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.
MSC: 35F25, 93C20.
Keywords:nonlinear evolution equation; monotone operator; approximate controllability; regularity; hemicontinuous
In this paper, we deal with the approximate controllability for the semilinear equation in a Hilbert space H as follows:
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.
First, we consider the following initial value problem of a semilinear equation:
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 .
For , we denote by the value of l at . The norm of l as element of is given by
Therefore, we assume that V has a stronger topology than H and, for brevity, we may regard that
Let be a bounded sesquilinear form defined in and satisfying Gårding’s inequality
where and is a real number. Let A be an operator associated with this sesquilinear form:
Then −A is a bounded linear operator from V to by the Lax-Milgram theorem. The realization of A in H, which is the restriction of A to
is also denoted by A. It is well known that A is positive definite and self-adjoint and generates an analytic semigroup in both H and . From the following inequalities:
is the graph norm of , it follows that there exists a constant such that
Thus we have the following sequence:
where each space is dense in the next one, which is continuous injection.
Lemma 2.1With notations (2.3), (2.4), we have
If X is a Banach space, is the collection of all strongly measurable square integrable functions from into X, and is the set of all absolutely continuous functions on such that their derivative belong to . will denote the set of all continuous functions from into X with the supremum norm. If X and Y are two Banach spaces, is the collection of all bounded linear operators from X into Y, and is simply written as . Here, we note that by using interpolation theory, we have
First, we consider the following linear system:
Lemma 2.2 (1) For (see Lemma 2.1) and , , there exists a unique solutionxof (2.6) belonging to
where is a constant depending onT.
(2) Let and , . Then there exists a unique solutionxof (2.6) belonging to
where is a constant depending onT.
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 .
Definition 2.4 Let L be a generally multi-valued mapping from a Hilbert space X into itself. If
for all , and , then L is called a monotone operator. Sometimes L is also called a monotone operator if
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 .
Let X be a Banach space and its conjugate. For any , we set
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.
Lemma 2.3We have briefly explained the theory of monotone operators (see [, Section 6.6]).
(1) In Definition 2.3 above, is seen by taking .
(2) Hemicontinuous monotone mappings from a Banach spaceXinto are pseudo-monotone.
(3) LetXbe a reflexive Banach space, and let bothXand be strictly convex. Further, let be monotone andΛbe a duality mapping fromXinto . If , thenMis maximal monotone.
(4) LetXbe a reflexive Banach space andLbe a closed monotone linear operator fromXand . If the dual operator is monotone, thenLis maximal monotone.
(5) LetXbe a reflexive Banach space, be maximal monotone andLbe 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)
Let , and , where and . Then, for every , one has
Lemma 2.5LetHbe a Hilbert space and V be a reflexive Banach space. Suppose thatVis a dense subspace ofHand thatVhas a stronger topology thanH. Therefore, . Let and with . Then the operatorLdefined by
is maximal monotone linear.
Proof Since with , noting that and by Young’s inequality, we have
so that x belongs to . Therefore, we find that is well defined as an element of H. It is easily shown that L is a closed linear operator from X into and is dense in X. Further, since
we know that L is monotone. It is also easily seen that the adjoint operator of L is given by
Hence, is also monotone. Therefore, from (4) of Lemma 2.3, it is concluded that L is maximal monotone linear. □
3 Nonlinear equations
We consider the following initial value problem of a semilinear equation:
Lemma 3.1Letfbe a demicontinuous bounded mapping from intoH. Assume that is monotone for each :
Assume further that −Ais a generator of a contraction semigroup ( ). Then, for any , there exists a solution of the integral equation
corresponding to (3.1), and it is unique. Let and be the solutions with initial values and , respectively. Then the estimate
holds on . Hence, the mapping which carries the initial value to the solutionxis a continuous mapping fromHinto .
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 .
Theorem 3.1Let Assumption (F) be satisfied, and let the assumptions on the principal operatorAstated in Section 2 be satisfied. Assume that is an arbitrary element ofHand . Then there exists a solution , satisfying , of
and it is unique. Moreover, there exists a constant such that
where is a constant depending onTand the mapping
is Lipschitz continuous.
Proof As far as the existence of the solution is concerned, we may put . By Lemma 2.5, the operator defined by
is a maximal monotone linear operator from into . Let us write and for each . Then both A and F are monotone operators from into and . Since we assumed , equation (3.2) is equivalent to
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 .
Let h be an arbitrary element of , and in . Since Λ is positive definite and self-adjoint in both H and , the domain coincides with V and H, respectively. Hence, is a contraction operator in both H and V, and it converges strongly to I as . It is also easy to see that holds for and . Let us define . Then the mapping satisfies the assumption for f in Lemma 3.1. Hence, Lemma 3.1 can be applied to the initial value problem
Let us denote the semigroup generated by Λ by . Then it ensures the existence of a solution of the equation
Multiplying by on (3.3) and integrating over , we have
By using Young’s inequality and the monotonicity of f, the following holds:
So, we obtain
where is a constant, so that is bounded in . Therefore, is bounded in . By replacing them by their subsequence, we may assume in . By letting in (3.4), we have
from multiplying by and the monotonicity of , it follows that
Thus, noting that and converge strongly to h and in , respectively, we see that x is a solution of the equation . Finally, to prove the uniqueness of the solution, suppose that and are solutions with initial conditions and and forcing terms and , respectively. Then it is easy to see that there exists such that
This completes the proof of Theorem 3.1. □
Remark 3.1 In a similar way to Theorem 3.1, we also obtain the existence of solutions of (3.2) in the case where f is a demicontinuous bounded mapping from into H. Moreover, assume that for each t is monotone as a mapping from into H and , , then there exists a unique solution x of (3.1) such that
Moreover, there exists a constant such that
where is a constant depending on T and the mapping
is Lipschitz continuous.
4 Approximate controllability
In this section, we deal with the approximate controllability for the semilinear equation in H as follows.
In (4.1), the principal operator −A generates an analytic semigroup as stated in Section 2. Let U be a Hilbert space of control variables, and let B be a bounded linear operator from U to H, which is called a controller. The mild solution of initial value problem (4.1) is the following form:
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 .
For each , let us define . Then from Theorem 3.1 it follows that solution (4.1) exists and is unique in . Let be a state value of system (4.1) at time T corresponding to the nonlinear term f and the control u. We define the reachable sets for system (4.1) as follows:
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.
Definition 4.2 Let L be a mapping from a Banach space X into its conjugate space . T is called coercive if there exists such that
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 .
First, we consider the approximate controllability of system (4.1) in the case where the controller B is the identity operator on H under Assumption (F1) on the nonlinear operator f. So, obviously. Consider the linear system given by
and the following semilinear control system:
Lemma 4.1Let Assumption (F1) be satisfied, and let be the solution of (4.2) corresponding to a controlu. Then there exists such that
Let . Then equation (4.4) is equivalent to
It is easy to see that G is monotone as an operator from to , and is a demicontinuous bounded mapping as an operator from into . Let the collection of all finite dimensional subspaces of H be denoted by , and when , let the orthogonal projection on Y be denoted by . For , let us define ; thus also denotes the orthogonal projection in . According to Assumption (F1), we have that the operator is a coercive monotone operator from Y into itself. In general, any demicontinuous operator is hemicontinuous. Therefore, by Remark 4.1, we have , which (4.5) implies the existence of a solution to
Hence, the solution of (4.6) is bounded on . Let w be an arbitrary element of . Then we can take satisfying (4.6) such that in . Since G is monotone as an operator from to , and is a demicontinuous bounded mapping, we have in . Hence, we obtain that for ,
as , so that
If v is replaced by in (4.7), we have
which, by the demicontinuity of G, leads to
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.
Theorem 4.1Under Assumption (F1) and , we have
Therefore, if linear system (4.2) with is approximately controllable at timeT, then so is semilinear system (4.3).
Proof Let y, x be the solutions of (4.2) and (4.3), respectively. Let in the sense of Lemma 4.1. Then, since
Acting on both sides of the above equation, by , from the monotonicity of f, it follows
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.
Theorem 4.2Let Assumption (F1) and be satisfied. Assume that the inverse mapping of the controllerBexists and is monotone. Then the linear system
is approximately controllable at timeT, so is nonlinear system (4.1).
Proof Let y be a solution of (4.8) corresponding to a control u. Consider the following semilinear system:
Equation (4.10) is equivalent to
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.
Lemma 4.2If and
then for almost all .
Theorem 4.3Let Assumption (F2) and be satisfied. Assume thatBis a hemicontinuous monotone mapping fromVinto ; moreover, if it is coercive, then linear system (4.8) is approximately controllable at timeT, so is semilinear system (4.1).
Proof Let . We define the linear operator from to H by
for . As , there exists such that
for instance, take . By expressing for all . By Remark 4.1, since , there exists such that
Since p is an arbitrary element of , and in . This implies that linear system (4.8) is approximately controllable.
To prove the approximate controllability of (4.1), we will show that , i.e., for given and , there exists such that
Let . Then we write for each . Then we rewrite (4.12) as
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.
Let Ω be a bounded region in with smooth boundary ∂Ω. We define the following spaces:
where and are derivatives of u in the distribution sense. The norm of is defined by
Hence is a Hilbert space.
The norm and inner product of are defined by
for any . We put . Define the operator A by
The operator A in is defined so that for any , there exists such that
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.
Let u be fixed if we consider the functional , this function is continuous linear. For any , it follows that . We denote that for any ,
that is, . The operator is a one-to-one mapping from to . The relation of operators A and satisfy that
From now on, both A and are denoted simply by A.
We introduce a simple example of the control operator B which satisfies the condition in Theorem 4.2. Consider the case , and define the intercept operator on by
Then B is a continuous monotone mapping such that there exists a constant such that
Let . Then by Sobolev’s imbedding theorem, we have
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.
The authors declare that they have no competing interests.
YHK drafted the manuscript and corrected the main results, JMJ carried out the main proof of this paper, and HHR participated in its design and coordination.
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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