W Michiels
email: wim.michiels@cs.kuleuven.ac.be

V Van Assche

S.-I Niculescu
email: silviu.niculescu@hds.utc.fr

Abstract-We study the stability of a linear system with a pointwise, time-varying delay. We assume that the delay varies around a nominal value in a deterministic way and investigate the influence of this variation on stability. More precisely we are interested in characterizing situations where the time-varying delay system is stable, whereas the system with constant delay is unstable. Our approach consists of relating the stability properties of a system with a fast varying point-wise delay with these of a time-invariant system with a distributed delay. Then we can use frequency domain methods to analyze the problem and to derive stability criteria. The results are first illustrated with two theoretical examples. Then, we study a model of a variable speed rotating cutting tool. Based on the developed theory, we thereby provide both a theoretical explanation and a quantitative analysis tool for the beneficial effect of a variation of the machine speed on enhancing stability properties, which was reported in the literature. Index Terms-Delay, nonautonomous systems, stability of variable speed machines.

I. INTRODUCTION

T HIS paper concerns the stability analysis of a linear system with a time-varying delay, of the form [START_REF] Breda | Computing the characteristic roots for delay differential equations[END_REF] under appropriate initial conditions. We assume that the timedelay varies around a nominal value in a deterministic way [START_REF] Cooke | Discrete delay, distributed delay, and stability switches[END_REF] where is a periodic function with zero mean and and are parameters, determining the amplitude and frequency of the variation. We investigate the influence of the variation of the time-delay on the stability of [START_REF] Breda | Computing the characteristic roots for delay differential equations[END_REF].

This problem is inspired by applications in mechanical engineering. More precisely, in the study of the dynamics of rotating cutting machines one encounters models of the form [START_REF] Breda | Computing the characteristic roots for delay differential equations[END_REF], where the time-delay represents the time taken by the cutting inserts for one revolution of the workpiece. Therefore, the time-delay is proportional to with the rotational speed of the machine. A commonly used way to enlarge the stability region of the steady-state solution of rotational machines and, in this way, preventing undesirable oscillations or chatter, consists of fast modulating the speed around its nominal value [START_REF] Jayaram | Analytical stability analysis of variable spindle speed machines[END_REF], [START_REF] Sexton | The stability of machining with continuously varying spindle speed[END_REF]. Such a modulation of the speed precisely corresponds to a modulation of the time-delay in (1) of the form (2). Both experimental results and numerical computations have demonstrated the beneficial effect of the variation of the machine speed; see, e.g., [START_REF] Jayaram | Analytical stability analysis of variable spindle speed machines[END_REF] and the references therein.

Whereas the effects of constant delays on the systems' dynamics have been largely treated in the literature (see, for instance, [START_REF] Niculescu | Delay Effects on Stability. A Robust Control Approach[END_REF], [START_REF] Gu | Stability of Time-Delay Systems[END_REF], and the references therein), the time-varying delay needs a deeper analysis since its presence may induce complex behaviors: see, for example, the so-called quenching phenomenon, as suggested and discussed by Louisell in [START_REF] Louisell | New examples of quenching in delay differential equations having time-varying delay[END_REF], where he proposed some examples of delay intervals for which the constant and time-varying delays have "opposite" effects: stable for the time-varying delay and unstable for the constant one, and vice versa.

The aim of this paper is to handle the time-varying delay systems mentioned above using an appropriate comparison system combined with some frequency-domain techniques. To the best of the authors' knowledge, there does not exist any similar analysis result in the literature. More explicitly, the stability analysis of (1)-( 2) and the comparison with the stability properties of the associated constant delay system

(3)

-to analyze the effect of the delay variation-are performed in two steps.

• First, we show that from a stability point of view, the system (1)-( 2) behaves as a time-invariant system with a distributed delay, provided the frequency of variation of the delay, , is large compared to the system's dynamics. Such a condition is satisfied in the applications mentioned previously.

•

The second step then consists of comparing the stability of this distributed delay system, whose analysis is more tractable (since it allows, e.g., frequency domain techniques), with the stability of the associated constant delay system.

While the variation of the delay in (1)-( 2) could even be interpreted as a control action, as in the inspiring mechanical application, in the existing literature on time-delay systems a delay variation is always treated as an uncertainty, using an appropriate model transformation of the original system or the definition of some "fictitious" interconnected systems (see, for instance, [START_REF] Niculescu | Delay Effects on Stability. A Robust Control Approach[END_REF], [START_REF] Gu | Stability of Time-Delay Systems[END_REF], and the references therein). For a nominally stable system, typically upper/lower bounds on the delay variation and/or its derivative guaranteeing stability are determined. In the time-varying delay case, we mention the Lyapunov based approach [START_REF] Kharitonov | On the stability of linear systems with uncertain delay[END_REF], some comparison system methodology [START_REF] Niculescu | Robust exponential stability of uncertain systems with time-varying delays[END_REF], [START_REF] Goubet-Bartholoméüs | Stability of perturbed systems with time-varying delays[END_REF], and the integral quadratic constraint approach [START_REF] Gu | Stability of Time-Delay Systems[END_REF] to derive such (upper, lower, or derivative) bounds.

A deliberate fast periodic forcing of the delay (2) can be seen as an application of vibrational control. In [START_REF] Lehman | Vibrational control of nonlinear time lag systems with bounded delay: Averaging theory, stabilizability, and transient behavior[END_REF] and the references therein, extensions of the classical averaging theory for time-varying ordinary differential equations toward classes of time-delay systems and the application to vibrational control are considered. In [START_REF] Michiels | Using delays and timevarying gains to improve the output feedback stabilizability of linear systems: A comparison[END_REF], output feedback controllers with oscillatory gains are analyzed. These works do however not treat the case where the time-dependence of the vector-field precisely comes from a variation of the delay.

The structure of the paper is as follows. In Section II, we relate stability properties of ( 1)-( 2) with those of a time-invariant distributed delay system. In Section III, we study the stability of this distributed delay system and compare its properties with the constant delay system (3). In Section IV, we present some illustrations of the main results. After analyzing two theoretical examples, we study a model for a variable speed cutting machine. Section V summarizes the main contributions of the paper. The notations are standard.

II. DISTRIBUTED DELAYS AS APPROXIMATION OF FAST TIME-VARYING POINT-WISE DELAYS

The main result of the paper relates the stability of a system with a fast varying point-wise delay with the stability of an autonomous system with distributed delay.

Theorem 1:

Consider (4)
where is a periodic function, with zero mean and period , and . Further, let , and . Let the integrable function 1 be defined by the relation [START_REF] Gu | Stability of Time-Delay Systems[END_REF] for every continuous function . If the comparison system (6) 1 In fact, w represents a positive density measure. is asymptotically stable, then ( 4) is globally uniformly asymptotically stable for large values of the parameter .

For sake of conciseness we postpone a detailed proof to the Appendix. However, because of the nonstandard proof technique, we now outline its main elements.

Sketch of Proof: The existence of an integrable function satisfying ( 5) is shown using the change of measure property and the Radon-Nikodym theorem.

The stability assertion follows from an application of the trajectory based proof technique, developed in [START_REF] Moreau | Practical stability and stabilization[END_REF] and [START_REF] Moreau | Trajectory-based local approximations of ordinary differential equations[END_REF] for the stability analysis of ordinary differential equations and extended toward classes of delay differential equations in [START_REF] Moreau | Robustness of nonlinear delay equations w.r.t. bounded input perturbations: A trajectory based approach[END_REF]. This proof technique relates closeness results for trajectories (in the sense of uniform convergence of trajectories on compact time-intervals) with stability results (which involve the behavior of trajectories on infinite time-intervals). The main steps are as follows.

1)

We prove that trajectories of ( 4) and ( 6) with matching initial conditions uniformly converge to each other on compact time-intervals as the parameter tends to infinity. This is done by estimating the difference of solutions of ( 4) and ( 6) at time-instants later than the initial time, and involves the application of a generalization of the celebrated Gronwall Lemma.

2)

This closeness result of trajectories of ( 6) and ( 4) is linked with stability assertions. By a slight generalization of [18, Th. 1], one can conclude from the exponential stability of ( 6) and the closeness result that the null solution of system ( 4) is practically uniformly asymptotically stable (see the Appendix for a precise definition).

3)

Practical uniform asymptotic stability of the null solution of (4) implies global uniform asymptotic stability by a scaling property of its solutions. Remark: From (5), one can interpret as the probability distribution of , where is uniformly distributed over the interval . Remark: Under the conditions of the theorem, the stability of ( 4) is only guaranteed if is sufficiently large. Explicit bounds on may be obtained from theoretical considerations (see, e.g., [START_REF] Moreau | Trajectory-based global and semi-global stability results[END_REF]), but these bounds are typically conservative. Therefore, we advise to determine a threshold based on numerical simulation.

The stability of the time-invariant comparison system ( 6) is determined by the roots of its characteristic equation [START_REF] Halmos | Measure Theory, ser. The University Series in Higher Mathematics[END_REF] where [START_REF] Jayaram | Analytical stability analysis of variable spindle speed machines[END_REF] We now list some general properties of the functions and , which will be useful in the rest of this paper. The weight function always satisfies

Furthermore, when is symmetric around the origin, i.e., , then is real, following from: [START_REF] Kolmanovskii | Stability of Functional Differential Equations[END_REF] As an illustration we show for different types of functions in (4) the corresponding weight function of the distributed delay of comparison system (6) and the function in Table I.

III. STABILITY OF THE COMPARISON SYSTEM

The next step consists of relating the stability of the comparison system [START_REF] Hale | Introduction to functional differential equations[END_REF] with the stability of the constant delay system [START_REF] Lehman | Vibrational control of nonlinear time lag systems with bounded delay: Averaging theory, stabilizability, and transient behavior[END_REF] Notice that (6) can be seen as a modification of (11) by "spreading out" the point-wise delay over an interval of length . If this leads to larger stability regions in a parameter space, then, by virtue of Theorem 1, a fast forcing of the delay will also lead to larger stability regions.

First, we investigate the stability of ( 6) for small values of . Then, we relate the obtained result with the sensitivity of stability w.r.t. changes of the nominal delay . Finally, we comment on numerical tools for computing the right most eigenvalues of the comparison system [START_REF] Hale | Introduction to functional differential equations[END_REF].

A. Behavior of the Comparison System for Small

We investigate how the eigenvalues of ( 6) or, equivalently, the roots of [START_REF] Louisell | New examples of quenching in delay differential equations having time-varying delay[END_REF] behave for small values of .For , [START_REF] Louisell | New examples of quenching in delay differential equations having time-varying delay[END_REF] simplifies to [START_REF] Luzyanina | Computing stability of differential equations with bounded distributed delays[END_REF] the characteristic equation of [START_REF] Lehman | Vibrational control of nonlinear time lag systems with bounded delay: Averaging theory, stabilizability, and transient behavior[END_REF].

Because is a smooth function with , roots of ( 12) are continuous at each value of parameter . For a given simple root of ( 13), this implies the existence of a root function of ( 12), satisfying and

To compute the sensitivity of the root w.r.t. , we differentiate [START_REF] Luzyanina | Equations with distributed delays: Bifurcation analysis using computational tools for discrete delay equations[END_REF], leading to

(15) (16)
where . The property ( 15) is due to the zero mean assumption on the delay forcing function , which is expressed in As a consequence, the behavior of the eigenvalues for small is determined by the second-order derivative . A stability related corollary is as follows.

Proposition 1: Assume that the rightmost eigenvalues of ( 11) are simple and on the imaginary axis. Denote them by .If [START_REF] Moreau | Trajectory-based global and semi-global stability results[END_REF] then ( 6) is asymptotically stable for small values of .

Proof: Choose such that, except for the eigenvalues on the imaginary axis, (13) has no zeros in . In this half plane, the roots of (12) satisfy Therefore, there exists a compact set containing all the roots of ( 12) in the half plane . From the continuity of and , it follows that as , uniformly on . By [15, Th. A1], a variant of Rouché's theorem, this implies that for small both (entire) functions have the same number of zeros in , and as a consequence in . Combining this fact with [START_REF] Moreau | Trajectory-based global and semi-global stability results[END_REF], which expresses that the roots lying on the imaginary axis for move to the open left half plane when slightly increasing , yields that all the roots of ( 12) lie in the open left-half plane for sufficiently small values of .

In the next section, we will study classes of parametrized delay systems, for which condition ( 17) is always satisfied in case of eigenvalues on the imaginary axis. This means that stability regions in the parameter space become larger when increasing from zero. Indeed, internal points of a stability region in the parameter space correspond to an asymptotically stable system, which remains asymptotically stable when increasing from zero (continuity argument), whereas points on the boundaries of a stability region, if any, correspond to a system with rightmost eigenvalues on the imaginary axis, which becomes asymptotically stable under the conditions of Proposition 1.

Finally, notice that parameter values corresponding to a zero eigenvalue are invariant w.r.t. , following from .

B. Relation With Delay Sensitivity

To reveal a relation between the sensitivity of a root of (12) for , i.e., an eigenvalue of (11), w.r.t. changes of and its sensitivity w.r.t. the nominal delay value , extensively studied in [START_REF] Niculescu | Delay Effects on Stability. A Robust Control Approach[END_REF] and the references therein, one can study the root function satisfying [START_REF] Louisell | New examples of quenching in delay differential equations having time-varying delay[END_REF] and . A simple computation shows that [START_REF] Moreau | Practical stability and stabilization[END_REF] Since is positive, a consequence for the case of roots on the imaginary axis is as follows.

Proposition 2: Assume that . Then

Thus, the behavior of the real parts of eigenvalues on the imaginary axis when changing is determined by the behavior of their imaginary parts when changing the nominal delay. This proposition and ( 18), along with Theorem 1, allow to retrieve stability information for the time-varying delay case directly from known results concerning sensitivity of stability w.r.t. the nominal delay value. This will be illustrated with the second example of the next section.

C. Computational Tools

Whereas the aforementioned analytical computations allow to study the stability of the comparison system ( 6) for small values of and to gain insight in the problem, quantitative stability information for any value of can be obtained by numerically computing the rightmost eigenvalues.

When the weight function of the distributed delay in ( 6) is bounded, both the numerical scheme proposed in [START_REF] Breda | Computing the characteristic roots for delay differential equations[END_REF], which relies on a discretization of the infinitesimal generator of the delay equation, as the numerical scheme of [START_REF] Luzyanina | Computing stability of differential equations with bounded distributed delays[END_REF], which involves a discretization of the time-integration operator, can be used to compute the rightmost eigenvalues. For the specific weight functions , respectively ,d e fined in Table I, tools for delay equations with point-wise delays, such as the software package DDE-BIFTOOL [START_REF] Engelborghs | DDE-BIFTOOL v. 2.00: A Matlab package for bifurcation analysis of delay differential equation[END_REF] are applicable. In the latter case, the distributed delay with constant weight function in [START_REF] Hale | Introduction to functional differential equations[END_REF] can be removed by a model transformation, yielding an equation with only point-wise delays, as explained in [START_REF] Luzyanina | Equations with distributed delays: Bifurcation analysis using computational tools for discrete delay equations[END_REF]. A nice feature of DDE-BIFTOOL package is its continuation facility which allows to compute stability regions in a two-parameter space semiautomatically.

IV. ILLUSTRATIVE EXAMPLES

A. First-Order System

Consider the scalar equation [START_REF] Moreau | Trajectory-based local approximations of ordinary differential equations[END_REF] whose stability regions in the plane for were determined in, e.g., [START_REF] Cooke | Discrete delay, distributed delay, and stability switches[END_REF]. The distributed delay comparison system is given by [START_REF] Moreau | Robustness of nonlinear delay equations w.r.t. bounded input perturbations: A trajectory based approach[END_REF] with characteristic equation The analytical expressions are given by ( 21)- [START_REF] Rudin | Real and Complex Analysis[END_REF].

Applying [START_REF] Michiels | Using delays and timevarying gains to improve the output feedback stabilizability of linear systems: A comparison[END_REF] to an imaginary root of and taking the real part give This quantity is negative for purely imaginary eigenvalues corresponding to values of and on the stability boundary, since such parameter values satisfy , see [START_REF] Cooke | Discrete delay, distributed delay, and stability switches[END_REF]. Therefore, the stability region of (20) in the plane enlarges when is increased from zero. Under the assumption of a symmetric weighting function, i.e., , this stability region can easily be computed analytically using the method of -subdivision [START_REF] Kolmanovskii | Stability of Functional Differential Equations[END_REF], to yield [START_REF] Niculescu | Delay Effects on Stability. A Robust Control Approach[END_REF] where the relation is implicitly defined by the curve [START_REF] Niculescu | Some remarks on the delay stabilizing effect in SISO systems[END_REF] with ( 23) and [START_REF] Rudin | Real and Complex Analysis[END_REF], as shown at the bottom of the page. Notice that for any is a positive real number, smaller than one, following from (10), (9), and . Furthermore, the specific choices of ,defined in Table I, satisfy [START_REF] Sexton | The stability of machining with continuously varying spindle speed[END_REF] From ( 23) and ( 25), one can conclude that the largest stability region is obtained for the case , where all the weight of the distributed delay lies in end-points of the interval.

In Fig. 1, we show the stability region of (20) in the plane for different values of and , respectively . In the latter case and for ,w eh a v e for . In Fig. 2(a), we plot the real parts of the rightmost eigenvalues of ( 20 show simulated trajectories of the corresponding system with time-varying delay [START_REF] Moreau | Trajectory-based local approximations of ordinary differential equations[END_REF].

B. Delayed Feedback Stabilized Oscillators

Consider a single-input-single-output system with rational transfer function , this stabilization problem was studied in [START_REF] Niculescu | Some remarks on the delay stabilizing effect in SISO systems[END_REF] and a procedure to determine a stabilizing pair (if any) was presented. The characteristic equation of the distributed delay comparison system of ( 26)-( 27) is given by When applying [START_REF] Michiels | Using delays and timevarying gains to improve the output feedback stabilizability of linear systems: A comparison[END_REF] to an imaginary root of , some simple computations yield (28) For more details, see the proof of [START_REF] Niculescu | Some remarks on the delay stabilizing effect in SISO systems[END_REF]Th. 5], where one computes the sensitivity of eigenvalues to changes of the nominal delay value , and apply property [START_REF] Moreau | Practical stability and stabilization[END_REF].

When is constant and even, expression (28) simplifies to thus increasing from zero leads to larger stability regions. As an illustration, we show in Fig. 3 stability regions of (29) in the space for two different values of . Notice that (29) concerns the case of an output feedback controlled second-order oscillator, . The nominal delay is considered also as a system parameter and concerns the relative width of the interval over which the delay is varied.

C. Variable Spindle Speed Cutting Machines

The following equation, taken from [START_REF] Jayaram | Analytical stability analysis of variable spindle speed machines[END_REF]:

(30) models one mode of a mechanical rotational cutting process, where represents the deflection of the machine tool and/or workpiece, the natural frequency, the damping ratio and the modal mass. The term , with the cutting force coefficient, models the cutting force, which depends on the time , taken by the cutting insert for one revolution of the workpiece. Clearly the time-delay is proportional to with the rotational speed of the machine. In [START_REF] Jayaram | Analytical stability analysis of variable spindle speed machines[END_REF] one assumes that this speed is varied around a nominal value in the following way: which precisely corresponds to a modulation of the time-delay of the form (31)

Based on the theory developed in this paper and the model (30), we now give an explanation for the beneficial effect of a high frequency modulating of the machine speed on increasing stability regions.

The characteristic equation of the distributed delay comparison system of (30)-( 31) is given by Fig. 4. Dashed curve separates the stable and unstable regions in the (k; )-space for (30) and (32) with m =1 0 0 ;! = 632:45; =0 :039 585, and =0 . The solid curve separates the stable and unstable regions for the distributed delay comparison system with =0 :05 .

Applying the sensitivity formula ( 16) to an imaginary root yields which is strictly negative for any value of the system parameters and any delay forcing function . Therefore, the stability region of the steady state solution can always be enlarged by "distributing" the point-wise delay over an interval, or, by virtue of Theorem 1, by modulating the point-wise delay (speed).

To illustrate this, we compute the stability limits for the system (30) and for the comparison system corresponding to a modulation of the delay (32) where is the sawtooth function described in Table I. With this choice, the kernel of the distributed delay of the comparison system is a constant. Then, as explained in [START_REF] Luzyanina | Equations with distributed delays: Bifurcation analysis using computational tools for discrete delay equations[END_REF], we can use the DDE-BIFTOOL package [START_REF] Engelborghs | DDE-BIFTOOL v. 2.00: A Matlab package for bifurcation analysis of delay differential equation[END_REF] to compute accurately the stability limits which appear in Fig. 4. Note that the coefficient depends on the width of the cutting tool and on the nominal depth of cut. Therefore, the variation of the rotating speed allows to use a larger tool and/or to remove more material at once, especially at lower speed (i.e., larger delay).

V. C ONCLUDING REMARKS

We investigated the effect of a delay variation on the stability of linear time-delay systems. Theorem 1 related the stability of a time-varying point-wise delay system with the stability of a time-invariant distributed delay system. Its proof consisted of an application of the recently developed trajectory-based proof technique, which is time-domain based and relies on a comparison of trajectories. As a mathematical tool, an extension of the Gronwall Lemma to distributed delay systems played an important role.

The stability of the distributed delay comparison system was analyzed by studying the roots of its characteristic equation. For sake of generality, the derived stability criteria, as Proposition 1, were formulated using the correction function and are thus not restricted to a particular class of delay variations (function ). The same holds for the stabilizability assertions in the examples. Only for the graphical illustrations particular choices of Table I were used.

We provided three examples where a delay variation always leads to larger stability regions in a parameter space. For the rotating cutting machine application, our results confirmed that a variation of the rotational speed can be used to allow a larger tool and/or to remove more material per turn. Furthermore, we believe that studying the eigenvalues of the distributed delay comparison systems and their numerical computation are useful new quantitative and qualitative tools for analysis and parameter tuning in such problems.

Fig. 1 .

 1 Fig. 1. Stability region of (20) in the (a; b) plane for different values of . The solid curves correspond to g(s)=g (s); the dashed curves to g(s)=g (s).The analytical expressions are given by (21)-[START_REF] Rudin | Real and Complex Analysis[END_REF].

  ) with as a function of . The solid curves correspond to , the dashed curves to . In Fig.2(b)

Fig. 2 .

 2 Fig. 2. (a) Real parts of the rightmost eigenvalues of (20) with (a; b)=( 01; 04) as a function of . The solid curves correspond to g = g (s), the dashed curves to g = g (s). (b) Simulated trajectories of the original system (19) with (a; b)=( 01;04); =1; =20;x 1, and f = f ;f ;f (above, middle, below).

Fig. 3 .

 3 Fig. 3. Solid curves separate stability-instability regions of (29) in the (k; )-plane for =0 . Stable regions are indicated with "S." For =0 :05 the stability regions of (29) are separated with the dotted curves.

  Stabilization of Time-Delay Systems With a Controlled Time-Varying Delay and Applications

Wim Michiels, Member, IEEE, Vincent Van Assche, and Silviu-Iulian Niculescu, Member, IEEE

TABLE I FOR

 I THREE TYPES OF FUNCTIONS f IN (4), THE WEIGHT FUNCTION OF THE DISTRIBUTED DELAY COMPARISON SYSTEM (6), AS WELL AS THE CORRECTION TERM g(s) IN THE CHARACTERISTIC (8) ARE SHOWN. J ( 1 ) DENOTES THE BESSEL FUNCTION OF THE FIRST KIND OF ORDER ZERO, h( 1 ) ISTHEDIRAC IMPULSE FUNCTION

	take	, respectively	in (5). Since is integrable,
	the function	is smooth, and

On the imaginary axis, has a modulus smaller than one since for all

PROOF OF THE MAIN THEOREM

We give a detailed Proof of Theorem 1. This requires a generalization of the Gronwall Lemma, which we consider first.

A. Generalization of the Gronwall Lemma

B. Proof of Theorem 1

First, we show that there always exists an integrable function satisfying [START_REF] Gu | Stability of Time-Delay Systems[END_REF]. Intuitively, for fixed , the right-hand side of (5) can be considered as the image of the function under a continuous linear functional over the space . Next, following from the Riesz Representation Theorem this image can always be expressed as a weighted integral of the function over the space (left-hand side).

In a different way, since defined as in the Theorem's statement is measurable (a bounded function on some compact), then for any -finite measure for which the right-hand integral in (5) is properly defined in the sense (see, for instance, [START_REF] Rudin | Real and Complex Analysis[END_REF]), the quantity will define a "new" measure, and we will have the equality (change of measure) with respect to the measure for any continuous function , since , and there exists such that , and . Here, is defined as follows: for any set (the measure of the inverse image ; see, for instance, [START_REF] Halmos | Measure Theory, ser. The University Series in Higher Mathematics[END_REF]). Notice that the result is still valid if is integrable, and not necessarily continuous. Next, since has zero mean, and using the previous definition, we may restrict the class of measures to those measures for which is positive. Finally, for any measure -finite which is absolutely continuous with respect to (see [START_REF] Rudin | Real and Complex Analysis[END_REF] for the definitions), there exists a (positive) measure (integrable function) such that the following condition holds:

The existence of a such a measure is guaranteed by the Radon-Nikodym theorem (see [START_REF] Rudin | Real and Complex Analysis[END_REF]). The last term can be rewritten as follows (density measure properties): which simply says that the density will define the integrable in [START_REF] Gu | Stability of Time-Delay Systems[END_REF].

We now come to the stability assertion, where we use the trajectory-based proof technique, developed in [START_REF] Moreau | Practical stability and stabilization[END_REF]- [START_REF] Moreau | Robustness of nonlinear delay equations w.r.t. bounded input perturbations: A trajectory based approach[END_REF].

For a given and , where is the Banach space of continuous functions mapping the interval into and equipped with the supremum norm , denote by and define as the unique forward solution of (4) with initial condition . Analogously, define as the unique forward solution of (6) with initial condition . For technical reasons, we temporarily restrict the state-space to ,defined as

This is possible because the following invariance property holds along the solutions of ( 4) and ( 6):

following from the fact that for all (36) and (37) where we used .

The crucial step of the trajectory based approach consists of proving uniform convergence of solutions of ( 4) and ( 6 This result can be strengthened. From (47) and (46), one can conclude that (with initial conditions restricted to ) the origin of ( 4) is globally uniformly asymptotic stability for large , by making use of the following scaling property: if is a solution of (4), then is a solution of (4), for any . Notice that the space is invariant w.r.t. a multiplication of elements with a scalar. 2 The only difference is that initial conditions are now taken in a function space. This does however not change the proof substantially. See [START_REF] Moreau | Robustness of nonlinear delay equations w.r.t. bounded input perturbations: A trajectory based approach[END_REF], where such a generalization is made.

Finally, global asymptotic stability carries over to the space , because of the following property: if then and This completes the proof.


Andrea Mattioni
email: andrea.mattioni@studio.unibo.it

Yongxin Wu
email: yongxin.wu@femto-st.fr

Hector Ramirez
email: hector.ramirez@femto-st.fr

Yann Le Gorrec
email: yann.le.gorrec@ens2m.fr

Alessandro Macchelli
email: alessandro.macchelli@unibo.it

Modelling and control of a class of lumped beam with distributed control

Keywords: Port-Hamiltonian system, IDA-PBC, medical endoscope, actuated beam Main body Flexible beam Electro-active polymer actuators

A simple lumped port-Hamiltonian model for an actuated flexible beam is proposed. The flexible beam is modelled as a n-DOF actuated beam, and the port-Hamiltonian model is constructed by a systematic interconnection of the links of the beam. The proposed model is then instrumental to derive a stabilizing controller using interconnection and damping assignment -passivity based control considering an underactuated scenario. The work has been developed motivated by the practical application to a medical endoscope with distributed actuation by electro-active polymers. The lumped parameter model offers the possibility of having input/output ports in every joint between successive links, this permits to easily model the action of the actuators as an input force applied to a specific joint.

 (Maschke andvan der Schaft, 1992, 1994;[START_REF] Van Der Schaft | L2-Gain and Passivity Techniques in Nonlinear Control[END_REF]have proven to be powerful for the modelling and control of complex physical control systems [START_REF] Duindam | Modeling and Control of Complex Physical Systems -The Port-Hamiltonian Approach[END_REF], such as multiphysical [START_REF] Falaize | Passive simulation of the nonlinear port-Hamiltonian modeling of a rhodes piano[END_REF][START_REF] Doria-Cerezo | Passivity-based control of a wound-rotor synchronous motor[END_REF], non-linear [START_REF] Ramirez | On the passivity based control of irreversible processes: A port-Hamiltonian approach[END_REF], described by partial differential equations [START_REF] Macchelli | Port-based modeling and simulation of mechanical systems with rigid and flexible links[END_REF][START_REF] Jacob | Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces[END_REF][START_REF] Ramirez | Exponential stabilization of boundary controlled port-Hamiltonian systems with dynamic feedback[END_REF] or with irreversible thermodynamic behaviour [START_REF] Ramirez | Irreversible port-Hamiltonian systems: A general formulation of irreversible processes with application to the CSTR[END_REF]. Modelling by the PHS approach is based on the characterization of energy exchanges between components of a system. The framework permits in a quite straightforward and elegant manner to interconnect the different parts of a system through energy exchange ports, hence it is well suited

The authors gratefully acknowledge the support of the ANR-DFG (French-German) project INFIDHEM with Reference Code ANR-16-CE92-0028. The first and second authors acknowledge the support of the ENSMM BQR project with Reference Code BQR ENSMM N o 06.2017 for the modelling of the medical endoscope interconnected with the ionic polymer metal composites (IPMC) actuators. On the other hand, the PHS approach is well adapted for the application of powerful passivity based control tools with clear physical interpretation, such as energy shaping and control by interconnection and damping assignment (IDA-PBC) [START_REF] Ortega | Putting energy back in control[END_REF][START_REF] Ortega | Interconnection and damping assignment: passivity-based control of port-controlled Hamiltonian systems[END_REF][START_REF] Macchelli | On the synthesis of boundary control laws for distributed port-Hamiltonian systems[END_REF].

In this paper we use the PHS formalism to model a class of n-degree of freedom (DOF) actuated beam. The class of actuated beam proposed in this work is a simple but realistic approximation of an actuated medical endoscope [START_REF] Chikhaoui | Kinematic Modeling of an EAP Actuated Continuum Robot for Active Micro-endoscopy[END_REF]. The principle of the actuated medical endoscope (Fig. 1) is that the bending of the main body is achieved with IPMC actuators, which are a class of electro-active polymer actuators. The main body of the endoscope is a flexible structure and in this work we shall consider a lumped model of interconnected links to represent it. An IDA-PBC for the PHS model of the endoscope is proposed, assuming an underactuated scenario, and numerical simulation results provided. This paper is organized as follows. Section 2 presents the PH model of the endoscope. A position controller is proposed in Section 3 by using the IDA-PBC. Simulation results are shown in Section 4. Finally, Section 5 gives some final remarks and perspectives of future work.

PORT HAMILTONIAN MODEL OF A N -DOF FLEXIBLE BEAM

The beam is modelled as n different elements connected through n joints. Between every joint there is an angular spring and a damper as shown in Figure 2. We shall assume a planar model, and so all the links are allowed to move only in the x-y plane.

Fig. 2. Lumped parameters Beam

The parameters of the n-degree of freedom mechanism, with i = 1, 2, ..., n, are:

• q i the i-th joint angular displacement;

• m i the i-th link's mass;

• I i the moment of inertia about the axis passing through the Center of Mass (CoM) of the i-th link; • a i length of the i-th link;

• a Ci distance between the i-th Joint and the CoM of the i-th link; • τ i applied torque on the i-th joint;

• K i stiffness of the i-th joint;

• c i ( qi ) viscous non-linear damping at the i-th joint; • P i , E i Potential and Kinetic energy of the i-th link.

• F 0 is the inertial frame.

• F i is the reference frame attached to the CoM and with axis parallel to principal axis of inertia of the i-th link .

The Hamiltonian function

In this subsection we derive the Hamiltonian function of the system with respect to the coordinate frame in which the system has been set up. The Hamiltonian corresponds to the total physical energy of the system which is the sum of the kinetic and potential energy. The kinetic energy of the i-th link has the form

E i = 1 2 m i v T Ci v Ci + 1 2 w T i R i Ĩi R T i w i
, where v Ci is the speed of the center of mass (CoM) of the ith link, w i is the angular speed of the i-th link with respect to F 0 , Ĩi is the inertia matrix of the i-th link with respect to F i and R i is the Rotational matrix between F i and F 0 . The goal is to express the kinetic energy of every link only with respect to the derivatives of the angular displacements. Thanks to the rigidity of the links, it is possible to relate both the speed of the CoM and the angular speeds to the derivative of the angular displacement of every joint. The relation that links angular displacements derivative to angular speeds is trivial w i = q1 + q2 + ... + qi .

Then, this relation can be expressed through the use of the so called angular Jacobian, w i = J i w q, where q is the vector containing all the angular displacement and q is the one containing all the derived angular displacement. In this case it can be seen that the angular velocity Jacobian does not depend on the angular displacements. This is not the case for the Jacobian related to the velocities of the center of mass. The velocity Jacobian of the i-th link can be found differentiating with respect to time the position of the i-th center of mass in the F 0 frame,

q Ci = x Ci y Ci = f xi (q) f yi (q) = f i (q),
where,

f xi (q) = i-1 k=1 a k cos( k j=1 q j ) + a Ci cos( i k=1 q k ), f yi (q) = i-1 k=1 a k sin( k j=1 q j ) + a Ci sin( i k=1 q k ).
Differentiating q Ci with respect to time, we obtain qCi = v Ci = dfi(q) dq q, hence the velocity Jacobian is

J i v = df i (q) dq .
Now it is possible to express the kinetic energy of every link with respect to the derivative of the displacement vector

E i = 1 2 qT (m i J iT v (q)J i v (q) + J iT w (q)R i Ĩi R T i J i w (q)) q.
The total kinetic energy of the beam is then

E = 1 2 qT M (q) q,
where M (q) is the mass matrix of the system, given by

M (q) = n i=1 [m i J iT v (q)J i v (q) + J iT w (q)R i I i R T i J i w (q)].
The mass matrix allows to relate the generalized speed with momentum of the mechanical system p = M (q) q,

where p = [p 1 p 2 • • • p n ]
T . The kinetic energy expressed as a function of the momentum is then E(q, p) = p T M -1 (q)p. In our framework we are supposing that the work plane is parallel to the ground, therefore we ignore the effect of the gravity on the dynamic of the system. Then, the potential energy is only due to the springs deformation. To find the potential energy we first define the stiffness matrix of the system

K = diag [K 1 , K 2 , • • • , K n ] .
The constitutive relation between elastic torques and springs deformation is given by τ e = Kq, hence the total potential energy can be trivially expressed as

P (q) = 1 2 q T Kq.
Finally, define the Hamiltonian as the total energy of the system, i.e., the sum of the total kinetic and total potential energy of the system H(q, p) = E(q, p) + P (q) = p T M -1 (q)p + 1 2 q T Kq. (1)

The input matrix

In this case study the beam is force actuated. Hence, it is necessary to find the input matrix which maps the normal forces at every joint into torques at each joint. Define by u ∈ R m the vector of magnitudes of the input forces applied to the beam, where m is the number of actuated joints. Define an arbitrary point of the lumped beam as δ i . The velocity of point δ i is mapped to the joint's velocity using the Jacobian matrix as follow δi = J δi v q, where J δi v is the Jacobian matrix related toan arbitrary point δ i . By conservation of energy, we have that F T δi = τ T q, where F is the vector containing the x and y components of every applied force

F = [F 1x F 1y • • • F mx F my ] T .
The velocity Jacobian matrix is constructed as the composition of all the Jacobian of every force application point

J v = J δ1 v J δ2 v • • • J δm v T ,
and it is direct to show that τ = J T v F (q), where τ is the vector of torque applied to the n different joints. Since the direction of every application force is assumed to be always perpendicular to the joint itself, the F ix and F iy components of every force can be written with respect to the force magnitude and the joint configuration. For this reason we define

f F xi (q) = -sin( lapp(i) k=1 q i ), f F yi (q) = + cos( lapp(i) k=1 q i )
where l app (i) gives the link in which is applied the i-th force. Then one has that

F xi (q) = -|F i (t)|f F xi (q), F yi (q) = +|F i (t)|f F
yi (q). Since the vector of input is composed by the magnitude of every applied force u

= |F | = [|F 1 (t)| • • • |F m (t)|]
T , the vector of applied forces can be written as F (q) = L F (q)u where,

L F (q) = diag F x1 (q) F y1 (q) , F x2 (q) F y2 (q) , • • • , F xm (q) F ym (q) .
Finally, the input matrix of the system is

g(x) = 0 n×m J T v (q)L F (q)
.

The upper part of the g(x) matrix is null because the torques don't affect the first n equations which correspond to the displacement dynamics.

The port-Hamiltonian model

Define as state vector x = [q p] , then it is straightforward to define the following port-Hamiltonian representation of the system

ẋ = (J -R) ∂H(x) ∂x + g(x)u y = g(x) T ∂H(x) ∂x (2)
with

J = 0 I n -I n 0 , R = 0 0 0 C n , with C n = diag [c 1 ( q1 ), c 2 ( q2 ), • • • , c n ( qn )]
, a positive diagonal matrix containing the viscous friction coefficients of the dampers between every joint. The structural matrix J = -J represent how the energy is internally exchanged within the system while the damping matrix R = R > 0 captures the internal dissipation of the system. It is direct to verify that the system is passive, indeed noticing that H > 0 and H(0) = 0, and taking the time derivative of the Hamiltonian

Ḣ = - ∂H ∂x R ∂H ∂x + y u ≤ y u.
For further details and considerations concerning stability and stabilization properties the reader is referred for instance to van der Schaft (2000); [START_REF] Duindam | Modeling and Control of Complex Physical Systems -The Port-Hamiltonian Approach[END_REF] 3. IDA-PBC DESIGN

In this section we design a controller for the system which allows to change configurations in an stable fashion.

To this end we employ interconnection and damping assignement passivity based control (IDA-PBC) [START_REF] Ortega | Putting energy back in control[END_REF][START_REF] Ortega | Interconnection and damping assignment: passivity-based control of port-controlled Hamiltonian systems[END_REF]. The main idea is to match the openloop dynamic of the system with the one defined by a target system and solve the resulting PDE. Define an asymptotically stable PHS target system

ẋ = (J d -R d ) ∂H d ∂x (3)
with H d a positive definite Hamiltonian function with strict minimum at the desired equilibrium. Then the feedback law

β(x) = g gg -1 (J d -R d ) ∂H d ∂x -(J -R)
∂H ∂x asymptotically stabilizes the closed-loop system provided that the matching condition

g ⊥ (J d -R d ) ∂H d ∂x = g ⊥ (J -R) ∂H ∂x , (4)
with g ⊥ a full rank left annihilator of g, i..e, g ⊥ g = 0, is satisfied. The closed-loop dynamic will then behave as (3). We shall consider that the beam is of degree n = 3, i.e., Fig. 3. Three elements force actuated beam that it contains three links, and that two of the links are actuated. The inputs the force magnitudes applied to the first and third links at a distance of a f from respectively the first and third joint. The direction of the forces is supposed to be always perpendicular to the respective link as can be seen in the Figure (3). It is important to underline that the springs and the dampers are considered as torsional, i.e., they act punctually at each joint. The input matrix is in this case

g(x) =        0 0 0 0 0 0 a f a f + acos(q 3 ) + acos(q 2 + q 3 ) 0 a f + acos(q 3 ) 0 a f       
Lets assume the desired Hamiltonian H d in the form H d (x) = H(x) + H a (x). In order to guarantee that it is strictly positive with a strict minimum at the desired equilibrium x * it should satisfy

∂H d ∂x (x * ) = 0, (5)
∂ 2 H d ∂x 2 (x * ) > 0. ( 6
)
We shall not modify the closed-loop interconnection and damping matrices, hence J d = J and R d = R. The matching condition then becomes

g ⊥ (J -R) ∂H a ∂x = 0 A possible annihilator of g is g ⊥ (x) =    1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -a f a f + acos(q 3 )   
which leads to the following matching equations

∂H a ∂p 1 = ∂H a ∂p 2 = ∂H a ∂p 3 = 0 a f ∂H a ∂q 2 -(a f + a cos(q 3 )) ∂H a ∂q 3 = 0 (7)
The first line of ( 7) implies that H a cannot depend on the momentum variables, while the last is a partial differential equation imposes a relation between the displacement variables q 2 and q 3 . Since H a cannot depend on momentum variables, the only part of the total energy that we can modify is the one depending only on displacement variables, i.e. the potential energy. From (1), we define the desired Hamiltonian as the composition of the desired potential energy and the desired kinetic energy H d (q, p) = E d (q, p) + P d (q) From the first equation of ( 7) we obtain E d (q, p) = E(q, p). Since the only part of the energy that can be modified is P d (q), ( 5) and ( 6) become ∂P d ∂q (q * ) = 0, (8)

∂ 2 P d ∂q 2 (q * ) > 0. ( 9
)
The general solution of ( 7) is

H a = H a     q 1 , -q 2 - 2a f tanh -1 (a-a f ) tan( q 3 2 ) a 2 -a 2 f a 2 -a 2 f     (10)
A simple form of (10) leads to the following solution

∂H a ∂q 2 = κq * 2 ∂H a ∂q 3 = a f a f + a cos(q 3 ) κq * 2 (11)
The part of H a which depends on q 1 , i.e., H a (q 1 ), can be freely chosen, since there is no condition on ∂Ha ∂q1 . Then, H a (q 1 , q 2 , q 3 ) can be found by direct integration of ( 11). Once H a has been chosen it remains to check that P d is a Lyapunov function for the closed-loop system. This is achieved verifying that (8) and ( 9) are satisfied. The forced equilibrium position, and the corresponding steady state input, is computed from the dynamic model of the system (2). Since the system has two inputs we shall impose q * 1 and q * 2 , such that q * 3 is a function of q * 1 and q * 2 . Choosing H a such that ∂P d ∂q1 = K 1 (q 1 -q 1 * ), and making κ = -K 2 we obtain

∂P d ∂q =    K 1 (q 1 -q * 1 ) K 2 (q 2 -q * 2 ) K 3 q 3 - a f K 2 q * 2 a f + acos(q 3 )    It is straightforward to verify (8), ∂P d ∂q (q * ) = 0 ∀q * 1 , q * 2 ∈ [0, 2π].
On the other hand, condition (9) leads to

∂ 2 P d ∂q 2 (q * ) =    K 1 0 0 0 K 2 0 0 0 ∂ 2 P d ∂q 2 3 (q * )    where ∂ 2 P d ∂q 2 3 (q * ) = K 3 -K 2 a f q * 2 asin(q 3 ) (a f + acos(q 3 )) 2
Numerically, using the parameters in Table 1, we obtain that

∂ 2 P d ∂q 2 3 | x * > 0 ∀q * 1 ∈ R, q * 2 ∈ [-0.75π, 0.75π].
The stable configuration of q * 3 is computed by q * 1 and q * 2 . Hence, we obtain that the stable configuration space that can be reached with the proposed controller is given by q

* 1 q * 2 q * 3 ∈ -π, π -0.75π, 0.75π -0.33π, 0.33π .
The controlled beam with proposed method cannot move over all the x -y plane. However, from a practical point of view, this stable configuration space is large enough for the clamped medical endoscope beam.

SIMULATION RESULTS

In this section, simulation results are shown using a beam modelled with 3 elements. The parameters used in the simulation are resumed in Table 1.

Open-loop response

The free oscillations of the beam starting from an initial condition q 0 = [+ π 8 , -π 8 , + π 6 ] and p 0 = [0, 0, 0] are q 2 q 3 Fig. 4. Free Beam oscillations illustrated in Figure 4. The beam goes back to its natural equilibrium position q = [0, 0, 0] T with external forces equal to zero. We can observe that the beam is badly damped and the presence of large oscillations.

m 1 = m 2 = m 3 = 0, 0017 [kg] I 1 = I 2 = I 3 = 1, 3801x10 -6 [kg * m 2 ] a 1 = a 2 = a 3 = 0, 1 [m] a C1 = a C2 = a C3 = 0, 05 [m] K i = 10 i = 1, 2, 3 [ N m ] c i ( qi ) = 0.05| tan -1 ( qi * 4)| + 0.03 i = 1, 2, 3 [P a * s]

Closed-loop response

In this sub-section the IDA-PBC feedback law is illustrated. In a first instance a controller that doesn't modify the closed-loop stiffness of the first spring is employed, while in a second instance a controller which modifies this stiffness is used.

Unmodified stiffness

The controller is defined setting

H a = + 1 2 K 1 q * 2 1 -q * 1 q 1 K 1 + 1 2 K 2 q * 2 2 -q * 2 q 2 K 2 - 2a f q * 2 K 2 tanh -1 ( (a-a f ) tan( q 3 2 ) a 2 -a 2 f ) a 2 -a 2 f (12)
where one has the possibility of choosing q * 1 and q * 2 , begin q * 3 a function of the other equilibria. This leads to

∂H a ∂x =         -K 1 q * 1 -K 2 q * 2 - a f a f + acos(q 3 ) K 2 q * 2 0 0 0        
.

The time responses of the controlled three element beam with different initial positions are presented. In the first simulation we set the natural equilibrium position as the initial condition, i.e. all the angular displacements equal to zero and the final equilibrium position set as q * = [q * 1 q * 2 q * 3 ] = [0.5 0.52 0.15]. The initial position and the final position are shown in Fig. 5 and Fig. 6 respectively. The time responses of the angular displacement of every joint is shown in the Fig. 7. We observe that the response time is faster than the for the open-loop system and with less oscillations. The angle of every joint converges to the desired equilibrium position as expected.

In the second simulation, we consider a "non-natural" initial condition, q 0 = [-0.4 -0.8 0.7], shown in Fig. 8. The desired equilibrium position is always the same as in q 2 q 3 Fig. 9. Beam time response with worse initial condition appreciate from Fig. 9 that every joint reach the desired equilibrium position. Moreover, compare to the previous simulation, the response time is the same, however, one can see that the overshoot of the response is larger. That is because in this case, the equilibrium position is further away to the initial position than in the previous case.

Modified Stiffness Now we define the controller Hamiltonian such that

H a = - 1 2 K 1 q 2 1 + 1 2 K 1 (q 1 -q * 1 ) 2 + 1 2 K 2 q * 2 2 -q * 2 q 2 K 2 - 2a f q * 2 K 2 tanh -1 ( (a-a f ) tan( q 3 2 ) a 2 -a 2 f ) a 2 -a 2 f
where the stiffness of the first link in the closed-loop system has been modified to

K 1 = 50 [ N m ].
This lead to

∂H a ∂x =         -K 1 q 1 + K 1 (q 1 -q * 1 ) -K 2 q * 2 - a f a f + acos(q 3 ) K 2 q * 2 0 0 0        
.

The initial and final configurations are the same as before.

The time response of the angular displacement can be seen in Fig. 10. Comparing Fig. 9 with Fig. 10, it is possible to notice that the response time has been significantly decreased this case. This is due to the augmentation of the stiffness of the first joint. 0 0.02 0.04 0.06 0.08 -1 -0.5 0 0.5 1 q 1 q 2 q 3 Fig. 10. Beam time response with modified stiffness.

CONCLUSION

The PHS framework has been used to model and control an actuated medical endoscope system. The endoscope is modelled as a lumped parameter flexible beam. A position control, that uses forces as inputs, has been proposed for this model using the IDA-PBC. The proposed model and control law have been illustrated by means of numerical simulations. The closed-loop system behaves in a satisfactory manner using the quite simple approach proposed in this work. One of the interesting aspect of the stability analysis is that it is possible to characterize the stable configuration space a priori. Ongoing work deals with potential non-linearity of the stiffness of the beam, the robustness of the proposed control and the experimental implementation of the controller.

Fig. 1 .

 1 Fig. 1. Medical endoscope and its simplified model 1. INTRODUCTION Port-Hamiltonian systems (PHS)(Maschke and van der Schaft, 1992, 1994;[START_REF] Van Der Schaft | L2-Gain and Passivity Techniques in Nonlinear Control[END_REF] have proven to be powerful for the modelling and control of complex physical control systems[START_REF] Duindam | Modeling and Control of Complex Physical Systems -The Port-Hamiltonian Approach[END_REF], such as multiphysical[START_REF] Falaize | Passive simulation of the nonlinear port-Hamiltonian modeling of a rhodes piano[END_REF][START_REF] Doria-Cerezo | Passivity-based control of a wound-rotor synchronous motor[END_REF], non-linear[START_REF] Ramirez | On the passivity based control of irreversible processes: A port-Hamiltonian approach[END_REF], described by partial differential equations[START_REF] Macchelli | Port-based modeling and simulation of mechanical systems with rigid and flexible links[END_REF][START_REF] Jacob | Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces[END_REF][START_REF] Ramirez | Exponential stabilization of boundary controlled port-Hamiltonian systems with dynamic feedback[END_REF] or with irreversible thermodynamic behaviour[START_REF] Ramirez | Irreversible port-Hamiltonian systems: A general formulation of irreversible processes with application to the CSTR[END_REF]. Modelling by the PHS approach is based on the characterization of energy exchanges between components of a system. The framework permits in a quite straightforward and elegant manner to interconnect the different parts of a system through energy exchange ports, hence it is well suited

Fig. 7 .

 7 Fig. 5. Initial position with q 0 = [0, 0, 0]

Fig. 6 .Fig. 8 .

 68 Fig. 8. Initial position with q 0 = [-0.4, -0.8, 0.7]

Table 1 .

 1 Actuated beam parameters

	0.4
	0.2
	0
	-0.2				q 1
	-0.4
	-0.6	0	0.02	0.04	0.06	0.08


Christophe Quéva: Returns to the East: German internal migrations and local development in East Germany Returns to the East: German internal migrations and local development in East

Keywords: East Germany, Internal migrations, Demographic dynamics, Economic transformations, Spatial planning Zusammenfassung Binnenwanderung und lokale Entwicklung in Ostdeutschland Der Artikel betrachtet das Phänomen der Binnenwanderung in Deutschland nach der Wiedervereinigung Potenzial betrachtet werden? Ziel dieses Artikels ist, die Nachhaltigkeit von Rückkehrbewegungen im Osten auf drei Ebenen zu

The paper focuses on the phenomenon of internal migrations in Germany after German Reunification. The eastern part of Germany (the former GDR) lost a large share of its population due to internal migration from East to West Germany, resulting in a concomitant process of demographic and economic decline. However, in recent years, some regions such as rural areas and territories near the former inner-German Border are experiencing a re-migration process (from West to East Germany), with an economic renewal. To what extent could the re-migration to East Germany be considered as a potential demographic and economic resource? The aim of this article is to question the sustainability of return movements in the East at three levels: demographic and migratory dynamics, economic transformations, and public policies which support these mutations.

Introduction

"Nearly half the people leaving the West to settle in East Germany are originally from the East. Their return stabilises demographic dynamics and the labour market in the new Länder", personal translation 1 [START_REF] Matuschewski | Stabilisierung der Regionalentwicklung durch Rückwanderung? Theoretische Konzeptionalisierung und empirische Umsetzung am Beispiel von Ostdeutschland[END_REF].

This statement summarises Anke Matuschewski's article (2010) from the perspective of an economic stabilisation in the new Länder, partly linked to the returns to the East. It illustrates what is at stake in terms of migrations: in the current context, East Germany is being marked by demographic decline and population ageing, as well as by high economic fragility. Migrations from West to East Germany are not a new phenomenon, and since the Reunification they have occurred in parallel to the numerous migrations from the East to the West, although to a lesser degree. We can, however, note that there has recently been a double shift in migration dynamics. Statistically, movements to the West have kept decreasing since 2001, whereas movements to the East are stable; indeed, recently they have even tended to increase very slightly. Thus, the gap between migrations to the West and those to the East is continuously narrowing. Moreover, in qualitative terms, going towards the East (Zurück nach Osten) or returning to the East (Zurück in den Osten) is associated with new motivations; these include family or work migration patterns. Such patterns are much encouraged by public policies, which consider these return movements as a potential demographic and economic resource.

The aim of this article is to analyse the sustainability of these return movements and their implications for the concerned territories. In light of migration statistics, ongoing research projects in Germany, and an increasing number of scientific publications on the returns to the East, we will attempt to identify the extent to which these return movements can be considered as vectors 1 "Etwa die Hälfte der Menschen, die von West-nach Ostdeutschland umziehen, sind gebürtige Ostdeutsche. Ihre Rückwanderung stabilisiert Bevölkerungsentwicklung und Arbeitsmarkt in den neuen Ländern" (Matuschewski 2010, p. 81).

of a progressive readjustment of the demographic and economic fragilities that East Germany has experienced since the Reunification. Given the continuing works undertaken in Germany to address these issues, my hypothesis is that the current dynamics could arise from a structural rather than cyclical logic; this could therefore contribute to the demographic and economic readjustment of a certain number of areas in East Germany that have been weakened by departures to the West and population ageing. However, this logic can only be applied if local stakeholders (migrants, elected representatives, associations, etc.) demonstrate the will and the local capacity to support these return movements. This study thus aims to investigate the demographic sustainability of the returns to the East; this topic implies, as a backdrop, issues of socio-economic sustainability, and even of territorial sustainability. The latter issue involves the intergenerational transmission of socially and economically balanced territories, given that the environmental dimension of sustainability is not directly addressed in this article.

Understanding returns to the East: time, places and stakeholders

The return movement to the East -Rückwanderung -is a recent concept that is complex to comprehend. As a part of global migration dynamics, it is marked by regular and massive population movements between East and West Germany since the Reunification. Moreover, returns to the East are far from concerning all new Länder, and are characterised by a fragmentary and complex logic. The process is also generated by individual patterns and motivations; thus, we will attempt to represent these logics in order to grasp their potential similarities.

An insight into German migration dynamics

The Reunification is associated with a vast migration movement from the East to the West, with a million and a half individuals leaving the new Länder for the West between 2000 and 2007. Yet, a million people simultaneously went the opposite way, to (re)settle in the East [START_REF] Herfert | Campagnes et villes face au déclin démographique de l' Allemagne orientale: de nouveaux enjeux pour l'aménagement du territoire[END_REF]. In this context, and without considering the specific situation of the capital, Berlin, we can observe in the graph below (Fig. 1) that, although East-West migrations remained higher than West-East migrations between 1990 and 2015, the total volume of migrants nevertheless tended to converge during the same time period. Moreover, in terms of the balance between East-West and West-East migrants, the fluctuations seen in 1990-1991 and 2000-2001 were replaced with more regular patterns and a progressive decline of this difference up until 2014 (a balance of -4,804 persons in 2015 (-10,497 in 2013) The tendencies identified could demonstrate a structural change in migration dynamics, in terms of the reduction of departures to the West and a parallel movement of stabilisation or minor increase of returns to the East; though the latter numbers are very low, they are worth noting. According to Anke [START_REF] Matuschewski | Stabilisierung der Regionalentwicklung durch Rückwanderung? Theoretische Konzeptionalisierung und empirische Umsetzung am Beispiel von Ostdeutschland[END_REF], a fifth of the individuals who went West after the Reunification have now come back to the East, in or near their region of origin. We shall now try to spatially frame this logic of returns.

The spatial logic of return movements

Although several studies started to address the subject of returns to the East as early as the mid-2000s [START_REF] Beck | Wandern gegen den Strom: West-Ost-Migration in Deutschland[END_REF][START_REF] Schulz | Wandern und Wiederkommen? Humankapitalverlust und Rückkehrpotenzial für Mecklenburg-Vorpommern[END_REF][START_REF] Dienel | Zu-und Rückwanderung nach Ostdeutschland. Bevölkerungsstabili sierung ostdeutscher Städte durch West-Ost-Wanderung[END_REF][START_REF] Jain | Motive für die Wanderung von West-nach Ostdeutschland und Rückkehrtypen[END_REF][START_REF] Wiest | Rückwanderung nach Ostdeutschland: Erfolg bremst Heimkehrneigung[END_REF]Kubis a. Schneider 2010;[START_REF] Matuschewski | Stabilisierung der Regionalentwicklung durch Rückwanderung? Theoretische Konzeptionalisierung und empirische Umsetzung am Beispiel von Ostdeutschland[END_REF]Lacquement 2001), the spatial and temporal logics of return movements to the whole of East Germany were largely formalised for the first time in the recent works of a team of researchers from the Leibniz Institute for Regional Geography in Leipzig (Lang a. Hämmerling 2013; Nadler a. Wesling 2013; Nadler a.

Wesling 2014), in the framework of a European research programme (Re-turn/Regions benefiting from returning migrants, http://www.re-migrants.eu/). Based on a statistical study and a survey of 400 workers who had migrated to the West before resettling in the East between 2000 and 2010, their research was greatly disseminated in the scientific sphere via Nationalatlas Deutschland and the national press (Die Welt, Die Zeit, etc.). The authors first highlighted the transformation of the areas from which migrations to the East originated. Between 2000 and 2005, the issuing Länder mainly bordered East Germany (particularly Schleswig-Holstein, Bremen and Bavaria), whereas between 2006 and 2011, the Länder concerned were further away, such as Rhineland-Palatinate and Saarland. The comparative map (2001-2005 and 2006-2010) of workers' return movements to the East at a district level (Fig. 2) enables us to pinpoint the settlement patterns of migrants from the West. Besides Berlin, the Saxony and Thuringia Länder welcomed the largest numbers of returning migrants, as did the most touristic rural regions, more specifically on the Baltic coast. As for the places of arrival of these return movements, it may be noted that migrants' main destinations in East Germany are the rural areas: "Basically, the return rate in cities is much lower than in rural areas (...). Whether this is due to different framework conditions such as significantly higher cost of living in the cities than in the countryside, or to different individual preferences such as a greater attachment to the region of origin in the countryside, cannot be derived from the data. An independent survey of the returnees would be necessary to establish this" (personal translation)2 [START_REF] Fuchs | Rückwande rung von Erwerbspersonen -aktuelle Deutschlandzahlen im regionalen Vergleich. In: Leibnitz-Institut für Länderkunde[END_REF].

Between 2001 and 2005, as Robert Nadler and Mirko Wesling remarked, the districts that border older Länder were the most attractive, especially in Thuringia, Saxony-Anhalt and Mecklenburg-Western Pomerania (Nadler a. Wesling 2013, p. 2). These areas are linked to significant daily commuting movements between East and West Germany [START_REF] Quéva | Mobilités quotidiennes et construction métropolitaine en Allemagne : le rôle structurant de l'ancienne frontière interallemande[END_REF]. For instance, in 2012, 15 % of Thuringian employees worked in another Land (12 % in 1999, or one out of seven Thuringian workers). These mobilities mainly tended towards older Länder: Bavaria first, then Hessen [START_REF] Quéva | Mobilités quotidiennes et construction métropolitaine en Allemagne : le rôle structurant de l'ancienne frontière interallemande[END_REF]. However, as illustrated in Figure 2, between 2006 and 2011 the patterns became more complex, and return movements progressed further than the former inner-German border (Nadler a. Wesling 2013, p. 1). Return movements therefore seemed to spatially expand to a growing part of East Germany, especially in rural areas [START_REF] Fuchs | Rückwande rung von Erwerbspersonen -aktuelle Deutschlandzahlen im regionalen Vergleich. In: Leibnitz-Institut für Länderkunde[END_REF]. Beyond these areas marked by returns to the East and the tendencies mentioned, there are

East-West and West-East migrations between 1990 and 2015

Sources: Statistisches Bundesamt (1990Bundesamt ( -2011)); R. Nadler et M. Wesling (2013, p. 8);Statistisches Jahrbuch 2016 (p.45) IfL 2018 Author, contents: C. Queva Design: N. Amberg individual trajectories and various motivations that are worth mentioning.

* excluding Berlin

Profile and motivations of Rückwanderer

The profile of Rückwanderer -a notion that could be translated as "migrants returning to their region of origin" -has been analysed in several studies and on different time scales [START_REF] Beck | Wandern gegen den Strom: West-Ost-Migration in Deutschland[END_REF] she explains that the studied employers tried to attract Rückwanderer because of their knowledge of the region, of their involvement in the local economy and life, and because there is a lesser chance of them re-emigrating to the West after returnin to the East [START_REF] Matuschewski | Stabilisierung der Regionalentwicklung durch Rückwanderung? Theoretische Konzeptionalisierung und empirische Umsetzung am Beispiel von Ostdeutschland[END_REF]. In this respect, we need to assess the economic and political strategies that are implemented to encourage and support return migrations, both before and after people return to the East.

Supporting returns to the East: the role of public policies

In addition to the Rückwanderer, it is important to consider the role of public and private stakeholders who try to benefit from return migrations. In order to understand their strategies, we will first examine how return migrations can be a potential resource for demography and the economy. We will then be able to

Migration as a potential resource

In terms of population, Figure 1 illustrates how returns to the East have grown to balance internal migrations in Germany: we have seen that the number of arrivals in older Länder tends to fall closer to that of arrivals in new Länder, which could demonstrate an ongoing structural trend of rebalancing. Furthermore, as we have also already seen, migrants settling in the East are mostly workers, and fairly young (25-35 years old); this suggests that the population ageing phenomenon, which is often mentioned in studies dealing with East Germany, could be reduced in the long term. However, although East/West inequalities in terms of net migration may diminish, demographic differences are still significant, especially in terms of population ageing. The two maps below (Fig. 3 and4) illustrate how complicated the generational rebalancing appears to be, with an under-representation of the youth (Fig. 3) and an overrepresentation of the elderly (Fig. 4) in East Germany. Returns to the East must thus be viewed in a demographic perspective.

In economic terms, returns to the East could potentially constitute a real opportunity to maintain employment and to create new activities. As the report by the Federal Ministry for Economic Affairs and Energy explains in relation to the differences in representation between East and West Germany, young companies could benefit from a more favourable context in the East, due to local and family support: "In regions of origin, social networks and local support can be activated more easily for young companies. This is a typical East German factor that is triggered during the setting-up of new companies. Family plays a more important role than in the West", personal translation 4 (Bundesministerium für Wirtschaft und Energie 2014). This explanation must naturally be nuanced, as the East German context presents an increasingly spatially contrasted labour market: Figures 5 and 6 illustrate the unemployment rates per district at two different points in time. Although in 2006, all districts in the new Länder were marked by very high unemployment rates, above 11.4 % and even sometimes over 16 %, in 2014 these rates fell to less than 6.1 % in the western districts of Thuringia (located near the former inner-German border). These numbers must obviously be viewed in the context of the whole country, where the unemployment rate overall decreased between 2006 and 2014 -diminishing during this period from 12 % to 7.5 % in Germany, and from 19.2 % to 11 % in the new Länder. In the of East Germany, where the unemployment rate was 11 % in 2014 for all new Länder, these particularly low rates can largely be explained by the high numbers of return migrations to these districts (see Figure 2) and by daily commuting mobilities between both sides of the former inner-German border [START_REF] Quéva | Mobilités quotidiennes et construction métropolitaine en Allemagne : le rôle structurant de l'ancienne frontière interallemande[END_REF]. For instance, in 2016, the unemployment rates in the "border" districts of Thuringia were 5.1 % for Wartburgkreis, located near Hessen; 4.4 % for the Hildburghausen district (near Bavaria); and even as low as 3.9 % for the Sonneberg district (also near Bavaria). As for the districts in the north of the Land, far from the former border and marked by a lower number of returns (Fig. 2), their unemployment rates exceed 9 % in 2016 (9.2 % in Altenburger Land and 9.9 % in Kyffhaüserkreis). Besides these economic advantages of maintaining or creating employment, there are also territorial issues at stake, such as reinforcing access to services, which requires the involvement of public authorities in supporting these return movements.

Towards strategies to encourage returns to the East?

Numerous mechanisms have been implemented in Germany to accompany the country's demographic changes. On a national scale, a financial support scheme (Modellvorhaben "Demografischer Wandel -Region schafft Zukunft"), which focused on service provision in regions marked by demographic weaknesses, was implemented between 2007 and 2011. It is still ongoing today in the rural regions, through the "Aktionsprogramm Daseinsvorsorge", which selected 21 model regions (including nine in East Germany) to receive between 70,000 and 180,000 euros, in order to carry out a sustainable strategy to strengthen service provision (schools, nurseries, healthcare, assistance for seniors, etc.). Concerning the more specific issue of returns to the East, it is often the Länder or local authorities themselves who are in charge of implementing projects and strategies to welcome and integrate newcomers, especially the Rückwanderer. This is, for example, the case for the Dresden Chamber of Commerce and Industry, in Saxony (Fig. 7): it has created the "Sachse komm zurück" portal, which explicitly refers to return migrations, to "cross-border" mobilities to older Länder, and even to the Heimat. The region is presented as a booming area, partly due to community relationships.

There are also projects of action research which are directly focused on returns to the East, such as the portal "Menschen für Ostdeutschland" (Fig. 8). This portal reports research carried out in the framework of a programme entitled "Rückwanderung als dynamischer Faktor für ostdeutsche Städte" (Return In this perspective, these initiatives attempt to constitute a network of relations and dialogue between migrants who have gone to the West and those who have re-

Conclusion

The recent intensification of scientific study of return movements to the East, combined with action research programmes to support and encourage these returns, underline the importance of this issue in terms of comprehending the current demographic and economic changes in a country in transition, especially in the East and in rural areas. labour market, which can explain why certain individuals are reluctant to do so: "Only a quarter of the persons who have already returned to the East assert that their return was difficult or very difficult; but three quarters of individuals ready to go back to the East expect their return to be difficult or very difficult. The main reason mentioned by both groups is the labour market in East Germany", personal translation 7 (Lang a. Hämmerling 2013). The demographic and economic sustainability of the returns to the East will therefore be highly nuanced, and must be considered in a long time-frame. We will only be able to determine if a genuine rebalancing is tak-

  from East to West Number of migrants from West to East Balance between migrants going West and those going East

Fig. 1 :

 1 Fig.1:East-West and West-East migrations in Germany between 1990 and 2015

Fig. 3 :

 3 Fig. 3: Proportion of young persons under 17 years old in 2015

  4 "In der Heimat können soziale Netze für das junge Unternehmen besser aktiviert und familiäre Unterstützung mobilisiert werden. Damit ist ein typisch ostdeutsches Element der Gründungsphase angesprochen. Die Familie spielt eine ungleich wichtigere Rolle als im Westen" (BundesMinisteriuM Für wirtschaFt und energie 2014t h R h i n e -W e s t p h a l i a P a l a t i n a t e W e s t e r n P o m e r a n i a Source: Regionalatlas Deutschland https://www-

Fig. 4 :

 4 Fig. 4: Proportion of persons over 65 years old in 2015

  t h R h i n e -W e s t p h a l i a P a l a t i n a t e W e s t e r n P o m e r a n i a in %

Fig. 5 :

 5 Fig. 5: Unemployment rate in 2006

  t h R h i n e -W e s t p h a l i a P a l a t i n a t e W e s t e r n P o m e r a n i a

  Workers going back to the East between 2001 and 2005Workers going back to the East between 2006 and 2010

	these different categories of Rückwan-The "returns to the East" of workers between 2001-2005 and 2006-2010 derer settle: large cities for the first group, according to NADLER and WESLING 2013 former inner-German border regions for
	the second group, and regions of origin
	(urban or rural) for the last category.
	However, according to the different sur-
	Schleswig-Holstein Hamburg veys carried out in the works mentioned Mecklenburg-Western Pomerania above, the motivations for migration are similar for all three categories. Going to			Schleswig-Holstein Hamburg	Mecklenburg-Western Pomerania
	Bremen the West is mainly motivated by reasons				Bremen
	of salary, career opportunities and train-
	Lower Saxony ing courses, whereas individuals return to	Berlin		Lower Saxony	Berlin
	Saxony-Anhalt the East to reunite with family or friends, or for a better quality of life (cost of liv-	Brandenburg			Saxony-Anhalt	Brandenburg
	North Rhine-Westphalia ing especially). The relationship with the region of origin (the Heimat) is thus an			North Rhine-Westphalia
	Thuringia important key to understand these return	Saxony			Thuringia	Saxony
	Hesse movements; Eastward migrations mostly				Hesse
	demonstrate a logic of integration, if not
	Rhineland-of grounding. This integration is spatial			Rhineland-
	Palatinate (individuals return to their place of birth,			Palatinate
	Saarland of study, or near where their families live),			Saarland
	Bavaria but also social: "Social networks are of-				Bavaria
	Baden-Württemberg ten important for individuals who have			Baden-Württemberg
	'socialised' in the East, and perhaps even
	more important for West Germans who	0	100 km
	appear to be more individualistic, accord-
	Lake Constance ing to surveyed participants", personal			Lake Constance
	Proportion of workers having moved translation 3 (Bundesministerium für		Proportion of workers having moved
	to one of the older Länder before returning to a new Länder Wirtschaft und Energie 2014).		to one of the older Länder before returning to a new Länder
	in % This differentiated perception between 28.2		in %	18.6
	12 9 6 East and West Germans is paradoxi-cal, as returning migrants also insist on			12 9 6
	how the West-East dichotomy has now
	international boundary been overcome, in view of the patterns of
	state boundary mutual integration and adaptation (Ma-regional boundary tuschewski 2010; Bundesministerium				IfL 2018 Author: C. Queva, according to NADLER & WESLING 2013 Cartography: A. Kurth
	für Wirtschaft und Energie 2014).
	Fig. 2: The "returns to the East" of workers between 2001-2005 and 2006-2010 Nonetheless, returning movements to the
	East are based on a perception that life in
	the West is unsatisfactory. Similarly, dif-	2013). Three main categories identified	(surveys) and on limited sample groups.
	ferences in mentality between both sides	in these different works are as follows:	For instance, the survey "Re-Turn" (on-
	of Germany are highlighted in the surveys	young graduates (25-35 years old) who	line questionnaire) implemented by the
	implemented by Anke Matuschewski;	left to study in the West before return-	Leibniz-Institut, Leipzig Institute of Geog-
		ing to work in the East; people working	raphy, concerned a sample group of 400
		in the West but living in the East; and	individuals. As for the survey of reference
		retired persons who had left to work in	in Anke Matuschewski's article (2010), it
		the West but who returned to the East	focused on 50 individuals who had reset-
		for their retirement. It is difficult to num-	tled in Mecklenburg-Western Pomerania
	; Jain a.	ber the proportion of these categories,	and Saxony, and on 50 businesspersons
	Schmithals 2009; Matuschewski 2010;	as the different works dealing with this	from East Germany. We can nevertheless
	Nadler a. Wesling 2013; Scheffel	subject are based on qualitative analyses	note differences in the locations where

'Est: migrations de retour et développe- ment local à l'Est de l'Allemagne Dans

  статье рассматривается феномен внутренней миграции после объединения Германии. Восточная часть Германии (бывшая ГДР) потеряла большую часть населения в результате оттока жителей из Восточной в Западную Германию. Отток населения имел серьезные последствия для экономики (нехватка специалистов, снижение спроса на потребительские товары), а экономический спад в годы после объединения Германии, повлекший за собой безработицу, побудил многих людей мигрировать. Однако в последние годы в некоторых регионах, например, в сельских районах и в районах, расположенных рядом с бывшей внутригерманской границей, наблюдается процесс обратной миграции (из Западной Германии в Восточную), сопровождающийся экономическим подъемом. Насколько можно рассматривать обратную миграцию в Восточную Германию как явление, обладающее демографическим и экономическим потенциалом? Цель данной статьи заключается в анализе обратной миграции на восток страны с трех точек зрения: демографической и миграционной динамики, изменения экономической структуры и государственной политики, поддерживающей эти изменения. un contexte de fragilité démographique et économique à l'Est de l' Allemagne (ex-RDA), un certain nombre d'espaces connaissent paradoxalement un ralentissement du déclin démographique, parfois accompagné d'un renouveau économique. En effet, si la Réunification s'est associée à un vaste mouvement migratoire de l'Est vers l'Ouest, c'est, en parallèle, un million de personnes qui ont suivi un mouvement inverse, en venant se (ré)installer à l'Est. Dans ce cadre, le mouvement de retour (Rückwanderung) concerne deux principaux types d'espaces: les espaces ruraux et les espaces situés à proximité de l'ancienne frontière interallemande. Ces »retours à l'Est« peuvent-ils être envisagés comme des vecteurs de rééquilibrage progressif des pertes démographiques subies depuis la Réunification? Pourraient-ils également contribuer à restructurer économiquement une partie de ces territoires? L'enjeu de cet article est d'interroger la durabilité des mouvements de retour à l'Est à trois niveaux: celui des dynamiques démographiques et migratoires, celui des transformations économiques, et enfin celui des politiques publiques qui accompagnent l'ensemble de ces mutations.

	Peзюме	Résumé
	Кристоф Кева	Retours à l
	Назад на Восток: внутренняя миграция и локаль-
	ное развитие Восточной Германии
	Восточная Германия; внутренняя миграция; демографи-
	ческие изменения; экономические изменения; региональное
	развитие

7 "Lediglich ein Viertel der bereits Zurückgekehrten gibt an, dass die Rückkehr (sehr) schwer war. Demgegenüber erwarten fast drei Viertel der Rückkehrbereiten, dass die Rückkehr (sehr) schwer wird. Als Hauptursache für die Schwierigkeiten bei der Rückkehr nennen beide Gruppen die Arbeitsmarktsituation in Ostdeutschland" (Lang a. häMMerLing 2013).

ing place if we continuously examine these dynamics in the following years Fig. 8: "Menschen für Ostdeutschland" portal homepage, created by the Nexus Institute (Institut für Kooperationsmanagement und interdisziplinäre Forschung GmbH), Leipzig University and the Stendhal School of Magdeburg Source: http://nexusinstitut.de/de/arbeits-forschungsbereiche/buergergesellschaft-globalisierung-und-identitaet/41-abgeschlosseneprojekte/204-rueckwanderung. Consulted on September 26, 2018 В Allemagne de l'Est; migrations intérieures; dynamiques démographiques; transformations économiques; aménagement

"Grundsätzlich ist die Rückkehrquote in Städten deutlich geringer als in ländlichen Gebieten (...). Ob dies an unterschiedlichen Rahmenbedingungen wie deutlich höheren Lebenshaltungskosten in den Städten als auf dem Land oder an unterschiedlichen individuellen Präferenzen wie eine größere Bindung an den Ursprungskreis auf dem Land liegt, lässt sich aus den Daten nicht ableiten. Hierfür wäre eine eigenständige Befragung der Rückkehrer notwendig)"(Fuchs et al. 2017, p. 2)

"Soziale Netze sind für Menschen, die im Osten sozialisiert wurden, offenbar sehr wichtig möglicherweise auch wichtiger als für Westdeutsche, die nach Eindruck mehrerer Interviewpartnerinnen und -partner stärker individualistisch geprägt sind" (BundesMiniste-riuM Für wirtschaFt und energie 2014).