Controllability for nonlinear evolution equations with monotone operators
© Kang et al.; licensee Springer. 2013
Received: 7 August 2013
Accepted: 22 October 2013
Published: 12 November 2013
In this paper, we investigate the approximate controllability for nonlinear evolution equations with monotone operators and nonlinear controllers according to monotone operator theory. We also give the regularity for the nonlinear equation. Finally, an example, to which our main result can be applied, is given.
In (1.1), the principal operator −A generates an analytic semigroup . Let U be a Hilbert space of control variables, and let B be a linear (or nonlinear) operator from U to H, which is called a controller.
If is an unbounded operator, Di Blasio et al.  proved -regularity for a retarded linear system in Hilbert spaces, and Jeong  (also see ) considered the control problem for retarded linear systems with -valued controller and more general Lipschitz continuity of nonlinear terms.
For the theory of monotone operators, there are many literature works; for example, see Lions , Stampacchia , Browder , and the references cited therein. Kenmochi  derived new results on monotone operator equations, and Ouchi  proved the analyticity of solutions of semilinear parabolic differential equations with monotone nonlinearity. For the existence of solutions for a class of nonlinear evolution equations with monotone perturbations, one can refer to [9–11]. We refer to Pascali and Sburlan , Morosanu  to see the applications of nonlinear mapping of monotone type and nonlinear evolution equations. The classical solutions of (1.2) were obtained by Kato  under the monotonicity condition on the nonlinear term f as an operator from to H.
In the first part of this note, we apply results of  to find -regularity of solutions in the wider sense of (1.2) under the more general monotonicity of a nonlinear operator f from to , which is related to the results of Tanabe [, Theorem 6.6.2].
Next, we extend and develop control problems on this topic. In recent years, as for the controllability for semilinear differential equations with Lipschitz continuity of a nonlinear operator f, Naito  and [17–19] proved the approximate controllability under the range conditions of the controller B. However, we can find few articles which extend the known general controllability problems to nonlinear evolution equation (1.1) with monotone operators and nonlinear controllers.
In this paper, based on the regularity for solutions of equation (1.2), we obtain the approximate controllability for nonlinear evolution equation (1.1) with monotone operators and nonlinear controllers.
The paper is organized as follows. In Section 2, we explain several notations of this paper and state results about -regularity for linear equations in the sense of [1, 15, 20]. In Section 3, we give the regularity for nonlinear equation (1.2). In Section 4, we obtain the approximate controllability for nonlinear evolution equation (1.1) with hemicontinuous monotone operators by using the theory of monotone operators. In the end, an example is provided to illustrate the application of the obtained results.
If H is identified with its dual space, we may write densely and the corresponding injections are continuous. The norm on V, H and will be denoted by , and , respectively. The duality pairing between the element of and the element of V is denoted by , which is the ordinary inner product in H if .
where each space is dense in the next one, which is continuous injection.
where is a constant depending on T.
where is a constant depending on T.
Throughout this paper, strong convergence is denoted by ‘→’ and weak convergence by ‘⇀’.
Definition 2.1 Let X and Y be Banach spaces and L be a mapping from X into Y. The domain of L is assumed to be convex. L is called hemicontinuous if for any is continuous in in weak topology of Y.
The linear operator is obviously hemicontinuous.
Definition 2.2 Let X and Y be Banach spaces and L be a single-valued mapping from X into Y. L is called demicontinuous if and imply that .
Definition 2.3 Let L be a mapping from a Banach space X into its conjugate space . L is said to be pseudo-monotone if the following condition is satisfied. If is a directed family of points, contained in , which converges weakly to an element x of and if , then for all .
holds instead of (2.9).
Definition 2.5 A real-valued continuous function j is called a gauge function defined on if it is strictly monotone increasing and satisfies and .
The multi-valued operator is called the duality mapping of X with a gauge function j.
Let us denote by Λ the operator determined by an inner product on . Then it is immediate that Λ is a duality mapping from V into with a gauge function . It is also known that the duality mapping is monotone and hemicontinuous, and hence it is pseudo-monotone.
In Definition 2.3 above, is seen by taking .
Hemicontinuous monotone mappings from a Banach space X into are pseudo-monotone.
Let X be a reflexive Banach space, and let both X and be strictly convex. Further, let be monotone and Λ be a duality mapping from X into . If , then M is maximal monotone.
Let X be a reflexive Banach space and L be a closed monotone linear operator from X and . If the dual operator is monotone, then L is maximal monotone.
- (5)Let X be a reflexive Banach space, be maximal monotone and L be a pseudo-monotone bounded mapping from into . If there exists such that
then , that is, for every , has a solution .
The following inequality is referred to as Young’s inequality.
Lemma 2.4 (Young’s inequality)
is maximal monotone linear.
Hence, is also monotone. Therefore, from (4) of Lemma 2.3, it is concluded that L is maximal monotone linear. □
3 Nonlinear equations
holds on . Hence, the mapping which carries the initial value to the solution x is a continuous mapping from H into .
Next, we apply Lemma 3.1 to find a solution in the wider sense of (3.1) under somewhat different assumptions. Concerning the nonlinear mapping f, assume the following hypothesis.
Assumption (F) The mapping f is demicontinuous bounded from into , and for each t is monotone as a mapping from V into .
The following theorem is a part of Theorem 6.6.2 due to Tanabe .
is Lipschitz continuous.
Note that is monotone, if it is shown to be maximal monotone, assumption (5) of Lemma 2.3 is satisfied with , and . Then we have , which implies the existence of the solution. Thus, from now on, we prove the maximal monotonicity of . Since Λ determined by an inner product on V is a duality mapping from V into , as seen under Definition 2.5, to see the maximal monotonicity of , on account of (3) of Lemma 2.3, it is enough to verify that .
This completes the proof of Theorem 3.1. □
is Lipschitz continuous.
4 Approximate controllability
Let f be a nonlinear mapping satisfying the following.
Assumption (F1) The mapping f is demicontinuous bounded from into . Assume that for each is monotone as a mapping from V into with , and for each is monotone as a mapping from U into .
Definition 4.1 System (4.1) is said to be approximately controllable at time T if for every desired final state and , there exists a control function such that the solution of (4.1) satisfies , that is, , where is the closure of in H.
Remark 4.1 [, Theorem 1.3]
It is well known that if X is a reflexive Banach space and L is monotone, everywhere defined and hemicontinuous from into , then L is maximal monotone. If in addition L is coercive monotone, then .
in the limit as . Since v is arbitrary, we obtain . □
Remark 4.2 As seen in , we know that if X is a Hilbert space and is maximal monotone, then . So, (4.5) is easily obtained if the operator G in Theorem 4.1 is maximal monotone.
Therefore, if linear system (4.2) with is approximately controllable at time T, then so is semilinear system (4.3).
By using Gronwall’s inequality, we get in . Noting that , every solution of the linear system with control u is also a solution of the semilinear system with control v, that is, we have that . □
From now on, we consider the initial value problem for semilinear parabolic equation (4.1). Let U be a Hilbert space, and let the controller operator B be a nonlinear operator from U to H.
is approximately controllable at time T, so is nonlinear system (4.1).
Here, similarly to the proof of Lemma 4.1, we have that there exists an element satisfying (4.10), that is, . In a similar way to the proof of Theorem 4.1, we get . Since system (4.1) is equivalent to (4.9), we conclude that . □
Now we consider the control problem of (4.1) when the controller B is a nonlinear mapping in the case where . In this case, we suppose that Assumption (F) and the next additional assumption are satisfied.
Assumption (F2) Assume that for each is maximal monotone as a mapping from U into .
The following result is well known from semigroup properties.
then for almost all .
Theorem 4.3 Let Assumption (F2) and be satisfied. Assume that B is a hemicontinuous monotone mapping from V into ; moreover, if it is coercive, then linear system (4.8) is approximately controllable at time T, so is semilinear system (4.1).
Since p is an arbitrary element of , and in . This implies that linear system (4.8) is approximately controllable.
Thus, in view of Lemma 4.2, it is enough to verify that there exists an arbitrary element u of such that . By (2) of Lemma 2.3, B is pseudo-monotone and satisfies the condition (5) of Lemma 2.3. Thus, we have . Since p is an arbitrary element of , and in . This implies inequality (4.3) and completes the proof of the theorem. □
Remark 4.3 We know that by Assumption (F1) and (4.8), is monotone, hemicontinuous and coercive from U into . Therefore, as seen in Remark 4.1, we have , that is, system (4.1) is approximately controllable.
Then, for any , and A is a positive definite self-adjoint operator.
Let be a dual space of . For any and , the notation denotes the value l at v.
From now on, both A and are denoted simply by A.
where and . Assume that is a continuous and increasing function defined on such that as . If we put for each , then , so that it is clear that is monotone as an operator into . To show that is a demicontinuous mapping from H into , let in H. Since is bounded in H, so is in . Hence, there exists a subsequence such that almost everywhere in Ω and there exists an element such that . Since is a continuous function of real variables λ, almost everywhere in Ω. Otherwise, one can find an appropriate convex combination , where , which is strongly convergent to g in . This says that for all y for which . Therefore, we obtain , that is, . Thus, all the conditions stated in Theorem 4.2 are satisfied. Therefore, nonlinear system (4.1) with monotone operators is approximately controllable at time T.
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0007560).
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