Abstract
In this article, we deal with the existence, uniqueness, and a variation of solutions of the nonlinear control system with nonlinear monotone hemicontinuous and coercive operator. Moreover, the approximate controllability for the given nonlinear control system is studied.
Keywords:
nonlinear differential equation; regularity; reachable set; degree theory; approximately controllable1 Introduction
Let H and V be two real separable Hilbert spaces such that V is a dense subspace of H. We are interested in the following nonlinear differential control system on H:
where the nonlinear term, which is a Lipschitz continuous operator, is a semilinear version of the quasilinear form. The principal operator A is assumed to be a single valued, monotone operator, which is hemicontinuous and coercive from V to V*. Here, V* stands for the dual space of V . Let U be a Banach space of control variables. The controller B is a linearbounded operator from a Banach space L^{2}(0, T; U) to L^{2}(0, T; H) for any T > 0. Let the nonlinear mapping k be Lipschitz continuous from ℝ × [ h, 0] × V into H. If the righthand side of the equation (SE) belongs to L^{2}(0, T; V* ), then it is well known as the quasiautonomous differential equation(see Theorem 2.6 of Chapter III in [1]).
The problem of existence for solutions of semilinear evolution equations in Banach spaces has been established by several authors [13]. We refer to [2,4,5] to see the existence of solutions for a class of nonlinear evolution equations with monotone perturbations
First, we begin with the existence, and a variational constant formula for solutions of the equation (SE) on L^{2}(0, T; V ) ∩ W^{1,2}(0, T; V* ), which is also applicable to optimal control problem. We prove the existence and uniqueness for solution of the equation by converting the problem into a fixed point problem. Thereafter, based on the regularity results for solutions of (SE), we intend to establish the approximate controllability for (SE). The controllability results for linear control systems have been proved by many authors, and several authors have extended these concepts to infinite dimensional semilinear system (see [68]). In recent years, as for the controllability for semilinear differential equations, Carrasco and Lebia [9] discussed sufficient conditions for approximate controllability of parabolic equations with delay, and Naito [10] and other authors [68,11] proved the approximate controllability under the range conditions of the controller B.
The previous results on the approximate controllability of a semilinear control system have been proved as a particular case of sufficient conditions for the approximate solvability of semilinear equations, assuming either that the semigroup generated by A is a compact operator or that the corresponding linear system (SE) when g ≡ 0 is approximately controllable. However, Triggian [12] proved that the abstract linear system is never exactly controllable in an infinite dimensional space when the semigroup generated by A is compact. Thus, we will establish the approximate controllability under more general conditions on the nonlinear term and the controller.
Our aim in this article is to establish the approximate controllability for (SE) under a stronger assumption that {y : y(t) = (Bu)(t), u ∈ L^{2}(0, T; U)} is dense subspace of L^{2}(0, T, H), which is reasonable and widely used in case of the nonlinear system. We show that the input to solution (control to state) map is compact by using the fact that L^{2}(0, T; V ) ∩ W^{1,2}(0, T; V* ) furnished with the usual topology is compactly embedded in L^{2}(0, T, H), provided that the injection V ⊂ H is compact.
In the last section, we give a simple example to which the range conditions of the controller can be applied.
2 Nonlinear functional equations
Let H and V be two real Hilbert spaces. Assume and V is dense subspace in H and the injection of V into H is continuous. If H is identified with its dual space, then we may write V ⊂ H ⊂ V* densely, and the corresponding injections are continuous. The norm on V (resp. H) will be denoted by  ·  (resp. · ). 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  · _{* }is the norm of the element of V*. If an operator A is bounded linear from V to V* and generates an analytic semigroup, then it is easily seen that
for the time T > 0. Therefore, in terms of the intermediate theory we can see that
where denotes the real interpolation space between V and V*.
We note that a nonlinear operator A is said to be hemicontinuous on V if
for every x, y ∈ V where "w  lim" indicates the weak convergence on V*. Let A : V → V* be given a singlevalued, monotone operator and hemicontinuous from V to V* such that
(A1) A(0) = 0, (Au  Av, u  v) ≥ ω_{1 }u  v^{2 } ω_{2 }u  v^{2} ,
(A2) Au_{* }≤ ω_{3}(u + 1)
for every u, v ∈ V where ω_{2 }∈ ℝ and ω_{1}, ω_{3 }are some positive constants.
Here, we note that if 0 ≠ A(0), then we need the following assumption:
for every u ∈ V . It is also known that A is maximal monotone, and R(A) = V* where R(A) denotes the range of A.
Let the controller B is a bounded linear operator from a Banach space L^{2}(0, T; U) to L^{2}(0, T; H) where U is a Banach space.
For each t ∈ [0, T], we define x_{t }: [ h, 0] → H as
We will set
Let ℒ and ℬ be the Lebesgue σfield on [0, ∞) and the Borel σfield on [ h, 0] respectively. Let k : ℝ^{+ }× ℝ^{+ }× ∏ → H be a nonlinear mapping satisfying the following:
(K1) For any x. ∈ ∏ the mapping k(·, ·, x.) is strongly ℒ × ℬ measurable;
(K2) There exist positive constants K_{0} , and K_{1 }such that
for all (t, s) ∈ ℝ^{+ }× [ h, 0] and x., y. ∈ ∏.
Let g : ℝ^{+ }× ∏ × H → H be a nonlinear mapping satisfying the following:
(G1) For any x ∈ ∏, y ∈ H the mapping g(·, x., y) is strongly ℒ measurable;
(G2) There exist positive constants L_{0}, L_{1}, and L_{2 }such that
for all t ∈ ℝ^{+}, x, , and y, .
Remark 2.1. The above operator g is the semilinear case of the nonlinear part of quasilinear equations considered by Yong and Pan [13].
For x ∈ L^{2}(h, T; V ), T > 0 we set
Here, as in [13], we consider the Borel measurable corrections of x(·).
Lemma 2.1. Let x ∈ L^{2}( h, T; V ). Then, the mapping t ↦ x_{t }belongs to C([0, T ]; ∏) and
Proof. It is easy to verify the first paragraph and (2.1) is a consequence of the estimate:
Lemma 2.2. Let x ∈ L^{2}( h, T; V ), T > 0. Then, G(·, x) ∈ L^{2}(0, T; H) and
Moreover, if x_{1}, x_{2 }∈ L^{2}( h, T; V ), then
Proof. It follows from (K2) and (2.1) that
and hence, from (G2), (2.1), and the above inequality, it is easily seen that
Similarly, we can prove (2.3).
Let us consider the quasiautonomous differential equation
where A satisfies the hypotheses mentioned above. The following result is from Theorem 2.6 of Chapter III in [1].
Proposition 2.1. Let Φ ^{0 }∈ H and f ∈ L^{2}(0, T; V* ). Then, there exists a unique solution x of (E) belonging to
and satisfying
where C_{1 }is a constant.
Acting on both sides of (E) by x(t), we have
As is seen Theorem 2.6 in [1], integrating from 0 to t, we can determine the constant C_{1 }in Proposition 2.1.
We establish the following result on the solvability of the equation (SE).
Theorem 2.1. Let A and the nonlinear mapping g be given satisfying the assumptions mentioned above. Then, for any (Φ^{0}, Φ^{1}) ∈ H × L^{2}( h, 0; V ) and f ∈ L^{2}(0, T; V*), T > 0, the following nonlinear equation
has a unique solution x belonging to
and satisfying that there exists a constant C_{2 }such that
Proof. Let y ∈ L^{2}(0, T; V ). Then, we extend it to the interval (h, 0) by setting y(s) = Φ^{1}(s) for s ∈ (h, 0), and hence, G(·, y(·)) ∈ L^{2}(0, T; H) from Lemma 2.2. Thus, by virtue of Proposition 2.1, we know that the problem
has a unique solution x_{y }∈ L^{2}(0, T; V ) ∩ W^{1,2}(0, T; V* ) corresponding to y. Let us fix T_{0 }> 0 so that
Let x_{i}, i = 1, 2, be the solution of (2.8) corresponding to y_{i}. Multiplying by x_{1}(t)  x_{2}(t), we have that
and hence it follows that
From Lemma 2.2 and integrating over [0,t], it follows
where c is a positive constant satisfying 2ω_{1 } c > 0. Here, we used that
for any pair of nonnegative numbers a and b. Thus, from (2.3) it follows that
By using Gronwall's inequality, we get
Taking c = ω_{1}, it holds that
Hence, we have proved that y ↦ x is a strictly contraction from L^{2}(0, T_{0}; V ) to itself if the condition (2.9) is satisfied. It shows that the equation (2.6) has a unique solution in [0, T_{0}].
From now on, we derive the norm estimates of solution of the equation (2.6). Let y be the solution of
Then,
by multiplying by x(t)  y(t) and using the assumption (A1), we obtain
By integrating over [0, t] and using Gronwall's inequality, we have
and hence, putting
it holds
for some positive constant C_{2}. Since the condition (2.9) is independent of initial values, the solution of (2.6) can be extended to the internal [0, nT_{0}] for natural number n, i.e., for the initial value (x(nT_{0}), x_{n}T_{0 }) in the interval [nT_{0}, (n + 1)T_{0}], as analogous estimate (2.11) holds for the solution in [0, (n + 1)T_{0}].
Theorem 2.2. If (Φ^{0}, Φ^{1}) ∈ H × L^{2}(h, 0, V )) and f ∈ L^{2}(0, T; V* ), then x ∈ L^{2}(h, T; V ))∩ W^{1,2}(0, T; V* ), and the mapping
is continuous.
Proof. It is easy to show that if (Φ ^{0}, Φ^{1}) ∈ H × L^{2}(h, 0; V )) and f ∈ L^{2}(0, T; V* ) for every T > 0, then x belongs to L^{2}( h, T; V )∩W^{1,2}(0, T; V*). Let
and x_{i }be the solution of (2.6) with in place of for i = 1,2.
Then, in view of Proposition 2.1 and Lemma 2.2, we have
If ω_{1 } c/2 > 0, we can choose a constant c_{1 }> 0 so that
and
Let T_{1 }< T be such that
Integrating on (2.12) over [0, T_{1}] and as is seen in the first part of proof, it follows
Putting that
we have
Suppose that
and let x_{n }and x be the solution (2.6) with and respectively.
By virtue of (2.13) with T being replaced by T_{1}, we see that
This implies that in H × L^{2 }(h, 0; V). Hence, the same argument shows that
Repeating this process, we conclude that
Remark 2.2. For x ∈ L^{2}(0, T; V ), we set
where k belongs to L^{2}(0, T) and g : [0, T] × V → H be a nonlinear mapping satisfying
for a positive constant L. Let x ∈ L^{2}(0, T; V ), T > 0. Then, G(·, x) ∈ L^{2}(0, T; H) and
Moreover, if x_{1}, x_{2 }∈ L^{2}(0, T; V ), then
Then, with the condition that
in place of the condition (2.9), we can obtain the results of Theorem 2.1.
3 Approximate controllability
In what follows we assume that the embedding V ⊂ H is compact, and A is a continuous operator from V to V* satisfying (A1) and (A2). For h ∈ L^{2}(0, T; H) and let x_{h }be the solution of the following equation with B = I:
where
We define the solution mapping S from L^{2}(0, T; V*) to L^{2}(0, T; V ) by
Let and be the Nemitsky operators corresponding to the maps A and G, which are defined by and , respectively. Then, since the solution x belongs to L^{2}(h, T; V ) ∩ W^{1,2}(0, T; V*) ⊂ C([0, T]; H), it is represented by
and with aid of Lemma 2.2 and Proposition 2.1
Hence, if h is bounded in L^{2 }(0, T; V*), then so is x_{h }in L^{2}(0, T; V)∩W^{1,2}(0, T; V*). Since V is compactly embedded in H by assumption, the embedding L^{2}(0, T; V) ∩ W^{1,2 }(0, T; V*) ⊂ L^{2 }(0, T; H) is compact in view of Theorem 2 of Aubin [14]. Hence, the mapping h ↦ Sh = x_{h }is compact from L^{2}(0, T; V*) to L^{2}(0, T; H). Therefore, is a compact mapping from L^{2}(0, T; V*) to L^{2}(0, T; H) and so is from L^{2}(0, T; V*) to itself. The solution of (SE) is denoted by x(T; g, u) associated with the nonlinear term g and control u at time T.
Definition 3.1. The system (SE) is said to be approximately controllable at time T if Cl{x(T; g, u): u ∈ L^{2}(0, T; U)} = V* where Cl denotes the closure in V*.
We assume
(B) Cl{y : y(t) = (Bu)(t), a.e. u ∈ L^{2}(0, T; U)} = L^{2}(0, T; U)}. Here Cl is the closure in L^{2}(0, T; H).
Theorem 3.1. Let the assumptions (T) and (B) be satisfied. Then,
Therefore, the following nonlinear differential control system
is approximately controllable at time T.
Proof. Let z ∈ L^{2}(0, T; V*) and r be a constant such that
Take a constant d > 0 such that
where
Taking scalar product on both sides of (3.1) with G = 0 by x(t)
where c is a positive constant satisfying 2ω_{1 } c > 0. Integrating on [0, t], we get
and hence,
By using Gronwall's inequality, it follows that
that is,
Let us consider the equation
Let w be the solution of (3.9). Then z ∈ U_{d }and taking c = ω_{1}, from (3.7), (3.8)
and hence
it follows that w ∉ ∂U_{d }where ∂U_{d }stands for the boundary of U_{d}. Thus, the homotopy property of topological degree theory there exists w ∈ L^{2}(0, T; V*) such that the equation (3.9) holds. Based on the assumption (B), there exists a sequence {u_{n}} ∈ L^{2}(0, T; U) such that Bu_{n }↦ w in L^{2}(0, T; V*). Then, by the last paragraph of Theorem 2.1, we have that x(·; g, u_{n}) ↦ x_{w }in L^{2}(0, T; V ) ∩ W^{1,2}(0, T; V*) ⊂ C([0, T]; H). Hence, we have proved (3.5). Let y ∈ V*. Then, there exists an element u ∈ L^{2}(0, T; U) such that
Thus
Therefore, the system (3.6) is approximately controllable at time T.
In order to investigate the controllability of the nonlinear control system, we need to impose the following condition.
(F) g is uniformly bounded: there exists a constant M_{g }such that
for all x, y ∈ V.
By (F) it holds that
and for every h ∈ L^{2}(0, T; V*)
Theorem 3.2. Let the assumptions (T), (B), and (F) be satisfied. Then, we have
Thus, the system (SE) is approximately controllable at time T.
Proof. Let U_{r }be the ball with radius r in L^{2}(0, T; V*) and z ∈ U_{r}. To prove (3.11), we will also use the degree theory for the equation
in open ball U_{d }where the constant d satisfies
where the constant N_{2 }is in Theorem 3.1. If w is the solution of (3.12), then z ∈ U_{d }and from Lemma 2.1
and hence
it follows that w ∉ ∂U_{d}. Hence, there exists w ∈ L^{2}(0, T; V*) such that the equation (3.12) holds. Using the similar way to the last part of Theorem 3.1 and the assumption (B), there exists a sequence {un} ∈ L^{2}(0, T; U) such that Bu_{n }↦ w in L^{2}(0, T; V*) and x(·, g, u_{n}) ↦x_{w }in L^{2}(0, T; V ) ∩ W^{1,2}(0, T; V*) ⊂ C([0, T]; H). Thus, we have proved (3.11), and the system (1.1) is approximately controllable at time T.
4 Example
Let A be an operator associated with a bounded sesquilinear form a(u, v) defined in V × V and satisfying Gårding inequality
for any u ∈ V. It is known that A generates an analytic semigroup in both H and V*. By virtue of the RieszSchauder theorem, if the embedding V ⊂ H is compact, then the operator A has discrete spectrum:
which has no point of accumulation except possibly when μ = ∞. Let μ_{n }be a pole of the resolvent of A of order k_{n }and P_{n }the spectral projection associated with μ_{n}
where Γ_{n }is a small circle centered at μ_{n }such that it surrounds no point of σ(A) except μ_{n}. Then, the generalized eigenspace corresponding to μ_{n }is given by
and we have that from and H_{n }⊂ V ; it follows that
Definition 4.1. The system of the generalized eigenspaces of A is complete in H if Cl {span{H_{n }: n = 1, 2,...}} = H where Cl denotes the closure in H.
We need the following hypotheses:
(B1) The system of the generalized eigenspaces of A is complete.
(B2) There exists a constant d > 0 such that
We can see many examples which satisfy (B2)(cf. [8,11]).
Consider about the intercept controller B define d by
where
Hence, we see that u_{1}(t) ≡ 0 and u_{n}(t) ∈ Im P_{n}.
First of all, for the meaning of the condition (B) in section 3, we need to show the existence of controller satisfying Cl{Bu : u ∈ L^{2}(0, T; U)} ≠ L^{2}(0, T; H). In fact, by completion of the generalized eigenspaces of A, we may write that for ∈L^{2}(0, T; H). Let us choose f ∈L^{2}(0, T; H) satisfying
Then, since
the statement mentioned above is reasonable.
Let f ∈ L^{2}(0, T; H) and α = T/(T  T/n). Then we know
where K_{[T,T/n] }is the characteristic of [T,T/n]. Define
Thus Since the system of the generalized eigenspaces of A is complete, it holds that for every f ∈ L^{2}(0, T; H) and ∈ > 0
Thus, the intercept controller B define d by (4.1) satisfies the condition (B).
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
JMJ carried out the main proof of this manuscript, JRK drafted the manuscript and corrected the main theorems, EYJ conceived of the study, and participated in its design and coordination.
Acknowledgements
This study was supported by the Korea Research Foundation(KRF) grant funded by the Korea government (MOEHRD, Basic Research Promotion Fund) (KRF351C00102).
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