Exact Boundary Controllability on a Tree-Like Network of Nonlinear Planar Timoshenko Beams?

Qilong GUG¨unter LEUGERINGTatsien LI

1 Introduction

We consider a planar network of initially straight nonlinear Timoshenko beams under control at some external boundary nodes.The corresponding linear system has been modeled and analyzed with respect to wellposedness and controllability,observability and stabilizability,optimal control and domain decomposition methods by Lagenese,Leugering and Schmidt[5–6]and Lagnese and Leugering[8].Nonlinear Timoshenko beams in three spatial dimensions including thermal effects have been introduced by the same authors in[7].Modeling and wellposedness for nonlinear Timoshenko beams and in particular for networks of such beams,to the best knowledge of the authors,have not been studied by many authors.A planar couple-stress modeling has been recently described by Asghari et al.[1].Other planar models have been given by Zhong and Guo[17]and Racke and Riviera[12],the latter with thermal effects but without longitudinal displacement.We will show below that these models are included in the approach described in[7].Global wellposedness for nonlinear Timoshenko beams in a general framework does not seem to be available within the literature so far.Problems of exact controllability,observability and even stabilizability for single nonlinear Timoshenko beams has,to the best knowledge of the authors,not been considered in the literature,and this is particularly so for networks of nonlinear Timoshenko beams.This paper provides a first attempt to solve these problems.We concentrate here our attention to planar networks of such kind.Networks in three-spatial dimensions are subject to a forthcoming publication.

The plan of the paper is as follows.We first brie fly describe the modeling procedure outlined in[7].The focus here,however,is on planar,initially straight and isothermal shearable beams.These assumptions drastically reduce the complexity of the modeling procedure and,therefore,the description is of independent interest.We then formulate the corresponding initial-boundary value problem for a single beam under gravity.In the next step,we look for equilibria under a given set of boundary conditions.In order to discuss well-posedness,we rewrite the system in quasilinear form followed by a representation as a first order system.We notice that in doing this,an arti ficial zero eigenvalue appears related to the shear angle.Then,the first order format allows the application of the concept of semi-global classical solutions in the sense of Li Tatsien[13].Having established semi-global existence for the solution of the problem and the problem resulting in interchanging the space and time variable a transformation that has to be veri fied in due course the exact controllability can be shown as in[13].The remaining part of the paper is then devoted to extend the method to networks of such nonlinear Timoshenko beams.Indeed,as is well-known that this procedure applies to tree-like networks using the so-called peeling method.

2 Modeling of Nonlinear Beams

Let ? be the domain of the undeformed planar beam:

where e1,e2,e3constitute an orthonormal basis in R3,L and A(x1)denote the length of the beam and its cross section at x1,respectively.As we consider a planar and initially straight reference con figuration,e1,e3describe the plane of deformation,while e2,pointing into the plane,is the axis of rotation.We may,in fact,assume that A(x1)=A is constant along the beam.It is clear that more general cases can also be dealt with.The reference con figuration ?is subject to deformation,and thus we consider R(x)as the vector pointing into the deformed con figuration.Consequently,V(x):=R(x)?r(x)is the displacement.As in all beam theories,one finally wants to express everything in terms of variables related to the center line x3=0 of the beam.Thus,the displacement of the center line is introduced as W(x1):=V(x1,0),where we suppress the variable x2,which is zero in the planar case.The tangents are given byWe then can write down the strains

and

Using standard arguments,see[7,16],for a shearable beam one can establish the following representation of the potential energy of the beam under deformation:

whereis the curvature due to bending and E,A,I,G are Young’s modulus,the area of cross section,the inertial moment around the e2axis,and the shear modulus,respectively.In order to derive the equations governing the motion of the beam,we need to express the strains(2.2)and the curvaturein terms of primitive variables such as displacements and rotation.We consider the deformation process as being composed of the mappings,a rotation with the angle Θ about the e2axis carrying the orthonormal system e1,e3intofollowed by a deformation ofinto the non-orthogonal system E1,E3,described by the strainsThe first smallness assumption concerns the rotation angle Θ such that the rotation takes the form

By de finition,the curvature is given by

It is obvious from the de finition(2.2)that E1,E3take the form

Using(2.4)we find

(2.7)describes the overall deformation from e1,e3into E1,E3and the second line suggests the introduction of a total rotation angle ? := Θ+which accounts for rotation due to both bending and shear.We are now going to expressin terms of a linear symmetric part and a nonlinear part based on anti-symmetric quantities.Let

With this notation,we can expressgiven by(2.2),as

At this point,we introduce the second hypothesis on the smallness,namely,we assume that the strainsare small with respect to the rotationsin other words,we set

We have W,1·e3=(E1?e1)·e3= ?Θ =W3,1and,therefore,

With(2.11)we can now express the strainsas follows:

It depends now on how one handles Θ versus ? in connection with W3,1in order to obtain different models of Timoshenko beams and Euler-Bernoulli beams.If there is no shear strain,then upon Θ = ? = ?W3,1one obtains

This leads to a nonlinear Euler-Bernoulli beam as discussed in[7].If one keeps the shear strain,one arises at what has come to be known as von Karman relation,namely,

We now express the quantities W1,W3,? by u,w,?ψ,respectively,where the minus sign is introduced only for easier comparison with the traditional notation for linear models.We introduce then the following potential energy:

If we consider the deformation of the beam under its own weight,we have to add to the potential energy the corresponding gravitational effect:

Obviously,axial forces no longer couple to rotation(shear),and we then derive the following nonlinear Timoshenko beam model:

Remark 2.1 Such a system(including higher order terms)has been investigated by Asghari et al.[1].If one considers very thin beams,then their model reduces to(2.17).It should also be mentioned that upon neglecting the longitudinal displacement in(2.17),one derives a model that can be written in the format of Riviera and Racke[12],where the thermal coupling is also present.These models are included in the framework of Lagnese,Leugering and Schmidt[7].While in[12]a wellposedness result is derived for the thermoelastic Timoshenko beam,a global in time existence and uniqueness result does not seem to be known in the literature as regards systems(2.17).Finally,if one assumes no shear,i.e.,ψ=then one obtains from the Hamilton principle,after suitably adjusting the potential and kinetic energies,a nonlinear Euler-Bernoulli-beam equation coupled to longitudinal motion.If the latter,in turn,is neglected one arrives at a 1-d version of the von Karman-system.See Langese and Leugering[4]and Horn and Leugering[3]for stabilizability results for the latter beam equations.

We use the model(2.17)in this article.We have the following boundary conditions.

(i)Dirichlet conditions at x=0:

(ii)Neumann conditons at x=L:

Remark 2.2It is clear that with homogeneous boundary data,u=w=ψ=0,x∈[0,L]is an equilibrium.The determination of all non-zero constant and also nonconstant equilibria is beyond the scope of the article.This will be considered in a forthcoming publication.Here we restrict ourselves only with some examples.

We are now in the position to formulate the initial-boundary value problem for a planar nonlinear Timoshenko beam.

Here(2.22),(2.23)and(2.24)represent the state equations,the boundary conditions and the initial conditions,respectively.

2.1 Equilibrium solutions

We now consider equilibrium solutions of(2.22).Clearly,for the homogeneous system,the zero-state is an equilibrium.However,in the context of mechanics,we always have to deal with gravitational forces.

Example 2.1The first case concerns a horizontal beam that is clamped at x=0 and free at the other end,i.e.,e1=(1,0)T,e3=(0,1)T=e.Therefore,looking for an equilibrium solution in the context of gravitation leads to the following ordinary differential system:

The solution of(2.25)and(2.26)is given by

Example 2.2In the case of a hanging beam,where the top end x=0 is clamped and the end x=L is free,we have e1=?(0,1)T=?e,e3=(1,0)T.Thus,

The boundary conditions are the same as in(2.26).In this case the first equation gives a tangential load

Obviously,the unique solution is the one,where w(x)=ψ(x)=0,?x∈[0,L],while

which clearly shows the stretching due to gravitation.

2.2 Quasilinear form

We proceed to derive the quasilinear form of the system(2.17)or of the initial-boundary value problem(2.22)–(2.24).

We rewrite(2.33)as a system of second order equations in vectorial form as follows.We introduce the vectorial state as Φ :=(u,w,ψ)Tand de fine

The system(2.33)can be written as

System(2.35)is a quasilinear system of second order in space and time.Now,given an equilibriumwe look for states Φ=+for possibly smallClearly,we can reformulate the quasilinear system in terms of the perturbationTo this end,we de fine

Because theis an equilibrium solution,we have

The system(2.33)takes now the form

Now,

is a symmetric matrix.Hyperbolicity is then a matter of showing that the eigenvalues of M?1G are uniformly positive in a neighborhood of the equilibrium solution.To this end,we introduce

Then the eigenvalues ofwith

As the general discussion on the hyperbolicity of the system depending on the magnitudes of E and G is a bit involved,for the sake of simplicity,in this article we resort to equilibria such that the eigenvaluesμiare uniformly positive and smooth in a sufficiently small neighborhood of the equilibrium solution

For the analysis of(2.35),in particular for its controllability,it is important to consider the invertibility of the matrixin a neighborhood ofIndeed,we formally have

where It is obvious that the matrixis indeed invertible in a neighborhood of the equilibrium statethat satis fies the requirements above.Under these conditions,one may then multiply the system(2.38)by G?1and obtain the second order derivative in the spatial variable with the identity matrix as coefficient.Then,one may interchange x and t.Moreover,in this case,the transformed system is again a hyperbolic system of second order.Formally,the original system and the one obtained after interchanging x and t are not of the same type,as first order spatial derivatives in the coefficients of the x?t version are now time derivatives in the t?x version of the problem.In order to fully symmetrize the situation,one can use the format discussed by Wang[14].

3 First Order System

In order to analyze the well-posedness of system(2.33),we transform the second order equations into a quasilinear hyperbolic system of first order.There are a number of equivalent ways to do that.In order to avoid thee·-notation,we write the perturbations without thee·-sign.To this end,we introduce the following variables U=(u1,···,u7):

and write(2.17)in the following form:

We rewrite(3.2)into matrix format and introduce

With(3.3)and(3.4),system(3.2)can be written in standard form as follows:

In order to verify the hyperbolicity of(3.5),we need the eigenvalues and eigenvectors of A(U).We de fine

There are three cases to deal with:

(1)G

(2)G=E,

(3)G>E.

We first consider the case E>G.Then the eigenvalues are given in increasing order by

It is clear that the eigenvalueswith(2.42)of the second order system.In order to establish the relation of the nonlinear model under gravity with linear Timoshenko model without gravity,we set for the moment g=0,i.e.,we look at the case=0.In this case,we have as(u1,u3)→(0,0),

Remark 3.1 It should be noted that for=U=0 the eigenvalues λi(i=1,···,7)coincide with those of the linear Timoshenko beam system,namely,

ordered by magnitude for E>G.Notice that forthe eigenvalueshave double multiplicity,while the system is strictly hyperbolic for=0 and U such that(u1,u3)≠The zero eigenvalue is an arti ficial one which can be avoided by directly considering the original system(2.35)of second order.It is,however,necessary to use the first order format in order to utilize the concept of semi-global classical solutions in the sense of Li[13].

For the case E>G we find the following right-eigenvectors:

while for the case G>E,we have to interchange the role of v2,v3and v5,v6,according to the change in the magnitude of G and E.E=G is a degenerate case with the additional property that now,in the limit,we have three double eigenvalues.Nevertheless,even in this case the system is strictly hyperbolic in a neighborhood of the origin.We need to evaluate the left-eigenvectors.These,for the case G

We suppress the analogous cases for G>E and E=G.

Clearly,for E>G(E0 when t→0.Clearly,on t(u1,0),t>0 the expressions are not de fined.For this reason,that situation is not considered in this article further.We introduce

We have

We can also express the boundary conditions at x=0 and x=L in terms of U.To this end,we introduce the matrices

and

With(3.11)and(3.12),we can express nonhomogeneous Dirichlet conditions at x=0 and Neumann conditions at x=L as

We are now in the position to formulate the initial-boundary value problem(2.22)–(2.24)as a first order hyperbolic system of equations in the classical format:

In order to apply the theory for semi-global classical solutions of Li[13],we need to express the boundary conditions in terms of the variables

Namely,

together with

We analyze the situation for(3.10);hence for G≤E,we leave the case to the reader.Here we obtain

This can be written in matrix form as follows:

and

In order to simplify the notation,we order the system variables in a different way.Namely,the vector U is now ordered according to=(u1,u3,u5,u2,u4,u6,u7)=:(w1,w2,w3),where w1=(u1,u3,u5),w2=(u2,u4,u6),w3=u7.We also introduce ξ+:=(v7,v6,v5)Tand ξ?:=(v1,v2,v3)T.With this notation,we can reformulate(3.17)and(3.18)as follows:

The derivative

is invertible and,therefore,we can apply the implicit function theorem and conclude that there exists a function Θ such that

Inserting this into the first equation in(3.19),we obtain

We then de fine

Obviously,we have

Thus,we can apply the implicit function theorem again in order to solve for ξ+.There exists a vector function GD,0(·,·)such that

Upon de fining Dirichlet controls at x=0 as

we rewrite(3.20)with

and VD(t)=GD,0(0,h0(t))as

where now GD,0(0;t)=0.We go back to the previous notation and conclude

This is the format required in(3.15).As for the Neumann boundary conditions(2.19)–(2.21),we have,after dividing through the constants in each condition and renaming the controls(i=1,2,3)

Using the first equation of(3.23)in the second condition,we arrive at

We are going to use(3.17)–(3.18).We notice that v4=u7.According to the second equation in(3.24),u3(L,t)can be expressed in terms of u7(L,T)and,hence,in terms of v4(L,T).By the first equation in(3.24),this is true also for u1(L,T).Thus,

and

Aswe can express the third condition in(3.23)as v1(L,t)=Now,the last two equations of(3.16)contain on their right-hand sides terms in the variables u1,u3,only.As seen above,these can be expressed in terms of v4.Thus,v2,v6can be expressed in terms of v4and terms involving the controls(i=1,2,3).This shows that we can express(3.23)in the form

Similarly,Dirichlet boundary conditions at x=0 and Neumann conditions at x=L can be shown to satisfy

Remark 3.2It should be remarked that the system matrices with=:(w1,w2,w3),where w1=(u1,u3,u5),w2=(u2,u4,u6),w3=u7,have now the following forms:

With(3.28)and(3.29),system(3.2)can be written in the following equivalent form:

The advantage of this form is that it reveals the typical block structure for wave equations.

4 Existence of Solutions

In order to study the well-posedness of(2.22)–(2.24)in the framework of semi-global classical solutions,we need to assume regular initial and boundary data,as well as compatibility conditions.

De finition 4.1We say that the initial conditions and boundary conditions satisfy C1-compatibility conditions for the Dirichlet case at x=0 and for the Neumann case at x=L,if the following conditions hold:

as well as

and

We say that the initial and boundary conditions satisfy C2-compatibility conditions if they satisfy the C1-compatibility conditions and,in addition,the partial differential equations(2.33)hold at x=0,L on time t=0,where the second order in time derivatives are replaced with the second order in time derivatives of the controls at x=0 and x=L,respectively.

Remark 4.1The representation as a first order system,say,in the format described in Remark 3.2,is useful.In particular,w1,w2,at t=0,are related to(the spatial derivatives of)the initial displacements u0,w0,ψ0and the initial velocities u1,w1,ψ1,respectively.

Theorem 4.1Let T>0 be given.Let the boundary controlsand the initial datasatisfy the C2compatibility conditions of De finition 4.1,such thatare sufficiently small.Thenthere exists a unique semi-global C2-solution Φ(x,t)=(u(x,t),w(x,t),ψ(x,t))with small C2-norm on R(T)de fined as

ProofThe proof is similar to the one given in[10]and follows the lines of[13].

5 Exact Controllability

We are now in the position to formulate the problem of local one-sided exact boundary controllability around an equilibrium solution=().We denote the perturbations ofby Φ.Let us recall the eigenvalues μi(i=1,2,3)given by(2.42).For the sake of brevity,we writeμi(x,Φ)(i=1,2,3)in order to indicate the dependence ofμion the spatial variable and the perturbation.Thus,μi(x,0)(i=1,2,3)signi fies the eigenvalues in(2.42)at the spatial point x at equilibrium.Both,the systems at equilibrium and at the actual position,de fine speeds of propagation.The maximum travel time in the beam can be estimated as follows:

For a given ?-neighborhood B?(Φ)of Φ,we can bound the travel time by

Indeed,by our assumptions,we find such an ?0>0 such that T>2T0implies T>2T1.Thus,we can bound the time that a signal needs to travel from the boundary,where controls that apply to the clamped origin and back can be estimated by the corresponding time for the system seen at equilibrium.

De finition 5.1Let a control time T>0,initial and final data(Φ0,Φ1),(Ψ0,Ψ1)where Φ0=(u0,w0,ψ0)T ∈ C2([0,L])3,Φ1:=(u1,w1,ψ1)T ∈ C1([0,L])3,Ψ0=()T ∈C2([0,L])3,Ψ1:=()T ∈ C1([0,L]3)be given.We say that the problem(2.22)–(2.24)is exactly controllable in time T with one-sdided controls,if there exist boundary controls VD with VN=0 or VNwith VD=0,satifying the compatibility conditions(4.1)–(4.3),such that the corresponding solution,satisfying the conditions of Theorem 4.1,admits the final values

Theorem 5.1Let T be given by T>2T0.There exist neighborhoods U0,U1of(bΦ,0)such that for each pair of initial states in U0and final states in U1satisfying the regularity and compatibility conditions given in Theorem 4.1,there exist C2(0,T;R3)-controls VDor VNsuch that the solutions of(2.38)satisfy the conditions in De finition 5.1.

ProofThe remaining part of the proof consists of applying method described in[13].For the sake of brevity,we refer to the proof of Theorem 7.1(Section 7 below)for n=1.

Remark 5.1We remark that we do not consider the exact controllability problem on the level of the first order system(3.5).Indeed,the exact controllability of the full state via boundary controls is generally impossible,due to the appearance of the zero eigenvalue.This eigenvalue would make it necessary to involve a distributed control(see[13]and a more recent discussion in Hu[2]).The controllability result is a local one.It can be extended to a localglobal result if one considers two distinct equilibria connected by a path of equilibria(see[10]for the analogous case in the context of nonlinear strings).

6 Networks of Nonlinear Timoshenko Beams

We now consider networks of planar initially straight nonlinear Timoshenko beams according to(2.17).We introduce some notation in order to describe the network.We suppose that there are n beams indexed by i ∈ I={1,···,n}.We let the i-th beam be parametrized by its rest arc length x with x∈[0,Li],Lithe length of that beam.The position and shear at time t of the point corresponding to the parameter x will be denoted by the vector((u,w,ψ)i(x,t))T.The positions and shear at the endpoints,which we refer to as nodes,are given by functions(()j)T(t)with j ∈ J={1,···,m}.A similar statement holds for the simple nodes where Neumann conditions are applied,see below.At multiple nodes which we denote by JM,where several beams meet there is a common location Nj.Simple nodes are those corresponding to the endpoints of only one beam.This set is split into nodes JSD,where Dirchilet conditions are satis fied,and JSNwhere Neumann conditions hold.We let Ij={i∈I:Njis an end point of the i-th beam},JMbe the subset of J corresponding to multiple nodes,while JScontains the indices of simple nodes.We assume that there are simple nodes so that JSis not empty.For j∈JSwe have Ij={ij}.For i∈Ijwe let xij=0 or xij=Lidepending on whether the beam begins or ends at the simple node,respectively.For purposes of integration by parts we also introduce ?ijto equal 1 or ?1 depending on whether xijis equal to Lior 0.

We only consider rigid joints such that the positions and,hence,the displacements as well as the angles between to adjacent beams before and after deformation coincide.Pinned joints will be treated elsewhere.The continuity of displacements is expressed as

Re flecting the meaning of W=ue1+we3,we can rewrite(6.1)as

where we notice that because the undeformed and the deformed con figurations ri,Risatisfy the same conditions,these conditions are also valid for the displacements at the centerline.As for the total angles ψi,we obtain at a rigid joint

We can now derive the following conditions on balance of forces and moment at a multiple node:

Remark 6.1We notice that the conditions(6.3)–(6.4)at multiple node are vectorial conditions coupling longitudinal motion and shearing,while the rotation around e2does not couple to other primitive variables.This is in contrast to scalar beam models which represent out-ofthe-plane dsiplacements,only.The fact that the multiple node conditions are vectorial implies that the network depends on the angels between the beams,thus,on the topology.In scalar networks,the angles do not matter.

Let e be the upright unit vector.The full initial-,boundary-,nodal-value network problem for initially straight planar nonlinear Timoshenko beams reads as follows:

Here(6.5)includes the equations governing the motion of the beams along the individual edges,(6.6)describes the boundary conditions at simple Dirichlet-and Neumann-nodes,(6.7)provides the conditions of continuity and force/moment balance at multiple nodes and(6.8)denotes the initial conditions.As for the single element case,we introduce the mass-matrices and stiffness operators for each individual beam:For each i∈I,we de fine

Then each Timoshenko-beam system can be written as

Now,given an equilibrium con figuration such that on the edge i we havewe look for statesfor possibly small Φi.Notice that we omit the tilde for the perturbations right away.We can reformulate the quasilinear system in terms of the perturbation

Because thecorrespond to an equilibrium solution we have

We thus have

Under precisely the same conditions as in the single link case,we may invertas this matrix is uniformly positive de finite in a sufficiently small neighborhood of the equilibrium.This fact will be important for the proof of controllability.In order to proceed with existence in the sense of[13],we need compatibility conditions both at the simple nodes and at the multiple nodes.At a simple node j∈there is only one edge incident and the location is denoted as above by xij.It is straightforward to reformulate the compatibility conditions(4.1)–(4.3)for such simple node conditions.We refrain from displaying the corresponding conditions.However,the new multiple node conditions(6.7)require the following new compatibility conditions:

6.1 Equilibrium solutions for the network

We assume from now on that we have an equilibrium solution of the entire tree-like network.That is a solutionof the following steady state problem:

It is clear that for the local equations(6.17)a similar analysis as for the single-link case(2.25)can be performed.The full analysis of the equlibirum problem is beyond the scope of this article.We provide,however,a simple example that makes the vectorial nodal conditions evident.

Example 6.1In order to elucidate the network setup,we give an example of a“carpenter square”.Here,we have a horizontal beam(labelled 1)and a second hanging beam(labelled 2)mounted at the end of the first beam.The constellation of local bases is as follows:=(1,0),=(0,1)and=(?1,0),=(1,0).

The boundary conditions at the simple nodes are

while the transmission conditions at the multiple node read as follows:

Figure 1 A carpenter’s square:Current,geometric exact and linear model.

The conditions at the multiple node are intuitive,as they clearly show that the longitudinal displacement of the horizontal beam converts to transversal displacements of the second beam and vice versa.The analogous observation is evident for the balance of forces.It can be shown that this system of ordinary differential equations has a unique solution.However,the analytical solution cannot be provided due to space limitations.See Figure 1,however,for a numerical comparison between the current beam model,the geometric exact model and the linear Timoshenko beam model.It is clearly seen that the two nonlinear models are in good agreement,while the classical linear model deviates signi ficantly.In the linear theory the downwards movement of the tip of the horizontal beam follows the vertical line,which is clearly non-intuitive for large displacements.

6.2 Network equations in first order format

By taking(3.2)into a matrix format,we denote the individual matrices for the i-th beam as follows:

Also for the multiple node,we introduce the representation of the local coordinate systems in terms of global coordinates in the 1-3-plane as follows:=(),=().Moreover,the continuity conditions have to be differentiated with respect to time,so that instead of ui,wi,ψi,?tui,?twi,?tψi,hence,are involed.Then

and

With this notation,the general network problem can be expressed as follows:

An alternative formulation is as follows:We rewrite the system in terms of w according to=:(w1,w2,w3),where w1=(u1,u3,u5),w2=(u2,u4,u6),w3=u7has now the following form:

where we have used the obvious condensed forms of the operatorswith respect to the partition of variables.We can now apply the same calculus as in the single-link case.This means that we may introduce,for each individual edge,a system of left and right eigenvalues of the matrices Ai(Ui),(j=1,···,7),(j=1,···,7)orrespectively.This makes it possible to rewrite the system(6.28)as a first order system in these new variables,analogous to the single-link case.For more details see[13].We introduce

According to(3.17)–(3.18),we obtain

We consider the case that xij=0,the other case is completely analogous.In a tree,and this is the case we consider,one can always arrange the multiple nodes in such a way that all incident edges either start at the node,i.e.,xij=0,?i∈ Ij,or end there,i.e.,xij=L,?i∈ Ij.For this,one has to work with a simple scaling in order to transform the lengths to a uniform quantity.Now,the second equation in(6.33),after applying the implicit function theorem at w=0,provides a function ?isuch that

We insert this in the first equation of(6.33)and get

In view of this,we can de fine the map

together with

We may,thus,apply the implicit function theorem again and obtain

so that at a controlled Dirichlet simple node we obtain

whereis the vector of Dirichlet controls at x=0.This is the format required in[13].Another way of writing(6.33)is

Now,(6.37)de fines a mapthe Jacobian of which is given by

which,in turn,is invertible.Notice that

Therefore,we may apply the implicit function theorem once again in order to find mappings ?i,ψiwith

With these expressions the Neumann node conditions at controlled simple nodes can be expressed in terms of wi(0,t):

The continuity condition at a multiple node(taken at x=0 for all incident edges)reads

where nj=|Ij|is the edge degree of the current multiple node j∈JMat xij=0.We can now use(6.38)and the lst equation to obtain

The final point is now to consider the transmission conditions involving the forces and moments.These conditions can be expressed as follows:

which turns into

This gives rise to the map

Now,H0(0,0,0)=0 and(0,0,0)is invertible.This finally shows thatis a function of

and this is the precisely format required in[13].

Theorem 6.1Consider a tree-like network of Timosheko beams as described by(6.5)–(6.8).For each T>0,there exist constants c0and cTsuch that for initial dataand boundary datasatisfying a uniform smallness condition,i.e.,

and the compatibility conditions(4.1)–(4.3),extended to all simple nodes,and(6.15)–(6.16)at the multiple nodes.Then,there exists a unique piecewise twice continuously differentiablesolutiondepending continuously on the data:

7 Exact Controllability on Star-Like Networks and Trees

In this section,we assume that n beams meet at one node,such that for each beam the junction is at x=0.Thus,|JM|=|J1|=1 and xi1=0,i∈I1.We de fine the travel times

Theorem 7.1Let{Φi}i∈Ibe an equilibrium solution of(6.5)–(6.8)and let T>2T0.Then there are neighborhoods U0and U1of({Φi}i∈I,0)such that given initial and final data

one can find Dirichlet-controlsvi∈ C2(0,T;R3)such that the corresponding solutions of(6.5)–(6.8)satisfy

ProofWe follow the spirit of the proof of Theorem 5.2 in[13].The principal idea in exact boundary controllability of 1D-hyperbolic systems is to solve a forward problem with given initial data,a backward problem with given final data and a corresponding initial-boundary value problem from “the left or the right”.In particular,for the latter it is convenient to interchange the spatial and time variables x and t,and then solve a mixed Cauchy-problem from the left or the right,once the corresponding boundary conditions have been reduced from the initial and final data.We assume equilibria for which Gi(0,0)andare positive de finite uniformly with respect to(x,t)for a sufficiently small neighborhood ofIf one has Cauchy-dataat a boundary point,say x=Liand “boundary data”one can solve the wave-type equation “from x=Lito x=0”.For this procedure,it is important to understand that the Cauchy data at t=0 and t=T can be converted to the proper boundary conditions for the system when the role of x and t is reversed.As the existence and uniqueness results are obtained on the level of first order equations,one needs to first invert on the second order level and then rewrite the resulting second order system as a first order system such that the corresponding boundary conditions are of the standard type.Due to the zero eigenvalue for the first order system,the direct inversion on the level of the first order system is impossible.We describe the idea of the proof as follows:There are five steps.Step 1:In the first step we proceed forward from t=0 to t=T1.We solve the initial-boundary value problem with arti ficial controls at x=Li.For each beam i∈ I,we de fine the setand for the network we setThe first beam to be fixed at x=L1.We may,for the sake of convenience,assume that we impose a homogenous Dirichlet condition there.This speci fies the corresponding compatibility conditions.We impose arti ficial inhomogeneous Dirichlet conditions atwhere vi(·)are small in C2(0,T1;R3).We also havesufficiently small initial datafor all beams.We apply the existence Theorem 6.1 and obtain a unique solution on RI.We can now take traces ofalongat the boundary of the first beam for all beams at{0}×[0,T1].It is clear thatsatisfy the nodal conditions at the common node.Moreover,all and of data is small in the appropriate spaces.

Step 2:We perform the same procedure,but now reversing the time and progressing from the final time T to T?T1.More precisely,we introduce the individual domainsand the global one By the same argument,a unique semi-global small solutionof the network problem exists,and we can take traces for the first beam beams labelled i=2,···,n.

In order to prepare Step 3,we extend the Cauchy-data at{{L1}×[0,T1]}∪{{L1}×[T?T1,T]}in the C2-sense to After that we can use these Cauchy-data along{L1}×[0,T]as“initial conditions”.for the

Step 3:We change the order of x and t as explained in the beginning of the proof.The Cauchy-data just constructed can be taken as “initial conditions” for the first beam “starting”at x=L1with ‘boundary conditions’at t=0 and t=T taken from the original initial and final data.Applying the semi-global existence Theorem 6.1 to that situation,we can evaluate thethis solution Φ1is identical toTherefore,at t=0 we have

At x=0 we have

The analogous uniqueness argument applies for the backward solution of Step 2,such that the final conditions are

while the evaluation at x=0 provides the Cauchy-data Step 4:We now extend the Cauchy-datat∈[0,T2]together with(t)),t∈[T?T2,T]to Cauchy-datat∈[0,T]such that corresponding solutions satisfy the nodal conditions.

Step 5:We now have Cauchy-data on{0}×[0,T]such that the nodal conditions are satis fied.Therefore,we can use these as compatible initial conditions for the beams labelled i=2,···,n after interchanging x and t.Thus,on the domains:={(x,t)∈[0,Li]×[0,T]}we solve the initial boundary value problems with Cauchy-data

and boundary conditions

By construction,the solutions are small in the sense described above.A similar uniqueness argument applies to the regionto the effect that

The analogous argument on the “upper” domain leads to

This gives the solution to the problem stated.

Remark 7.1It is clear that a local-global controllability theorem can be proved,provided that we have two non-identical equilibrium solutions connected by a path of equilibria.However,as the analysis of general network equilibria is still open,we refrain from stating the theorem here.Also,the analogous controllability holds for tree-like networks as usual.This is proved by the so-called peeling method(see[13]).Moreover,it should be remarked that the observability does not directly follow as in the linear case(see the corresponding remarks in the monograph of Li[13]).Also feedback stabilization is an open problem that will be addressed in a forthcoming publication.

[1]Asghari,M.,Kahrobaiyan,M.H.and Ahmadian,M.T.,A nonlinear Timoshenko beam formulation based on the modi fied couple stress theory,International Journal of Engineering Science,48,2010,1749–1761.

[2]Long Hu,Contr?ole fronti`ere,stabilisation et synchronisation pour des syst`emes de lois de bilan en dimension un d’espace,PhD Thesis,University Paris 6,2015.

[3]Horn,M.A.and Leugering,G.,An overview of modelling challenges for a nonlinear plate-beam model,Nonlinear Anal.,Theory Methods Appl.,Ser.A,Theory Methods,63(5–7),1529–1539.

[4]Lagnese,J.E.and Leugering,G.,Uniform stabilization of a nonlinear beam by nonlinear boundary feedback,J.Differ.Equations,91(2),1991,355–388.

[5]Lagnese,J.E.,Leugering,G.and Schmidt,E.J.P.G.,Modeling,Analysis and Control of Dynamic Elastic Multi-Link Structures,Birkhauser,Boston/Basel/Berlin,1994.

[6]Lagnese,J.E.,Leugering,G.and Schmidt,E.J.P.G.,Control of planar networks of Timoshenko beams,SIAM J.Control Optimization,31(3),1993,780–811.

[7]Lagnese,J.E.and Leugering,G.,Modelling of dynamic networks of thin thermoelastic beams,Mathematical Methods in the Applied Sciences,16,1993,327–358.

[8]Lagnese,J.E.and Leugering,G.,Birk¨auser Verlag,ISNM Series,1994.

[9]Leugering,G.and Schmidt,E.J.P.G.,On the control of networks of vibrating strings and beams,Proceedings of the 28th Conference on Decision and Control,Tampa,Florida,December,1989,2287–2290.

[10]Leugering,G.and Schmidt,E.J.P.G.,On exact controllability of networks of nonlinear elastic strings in 3-dimensional space,Chin.Ann.Math.Ser.B,33(1),2012,33–60.

[11]Gu,Q.L.and Li,T.T.,Exact bounmdary controllability for quasilinear wave equations in a tree-like planar network of strings,Ann.I.H.Poincae AN,26,2009,2373–2384.

[12]Munoz Rivera,J.E.and Racke,R.,Mildly dissipative nonlinear Timoshenko systems:Global existence and exponential stability,Konstanzer Schriften in Mathematik und Informatik Nr.167,Marz 2002.

[13]Li,T.T.,Controllability and Observability for Quasilinear Hyperbolic Systems,AIMS Ser.Appl.Math.vol 3,American Institute of Mathematical Sciences and Higher Education Press,2010.

[14]Wang,K.,Exact boundary controllability for a kind of second-order quasilinear hyperbolic systems,Chin.Ann.Math.Ser.B,32(6),2011,803–822.

[15]Wang,Z.Q.,Exact controllability for nonautonomous first order quasilinear hyperbolic systems,Chin.Ann.Math.Ser.B,27(6),2006,643–656.

[16]Wempner,G.,Mechanics of solids with applications to thin bodies,Monographs and Textbooks on Mechanics of Solids and Fluids,Mechanics of Elastic and Inelastic Solids,2.Alphen aan den Rijn,The Netherlands-Rockville,Maryland:Sijtho ff&Noordho ff.XVII,633 pages.

[17]Zhong,H.H.and Guo,Q.,Nonlinear vibration analysis of Timoshenko beams using the differential quadrature method,Nonlinear Dynamics,32,2003,223–234.