The approach used here is similar to that developed in [8, 9, 13, 15] on the general semilnear evolution equations. Moreover, in Section 3, we establish the null controllability of semilinear functional integrodifferential control systems (1.1) under the Lipschitz continuity instead of the uniform boundedness of the Frechet derivative of nonlinear term. It is useful for physical applications of the given equations.
2. Regularity for Solutions
If H is identified with its dual space we may write V ⊂ H ⊂ V^{*} densely and the corresponding injections are continuous. The norm on V, H and V^{*} will be denoted by ||·||, |·| and ||·||_{*}, respectively. The duality pairing between the element v_{1} of V^{*} and the element v_{2} of V is denoted by (v_{1}, v_{2}), which is the ordinary inner product in H if v_{1}, v_{2} ∈ H. For the sake of simplicity, we may consider
where (V, V^{*})_{1/2,2}denotes the real interpolation space between V and V^{*}(Section 1.3.3 of [21]).
It is also well known that A generates an analytic semigroup S(t) in both H and V^{*}. For the sake of simplicity, we assume that c_{1} = 0 and hence the closed half plane {λ: Re λ ≥ 0} is contained in the resolvent set of A.
Conversely, suppose that x ∈ V^{*} and ${\int}_{0}^{T}{\Vert A{e}^{tA}x\Vert}_{*}^{2}dt<\infty $. Put u(t) = e^{tA}x. Then since A is an isomorphism operator from V to V^{*} there exists a constant c > 0 such that
By virtue of Theorem 3.3 of [4](or Theorem 3.1 of [9]), we have the following result on the corresponding linear equation of (2.4).
Proposition 2.1
Suppose that the assumptions for the principal operator A stated above are satisfied. Then the following properties hold:
Let V = (D(A), H)_{1/2,2}where (D(A), H)_{1/2,2}is the real interpolation space between D(A) and H(see [[21]; section 1.3.3], or Lemma 2.1). For x_{0} ∈ V and h ∈ L^{2}(0, T;H), T > 0, there exists a unique solution x of (2.4) belonging to$${L}^{2}(0,T;D(A))\cap {W}^{1,2}(0,T;H)\subset C([0,T];V)$$
and satisfying$${\Vert x\Vert}_{{L}^{2}(0,T;D(A))\cap {W}^{1,2}(0,T;H)}\le {C}_{1}(\Vert {x}_{0}\Vert +{\Vert h\Vert}_{{L}^{2}(0,T;H)}),$$
where C_{1}is a constant depending on T.
Let x_{0} ∈ H and h ∈ L^{2}(0, T; V^{*}), T > 0. Then there exists a unique solution x of (2.4) belonging to$${L}^{2}(0,T;V)\cap {W}^{1,2}(0,T;{V}^{*})\subset C([0,T];H)$$
and satisfying$${\Vert x\Vert}_{{L}^{2}(0,T;V)\cap {W}^{1,2}(0,T;{V}^{*})}\le {C}_{1}(\mid {x}_{0}\mid +{\Vert h\Vert}_{{L}^{2}(0,T;{V}^{*})}),$$
where C_{1}is a constant depending on T.
For the sake of simplicity, we assume that solution semigroup S(t) generated by A is uniformly bounded:
Consider the following initial value problem for the abstract semilinear parabolic equation (1.1). Let U be a Banach space and the controller operator B be a bounded linear operator from U to H.
Let g: R^{+} × V → H be a nonlinear mapping satisfying the following:
(F1) For any x ∈ V, the mapping g(·, x) is strongly measurable;
(F2) There exist positive constants L_{0}, L_{1} such that $$\begin{array}{l}\mid g(t,x)-g(t,\widehat{x})\mid \hspace{0.17em}\le {L}_{1}\Vert x-\widehat{x}\Vert ,\\ \mid g(t,0)\mid \hspace{0.17em}\le {L}_{0}\end{array}$$
for all t ∈ R^{+}, and x, x̂ ∈ V.
For x ∈ L^{2}(0, T; V ), we set
$$f(t,x)={\int}_{0}^{t}k(t-s)g(s,x(s))ds,$$
where k belongs to L^{2}(0, T).
Lemma 2.4
Let x ∈ L^{2}(0, T; V ) for any T > 0. Then f(·, x) ∈ L^{2}(0, T;H) and
We are going to show that x ↦ y is strictly contractive from L^{2}(0, T_{0}; V ) to itself. Let y, ŷ belong to V with the same initial condition in [0, T_{0}]. Then from assumption (2.9), (2.11) and
So by virtue of the condition (2.13) the contraction mapping principle gives that the solution of (1.1) exists uniquely in [0, T_{0}]. Let x be a solution of (1.1) and x_{0} ∈ H. Then there exists a constant C_{1} such that
since the condition (2.13) is independent of initial values, the solution of (1.1) can be extended to the interval [0, nT_{0}] for every natural number n. An analogous estimate to (2.12) holds for the solution in [0, nT_{0}], and hence for the initial value x_{nT}_{0} in the interval [nT_{0}, (n + 1)T_{0}].
3. Null Controllability of Semilinear Systems
Let S(t) be the analytic semigroup generated by the principal operator A. We define the linear operator Ŝ from L^{2}(0, T;H) to H by
$${\widehat{S}}^{T}p={\int}_{0}^{T}S(T-s)p(s)ds$$
for p ∈ L^{2}(0, T;H). Let x(T; f, u) be a state value of the system (1.1) at time T corresponding to the nonlinear term f and the control u. Then the solution x(T; f, u) of (1.1) is represented by
Equation (1.1) is said to be null controllable at time T > 0 if for a given x_{0} ∈ H there exists a control u ∈ L^{2}(0, T;U) such that x(T; f, u) = 0.
Let G^{T} = Ŝ^{T}B. Here, we remark that G is a bounded linear operator(see Proposition 2.1 or Theorem 2.1) but necessary one-to-one. Denote the orthogonal complement in L^{2}(0, T;U) by [kerG^{T}]^{⊥}. Let G: [kerG^{T}]^{⊥} → ImG^{T} be the restriction of G^{T} to [kerG^{T}]^{⊥}. Then we know that G is necessary a one-to-one operator.
Let us assume the natural assumption that the embedding $$D(A)\subset V\hspace{0.17em}\text{is\hspace{0.17em}compact.}$$
For any p ∈ L^{2}(0, T;H) there exists a u ∈ L^{2}(0, T;U) such that $${\widehat{S}}^{T}p={\widehat{S}}^{T}Bu.$$
Remak 3.1
Denote the kernel of the operator Ŝ^{T} by N, which is a closed subspace in L^{2}(0, T;H), and its orthogonal space in L^{2}(0, T;H) by N^{⊥}. Let ℬ be defined by (ℬu)(·) = Bu(·). Denote the range of the operator ℬ by R(ℬ) and its closure by $\overline{R(\mathcal{B})}$ in L^{2}(0, T;H). As seen in [9, 16], it is easily known that the hypothesis (B) is equivalent to the following condition: ${L}^{2}(0,T;H)=\overline{R(\mathcal{B})}+N$.
Lemma 3.2
Let us assume the hypothesis (B). Then we have
$$D(A)\subset \text{Im\hspace{0.17em}}G.$$
Proof
Let x_{0} ∈ D(A) and put p(s) = (x_{0} − sAx_{0})/T. Then p ∈ L^{2}(0, T;H) and
Hence, from (B) we can choose a control u_{0} ∈ L^{2}(0, T;U) satisfying
$${\widehat{S}}^{T}p={\widehat{S}}^{T}B{u}_{0},$$
which implies D(A) ⊂ ImG.
Theorem 3.1
For u ∈ L^{2}(0, T;U), let x_{u} = G^{T}u with x_{u}(0) = 0. Under Assumption(A), we have the mapping G^{T}: u → x_{u} is compact from L^{2}(0, T;U) to L^{2}(0, T; V ) ⊂ L^{2}(0, T;H).
Proof
If u ∈ L^{2}(0, T;U), with the aid of Proposition 2.1 (or Lemma 2.3), we have x_{u} ∈ L^{2}(0, T;D(A)) ∩ W^{1,2}(0, T; V^{*}) and satisfy the following inequality:
where C_{1} is the constant in Proposition 2.1. Hence if u is bounded in L^{2}(0, T;U), then so is x_{u} in L^{2}(0, T;D(A)) ∩ W^{1,2}(0, T;H) by the above inequality. Since D(A) is compactly embedded in V by assumption, the embedding
be the restriction of G^{T} to [kerG^{T}]^{⊥}. G is necessarily a one-to-one operator. Here, we remark that by Lemma 3.2, S(T)x_{0} ∈ ImG since S(T)x_{0} ∈ D(A) for x_{0} ∈ D(A). Define
by W(x, h) ≡ (G)^{−1}N(x, h). From Theorem 3.1, it follows that ImG^{T} is closed and [kerG^{T}]^{⊥} is obviously closed. Hence, the inverse mapping theorem says that G^{−1} is a bounded linear operator, and so is W.
Since the operator BW is bounded, for the sake of simplicity, we assume that
where C_{0} is constant in (2.3). We are going to show that x → Fx is strictly contractive from L^{2}(0, T_{0}; V ) to itself if the condition (3.3) is satisfied. Let Fx_{1}, Fx_{2} be the solutions of the above equation with x replaced by x_{1}, x_{2} ∈ L^{2}(0, T_{0}; V) respectively. From (3.3) it follows that
Hence, by virtue of (3.4) the contraction mapping principle gives that the operator F has unique solution in [0, T_{0}], that is, x is the solution of the following equation:
Next we establish the estimates of solution. Let x(·) be the solution of (3.5) in the (0, T_{0}) and y(·) be the solution of (3.1) with the control u(t) = −W(x_{0}, k) as in (3.2), i.e., the solution y of (3.1) is represented by
since the condition (3.3) is independent of initial values, the solution of (1.1) can be extended to the interval [0, nT_{0}] for every natural number n. That is, an analogous estimate to (3.6) holds for the solution in [0, nT_{0}], and hence for the initial value x_{nT}_{0} in the interval [nT_{0}, (n + 1)T_{0}], which means that the system (1.1) is null controllable at time T > 0 with the control u = −W(x_{0}, f).
Theorem 3.3
Let the assumption (F1), (F2) be satisfied and (x_{0}, u) ∈ V × L^{2}(0, T;U), Then the solution x of the equation (1.1) belongs to x ∈ L^{2}(0, T;D(A)) ∩ W^{1,2}(0, T;H) and the mapping
It is easy to show that if x_{0} ∈ V and f(·, x) ∈ L^{2}(0, T;H), then x belongs to L^{2}(0, T;D(A)) ∩ W^{1,2}(0, T;H). Let $({x}_{0}^{i},{u}_{i})\in H\times {L}^{2}(0,T;U)$ and x_{i} be the solution of (1.1) with (x_{0}, u) in place of (${x}_{0}^{i}$, u_{i}) for i = 1, 2. Then
Suppose that ${x}_{0}^{n}\to {x}_{0}$ in V and let x_{n} and x be the solution (1.1) with ${x}_{0}^{n}$ and x_{0} respectively. Let 0 < T_{1} ≤ T be such that
Then, as seen in [12], we note that for every desired final state x_{1} ∈ H and ε > 0 there exists a control function u ∈ L^{2}(0, T;U) such that the solution x(T; u) of (1.1) satisfies |x(T; u) − x_{1}| < ε, i.e., the system (1.1) is said to be approximately controllable in the time interval [0, T].
Example 3.1
We consider an application of the results obtained in the preceding sections to a class of partial functional integrodifferential systems with delay terms dealt with by Naito [16] and Zhou [23]:
and k belongs to L^{2}(0, T). Here, Ω ⊂ ℛ^{n} is a bounded domain with smooth boundary ∂Ω. We set H = L^{2}(Ω) and $V={H}_{0}^{1}(\mathrm{\Omega})$. Let b(u, v) be the sesquilinear form in ${H}_{0}^{1}(\mathrm{\Omega})\times {H}_{0}^{1}(\mathrm{\Omega})$ defined by
Here, we assume that a_{ij} is a real-valued and smooth function for each i, j = 1, ···, n, and a_{ij}(x) = a_{ji}(x) for each x ∈ Ω̄ and {a_{ij}(x)} is positive definite uniformly in Ω, i.e., there exists a positive number c_{0} such that
for all x ∈ Ω̄ and all real vectors ξ. Let b_{i} ∈ L^{∈}(Ω) and c ∈ L^{∈}(Ω). As is well known this sesquilinear form a(·, ·) is bounded and satisfying the Gårding’s inequality (2.2)(see e.g. Tanabe [20]. Let
be the associated uniformly elliptic differential operator of second order. Then the realization of In L^{2}(Ω) under the Dirichlet boundary condition is exactly A, i.e.,
It is not difficult to verify that for $u\in {H}_{0}^{1}(\mathrm{\Omega})$ in the sense of distribution and u|_{∂Ω} = 0 for $u\in {H}_{0}^{1}(\mathrm{\Omega})$ also in the sense of distribution(see Lions and Magenes [14]), and
Then g is not uniformly bounded and satisfies hypotheses (F1)and (F2). Let U = H be the space of control variables and let us define the intercept controller operator B_{α}(0 < α < T) on L^{2}(0, T;H) by
Then w ∈ L^{2}(0, T;H) and Ŝp = ŜB_{α}w, which is that the controller B_{α} satisfies Assumption (B). Hence, for the initial data ${u}_{0}\in {W}^{2,2}(\mathrm{\Omega})\cap {H}_{0}^{1}(\mathrm{\Omega})$ the system (3.9) is null controllable.
Acknowledgements
Authors would like to thank the referees for their useful suggestions which have significantly improved the paper.
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