A Characterization of Semilinear Surjective Operators and Applications to Control Problems *

In this paper we characterize a broad class of semilinear surjective operators H G V Z   given by the following formula ( ) H G w Gw H w    , w V  where , V Z are Hilbert spaces, ( ) G L V Z   and H V Z   is a suitable nonlinear function. First, we give a necessary and sufficient condition for the linear operator G to be surjective. Second, we prove the following statement: If ( ) Rang G Z  and H is a Lipschitz function with a Lipschitz constant h small enough, then ( ) H Rang G Z  and for all z Z  the equation ( ) Gw H w z   admits the following solution 1 1 1 ( ) ( ( ) ) z w G GG I H G GG z           .We use these results to prove the exact controllability of the following semilinear evolution equation ( ) ( ( )) z Az Bu t F t z u t       , 0 z Z u U t      , where Z , U are Hilbert spaces, ( ) A D A Z Z    is the infinitesimal generator of strongly continuous semigroup 0 { ( )}t T t  in Z  B ( ), L U Z  the control function u belong to 2 (0 ) L U    and [0 ] F Z U Z       is a suitable function. As a particular case we consider the semilinear damped wave equation, the model of vibrating plate equation, the integrodifferential wave equation with Delay, etc.


Introduction
In this paper we characterize a broad class of semilinear surjective operators given by the following formula ( ) Where Z , V are Hilbert spaces, G V Z   is a bounded linear operator (continuous and linear) and is a suitable non linear function in general nonlinear.First, we give a necessary and sufficient condition for the linear operator G to be surjective.Second, we prove the following statement: If ( ) Rang G Z  and H is a Lipschitz function with a Lipschitz constant h small enough, then ( ) and for all z Z  the equation ( ) Gw H w z    admits the following solution We apply our results to prove the exact controllability of the following semilinear evolution equation where Z and U are Hilbert spaces, ( ) is the infinitesimal generator of strongly continuous semigroup is a suitable function.We give a necessary and sufficient condition for the exact controllability of the linear system Under some conditions on F, we prove that the controllability of the linear system (1.3) is preserved by the semilinear system (1.2).In this case the control 2 steering an initial state 0 z to a final state 1 z at time 0   (using the non linear system (1.2)) is given by the following formula: is non linear operator given by: 0 As an application we consider some control systems governed by partial differential equations, integrodifferential equations and difference equations that can be studied using these results.Particularly, we work in details the following controlled damped wave equation and the nonlinear term ( ) A physical interpretation of the nonlinear term (  ) could be as an eternal force like in the suspension bridge equation proposed by Lazer and McKenna (see [1]).
The novelty in this work lies in the following facts: First, the main results are obtained by standard and basic functional analysis such as Cauchy-Schwarz inequality, Hahn-Banach theorem, the open mapping theorem, etc.Second, the results are so general that can be apply to those control systems governed by evolutions equations like the one studied in [1][2][3] and [4].Third, we find a formula for a control steering the system from the initial state 0 z to a final state 1 z on time 0   , for both the linear and the nonlinear systems, which is very important from engineering point of view.Also, we present here a variational approach to find solutions of the semilinear equation ( ) Gw H w z   which is motivated by the one used to prove the interior controllability for some control system governed by PDE's, see [5].Finally, these results can be used to motivate the study of semilinear range dense operator in order to characterize the approximate controllability of evolution equations.

Surjective Linear Operator
In this section we shall presents a characterization of surjective bounded linear operator.To this end, we denote by ( ) L V Z  the space of linear and bounded operators mapping V to Z endow with the norm of the uniform convergence, and we will use the following lemma from [6] in Hilbert space: Then the following statements holds: In the same way as definition 4.1.3from [7] we define the following concept: Definition 1.The generalize controllability gramian of the operator G is define by: ( ) Moreover, this solution has minimum norm.i.e., Proof Suppose G is surjective.Then, from the foregoing Lemma there exists 0 This implies that W is one to one.Now, we shall prove that W is surjective.That is to say For the purpose of contradiction, let us assume that ( ) R W is strictly contained in Z .Using Cauchy Schwarz's inequality and (2.5) we get This w can be taking as follows On the other hand, Therefore, z w w  , and z w w defined by: Then, from Lemma 1 part (2) we have that For the purpose of contradiction, let us assume that Then, from Hanh Banach's Theorem there exists

Variational Method to Obtain Solutions
The Theorem 1 gave a formula for one solution of the system (2.2) which has minimum norma.But, it is not the only way allowing to build solutions of this equation.Next, we shall present a variational method to obtain solutions of (2.2) as a minimum of the quadratic functional It is easy to see that (2.9) is in fact an optimality condition for the critical points of the quadratic functional  define above.
Lemma 3. Suppose the quadratic functional  has a minimizer z Z   .Then, Proof.First, observe that  has the following form  is a point where  achieves its minimum value, we obtain that is a solution of (2.2).Remark 1.Under the condition of Theorem 1, the solution given by the formulas (2.10) and ( 2.3) coincide.
Theorem 2. The system (2.2) is solvable if, and only if, the quadratic functional  defined by (2.8) has a minimum for all z Z  .
Proof Suppose (2.2) is solvable.Then, the operator G is surjective.Hence, from Lemma 1 there exists 0 Consequently,  is coercive and the existence of a minimum is ensured.The other way of the proof follows as in proposition 1.

Surjective Semlinear Operators
In this section we shall look for conditions under which the semilinear operator given by: ( ), , is surjective.To this end, we shall use the following theorem from non linear analysis.
admits the following solution where the operator H G can be written as follows ( ) On the other hand, K is a Lipschitz function with a Lipschitz constant ( ) Then, in the same way as in the proof of Theorem 1 we get the result.
Corollary 1.Under the conditions of the above Theorems, the operator Z V    define by:

Controllability of Semilinear Evolution Equations
In this section we shall characterize the exact controllability of the semilinear evolution equation Where Z, U are Hilbert spaces, ( ) is a suitable function.

Linear Systems
First, we shall study the controllability of the linear system (1.3), and to this end, for all 0 z Z  and 2 (0 admits only one mild solution given by: 3) is is said to be exactly controllable on [0 ]    , if for all Consider the following bounded linear operator: Then, the gramian Then, the following Theorem from [7](pg.47, Theorem 4.17) is a characterization of the exact controllability of the linear system (1.3).Theorem 1.For the system (1.3) we have the following condition for exact controllability.
System (1.3) is exactly controllable on [0 ]    if, and only if, any one of the following condition hold for some 0 One can observe that the invertibility of the operator W is not proved in the foregoing theorem and, consequently, none formula for the control steering the system (4.2) from initial state 0 z to a final state 1 z on time 0   is given.Now, we are ready to formulate and prove a new result on exact controllability of the linear system (1.3).
Theorem 2. The system (1.3) is exactly controllable on [0 ]    if, and only if, the operator W is invertible.Moreover, the control 2 steering an initial state 0 z to a final state 1 z at time 0   is given by the following formula: Proof It follows directly from the above notation and applying Theorem 1.
Corollary 1.If the system (1.3) is exactly controllable, then the operator 2 (0 ) S Z L U      define by: is a right inverse of G .i.e., G S I   In this case the Equation (2.2) takes the following form 2 and the quadratic functional  given by (2.8) can be written as follows The following results follow from Proposition 1, Lemma 3 and Theorem 1 respectively.Proposition 1.For a given z Z  the Equation (4.9) has a solution 2 (0 It is easy to see that (4.11) is in fact an optimality condition for the critical points of the quadratic functional  define above.

Lemma 1. Suppose the quadratic functional
is a solution of (4.9).Theorem 3. The system (1.3) is exactly controllable if, and only if, the quadratic functional  define by (1) has a minimizer z  for all z Z  .
Moreover, under this condition we obtain that

Nonlinear System
We assume that F is good enough such that the Equation (4.1) with the initial condition admits only one mild solution given by: Definition 2. (Exact Controllability) The system (4.1) is said to be exactly controllable on [0 ]   Define the following operator: 2 (0 ) is the solution of (4.14) corresponding to the control u .Then, the following proposition is trivial and characterizes the exact controllability of (4.1).
Proposition 2. The system (4.1) is exactly controllable on [0 ]    if, and only if, ( ) So, in order to prove exact controllability of system (4.1)we have to verify the condition of the foregoing proposition.To this end, we need to assume that the linear system (1.3) is exactly controllable.In this case we know from corollary 1 that the steering operator S defined by (4.8) is a right inverse of G , so if we put  , we get the following representation: where ( ) z   is the solution of (4.14) corresponding to the control u define by: 1 the operator F G can be written as follows ( ) Now, we shall prove some abstract results making assumptions on the operator K .After that, we will put conditions on the nonlinear term F that imply condition on K .Theorem 4. If the linear system (1.3) is exactly controllable on [0 ]   and the operator K is globally Lipschitz with a Lipschitz constant 1 k   then the non linear system (4.1) is exactly controllable on [0 ]   and the control steering the initial state 0 z to the final state 1 z is given by: Proof It follows directly from Equation (3.6) and Theorem 1 or Theorem 2.
Theorem 5.If the system (1.3) is exactly controllable on [0 ]    and the operator K is linear with 0 K  , then the system (4.1) is exactly controllable on [0 ]    and the control ( ) u t steering the initial state 0 z to the final state 1 z is given by: Proof It follows directly from Equation (3).The proof of the following lemma follows as in lemma 5.1 from [1].
then the non linear system (4.1) is exactly controllable on [0 ]   and the control steering the initial state 0 z to the final state 1 z is given by: Proof From Lemma 2 we know that K is a Lipschitz function with a Lipschitz constant k given by: and from condition (4.19) we get that 1 k  .Hence, applying Theorem 4 we complete the proof.

Applications and Further Research
In this section we consider some control systems governed by partial differential equations, integrodifferential equations and difference equations that can study using these results.Particularly, we work in details the controlled damped wave equation.Finally, we propose future investigations an open problem.

The Controlled Semilinear Damped Wave Equation
Consider the following control system governed by a 1D semilinear damped wave equation ( ) ( 1) 0 2) The operator A has the following very well known properties: 1) The spectrum of A consists of only eigenvalues each one with multiplicity one.
2) There exists a complete orthonormal set { } n  of eigenvectors of A .
3) For all ( ) where    is the inner product in X and 2 2 and ( ) 2 sin( ) is a family of complete orthogonal projections in X and given by: and 5) The fractional powered spaces r X are given by: Hilbert Space with norm given by: Using the change of variables w v   , the system (5.1) can be written as a first order systems of ordinary differential equations in the Hilbert space is an unbounded linear operator with domain ( ) ( ) ] Z X Z Throughout this section, without lose of generality, we will assume that The following proposition follows from [8] and [1].Proposition 1.The operator A given by (5.8), is the infinitesimal generator of strongly continuous group   Z  given by: where   0 n n P  is a family of complete orthogonal projections on the Hilbert space 1 2 Z given by:   (5.15) The proof of the following theorem follows in the same way as the one for Theorem 4.1 from [1].
Theorem 1.The system Proof It follows from Theorem 6 one we observe that in this case 1 B  .

Future Research
These results can be applied to the following class of second order diffusion system in Hilbert spaces where  is a sufficiently smooth bounded domain in 2 IR ,

1 .
is a suitable function.Examples of this class are the following well known systems of partial differential equations: Example The n D wave equation with Dirichlet boundary