(x, t)=(Gu)(x, t) + f(t, w(x, t)),t > 0. (1) w(x, 0) = 0, in the domain x â [0, 2Ï] with periodic boundary condition. â k w. âxk. (0,t) = â k w. âxk. (2Ï, t),k = 0, 1, 2. (2).
Controllability of Semilinear Dispersion Equation Simegne Tafesse and N. Sukavanam Department of Mathematics, IIT Roorkee, Roorkee, India {wtsimegne,nsukavanam}@gmail.com
Abstract. In this paper, the exact controllability of the third order semilinear dispersion equation is established under simple sufficient conditions on the nonlinear term which is assumed to be the sum of three nonlinear functions. We use the integral contractor condition which is weaker than Lipschitz condition on one of the functions and imposing certain range conditions on the Nemskii operators of the other two nonlinear parts. Moreover, the Co-semigroup operator is assumed to be bounded. Keywords: Dispersion equation, Semilinear, Integral contractor, Nemtskii operator, Exact controllable, Third order.
1
Introduction
Let X = L2 (0, 2π) be Hilbert spaces and Z = L2 ((0, T ) × (0, 2π) = L2 ((0, T : X) be function space. Consider the third order dispersion equation given by ∂3w ∂w (x, t) + (x, t) = (Gu)(x, t) + f (t, w(x, t)), t > 0 ∂t ∂x3 w(x, 0) = 0,
(1)
in the domain x ∈ [0, 2π] with periodic boundary condition ∂kw ∂kw (0, t) = (2π, t), k = 0, 1, 2. ∂xk ∂xk
(2)
The state function w and the control function u are in Z. The bounded linear operator G is defined on Z by 2π g(s)u(s, t))ds, (3) (Gu)(x, t) = g(x)(u(x, t) − 0
where g(x) is a piecewise continuous nonnegative function on [0, 2π] such that [g] = 1 where [g] is defined as
2π
g(s)ds
[g] =
(4)
0 P. Balasubramaniam and R. Uthayakumar (Eds.): ICMMSC 2012, CCIS 283, pp. 310–315, 2012. c Springer-Verlag Berlin Heidelberg 2012
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f : [0, T ] × X → X is defined by f = f1 + f2 + f3 where fi : [0, T ] × X → X, is continuous nonlinear function for each i = 1, 2, 3. Many researchers have worked on the controllability problems of third order dispersion equation. R.K. George et al. [3] studied and obtained sufficient conditions for the exact controllability of the nonlinear third order dispersion equation by assuming the two standard types of nonlinearity, namely, monotone and Lipschitz continuous. In that paper, under each condition, the existence and uniqueness of the solution has been shown first and then the controllability was proved. In [1] Chalishajar studied the controllability of nonlinear integrodifferential third order dispersion system. David L. and Bing Yu Zhang [2] discussed the exact controllability and stabilizability of a system described by the Korteweg-de Vries (KdV) equation given by ∂w ∂3w ∂w (x, t) + αw(x, t) (x, t) + (x, t) = u(x, t), ∂t ∂x ∂x3
(5)
where u is a control function. D.L Russel and B.Y Zhang [6] studied the controllability and stabilizability of the third order linear dispersion equation on a periodic domain. The equation (1) is the same as KdV equation when α = 0. In each of the above papers, the control problems are focussing on the conserva 2π tion of the so called fluid-volume which is given by 0 w(x, t)dx. N.K. Tomar, [4] studied on the controllability of third order dispersion equation where some parts of the nonlinear term are Lipschitz continuous, and monotone. Almost all of the above mentioned researchers dealt with the exact controllability of third order dispersion system. However, recently R. Sakthivel et.al [7] studied the approximate controllability of nonlinear third order dispersion equation. In our paper, the exact controllability of the third order dispersion equation (1) is studied by splitting the nonlinear part into three nonlinear functions and each function is assumed to have different conditions.
2
Preliminaries and Basic Assumption
This paper is aimed to discuss the exact controllability of semilinear dispersion equation by assuming some simple conditions on each terms of the nonlinear function f. Consider the semilinear dispersion equation (1)-(2). Let A be an operator on X defined by Aw = −
∂ 3w ∂x3
(6)
with domain D(A) ⊆ H 3 (0, 2π) which is consisting of functions satisfying boundary condition (2). Then the abstract form of the dispersion equation (1)-(2) can be written as ∂w (x, t) = Aw(x, t) + (Gu)(x, t) + f (t, w(x, t)), t > 0 ∂t w(x, 0) = 0.
(7)
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By Lemma 8.5.2 of Pazy [5], A is infinitesimal generator of a Co-semigroup {T (t), t ≥ 0} of isometries on L2 (0, 2π). Then for all w ∈ D(A) Aw, wX = −w , w = w, w = −Aw, w where the middle equality is achieved by applying integration by parts three times. Also there exists a constant M > 0 such that sup{T (t) : t ∈ [0, τ ]} ≤ M . We define the mild solution of (7) by t t w(x, t) = T (t − s)(Gu)(s)ds + T (t − s)f (s, w(x, s))ds 0
0
Suppose the control operator G is defined as in (3). Then we have 2π 2π 2π (Gu)(x, s)dx = g(x)(u(x, t) − g(s)u(s, t)ds)dx 0
0
0
2π
2π
g(x)u(x, t)dx −
= 0
0
2π
g(x)u(x, t)dx(1 −
=
g(s)(
0
2π
g(x)u(s, t)dx)ds
0 2π
g(s)ds) = 0. 0
Now let w(x, t) be the solution of (1)-(2) then 2π ∂w d 2π (x, t)dx w(x, s)dx = dt 0 ∂t 0 2π ∂3w [− 3 (x, t) + (Gu)(x, t) + f (t, w(x, t))]dx = ∂x 0 2π 2π 2π ∂3w − 3 (x, t)dx + (Gu)(x, t)dx + f (t, w(x, t))dx = ∂x 0 0 0 2π 2π 2π ∂2w d[ 2 (x, t)] + (Gu)(x, t)dx + f (t, w(x, t))dx =− ∂x 0 0 0 2π ∂2w ∂2w (0, t) + f (t, w(x, t))dx = − 2 (2π, t) + ∂x ∂x2 0 2π = f (t, w(x, t))dx. 0
From this, we can see that the volume is conserved with periodic boundary 2π condition when 0 f (t, w(x, t))dx = 0. Let 2π V = {x ∈ X : [x] = x(s)ds = 0} 0
Then V is a Hilbert space with respect to L2 -norm (see [1]). Clearly Gu ∈ V . Definition 1. The system (1)-(2) is said to be exactly controllable over a time interval [0, τ ], if for any given wτ ∈ X with [wτ ] = 0, there exists a control u ∈ Z such that the mild solution w(x, t) of (1)-(2) with control u satisfies w(x, τ ) = wτ .
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Define the solution mapping W : Z → Z by (W u)(t) = w(x, t) where w(x, t) is the unique mild solution of the system (1)-(2) corresponding to the control u. Let C = C([0, τ ] : X) denote the Banach space of continuous functions on J = [0, τ ] with the standard norm wC = sup{w(t)X : 0 ≤ t ≤ τ } for w ∈ C. Definition 2. [3] Suppose Γ : J ×X → BL(C) is a bounded continuous operator and there exists a positive γ such that for any w, y ∈ C we have t sup {f (t, w(t)) + y(t) + T (t − s)(Γ (s, w(s))y)(s)ds) (8) 0≤t≤τ
0
−f (t, w(t)) − (Γ (t, w(t))y)(t)X } ≤ γyC Then we say that f has a bounded integral contractor {I + simply Γ ) with respect to the Co-semigroup T .
T Γ } (we may say
Definition 3. [3] A bounded integral contractor Γ is said to be regular if the integral equation t x(t) = y(t) + T (t − s)(Γ (s, w(s))y)(s)ds (9) 0
has a solution y in C for every w, x ∈ C. We assume Γ (t, w(t)) ≤ β1 for all t ∈ J and w ∈ C. Let the nonlinear function f is decomposed into three nonlinear functions written as f (t, w(x, t) = f1 (t, w(x, t)) + f2 (t, w(x, t)) + f3 (t, w(x, t)) First, let us consider the dispersion equation with the nonlinear function f3 term which is given by ∂3w ∂w (x, t) + (x, t) = (Gu)(x, t) + f3 (t, w(x, t)), t > 0 ∂t ∂x3 w(x, 0) = 0,
(10)
where x ∈ [0, 2π] and periodic boundary condition ∂kw ∂kw (0, t) = (2π, t), k = 0, 1, 2. k ∂x ∂xk
(11)
From [3] one can obtain the exact controllability of (10)(f = f3 ) by assuming that f has an integral contractor. We define the operator S : Z → V as t Sw = T (t − s)w(x, s)ds (12) 0
Define the Nemytskii operator Fi : Z → Z, i = 1, 2 by (Fi w)(x, t) = fi (t, w(x, t)), i = 1, 2 Notation: The range of F is denoted by R(F ). Now we consider the following conditions
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There exists a constant M such that T (t) ≤ M, t ∈ [0, τ ] R(F1 ) ⊆ N (S) R(F2 ) ⊆ R(G) f3 has a regular bounded integral contractor Γ with sufficiently small constant γ 5. f3 satisfies the monotone condition, that is there exists a constant β > 0 such that for all x, y ∈
1. 2. 3. 4.
f3 (t, x) − f3 (t, y), x − y ≤ −β|x − y|2 6. f3 satisfies the growth condition, that is there exists a constant a ≥ 0 and b > 0 such that for all x ∈
|f3 (t, x)| ≤ a|x| + b. Remark 1. The existence and uniqueness of the solution of the system (10)−(11) have been shown by Theorem 5.6 of [3] if the conditions (1) and (4) are satisfied. Moreover, if conditions (5) and (6) are fulfilled, then Lemma 2.2 of [3] shows that the solution mapping W is well defined and the uniqueness of the solution follows. 2π Remark 2. Clearly 0 f2 (t, w(x, t))dx = 0.
3
Main Results
In this section we assume the existence of the mild solution since it has been done by [3] and other researchers in detail. Theorem 1. Suppose the nonlinear terms f1 , f2 and f3 satisfy the conditions (2), (3) and (4) respectively, then the semilinear dispersion equation (1) − (2) is exactly controllable if the corresponding linear dispersion equation is exactly controllable. Proof. Let wτ be any given final state. From Theorem 6.1 of [3] the dispersion equation (10)-(11) is exactly controllable in V . That means there is a control u ∈ Z such that the mild solution w(x, t) given by
t
w(x, t) =
t
T (t − s)(Gu)(s)ds +
0
T (t − s)f3 (s, w(x, s))ds
(13)
0
satisfies w(x, τ ) = wτ . Using condition 2, (13) can be written as
t
w(x, t) = 0
+
0
T (t − s)(Gu)(s)ds +
t
T (t − s)f1 (s, w(x, s))ds
0 t
T (t − s)f3 (s, w(x, s))ds.
(14)
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By condition (3) there exists a control v in Z such that f2 (t, w(x, t)) = (F2 w)(x, t) = (Gv)(x, t). Then using a simple technique the mild solution w(x, t) in (14) can be written as t t T (t − s)(Gu − Gv)(s)ds + T (t − s)f1 (s, w(x, s))ds w(x, t) = 0 0 t t T (t − s)f2 (s, w(x, s))ds + T (t − s)f3 (s, w(x, s))ds + 0 0 t t T (t − s)G(u − v)(s)ds + T (t − s)f (s, w(x, s))ds, = 0
0
where f (t, w(x, t)) = f1 (t, w(x, t))+f2 (t, w(x, t))+f3 (t, w(x, t)) Hence the system (1)-(2) is exactly controllable with control v = u − v. Theorem 2. Suppose the nonlinear terms f1 , f2 and f3 satisfy the conditions (2), (3) and (5) respectively. Moreover, f3 satisfies condition (6) with sufficiently small growth constant a. Then the nonlinear system (1) − (2) is exactly controllable. Proof. From theorem 4.3 of [3] the system (10) − (11) is exactly controllable in V . By similar approach to the proof of theorem 3.1 we can conclude that the nonlinear dispersion system is exactly controllable. Remark 3. If f1 = 0 = f2 , then theorem 6.1 of [3] is a particular case of theorem 3.1. Similarly theorem 4.3 of [3] is exactly the same as theorem 3.2 when f1 = 0 = f2 .
References 1. Chalishajar, D.N.: Controllability of nonlinear integro-differential third order dispersion system. J. Math. Anal. Appl. 348, 480–486 (2008) 2. David, L., Zhang, B.Y.: Exact controllability and stability of the kortwegede vries equation. Amer. Math. Soc. 348, 3643–3672 (1996) 3. George, R.K., Chalishajar, D.N., Nandakumaran, A.K.: Exact controllability of the nonlinear third-order dispersion equation. J. Math. Anal. Appl. 332, 1028–1044 (2007) 4. Tomar, N.K.: A Note on controllability of semilinear system, Ph D Thesis (2008) 5. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Springer, Heidelberg (1983) 6. Russel, D.L., Zhang, B.Y.: Controllability and stabilizability of the third order linear dispersion equation on a periodic domain. SIAM J. Control Optim. 31(3), 659–672 (1993) 7. Sakthivel, R., Mohmudov, N.I., Ren, Y.: Approximate controllability of nonlinear third order dispersion equation. Appl. Math. Comput. 217, 8507–8511 (2011)
