O Kresse

L Truskinovsky

Mobility of lattice defects: discrete and continuum approaches

Keywords: A. Phase transformations, Defects, B. Dynamics, Lattice model, C. Kinetic relations, Pinning

In this paper, we study a highly idealized model of a moving lattice defect allowing for an explicit, "rst principles" computation of a functional relation between the macroscopic cong-urational force and the velocity of the defect. The discrete model is purely conservative and contains information only about elasticities of the constitutive elements. The apparent dissipation is due to the presence of microinstabilities and the nonlinearity-induced tunneling of the energy from long to short wavelengths. This type of "radiative damping" is believed to be generic and accounting for a considerable fraction of inelastic irreversibility associated with fracture, plastic-ity and phase transitions. The paper contains direct comparison of the exact lattice solution with various continuum and quasicontinuum approximations. Despite its simplicity, the model can be used directly for the description of dynamic phase transitions in thin lms.

Introduction

Irreversible processes in solids can usually be traced to the motion of various lattice defects. Typically, the migration of these defects is a dissipative process with the energy loss in the core region manifesting itself through the localized increase of temperature. In the classical continuum elasticity the drag forces restraining the motion of the defects are absent and they can propagate freely and uniformly with arbitrary velocity. This "freedom" has long been realized as a source of nonuniqueness in the mathematical structure of the theory, pointing to the necessity of additional constitutive modeling. After the pioneering insights of Eshelby and subsequent developments in the theories of fracture, plasticity and phase transitions it has became common to present the missing constitutive information in the form of a functional relation between the velocity of the defect and the conjugate energy release rate known in dierent contexts as driving or congurational force.

The function relating velocity of a defect with the corresponding congurational force (kinetic relation) is usually either postulated or obtained from direct experimental measurements. Both approaches bring into constitutive theory a set of parameters of often unclear physical nature. Thus, the most popular phenomenological kinetic relations simply assert that the dissipation is either a quadratic function of velocity or a homogeneous function of degree one. While some of these phenomenological continuum models provide reasonably good overall description, the details of the dissipation process can only be understood by studying the phenomena taking place at the microscopic level, where the discrete nature of the material cannot be neglected. The term discreteness here should be understood rather broadly as inferring to dierent scales from crystal lattice to grain structure.

In this paper, we investigate the consequences of the direct lattice modeling of a mobile crystal defect and compare the resulting picture with the one provided by various continuum and quasicontinuum approximations. To make the analysis completely transparent we do not attempt to model a particular defect but instead consider a prototypical model. The main requirement to such a model is that elastic energy at the microlevel allows for internal buckling; the associated microscopic instability may then be of quite dierent nature as suggested, for instance, by the phenomena of twinning or fracture. In the situation when no dissipation is assumed at the microlevel, the macromodel can still be dissipative if the nonlinearity is suciently strong to support an irreversible energy ux from macro-to microscales and if the dispersion is suciently rich to make the process of energy transmission between the scales possible. The associated "radiative damping" explains at least some of the drag experienced by dislocations, cracks and phase boundaries moving in discrete lattices. A review of the role of radiative damping in plasticity can be found in Al 'shitz and Indenbom (1975). A systematic study of the radiative damping (or wave resistance) in fracture mechanics was initiated by the pioneering paper of [START_REF] Slepyan | Dynamics of a crack in a lattice[END_REF].

Our model essentially reduces to an array of linearly coupled bistable elements (Fig. 1). This mechanical system, a simplied version of which was considered by [START_REF] Slepyan | Dynamic factor in impact, phase transition and fracture[END_REF], can be viewed as describing a thin lm undergoing "thickness" phase transition or as a model of a side motion of a step on the surface of a generic domain boundary. To drive the system we apply to the elements an external force which creates a bias towards one of the energy wells and makes the propagation of a longitudinal switching wave (kink), energetically favorable. Our goal then is to compute the energy release rate and relate it to the velocity of the kink through a kinetic relation.

This prototypical system represents a two-well adaptation of the discrete FK model [START_REF] Frenkel | On the theory of plastic deformation and twinning[END_REF]. In its original (purely longitudinal and spatially periodic) form the model was used repeatedly for the description of dynamic dislocations (see [START_REF] Braun | Nonlinear dynamics of the Frenkel-Kontorova model[END_REF] for a recent review). Following the work of Atkinson and Cabrera (1965), we chose the nonlinear potential to be two-parabolic with equal elastic moduli. The advantage of this "symmetric" piece-wise linear approximation is that it lends itself to fully analytical treatment through a direct Fourier transform. Closely related discrete piece-wise linear models in fracture mechanics, requiring for their analysis somewhat less transparent Wiener-Hopf technique, have been considered previously by [START_REF] Slepyan | Dynamics of a crack in a lattice[END_REF][START_REF] Slepyan | The relation between the solutions of mixed dynamical problems for a continuous elastic medium and a lattice[END_REF]Slepyan ( , 2001 a, c) a, c) and [START_REF] Marder | Origin of crack tip instabilities[END_REF]. A different model of a phase transition in a chain of bistable springs with "nonsymmetric" energy well structure, which also requires application of the Wiener-Hopf method, was studied by Slepyan and[START_REF] Slepyan | Fracture wave in a chain structure[END_REF]Slepyan (2001b). In the context of reaction diusion problems similar piece-wise linear models have been recently studied by [START_REF] Cahn | Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice[END_REF].

In the present paper, we focus on the comparison of the radiation pattern emerging from the discrete model of [START_REF] Atkinson | Motion of a Frenkel-Kontorova dislocation in a one-dimensional crystal[END_REF] with the patterns produced by various continuum and quasicontinuum approximations. We recall that in the classical continuum limit subsonic kinks can propagate without any damping: the disappearance of radiative damping in this case is a consequence of a lack of dispersion. To improve the classical model, we consider two dierent dispersive approximations, polynomial [START_REF] Boussinesq | Theorie des ondes etde remous qui se propagent le longd'un canal rectangulaire horizontal, en communiquant au liquide contenu dans cecanal des vitesses sensiblement parailles de la surface au fond[END_REF] and rational [START_REF] Rosenau | Dynamics of nonlinear mass spring chains near the continuum limit[END_REF]. Both models augment the classical continuum theory by bringing nonzero internal scales: in the Boussinesq model additional strain gradient terms are added to the free energy, while in the Rosenau model it is the kinetic energy which is modied through the introduction of "transversal inertia." We remark that similar Boussinesq approximation of the FK model has been recently considered by [START_REF] Abeyaratne | Propagation of a front by kink motion[END_REF]; a related but dierent model of the Rosenau type was studied by [START_REF] Theil | A study of a Hamiltonian model for martensitic phase transformations including micro-kinetic energy[END_REF].

As we show in this paper both Boussinesq and Rosenau quasicontinuum approximations produce simple analytic kinetic relations which are in good quantitative agreement with the discrete theory for near sonic and supersonic velocities, where the number of radiated waves in the wake of the defect is small. At small velocities, when the number of radiated waves becomes large and the generated waves have lengths comparable to the lattice spacing, the agreement is expectedly poor. In particular, while both quasicontinuum models predict a nonzero trapping domain, the values of the Peierls stress in the approximate models are signicantly lower than in the discrete model. Overall, since the Boussinesq model has a very limited domain of application, our analysis suggests that preference should be given to the Rosenau model and the corresponding higher order quasicontinuum approximations.

The paper is organized as follows. In Section 2, we introduce the original discrete model, nondimensionalize variables and formulate the boundary-value problem for the traveling waves. In Section 3, we use Fourier transform to obtain the exact solution of the discrete model. As a part of the solution, we explicitly construct the spectrum of generated waves. In Section 4, we compute the total microscopic energy ux at innity and relate it to the rate of dissipation at the macroscopic level. This gives a simple analytic expression for the congurational force as a function of the velocity of the defect. Various continuum and quasicontinuum approximations are juxtaposed in Section 5; in each case we obtain a dispersion spectrum and explicitly compute the corresponding kinetic relation. In Section 6 we summarize our results. For convenience, some technical derivations are presented in Appendices A and B.

The model

Consider an array of particles attached at equal distances to a rigid background by bistable springs with energy density w(ũ n ). By ũ n we denote the vertical displacement of the particle with index n. To bias one of the energy wells, we introduce an external force per unit length , acting in the direction of the displacement ũ n and independent of the position of the particle. Finally, we assume that the neighboring particles interact through standard harmonic forces, characterized by an elastic modulus E. The potential and kinetic energies of the chain can be written in the form

V = n 1 2 E ũ n+1 -ũ n 2 + [w(ũ n ) -ũ n ]; K = n 1 2 u2 n ; (1)
where is the mass density per unit spring length. The Euler-Lagrange equations are then ũ n -

E (ũ n+1 -2ũ n +ũ n-1 )+[w ′ (ũ n ) -]=0: (2)
To overcome considerable analytical diculties associated with solving Eq. (2) for the general w, we consider a special double-well potential represented by two parabolas (Fig. 2)

w(ũ n )= 1 2 c ũ2 n ; ũ n ¡ a 2 ; 1 2 c(ũ n -a) 2 ; ũ n ¿ a 2 (3)
and study a special class of steady-state motions of the chain given by the traveling wave ansatz ũ n (t)= ũ(nṽt). The associated particle motion is relatively simple as

-0.5 0 0.5 1 1.5 -3 -2 -1 0 1 2 u W ′ α β A 1 A 2
Fig. 2. The piece-wise linear force-elongation relation for the bistable spring (3) with a = 1 and c =4 .

The propagating defect corresponds to a transition between points and . The congurational force is proportional to the dierence in areas A 2 -A 1 .

the conguration of the chain reproduces itself in discrete moments of time. Without loss of generality we may assume that ũ(0)=a=2. Then in coordinate system x =n-ṽt moving with the velocity ṽ, all springs located at x¿0 are in one of the energy wells (phase 1) while all springs located at x¡0 are in the other energy well (phase 2). By focusing on this particular class of solutions we eliminate the possibility of repeated transition between the wells and restrict our attention to isolated defects. Eq. (2) can then be rewritten in the form

ṽ2 d 2 ũ dx 2 - E [ũ(x + ) -2ũ(x)+ ũ(x -)] + [c ũ(x) --acH (-x)]=0; (4)
where H is the Heaviside function. As we see the nonlinearity was replaced by an additional force of known magnitude added at each bond that has snapped due to the advancement of the defect.

To highlight the essential parameters, it is convenient to use nondimensional variables

x = x ;u = ũ a ;v =ṽ E ; = ac : (5)
In the dimensionless form the nonlinear dierential advance-delay equation ( 4) can be rewritten as

v 2 d 2 u d x 2 -[u(x +1)-2u(x)+u(x -1)] + 2 0 [u(x) --H (-x)]=0; (6)
where we introduced the main nondimensional parameter of the problem

0 = c E : (7)
For consistency of the model we must require that

u(0) = 1 2 (8) and u(x) ¡ 1 2 ;x ¿ 0; u(x) ¿ 1 2 ;x ¡ 0: (9)
The last two inequalities oblige the solution to stay inside the two designated energy wells and play the role of the admissibility conditions.

Observe that congurations of the chain at x = ±∞ must correspond to static equilibrium. By selecting the two limiting states in dierent energy wells we obtain the following boundary conditions:

u(x) → ; x →∞; +1;x →-∞: (10)
Since we expect a nonzero radiation of elastic waves at ±∞ these boundary conditions should be understood as describing only the average values of the displacement elds.

The boundary conditions ( 10) are compatible with admissibility conditions if ¡ 0 = 1 2 , which represents the ultimate (spinodal) strength of our nonlinear springs.

Solution of the discrete problem

The piece-wise linear discrete boundary-value problem ( 6)-( 10), can be solved by the complex Fourier transform. We leave the details to Appendix A where we extended the methodology introduced in the dislocation context by [START_REF] Atkinson | Motion of a Frenkel-Kontorova dislocation in a one-dimensional crystal[END_REF]. The solution can be written as a combination of plane waves

u(x)=              -2 0 k∈M+ e ikx kL ′ (k) ;x ¿ 0; +1+ 2 0 k∈M- e ikx kL ′ (k) ;x ¡ 0: (11)
Here

L(k)= 2 0 + 4 sin 2 k 2 -v 2 k 2 (12)
and the participating wave numbers, characterizing linear modes generated ahead and behind the moving kink, are arranged in sets M + and M -. We recall that a plane wave e i(kn-t) can be a part of the solution if and only if its frequency satises the dispersion relation for the linear problem inside the wells 2 = 2 0 + 4 sin 2 k=2. The same plane wave will also be a part of the traveling wave solution if (k)=vk which means that phase velocity is equal to the velocity of the traveling wave. By applying these two conditions we obtain that the wave numbers constituting the sets M + and M -must necessary satisfy the condition

L(k)=0; (13)
which, of course, also follows from the inversion of the Fourier transform (see Appendix A). At a given value of v Eq. ( 13) may have real and complex roots k (Fig. 3). An innite number of complex roots correspond to exponentially damped localized modes describing the core region. The complex roots appear in symmetric quadruples; among them there is always a couple of symmetric pure imaginary roots providing a monotone contribution. An additional nite number of real roots represent acoustic radiation emitted by the defect. These modes must satisfy radiation conditions: for the radiative waves ahead of the moving kink the group velocity must be larger than the phase velocity, while for the waves behind the kink, the group velocity must be smaller than the phase velocity. From the denition of the group velocity we obtain v g = ′ (k)=v+L ′ (k)=2vk. Now we are in the position to specify the sets M + and M -

M ± = {k: L(k)=0;{Im k ? 0}∪{Im k =0;kL ′ (k) ? 0}}: (14)
The radiative modes with real wave numbers are illustrated in Fig. 4 (curve D). One can see that for every value of v there exists an odd number of real roots k, describing symmetric couples of emitted waves ahead and behind the kink, and an additional wave in the wake. We also observe, that whenever L ′ (k) = 0, two distinct radiative modes merge and for these special values of v the group velocity of a radiated wave is equal to its phase velocity. The amplitudes of the corresponding resonant modes diverge, and the solution blows up because the energy cannot escape from the core region. One can show that the appearance of resonances is not generic and is related to the over-symmetric setting of the model. After the set of excited modes is known one can use the explicit relations (11)t o construct the proles of the displacement eld. The associated required to sustain the steady motion of the defect can be found from the matching condition (8) (see Appendix A). Several representative proles of the displacement eld for dierent values of v are shown in Fig. 5. Notice that in the interval v¿0:2243 the oscillations can be seen only in the wake of the defect, which is in agreement with the imposed radiation conditions: in this interval the only intersection of the line = vk with the curve =(k) satises kL ′ (k) ¡ 0 which means that the group velocity is smaller than the phase velocity. The picture changes as we cross the resonance velocity v =0:2243 because now the oscillations are present on both sides of the moving front. For instance, in the interval 0:1324 ¡v¡0:2243 Eq. ( 13) has three real roots, responsible for one propagating mode ahead (kL ′ (k) ¿ 0) and two propagating modes behind (kL ′ (k) ¡ 0) the defect.

From Fig. 5(c,d) one can see that the admissibility conditions (9) may not be satised below the rst resonance velocity and therefore the corresponding traveling wave solutions of form (11) do not exist. More systematic analysis shows that most probably the whole velocity interval v¡0:26 must be excluded (see similar conclusions in Here 0 =0:5. [START_REF] Marder | Origin of crack tip instabilities[END_REF]. Since in this excluded velocity range one cannot expect the steady motion of the defect, the description of the actual mode of propagation with prescribed average velocity may require a dierent ansatz.

Kinetic relation

As it has already been mentioned in the Introduction, the propagation of a defect through the lattice can be viewed as a succession of internal instabilities associated with the snapping of individual elastic bonds. The inherently dynamic nature of these internal instabilities leads to the generation of high frequency vibrations in the lattice and results in tunneling of the energy from long to short waves. For our special traveling wave solutions the energy transfer between the scales shows as the radiation at innity. To account for this loss at the macrolevel one can compute the rate of change of the total energy

d(V + K) dt - dA dt = dQ dt : ( 15
)
Here dA=dt is the power supplied by the loading device and dQ=dt is the associated "heat" generation. Due to the trivial thermodynamic nature of the system, the function R(v)=-dQ=dt may be interpreted as the rate of dissipation and expressed as a product of the congurational force G(v) and the generalized ux represented in our case by the velocity of the defect v

R(v)=G(v)v: (16)
The relation between the congurational force G and the velocity v will be called kinetic relation. Kinetic relations are routinely employed in the modeling of various lattice defects ranging from twin boundaries to crack tips. Our G can be viewed as an analog of the Peach-Koehler force in the theory of dislocations, of the J integral in the theory of cracks and of the Eshelby force in the theory of phase transitions.

To obtain an explicit kinetic relation in the present case, we notice that the rate of macroscopic energy loss can be computed as the sum of the microscopic energy uxes at innity (e.g. Slepyan, 2001a, b, c). We can write

R(v)=R + (v)+R -(v); ( 17
)
where R ± (v)-are the energy uxes at ±∞, respectively. To compute R ± (v)w eo bserve that outside the core region the governing equations are linear and therefore the energy ux associated with an individual mode is equal to the product of the averaged density of the total energy G k and the relative velocity of the energy transport with respect to the front |v g -v|. In particular

R + (v)= k∈N+ G k + (v g -v); R -(v)= k∈N- G k -(v -v g ): (18)
Here G k represents the sum of kinetic and potential energies per particle carried by a linear wave with the wave number k; denotes averaging over suciently large number of periods. In Eq. ( 18) we have used the fact that due to the asymptotic orthogonality of dierent modes, the mixed terms do not contribute to the total ux. Since the microscopic radiation involves only propagating modes with real wave numbers, the sets N + and N -can be dened as the subsets of M + and M -satisfying additional condition Im k =0.

The average energy density associated with the radiative modes can be computed directly from the denition

G k = G -G 0 k ; (19)
where

G(x)=                    1 2 v 2 ( du dx ) 2 + 1 4 [u(x +1)-u(x)] 2 + 1 4 [u(x) -u(x -1)] 2 + 1 2 2 0 u 2 -2 0 u(x);x ¿ 0; 1 2 v 2 ( du dx ) 2 + 1 4 [u(x +1)-u(x)] 2 + 1 4 [u(x) -u(x -1)] 2 + 1 2 2 0 [u(x) -1] 2 -2 0 u(x);x ¡ 0 (20)
is the total energy per particle and

G 0 = -1 2 2 0 2 ;x ¿ 0; -1 2 2 0 2 -2 0 ; x ¡ 0 (21)
is the corresponding energy of the homogeneous states. For an individual mode characterized by the real wave number k we obtain

G k ± = - 2v n-2=k n 1 2 v 2 du k d x 2 + 1 4 (u k (x +1)-u k (x)) 2 + 1 4 (u k (x) -u k (x -1)) 2 + 1 2 2 0 (u k (x)) 2 d x =2 4 0 v 2 [L ′ (k)] 2 : (22)
Now by using Eq. ( 18) we can rewrite the expressions for the rate of radiative damping in the form

R(v)=v 4 0 k∈N+ 1 kL ′ (k) -v 4 0 k∈N- 1 kL ′ (k) : (23)
Finally, comparing this formula with Eq. ( 16) we obtain an explicit representation for the congurational force

G = 4 0 k∈N± 1 k|L ′ (k)| : (24)
The right-hand side of this expression can in principle be computed for any given v, which furnishes a desired relation between the congurational force G and velocity v.

Before presenting the results of the numerical computations, it is instructive to check the consistency of our approach and compute the expression for congurational force in a dierent way. Indeed, since the macroscopic motion at x = ±∞ is absent and the potential energies of the limiting homogeneous congurations are dierent, a moving defect converts macroscopic energy of the system at a rate

R(v)=[G 0 (+∞) -G 0 (-∞)]v = {w 0 (+∞) -w 0 (-∞) -2 0 [u 0 (+∞) -u 0 (-∞)]}v = 2 0 v: (25)
By comparing the two expressions for R, we obtain the relation between the driving force G and the applied force per unit length

G = 2 0 : (26)
This formula can also be interpreted geometrically as the dierence between the shaded areas A 2 -A 1 in Fig. 2. The fact that the congurational force is a linear function of applied stress is not generic; in the case of dierent moduli of the phases this relation is quadratic. One can also see that due to (A.15) the two ways of computing the congurational force represented by Eqs. ( 24) and ( 26) are consistent.

We begin the analysis of the resulting kinetic relation by rst investigating two important limiting cases:

1. Fast waves: v →∞ . According to the dispersion relation in this limit k → 0 and kL ′ (k) →-2 2 0 , which gives

lim v→∞ G = 2 0 2 : (27)
One can see that the stress required to maintain the limiting motion approaches asymptotically the value = 1 2 which corresponds to the spinodal limit (ultimate strength); as we have already mentioned, no solution is possible for ¿1=2. 2. Slow waves: v → 0. In this limit, the solution exhibits a growing number of excited lattice waves and because of the accumulation of resonances the asymptotic behavior of G is not straightforward. However, one can show that

lim v→0 G = 2 0 D p : (28)
In this expression

D p = 0 2 4+ 2 0 ( 29
)
is exactly the Peierls force (or critical toughness) in the static problem specifying the upper limit of the "lattice trapping" (see Appendix B for details). The consistency of static and dynamic computations conrms that the traveling wave regimes originate exactly from the bifurcation of the static solutions.

In the intermediate range of velocities between v = 0 and ∞ the form of the kinetic relation can be obtained numerically from the precise determination of the sets N ± at each value of v. The results for 0 =0:5 are presented in Fig. 6. An interesting feature of the obtained kinetic curve is the presence of the innite number of resonances at low velocities. As we have mentioned in the previous section, the traveling waves in this area do not exist and the corresponding branches should be excluded. Notice also that the kinetic relation can be smoothly extended from subsonic into supersonic region, even though the use of the term "sonic velocity" in the present model is somewhat questionable. The complete analysis of stability of the traveling wave solutions depicted in Fig. 6 presents a considerable challenge.

Quasicontinuum approximations

Our analysis has been so far focused on the discrete model. When the size of the core of the moving defect is mesoscopic the more natural description is furnished by continuum mechanics with most microscopic degrees of freedom integrated out; a quasicontinuum approximation aims at incorporating the leading eects due to discreteness into the continuum model. Although our discrete model is conservative, the continuum and quasicontinuum approximations may be dissipative since, due to nonlinearity, the energy is not conned at long waves.

To construct continuum and quasicontinuum approximations for our discrete model we replace the advance-delay terms in the main equation ( 6) by the corresponding higher derivative approximation. We recall that

u(x +1)-2u(x)+u(x -1) = 4 sinh 2 1 2 D u(x); ( 30
)
where D is the operator of dierentiation. The right-hand side of Eq. ( 30) can be formally approximated by either polynomial or rational (Pade) expansions of the form (e.g. [START_REF] Wattis | Approximations to solitary waves on lattices: 2. Quasi-continuum methods for fast and slow waves[END_REF])

4 sinh 2 1 2 D ≈ D 2 I + 1 12 (D) 2 ≈ D 2 I - 1 12 (D) 2 -1 : (31)
We begin with the polynomial approximation and rewrite Eq. ( 6) in the form (see also [START_REF] Abeyaratne | Propagation of a front by kink motion[END_REF])

u xxxx -12(v 2 -1)u xx -12 2 0 (u -)=-12 2 0 H (-x): (32)
As before, the nondimensional displacement eld u(x) must satisfy additional admissibility conditions ( 8)-( 9). The nondispersive approximation, obtained by dropping in expansion (32) the terms with spatial derivatives higher than two, represents a classical continuum limit. In this limit Eq. ( 32) takes the form

(v 2 -1)u xx + 2 0 u = 2 0 [ + H (-x)]: (33)
By applying the Fourier transform (as in the discrete problem) we obtain

u(x)= - 2 0 2i e ikx kL c (k) d k; ( 34
)
where now

L c (k)= 2 0 + k 2 (1 -v 2 ): (35)
In the continuum model the dispersion relation L c (k) = 0 has only two roots

k 1; 2 = ± 2 0 v 2 -1 ; ( 36
)
which are real if v¿1 (supersonic region) and pure imaginary if v¡1 (subsonic region). By using the radiation conditions we can compute the integral in Eq. ( 34) and obtain in the subsonic case

u(x)=        + 1 2 e -x √ 2 0 =(1-v 2 ) ;x ¿ 0; +1- 1 2 e √ 2 0 =(1-v 2 ) ;x ¡ 0: (37)
This classical solution satises the admissibility conditions at = 0 only. The fact that in the classical theory the defect can propagate at zero force, is not surprising since in this approximation the radiative damping is completely absent. Now notice that the above subsonic branch of solutions is connected to a branch of sonic "shock waves" at v = 1. In fact, for v = 1 the following discontinuous solution can be constructed:

u(x)= if x¿0; +1 if x¡0: ( 38
)
This solution is dened for 0 ¡¡1=2. In the supersonic case, v¿1, the traveling wave solution satisfying the admissibility conditions takes the form

u(x)=          1 2 ;x ¿ 0; 3 2 -cos 2 0 v 2 -1 x; x ¡ 0: (39)
This solution, existing only at =1=2, describes the motion of a supersonic phase boundary at ultimate stress.

If we now restore the fourth spatial derivative in Eq. ( 32) we obtain the quasicontinuum Boussinesq model. By applying the Fourier transform we can again represent solution in the form

u(x)= - 2 0 2i e ikx kL B (k) d k; (40)
where now

L B (k)= 2 0 + k 2 (1 -v 2 ) - k 4 12 : (41)
In the Boussinesq approximation the dispersion relation L B (k) = 0 generates four wave numbers for each value of v: two purely imaginary, describing the structure of the core region and two real, describing the radiation pattern. Explicitly we can write

k 1; 2; 3; 4 = ± 6(1 -v 2 )   1 ± 1+ 2 0 3(1 -v 2 ) 2   : (42)
According to radiation conditions we must close the contour in the upper half-plane for x¿0 and include one purely imaginary root only; for x¡0 we must instead close the contour in the lower half-plane and include one purely imaginary root and two real roots. As a result we obtain

u ± =                                    + 1 4   1+ √ 3(1 -v 2 ) 2 0 +3(v 2 -1) 2   e -l±x ;x ¿ 0; +1- 1 4   1+ √ 3(1 -v 2 ) 2 0 +3(v 2 -1) 2   e l±x - 1 2   1 ∓ √ 3(1 -v 2 ) 2 0 +3(v 2 -1) 2   cos m ± x; x ¡ 0; (43)
where the choice of signs ± indicates the cases v ? 1, respectively, and where

l ± = ±6(v 2 -1)   1+ 2 0 3(v 2 -1) 2 ± 1   ; m ± = ±6(v 2 -1)   1+ 2 0 3(v 2 -1) 2 ∓ 1   : (44)
Solution ( 43) satises all the admissibility conditions and the corresponding kinetic relation can be obtained in an explicit form (cf. [START_REF] Abeyaratne | Propagation of a front by kink motion[END_REF]) 7. The congurational force as a function of velocity of the defect in the discrete model (D), continuum model (C) and quasicontinuum models (B) and (R). The horizontal arrow marks the instability limit for the lattice trapping region leading to a transition to a fully developed inertial regime. For all curves 0 =0:5.

G = 2 0 4 1 ± 3(v 2 -1) 2 3(v 2 -1) 2 + 2 0 ;v ? 1: (45) 0 0.5 1 1.5 2 0.04 0.08 0.12 v G G p D G p B,R C R D B C C D, B, R D B,R Fig.
Here if v →∞, we obtain the same spinodal limit (27) as in the discrete case. In the other limiting case v → 0 we obtain

lim v→0 G = 2 0 B p ; (46) where
B p = 1 4 1 - 3 3+ 2 0 : (47)
As we show in Appendix B this value of the applied force matches exactly the static Peierls force computed for the Boussinesq model. The resulting kinetic equation is presented in Fig. 7.

Although the kinetic relation for the traveling waves in the Boussinesq approximation matches the kinetic relation in the discrete problem reasonably well, the initial-value problem for the Boussinesq model is linearly ill-posed with unbounded growth rates at short wavelengths [START_REF] Pego | Oscillatory instability of solitary waves in a continuum model of lattice vibrations[END_REF][START_REF] Christov | Well-posed Boussinesq paradigm with purely spatial higher-order derivatives[END_REF]. The reason for this failure of the Boussinesq model is that the strain gradient part of the elastic energy in this model is not positive denite (e.g. [START_REF] Kunin | Elastic Media with Microstructure. 1. One-dimensional Models[END_REF]. An alternative (rational) approximation of the delay-advance operator mentioned in Eq. ( 31), has the same order as the Boussinesq model but has an advantage of always being stable at short waves [START_REF] Rosenau | Dynamics of nonlinear mass spring chains near the continuum limit[END_REF].

To compare all four models we x the value of the nondimensional parameter 0 and superimpose the resulting kinetic relations on one graph (Fig. 7). One can see that the kinetic curves for discrete (D) and both quasicontinuum models (B) and (R) are almost indistinguishable in the region v¿0:2243 where only one propagating mode is emitted. It is also immediately obvious that the classical continuum model (C) cannot be considered satisfactory since it predicts zero drag everywhere in the subsonic range.

Both dispersive quasicontinuum approximations capture the eects of radiative damping although with dierent level of detail. To understand the dierence between the radiation patterns in dierent models we present on one graph all four dispersion spectra (Fig. 8). As expected all models essentially agree in the domain of supersonic velocities and long waves where the dispersion spectra in all four theories reduce to two real and two purely imaginary branches. Consequently for v¿1 solutions generated by all four models are almost identical. At v¡1 purely imaginary branches in the models (B), (R) and (C) practically coincide with the corresponding branches of the discrete model (D). Therefore, in this velocity range the structure of the nonoscillatory part of the wave eld near the core of the defect is similar in all four models. Another general observation is that for real branches, the Boussinesq model provides a lower bound, while the Rosenau model gives an upper bound of the discrete dispersion spectrum (see Fig. 4).

By examining the slopes of the real branches of the dispersion relation in both (R) and (B) models we conclude that they contribute to the wave eld only behind the kink. Therefore, these quasicontinuum approximations can be considered appropriate only at v¿0:2243, because below this resonant velocity the original discrete model exhibits radiation both ahead and behind the kink. At subsonic velocities the Boussinesq model radiates waves with basically one wavelength (k ≈ 3:5), while both discrete and Rosenau models develop, as velocity tends to zero, radiation with smaller and smaller wavelengths.

Important conclusions concerning the adequacy of the approximate models can be drawn from the comparison of the static limits for the corresponding kinetic relations. We recall that due to lattice trapping the solutions describing stationary kinks in the discrete model exist in a nite range of applied forces -D p ¡¡ D p , where D p is the corresponding Peierls force. In the continuum approximation static solutions exist only at zero stress, which means that C p =0. In both Boussinesq and Rosenau approximations the lattice trapping phenomenon is preserved. The relation among the values of Peierls forces D p , B p and R p at dierent values of 0 is illustrated in Fig. 9. As one can see from this gure and from the comparison of the kinetic relations at dierent values of 0 , the agreement between the models improves as the reference length of the discrete chain gets smaller or the harmonic coupling between the snap-springs gets stronger ( 0 → 0).

It is also instructive to compare the displacement elds obtained in dierent models (see Fig. 10). As expected, in the supersonic and near sonic cases, the displacement eld obtained in discrete (D) and in dierent quasicontinuum models practically coincide (Fig. 10a,b): all models predict exponentially damped displacements at x¿0 and innite oscillatory tails at x¡0. The agreement is so good because only long waves are involved. Outside this range the discrete model basically agrees with the quasicontinuum models in the velocity interval bounded below by the largest resonant velocity for the discrete model. Below this critical velocity the discrete model exhibits simultaneous radiation ahead and behind the kink while the quasicontinuum models show oscillatory components only behind the moving defect. The detailed comparison in this range of velocities does not make much sense because no discrete solution seems to be admissible. It is interesting, however, that at extremely small velocities, the behavior of the three models is very dierent: the Boussinesq model has a conventional limit with a nite wavelength, the discrete solution converges to a piece-wise constant function with the discontinuities at lattice points, while the Rosenau model has a nontrivial limit represented by a stationary Young measure (cf. [START_REF] Theil | A study of a Hamiltonian model for martensitic phase transformations including micro-kinetic energy[END_REF].

Conclusions

In this paper we studied a simple example of radiative damping in discrete lattices. The goal was to simulate an isolated kink moving uniformly under the action of a given congurational force. We computed an explicit expression for the dissipative function in the discrete model and showed it to be of a dry friction type. We then studied two related quasicontinuous models built on the polynomial (Boussinesq) and rational (Rosenau) approximations of the nite-dierence part of the discrete model. Although both models produced excellent agreement with the discrete model at high velocities, they both failed at low velocities where due to lattice resonances the discreteness of the model cannot be ignored. Since the Cauchy problem for the Boussinesq model is ill-posed, preference should be given to the Rosenau and similar higher-order Pade approximations generating much more realistic short wave part of the dispersive spectrum.

One possibility which has not been explored in our paper is that the adequate continuum description of the discrete system, valid in the broad range of wavelengths and wave velocities, is thermodynamical. We recall that the generation of short wave lattice vibrations by the moving defect is in many respects similar to the localized increase of temperature. Moreover, continuum models of phase transitions that take latent heat and heat conductivity into account, generate kinetic relations which are in surprisingly good qualitative agreement with our "rst principle" kinetic relation obtained from the discrete model. Specically in both discrete and nonisothermal models the kinetic curves are nonmonotone and exhibit a nite range of trapping; other similarities include the possibility of supersonic regimes and the multivaluedness of the kinetic functions at suciently slow velocities (Ngan andTruskinovsky 1999, 2002). This intriguing similarity is worth pursuing. Applying the Fourier transform to Eq. (A.4) and using Eq. ( 8), we obtain

û + ()= 1 √ 2L() v 2 u ′ + (0) - i 2 v 2 -

Fig. 1 .

 1 Fig. 1. Schematic conguration of particles around the core of a defect moving with the velocity v.

  Fig. 3. A complete set of wave numbers generated by the kink moving with velocity v (solutions of the equation L(k)=0 at 0 =0:5).

Fig. 4 .

 4 Fig.4. Propagating waves with the real wave numbers radiated by a kink moving with velocity v. Waves marked by "+" and "-" propagate ahead and behind the front, respectively. The three dierent curves correspond to a discrete (D) and to the two quasicontinuum models (B) and (R) (see the text). Parameters are the same as in Fig.3.

Fig. 5 .

 5 Fig. 5. The displacement eld around the core of a moving defect at v =2:0 (a); v =1:0 (b); v =0:224 (c); v =0:22 (d). Here 0 =0:5.

Fig. 6 .

 6 Fig. 6. Congurational force as a function of velocity (kinetic relation) at 0 =0 :5. The low velocity domain around the resonances does not correspond to admissible traveling waves and must be excluded. The corrected picture is shown in Fig. 7.

Fig. 8 .

 8 Fig. 8. Comparison of the dispersion spectra for discrete (D), continuum (C), Boussinesq (B) and Rosenau (R) models. Here 0 =0:5.

Fig. 9 .

 9 Fig. 9. (a) Comparison of the discrete model with continuum and quasicontinuum models at dierent values of 0 : force-velocity relations at 0 =0:01 and 0 =1:0; (b) the Peierls force for discrete (D), Boussinesq (B) and Rosenau (R) models as a function of 0 .

Fig. 10 .

 10 Fig. 10. Displacement elds for the discrete (D), Boussinesq (B), Rosenau (R) and continuum (C) models at v =2:0 (a), (b), v =0:5 (c), (d) and v =0:22 (e), (f). In all graphs 0 =0:5.

  former for s¿0, the latter for s¡0. Then for some s + ¿ 0, s -¡ 0 the Fourier transform can be inverted and we obtain expression for û + (), we recall that at x¿0 the displacement eld u + (x) satises the equationv 2 d 2 u + d x 2 -[u + (x +1)-2u + (x)+u + (x -1)] + 2 0 u + (x)= 2 0 : (A.4)

  order to select the physically correct solution, we need to interpret the integral in Eq. (A.8) as a contour integral along the path which does not pass through singular points. Consider rst the singularity at k = 0. By choosing a contour close to the real there exists a piece-wise constant function u 0 (x) describing discrete solution of the static problem, which is the limit of the solution of the dynamic problem u(xvt)a t v = 0. By computing the Fourier transform of the function u 0 (x) we can write û that in the limit v → 0 the Fourier image of dynamic solution is û compatibility conditions are the same as in the dynamic case. The compatibility conditions at x = 0 take the form[u]=[u x ]=[u xx ]=[u xxx ]

Acknowledgements

This work was supported by the NSF grants DMS-9803572 and DMS-0102841.

In the Rosenau approximation the equation for traveling waves takes the form

For solutions describing an isolated kink, the singular nonlinear equation ( 48) can be rewritten as two linear equations

supplemented by the following matching conditions at x =0:

Here the square parentheses denote the jump discontinuity [ ] = ( ) + -() -. Notice that in the linearized Rosenau model the dispersion equation relating phase velocity of the plane waves to the velocity of the kink and completely characterizing the radiative pattern at innity, takes the form L R (k) = 0, where now

This equation has four roots: two real and two purely imaginary

The displacement eld representing solution of Eq. ( 49) and satisfying the matching conditions (50), is given by the formulas

Here k 3 and k 4 indicate purely imaginary roots (52) in the upper and lower half-planes, respectively, and k 1 is the positive real root. The explicit kinetic relation takes the form

In the limit v →∞we again obtain condition (27) which agrees with both the discrete and the Boussinesq models. In the alternative limit v → 0 we obtain

Appendix A

Here we show how solution (11) of the piece-wise linear problem ( 6)-( 10) can be obtained by the Fourier method. We begin by representing the displacement eld in the form u(x)=u + (x)+u -(x), where

axis and passing above the point k = 0 we obtain

To compute the integral in Eq. (A.9) we can use the method of residues. For x¿0 we need to complete the contour in the upper half-plane and for x¡0 in the lower half-plane. In this way, we automatically include into the eld only the waves which are exponentially damped away from the phase boundary. The real roots of equation L(k) = 0 require special analysis, because the corresponding waves do not decay at ±∞. By applying the radiation conditions and computing the residues we obtain ( 11), ( 14).

In order to determine the value of applied force which is necessary to sustain our dynamic displacement prole we need to use the matching condition u(0)=1=2. From Eq. (A.9) we obtain

Now by closing the contour of integration in Eq. (A.10) in the upper half-plane, we obtain .11) where

Alternatively, we can compute (A.10) by closing the contour of integration in the lower half-plane. We obtain

where now N -(k)={k: L(k)=0;Imk =0∪ kL ′ (k) ¡ 0} and A and B are the same as in Eq. (A.12) due to the symmetry of the complex roots. Since the function A(k 1 ;k 2 ) is even in both k 1 and k 2 , and since the complex roots are symmetric with respect to the planes Im k = 0 and Re k = 0, we obtain 2i

By substituting this expression into Eqs. (A.10)-(A.11) we obtain the desired relation for

In statics our Eq. ( 2) takes the form

where we have already substituted the piece-wise linear force law (3) and used the nondimensional variables ( 5), ( 7); the boundary conditions remain the same as in Eq. ( 10).

Introduce new functions u + n and u - n , describing the solutions of (B.1)a tn ¿ 0 and n¡0, respectively. The compatibility conditions can be written in the form

By solving the nite-dierence problem (B.1)-(B.2) we obtain a (stable) solution in the form

where

Another (unstable) solution representing the so-called Peierls barrier (e.g. [START_REF] Hobart | Peierls stress dependence on dislocation width. Peierls-Barrier minima[END_REF] can be obtained if instead of conditions (B.2) we consider a dierent set of matching conditions

where r 1 and r 2 are the same as in Eq. (B.4). The two solutions satisfy the admissibility conditions for || ¡ D p where

is the Peierls force required to start the motion of the kink; at this force the stable and unstable solutions annihilate each other.

Notice that solution (B.3) and formula (B.7) can also be obtained by direct discrete Fourier transform of Eq. (B.1). Indeed, by transforming Eqs. (B.1) we obtain

u n e -ikn (B.9) and

Now if we make an inverse discrete Fourier transform of û D (k)

we obtain

The next step is to extend the above solution analytically by zero from the interval -¡k¡ to the whole real axis and choose the contour passing above the pole at the origin. Then we can write

Since equation L 0 (k) = 0 does not have real roots, we can compute the integral in Eq. (B.13) by closing the contour in the upper half-plane for n¿0, and in the lower half-plane for n¡0. The relevant complex roots of the equation L 0 (k)=0 are: p 1 = 2i sinh -1 ( 0 =2) in the upper half-plane, and p 2 = -2i sinh -1 ( 0 =2) in the lower half-plane. We obtain

Now by noticing that e ip1n = r n 2 and e ip2n = r n 1 we can rewrite (B.14) in form (B.3), and from condition (B.5) obtain the expression for Peierls force (B.7). Now for consistency of our construction we need to show that the static solution (B.3), (B.12) is compatible with the dynamic solution discussed in the main body of the paper. The idea of the analysis below can be traced to the works of [START_REF] Celli | Motion of a screw dislocation in a crystal[END_REF] and [START_REF] Slepyan | The relation between the solutions of mixed dynamical problems for a continuous elastic medium and a lattice[END_REF]. First notice that we need to identify the limiting dynamic solution u(x) represented by a function of continuum argument with the static solution u n represented by a function of a discrete argument. Suppose, that at v =0 From the admissibility conditions we nd that || ¡ B p . Here Since it satises admissibility conditions only for = C p = 0, the corresponding Peierls force is equal to zero.


Jian Wang

Denis Efimov

A A Bobtsov

On robust parameter estimation in finite-time without persistence of excitation

The problem of adaptive estimation of constant parameters in the linear regressor model is studied without the hypothesis that regressor is Persistently Excited (PE). First, the initial vector estimation problem is transformed to a series of the scalar ones using the method of Dynamic Regressor Extension and Mixing (DREM). Second, several adaptive estimation algorithms are proposed for the scalar scenario. In such a case, if the regressor may be nullified asymptotically or in a finite time, then the problem of estimation is also posed on a finite interval of time. Robustness of the proposed algorithms with respect to measurement noise and exogenous disturbances is analyzed. The efficiency of the designed estimators is demonstrated in numeric experiments for academic examples.

I. INTRODUCTION

Estimation and identification of model parameters of linear and nonlinear systems is an important problem, whose solution forms a basis for posterior state estimation and control synthesis [START_REF] Ljung | System Identification: Theory for the User[END_REF]. One of the most popular problem statements is presented by the static linear regression model [START_REF] Ljung | System Identification: Theory for the User[END_REF], [START_REF] Sastry | Adaptive Control: Stability, Convergence and Robustness[END_REF] (the basic problem of on-line estimation of constant parameters of the n-dimensional linear regression):

x(t) = ω (t)θ, t ∈ R, (1)
y(t) = x(t) + w(t),
where x(t) ∈ R is the model output, θ ∈ R n is the vector of unknown constant parameters that is necessary to estimate, ω : R → R n is the regressor function (usually assumed to be bounded and known), y(t) ∈ R is the signal available for measurements with a measurement noise w : R → R (here R denotes the set of real numbers). The noise w(t) may also represent, for example, the time-varying deviations of θ(t) (if it is not a constant), then w(t) is proportional to the derivative θ(t). A conventional additional requirement, which is usually imposed on the regressor function ω, consists in its persistent excitation [START_REF] Sastry | Adaptive Control: Stability, Convergence and Robustness[END_REF], [START_REF] Narendra | Persistent excitation in adaptive systems[END_REF], i.e. it is frequently assumed that there exist > 0 and ϑ > 0 such that t+ t ω(s)ω (s)ds ≥ ϑIn for any t ∈ R, where In denotes the identity matrix of dimension n × n. It is a well-known fact [START_REF] Khalil | Nonlinear Systems[END_REF], that if ω is PE, then for the linear estimation algorithm θ(t) = γω(t) y(t) -ω (t) θ(t) , γ > 0,

where θ(t) ∈ R n is the estimate of θ, its estimation error e(t) = θ -θ(t) dynamics is globally exponentially stable at the origin provided that w(t) ≡ 0 for all t ∈ R; and it is input-to-state stable (ISS) for any essentially bounded noise w [START_REF] Narendra | Persistent excitation in adaptive systems[END_REF], [START_REF] Efimov | Design of impulsive adaptive observers for improvement of persistency of excitation[END_REF]. There are several recent results attempting to relax the requirement on persistence of excitation and imposing some nonuniform in initial time restrictions on the regressor ω, which lead to a (nonuniform) global asymptotic stability of (2) [START_REF] Praly | Convergence of the gradient algorithm for linear regression models in the continuous and discrete time cases[END_REF], [START_REF] Barabanov | On global asymptotic stability of ẋ = -φ(t)φ (t)x with φ(t) bounded and not persistently exciting[END_REF], and without introducing additional constraints they also guarantee the integral ISS property of e(t) with respect to w(t) [START_REF] Efimov | Robust stability under relaxed persistent excitation conditions[END_REF] (with additional restrictions, a kind of nonuniform in initial time ISS can be recovered). The linear estimation algorithm ( 2) is one of the most popular methods to solve [START_REF] Ljung | System Identification: Theory for the User[END_REF], but there are many other approaches, see for example [START_REF] Sastry | Adaptive Control: Stability, Convergence and Robustness[END_REF], [START_REF] Narendra | Persistent excitation in adaptive systems[END_REF] or a recent work [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF], and references there. However, almost all of these results are based on the assumption that the regressor ω(t) is PE or has an analogous property. For example, in the concurrent learning approach [START_REF] Chowdhary | Concurrent learning adaptive control of linear systems with exponentially convergent bounds[END_REF], [START_REF] Kamalapurkar | Model-based reinforcement learning for approximate optimal regulation[END_REF], it is assumed that a kind of PE is satisfied on a finite interval of time only.

In this work we will consider the problem (1) without an assumption on interval excitation of ω. Moreover, for our design we will implicitly assume that the norm |ω(t)| of the regressor ω(t) might be converging to zero (asymptotically or in a finite time), which implies that the norm of the model output |x(t)| ≤ |ω(t)||θ| is also converging to zero that leads to a necessity of estimation of θ during the time interval when |x(t)| ≥ |w(t)|, i.e. when initially the measured output y(t) disposes the information about x(t) and it is not hidden completely by the noise w(t). Therefore, we will consider the problem of finite-time estimation in (1) for t ∈ [0, T ], where T is fixed, without persistence of excitation. For this purpose, first, we will apply DREM [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF] to [START_REF] Ljung | System Identification: Theory for the User[END_REF] in order to decouple this vector θ estimation problem on a series of independent problems of estimation of scalar components θi for i = 1, . . . , n. Such a decomposition is important since for scalar estimation algorithms it is possible to accelerate the convergence by parameter tuning [START_REF] Efimov | Design of impulsive adaptive observers for improvement of persistency of excitation[END_REF], [START_REF] Efimov | Robust stability under relaxed persistent excitation conditions[END_REF]. And, second, several algorithms are proposed for the scalar estimation without the regressor excitation and with the convergence in a finite-time. Robustness abilities of these algorithms are investigated (a preliminary version of this work was presented in [START_REF] Wang | Finite-time parameter estimation without persistence of excitation[END_REF], where the case w(t) ≡ 0 for all t ≥ 0 was studied only). Efficiency of the proposed algorithms is demonstrated in simulations for regressors with exponentially converging upper bound and a bounded noise.

The outline of this note is as follows. Some preliminary results are introduced in section II. The problem statement is given in Section III. The estimation algorithms are designed in Section IV, where also the convergence and robustness conditions are established.

Simple illustrating examples are considered in Section V.

Notation

• R+ = {x ∈ R : x ≥ 0}, where R is the set of real number.

• |x| denotes the absolute value for x ∈ R or a vector norm for x ∈ R n , and the corresponding induced matrix norm for a matrix A ∈ R n×n is denoted by A . • Define e = exp(1).

• Denote s α = |s| α sign(s) for any s ∈ R and α ∈ R+.

II. PRELIMINARIES

Consider a time-dependent differential equation [START_REF] Khalil | Nonlinear Systems[END_REF]:

dx(t)/dt = f (t, x(t), d(t)), t ≥ t0, t0 ∈ R, (3)
where x(t) ∈ R n is the state vector, d(t) ∈ R m is the vector of external inputs and d ∈ L∞(R, R m ); f : R n+m+1 → R n is a continuous function with respect to x, d and piecewise continuous with respect to t, f (t, 0, 0) = 0 for all t ∈ R. A solution of the system (3) for an initial condition x0 ∈ R n at time instant t0 ∈ R and some d ∈ L∞(R, R m ) is denoted as X(t, t0, x0, d), and we assume that f ensures definiteness and uniqueness of solutions X(t, t0, x0, d) in forward time at least on some finite time interval [t0, t0 +T ), where T > 0 may be dependent on the initial condition x0, the input d and the initial time t0.

A. Stability definitions

Let Ω, Ξ be open neighborhoods of the origin in R n , 0 ∈ Ω ⊂ Ξ.

Definition 1.

[4], [START_REF] Haddad | Finite-time stability for time-varying nonlinear dynamical systems[END_REF] At the steady state x = 0 the system (3) with d = 0 is said to be (a) uniformly stable if for any > 0 there is δ( ) such that for any x0 ∈ Ω and t0 ∈ R, if |x0| ≤ δ( ) then |X(t, t0, x0, 0)| ≤ for all t ≥ t0;

(b) uniformly asymptotically stable if it is uniformly stable and for any κ > 0 and > 0 there exists T (κ, ) ≥ 0 such that for any x0 ∈ Ω and t0 ∈ R, if |x0| ≤ κ then |X(t, t0, x0, 0)| ≤ for all t ≥ t0 + T (κ, );

(c) uniformly finite-time stable 1 if it is uniformly stable and finite-time converging from Ω, i.e. for any x0 ∈ Ω and t0 ∈ R there exists 0 ≤ T t 0 ,x 0 < +∞ such that X(t, t0, x0, 0) = 0 for all t ≥ T t 0 ,x 0 . The function T0(t0, x0) = inf{T t 0 ,x 0 ≥ 0 : X(t, t0, x0, 0) = 0 ∀t ≥ T t 0 ,x 0 } is called the settling time of the system [START_REF] Narendra | Persistent excitation in adaptive systems[END_REF].

If Ω = R n , then the corresponding properties are called global uniform stability/asymptotic stability/finite-time stability of x = 0.

1 Another version of uniform finite-time stability has also been proposed in [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF].

An equivalent formulation of these stability properties can be given using the classes of functions KL and GKL [START_REF] Khalil | Nonlinear Systems[END_REF]: i.e. the steady state x = 0 of the system (3) is uniformly stable iff there is a function σ ∈ K such that |X(t, t0, x0, 0)| ≤ σ(|x0|) for all t ≥ t0, any t0 ∈ R and all x0 ∈ Ω; and uniformly asymptotically (finitetime) stable iff there is a function β ∈ KL (β ∈ GKL) such that |X(t, t0, x0, 0)| ≤ β(|x0|, t -t0) for all t ≥ t0, any t0 ∈ R and all x0 ∈ Ω.

In this work we will be also interested in a special stability notion defined not for all t0 ∈ R as in Definition 1, but for a compact interval of initial times t0 and only on a fixed interval of time [START_REF] Kamenkov | On stability of motion over a finite interval of time[END_REF], [START_REF] Lebedev | The problem of stability in a finite interval of time[END_REF], [START_REF] Dorato | Short-time stability in linear time-varying systems[END_REF], [START_REF] Weiss | On the stability of systems defined over a finite time interval[END_REF]: Definition 2. [START_REF] Ríos | Homogeneous time-varying systems: Robustness analysis[END_REF] At the steady state x = 0 the system (3) with d = 0 is said to be (a) short-time stable with respect to (Ω, Ξ, T 0 , T f ) if for any x0 ∈ Ω and t0 ∈ [-T 0 , T 0 ], X(t, t0, x0, 0) ∈ Ξ for all t ∈ [t0, t0 + T f ];

(b) short-finite-time stable with respect to (Ω, Ξ, T 0 , T f ) if it is short-time stable with respect to (Ω, Ξ, T 0 , T f ) and finite-time converging from Ω with the convergence time T t 0 ,x 0 ≤ t0 + T f for all x0 ∈ Ω and t0 ∈ [-T 0 , T 0 ];

(c) globally short-finite-time stable for T 0 ≥ 0 if for any bounded set Ω ⊂ R n containing the origin there exist a bounded set Ξ ⊂ R n , Ω ⊂ Ξ and T f > 0 such that the system is shortfinite-time stable with respect to (Ω, Ξ, T 0 , T f ).

The notions of Definition 2, as in the case of Definition 1, can be equivalently formulated using the functions from the classes K and GKL. If we substitute T 0 = +∞ and T f = +∞, then the stability properties given in Definition 2 can be reduced to their uniform counterparts from Definition 1.

In [START_REF] Kamenkov | On stability of motion over a finite interval of time[END_REF], [START_REF] Lebedev | The problem of stability in a finite interval of time[END_REF], [START_REF] Dorato | Short-time stability in linear time-varying systems[END_REF], [START_REF] Weiss | On the stability of systems defined over a finite time interval[END_REF] the short-time stability is considered for a fixed initial time instant t0 only.

Remark 1. In the literature, short-time stability [START_REF] Dorato | Short-time stability in linear time-varying systems[END_REF] is frequently called stability over a finite interval of time [START_REF] Kamenkov | On stability of motion over a finite interval of time[END_REF], [START_REF] Lebedev | The problem of stability in a finite interval of time[END_REF], [START_REF] Weiss | On the stability of systems defined over a finite time interval[END_REF], but following [START_REF] Ríos | Homogeneous time-varying systems: Robustness analysis[END_REF], we prefer here the former notion to avoid a confusion with finite-time stability from [START_REF] Roxin | On finite stability in control systems[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], since both concepts of stability are used in the paper.

Lemma 1. [19] Let the system in (3) with d = 0 possess a Lyapunov function V : R×Ω → R+, where Ω ⊂ R n is an open neighborhood of the origin, such that for all x ∈ Ω and t ∈ R α1(|x|) ≤ V (t, x) ≤ α2(|x|) α1, α2 ∈ K∞; V (t, x) ≤ -αV η (t, x) + k( t)V η (t, x) α > 0, ∈ R, η ∈ (0, 1)
for a continuous k : R → R, k(0) = 0. Then there exist ∈ R and T 0 > 0 such that the system (3) is short-finite-time stable with respect to (Ω , Ξ, T 0 , T f ) for some Ξ ⊂ R n with Ω ⊆ Ω ⊂ Ξ and T f > 0.

B. Robust stability definitions

Consider the following definition of robust stability for (3) with d = 0:

Definition 3. The system (3) is said to be (a) short-finite-time ISS with respect to (Ω, T 0 , T f , D) if there exist β ∈ GKL and γ ∈ K such that for all x0 ∈ Ω, all d ∈ L∞(R, R m ) with d ∞ < D and t0 ∈ [-T 0 , T 0 ]: |X(t, t0, x0, d)| ≤ β(|x0|, t -t0) + γ( d ∞) ∀t ∈ [t0, t0 + T f ] and β(|x0|, T f ) = 0;
(b) globally short-finite-time ISS for T 0 ≥ 0 if there exist β ∈ GKL and γ ∈ K such that for any bounded set Ω ⊂ R n containing the origin there is T f > 0 such that the system is shortfinite-time ISS with respect to (Ω, T 0 , T f , +∞).

As we can conclude from Definition 3, if the system (3) is shortfinite-time ISS, then for d = 0 there exists Ξ ⊂ R n , Ω ⊂ Ξ such that the system is short-finite-time stable with respect to (Ω, Ξ, T 0 , T f ). The difference of global short-finite-time ISS and a conventional (finite-time or fixed-time) ISS [START_REF] Hong | H∞ control, stabilization, and input-output stability of nonlinear systems with homogeneous properties[END_REF], [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF] is that in the former case the stability property is considered on a finite interval of time [t0, t0

+ T f ] only for t0 ∈ [-T 0 , T 0 ]. Theorem 1. Let the constants > 0, D > 0, T0, T f ∈ R+ and > 0 be given. Let the system in (3) possess a Lyapunov function V : R × Ω → R+, where Ω = {x ∈ R n : |x| ≤ }, such that for all x ∈ Ω, all |d| ≤ D and all t ∈ [-T0, T0 + T f ]: α1(|x|) ≤ V (t, x) ≤ α2(|x|) α1, α2 ∈ K∞; V (t, x) ≤ -u(t)V η (t, x) + κ(|d|) κ ∈ K, η ∈ (0, 1)
for a function u : R → R+ satisfying

t+ t u(s)ds ≥ υ > 0 for all t ∈ [-T0, T0 + T f ] . Then the system (3) is short-finite-time ISS with respect to (Ω , T 0 , T f , D) for β(s, t) = α -1 1 3 η 1-η 4 α 1-η 2 (s) - 1 -η 2 1-η υ 4 (t -) 1 1-η , γ(s) = α -1 1 3 η 1-η 4 4 υ -1 κ(s) 1 η + κ(s)
and

Ω = {x ∈ Ω : β(|x|, 0) + γ(D) ≤ } provided that T f ≥ [1 + 2 1-η 1 -η 4 υ sup x 0 ∈Ω α 1-η 2 (|x0|)].
The condition imposed on u is a version of PE for a finite interval of time, while the last inequality simply checks that the time of convergence to zero of the proposed function β ∈ GKL from Ω is less than given T f .

Proof. First of all note that for any t ∈ (0, T f ] such that t = ν + µ with an integer ν ≥ 1 and µ ∈ [0, ) we have:

t 0 +t t 0 u(t)dt = ν i=1 t 0 +i t 0 +(i-1) u(t)dt + t 0 +t t 0 +ν u(t)dt ≥ ν i=1 t 0 +i t 0 +(i-1) u(t)dt ≥ νυ = νυ t t = υ ν ν + µ t ≥ υ ν ν + 1 t ≥ υ 2 t.
Next, define two sets of instants of time:

T1 = {t ∈ [t0, t0 + T f ) : 4 υ κ( d ∞) < V η (t, x)}, T2 = {t ∈ [t0, t0 + T f ) : 4 υ κ( d ∞) ≥ V η (t, x)}, then obviously [t0, t0 + T f ) = T1 ∪ T2. Denote V (t) = V (t, X(t, t0, x0, d)) and consider t ∈ [t1, t2) ⊂ T1, then V (t) ≤ -u(t) - υ 4 V η (t),
and integrating this inequality we obtain:

V 1-η (t) ≤ V 1-η (t1) -(1 -η) t t 1 u(s) - υ 4 ds for all t ∈ [t1, t2). Thus, V 1-η (t) ≤ V 1-η (t1) -(1 -η) υ 4 (t -t1)
for all t ∈ [t1 + , t2). From the inequality

V (t) ≤ κ(|d|)
for all t ∈ [t1, t1 + ) we get:

V (t) ≤ V (t1) + κ( d ∞)(t -t1) or V 1-η (t) ≤ 2 1-η [V 1-η (t1) + 1-η κ 1-η ( d ∞)].
Therefore,

V 1-η (t) ≤ 2 1-η [V 1-η (t1) + 1-η κ 1-η ( d ∞)] -(1 -η) υ 4 (t -t1 -) for all t ∈ [t1, t2) ⊂ T1, then the estimate that is satisfied for all t ∈ [t0, t0 + T f ) = T1 ∪ T2 is V 1-η (t) ≤ 2 1-η max{V 1-η (t0) - 1 -η 2 1-η υ 4 (t -t0 -), 4 υ -1 κ( d ∞) 1-η η } + 2 1-η 1-η κ 1-η ( d ∞)
and the system is ISS as needed for the given β and γ. The constraint on T f follows from the estimate on the finite time of convergence.

Finally, let us formulate a useful lemma:

Lemma 2.
[24] Let x, y ∈ R and p > 0, then for any κ1 ∈ (0, 1) there exists κ2 > 0 such that

x x + y p ≥ κ1|x| p+1 -κ2|y| p+1 .
In particular, κ2 = max{1 + κ1,

κ 1 (1-κ 1/p 1 ) p }.

C. Dynamic regressor extension and mixing method

Consider the estimation problem of the vector θ ∈ R n in (1) under the following hypothesis:

Assumption 1. Let ω ∈ L∞(R, R n ) and w ∈ L∞(R, R).
As it has been proposed in [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF], in order to overcome the limitations imposed by the condition that ω is PE and also to improve the transient performance, the DREM procedure transforms [START_REF] Ljung | System Identification: Theory for the User[END_REF] to n new one-dimensional regression models, which allows the decoupled estimates of θi, i = 1, n to be computed under a condition on the regressor ω that differs from the persistent excitation.

For this purpose n -1 linear operators Hj : L∞(R, R) → L∞(R, R) are introduced for j = 1, n -1 (for instance an operator Hj can be chosen as a stable linear time-invariant filter with the transfer function Wj(s) = α j s+β j , where s ∈ C is a complex variable and αj = 0, βj > 0 are selected to filter the noise w in (1); or it can realize the delay operation with the transfer function Wj(s) = e -τ j s for τj > 0). Note that y ∈ L∞(R, R) under Assumption 1, then these operators are applied to the measured output y(t) of (1), and using the superposition principles (the operators Hj are linear) we obtain:

ỹj(t) = Hj(y(t)) = ω j (t)θ + wj(t), j = 1, n -1,
where ỹj(t) ∈ R is the j th operator output, ωj : R → R n is the j th filtered regressor function and wj(t) : R → R is the new j th noise signal, which is composed by the transformation of the noise w(t) by Hj and other exponentially converging components related to the initial conditions of the filters. By construction ωj ∈ L∞(R, R n ) and wj ∈ L∞(R, R) for all j = 1, n -1. Define new vector variables

Ỹ (t) = [y(t) ỹ1(t) . . . ỹn-1(t)] ∈ R n , W (t) = [w(t) w1(t) . . . wn-1(t)] ∈ R n
and a time-varying matrix

M (t) = [ω(t) ω1(t) . . . ωn-1(t)] ∈ R n×n ,
then stacking the original equation ( 1) with the n -1 filtered regressor models we construct an extended regressor system:

Ỹ (t) = M (t)θ + W (t).
For any matrix M (t) ∈ R n×n the following equality is true [START_REF] Gantmacher | The Theory of Matrices[END_REF]:

adj (M (t)) M (t) = det (M (t)) In, even if M (t)
is singular, where adj (M (t)) is the adjugate matrix of M (t) and det (M (t)) is its determinant. Recall that each element of the matrix adj (M (t)), adj (M (t)) k,s = (-1) k+s M k,s (t) for all k, s = 1, n, where M k,s (t) is the (k, s) minor of M (t), i.e. it is the determinant of the (n -1) × (n -1) matrix that results from deleting the k th row and the s th column of M (t). Define

Y (t) = adj (M (t)) Ỹ (t), W (t) = adj (M (t)) W (t), φ(t) = det (M (t)) ,
then multiplying from the left the extended regressor system by the adjugate matrix adj (M (t)) we get n scalar regressor models of the form:

Yi(t) = φ(t)θi + Wi(t) (4)
for i = 1, n. Again, by construction Y ∈ L∞(R, R n ), W ∈ L∞(R, R n ) and φ ∈ L∞(R, R).
For the scalar linear regression model (4) the estimation algorithm (2) takes the form:

θi(t) = γiφ(t) Yi(t) -φ(t) θi(t) , γi > 0 (5)
for all i = 1, n, where now, contrarily to (2), the estimation processes for all components of θ are decoupled, and the adaptation gain γi can be adjusted separately for each element of θ. However, all these estimation algorithms are dependent on the same regressor φ(t) (determinant of M (t)).

Define the parameter estimation error as e(t) = θ -θ(t), then its dynamics admits the differential equation:

ėi(t) = -γiφ(t) (φ(t)ei(t) + Wi(t)) , i = 1, n (6)
and the following result can be proven for the DREM method:

Proposition 1. Consider the linear regression system (1) under Assumption 1. Assume that for the selected operators Hj :

L∞(R, R) → L∞(R, R), j = 1, n -1: +∞ t 0 φ 2 (t)dt = +∞ (7)
for any t0 ∈ R, then the estimation algorithm (5) has the following properties:

(A) If W ∞ = 0, then the system (6) is globally asymptotically stable at the origin iff (7) is valid.

(B) For all W ∈ L2(R, R n ) we have e ∈ L∞(R, R n ), in addition, for all t0 ∈ R and ei(t0) ∈ R:

|ei(t)| ≤ e -γ i t t 0 φ 2 (τ )dτ |ei(t0)| + γi 2 1 -e -2γ i t t 0 φ 2 (τ )dτ t t 0 W 2 i (s)ds
for all t ≥ t0 and i = 1, n.

Proof. If W ∞ = 0, then the system (6) can be rewritten as follows:

ėi(t) = -γiφ 2 (t)ei(t), i = 1, n,
and its global uniform stability can be established considering a Lyapunov function V (e) = e diag{γ -1 i } n i=1 e. The equivalence of global convergence of e(t) to zero and ( 7) has been established in [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF]. Thus, the part (A) is proven.

If W ∞ = 0, then solutions of ( 6) can be calculated analytically for all t ≥ t0 and ei(t0) ∈ R:

ei(t) = e -γ i t t 0 φ 2 (τ )dτ ei(t0)-γi t t 0 e -γ i t s φ 2 (τ )dτ φ(s)Wi(s)ds
for all i = 1, n. Further, for all t ≥ t0 and ei(t0) ∈ R:

|ei(t)| ≤ e -γ i t t 0 φ 2 (τ )dτ |ei(t0)| + γi t t 0 |e -γ i t s φ 2 (τ )dτ φ(s)||Wi(s)|ds ≤ e -γ i t t 0 φ 2 (τ )dτ |ei(t0)| + γi t t 0 e -2γ i t s φ 2 (τ )dτ φ 2 (s)ds t t 0 W 2 i (s)ds,
where the Cauchy-Schwarz inequality was used on the last step. Note that ∂ ∂s e -2γ i t s φ 2 (τ )dτ = 2γie -2γ i t s φ 2 (τ )dτ φ 2 (s), hence,

2γi t t 0 e -2γ i t s φ 2 (τ )dτ φ 2 (s)ds = 1 -e -2γ i t t 0 φ 2 (τ )dτ ,
then the desired estimate of the part (B) follows.

Obviously, if the signal φ(t) is PE, then the error dynamics is ISS with respect to W ∈ L∞(R, R n ) and an exponential convergence rate can be guaranteed [START_REF] Narendra | Persistent excitation in adaptive systems[END_REF], [START_REF] Efimov | Design of impulsive adaptive observers for improvement of persistency of excitation[END_REF], [START_REF] Efimov | Robust stability under relaxed persistent excitation conditions[END_REF].

An interested reader is directed to [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF] for a comparison of the condition (7) imposed for φ(t) and the requirement that ω(t) is PE, and also for the discussion about a possibility to select the operators Hj, j = 1, n -1 in a way enforcing the condition [START_REF] Barabanov | On global asymptotic stability of ẋ = -φ(t)φ (t)x with φ(t) bounded and not persistently exciting[END_REF] for φ(t) while initially ω(t) does not admit a persistent excitation. Also an inverse question can be posed: assume that ω(t) is PE, is there a guarantee or restrictions to be imposed for the operators Hj, j = 1, n -1 that the condition ( 7) is satisfied, which is partially addressed in [START_REF] Belov | Enhanced parameter convergence for linear systems identification: The DREM approach[END_REF], [START_REF] Aranovskiy | Parameter identification of linear time-invariant systems using dynamic regressor extension and mixing[END_REF]. In other words, can additional filtering, which leads to a decoupled scalar regressor model, destroys good estimation abilities in (1)? But since in this work we do not need the conditions of persistence of excitation, then there is no obstruction for us to use DREM, as it is stated in the problem statement below.

III. PROBLEM STATEMENT

Consider the static linear regression model (1) under Assumption 1, and assume that the DREM method has been applied in order to reduce the initial problem of vector estimation to n scalar regressor models in the form [START_REF] Khalil | Nonlinear Systems[END_REF]. Note that Y ∈ L∞(R, R n ), W ∈ L∞(R, R n ) and φ ∈ L∞(R, R) under Assumption 1 and due to properties of the DREM approach. Relaxing the condition [START_REF] Barabanov | On global asymptotic stability of ẋ = -φ(t)φ (t)x with φ(t) bounded and not persistently exciting[END_REF], which is also imposed on an infinite interval of time, assume that ω(t) may not admit PE, and that φ(t) is a converging function of time (more precise properties will be formulated separately for the designed algorithms).

It is also assumed that the set of admissible values for θ is known:

Assumption 2. A constant θ > 0 is given such that θ ∈ Ω = [-θ, θ] n .
It is necessary to propose an algorithm generating an estimate θ(t) ∈ R n of the vector of unknown parameters θ ∈ R n , and for W ∞ = 0 providing the property of short-finite-time stability with respect to (Ω, Ω, T 0 , T ) (see Definition 2) of the estimation error e(t) = θ-θ(t) dynamics under assumptions 1 and 2 for some given T 0 and T . If W ∞ = 0 then short-finite-time ISS with respect to (Ω, T 0 , T, W) (see Definition 3) has to be guaranteed for some W > 0.

Since by applying DREM method the problem is decoupled on n independent ones, for brevity of notation, we will further omit the index i in (4) by assuming that n = 1:

Y (t) = φ(t)θ + W (t), (8)
then θ ∈ R, Y ∈ L∞(R, R), W ∈ L∞(R, R) and φ ∈ L∞(R, R).

IV. DESIGN OF ESTIMATION ALGORITHMS CONVERGING IN SHORT-FINITE-TIME

Three different solutions to the posed estimation problem are proposed in this section, whose difference consists in the requirements imposed on excitation of φ(t) and on the guaranteed robustness abilities with respect to W (t).

A. Algorithm 1

Consider an adaptive estimation algorithm proposed in [START_REF] Rueda-Escobedo | Non-linear gradient algorithm for parameter estimation[END_REF], [START_REF] Noack | Fixedtime parameter estimation in polynomial systems through modulating functions[END_REF], [START_REF] Ríos | Time-varying parameter identification algorithms: Finite and fixed-time convergence[END_REF], [START_REF] Ríos | An adaptive sliding-mode observer for a class of uncertain nonlinear systems[END_REF]: [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF] which admits the following properties: Proposition 2. Let assumptions 1 and 2 be satisfied, and

θ(t) = γφ(t) Y (t) -φ(t) θ(t) α , γ > 0, α ∈ [0, 1),
t+ t |φ(s)| 1+α ds ≥ υ ( 10
)
for all t ∈ [-T0, T0 + T ] and some ∈ (0, T ), υ > 0. Take

γ ≥ 2 5-3α 2 1 -α θ1-α T -1 υ ,
then the estimation error e(t) = θ -θ(t) dynamics for (9) with θ(t0) = 0 is short-finite-time ISS with respect to (Ω, T 0 , T, +∞).

Proof. The error dynamics for the estimation algorithm ( 9) can be written as follows:

ė(t) = -γφ(t) φ(t)e(t) + W (t) α .
Consider a Lyapunov function candidate V (e) = 0.5e 2 , whose derivative has an upper estimate for some κ1 ∈ (0, 1) and κ2 > 0 coming from Lemma 2:

V (t) = -γe(t)φ(t) φ(t)e(t) + W (t) α ≤ -γκ1|e(t)φ(t)| 1+α + γκ2|W (t)| 1+α = -2 1+α 2 γκ1|φ(t)| 1+α V 1+α 2 (t) + γκ2|W (t)| 1+α
for any e(t) ∈ R and W (t) ∈ R. Then under the imposed restrictions for φ, the system is short-finite-time ISS with respect to (Ω, T 0 , T, +∞) due to Theorem 1. The imposed restriction on γ guarantees that there exist κ1 ∈ (0, 1) such that all trajectories from Ω converge to the origin faster than

T if W ∞ = 0.
It is important to clarify the differences between the restrictions imposed on φ(t) in propositions 1 and 2 for the case W ∞ = 0. Note that [START_REF] Barabanov | On global asymptotic stability of ẋ = -φ(t)φ (t)x with φ(t) bounded and not persistently exciting[END_REF] allows us to establish the fact of asymptotic convergence of the estimation error e(t), but it does not permit to evaluate the rate of convergence. Of course, the condition ( 7) can be formulated on a finite interval of time:

t 0 +T t 0 φ 2 (t)dt = +∞
that implicitly implies unboundedness of φ(t) contrarily to its admissible convergence in [START_REF] Chowdhary | Concurrent learning adaptive control of linear systems with exponentially convergent bounds[END_REF]. The condition [START_REF] Barabanov | On global asymptotic stability of ẋ = -φ(t)φ (t)x with φ(t) bounded and not persistently exciting[END_REF] can also be strengthened to a usual PE condition (on a finite interval of time), then a practical stability can be obtained only.

B. Algorithm 2

To simplify the notation, suppose in this subsection that φ(t) > 0 for all t ∈ [t0, t0 +T ]. An appealing idea to design a finite-or fixedtime converging system using time-varying feedbacks is presented in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF]. This idea in our context has an interpretation in the form of the following estimation algorithm:

θ(t) = γ(φ(t) -β) -η Y (t) -φ(t) θ(t) , γ > 0, η > 1 (11)
for θ(t0) = 0, which has well-defined solutions while φ(t) is separated with β > 0.

Proposition 3. Let assumption 1 be satisfied and there exist ρ > 0 and κ > 0 such that for any t0 ∈ [-T 0 , T 0 ]

|φ(t)| ≥ κe -ρ(t-t 0 ) ∀t ∈ [t0, t0 + T ]
with β = κe -ρT , then dynamics of the estimation error e(t) = θ -θ(t) in [START_REF] Kamalapurkar | Model-based reinforcement learning for approximate optimal regulation[END_REF] for W ∞ = 0 is short-time stable with respect to (Ω, Ω, T 0 , T ) and asymptotically converging. In addition, if φ(t0 + T ) = β, then the estimation error e(t) of ( 11) is short-finite-time stable with respect to (Ω, Ω, T 0 , T ).

Proof. The error dynamics for the estimation algorithm (11) can be written as follows:

ė(t) = -γ(φ(t) -β) -η (φ(t)e(t) + W (t)) .
For W ∞ = 0 the global uniform stability can be established considering a Lyapunov function V (e) = 0.5e 2 . To prove convergence consider the new time variable τ = f (t), where

f (t) = t t 0 φ(s) (φ(s) -β) η ds.
Such a transformation of time is well-defined under the restrictions introduced on φ(t) for t ∈ [t0, t0 + T ) (i.e.

φ(t)

(φ(t)-β) η > 0 for all t ∈ [t0, t0 + T )). Define (τ ) = e(f -1 (τ )) as the estimation error variable in the new time τ , and recall the following equality

df -1 (τ ) dτ = φ(f -1 (τ )) (φ(f -1 (τ )) -β) η -1 , then d (τ ) dτ = de(f -1 (τ )) dτ = de(s) ds s=f -1 (τ ) df -1 (τ ) dτ = -γ φ(f -1 (τ ))e(f -1 (τ )) + W (f -1 (τ )) (φ(f -1 (τ )) -β) η φ(f -1 (τ )) (φ(f -1 (τ )) -β) η -1 = -γ e(f -1 (τ )) + W (f -1 (τ )) φ(f -1 (τ )) = -γ (τ ) + W (f -1 (τ )) φ(f -1 (τ )) .
Therefore, the variable (τ ) is asymptotically decreasing for W ∞ = 0:

e(f -1 (τ )) = (τ ) = e -γ(τ -f (t 0 )) (f (t0)) + τ f (t 0 ) e -γ(τ -s) W (f -1 (s)) φ(f -1 (s)) ds = e -γτ e(t0) + τ 0 e -γ(τ -s) W (f -1 (s)) φ(f -1 (s)) ds,
where the property f (t0) = 0 has been used in the last step, and returning back to the time t = f -1 (τ ) with a change of variables s = f (σ) in the integral we obtain: [START_REF] Wang | Finite-time parameter estimation without persistence of excitation[END_REF] for all t ∈ [t0, t0 + T ). If φ(t0 + T ) = β and W ∞ = 0 then limt→t 0 +T f (t) = +∞ and, consequently, e(t0 + T ) = 0.

e(t) = e -γf (t) e(t0) + t t 0 e -γ(f (t)-f (σ)) W (σ) (φ(σ) -β) η dσ
An advantage of this lemma is that its result is independent in Assumption 2. For implementation of [START_REF] Kamalapurkar | Model-based reinforcement learning for approximate optimal regulation[END_REF] it is enough that the condition φ(t0 + T ) = β is verified approximately, i.e. φ(t0 + T ) = β +ε for some ε > 0 being close to machine computation precision.

A disadvantage of [START_REF] Kamalapurkar | Model-based reinforcement learning for approximate optimal regulation[END_REF] consists in its weak robustness with respect to the measurement noise W (t) = 0. As we can conclude from [START_REF] Wang | Finite-time parameter estimation without persistence of excitation[END_REF], if φ(t0 + T ) = β and W (t0 + T ) = 0, then e(t0 + T ) = +∞ (recall that if φ(t0 + T ) > β then [START_REF] Kamalapurkar | Model-based reinforcement learning for approximate optimal regulation[END_REF] has an asymptotic convergence rate).

C. Algorithm 3

And, finally, let us introduce a united version of the algorithms ( 9) and [START_REF] Kamalapurkar | Model-based reinforcement learning for approximate optimal regulation[END_REF], which borrows the nonlinear paradigm of the former and uses the time-varying feedback of the latter, but in a nonlinear fashion:

θ(t) = γsign(φ(t)) Y (t) -φ(t) θ(t) |φ(t)| ςφmax , γ > 0, ς > 1, (13)
where φmax = max t∈[-T 0 ,T 0 +T ] |φ(t)| for given T 0 ∈ R+ and θ(t0) = 0. Proposition 4. Let assumptions 1 and 2 be satisfied, and ϑ ∈ L∞(R, R+) where ϑ(t) = W (t) φ(t) . Take

γ ≥ ς max{ θ ς-1 ς , θ}e (ςe) -1 T -1 2 min{1, φ ς -1 max}(ς -1)
, then the estimation error e(t) = θ -θ(t) dynamics of (13) is shortfinite-time ISS with respect to (Ω, T 0 , T, +∞) for the input ϑ.

Proof. The error dynamics for the estimation algorithm (13) can be written as follows:

ė(t) = -γsign(φ(t)) φ(t)e(t) + W (t) |φ(t)| ςφmax .
Consider a Lyapunov function V (e) = 0.5e 2 and observe that

r a(t) (t) ≥ r a min (t) r(t) ≥ 1 r amax (t) r(t) < 1 ≥ min{1, r amax min }
for any r : R → R+ and a : R → R+ such that rmin = inf t∈R r(t), amin = inf t∈R a(t) and amax = sup t∈R a(t) for some rmin, amin, amax ∈ R+, then the time derivative of V for the estimation error dynamics admits an upper estimate: where

V (t) = -
ξ(s) = √ 2γκ1 s 0.5 s ≥ 1 s 1+ς 2ς s < 1 , σ(s) = γκ2 max{1, φ ς -1 max } s 1+ς -1 s ≥ 1 s s < 1
and κ1 ∈ (0, 1), κ2 > 0 are from Lemma 2 (since φ(t) is upper bounded in amplitude such a κ2 exists). Note that 1+ς 2ς < 1 for ς > 1. The last inequality can be represented as follows:

V (t) ≤ -φ |φ(t)| ςφmax max |φ -1 max φ(t)| |φ(t)| ςφmax ξ (V (t)) + σ( ϑ ∞) ≤ -min{1, φ ς -1 max }fς (φ -1 max |φ(t)|)ξ (V (t)) + σ( ϑ ∞),
where

fς (x) = x x/ς for x ∈ [0, 1]. Since fς (0) = fς (1) = 1 and fς is a continuous function, then it has a critical point on the interval [0, 1]. Calculate ∂fς (x) ∂x = fς (x) ς (1 + ln(x)), ∂f 2 ς (x) ∂x 2 = fς (x) ς 1 ς (1 + ln(x)) 2 + 1 x ,
then it is straightforward to conclude that it should be a minimum at the point xmin = e -1 since it is the only point where ∂fς

V (t) ≤ -min{1, φ ς -1 max }e -e -1 ξ (V (t)) + σ( ϑ ∞)
and using the result of Theorem 1, the system is ISS as needed.

In order to evaluate more accurately the time of convergence to the origin assume that W ∞ = 0 then

V (t) ≤ -γκ1|φ(t)| |φ(t)| ςφmax |e(t)| 1+ |φ(t)| ςφmax = -γκ1|φ(t)| |φ(t)| ςφmax θ1+ |φ(t)| ςφmax | θ-1 e(t)| 1+ |φ(t)| ςφmax ≤ -γκ1|φ(t)| |φ(t)| ςφmax min{ θ, θ1+ς -1 }| θ-1 e(t)| 1+ς -1 = -2 1+ς 2ς γκ1 min{1, θ-ς -1 }φ |φ(t)| ςφmax max |φ -1 max φ(t)| |φ(t)| ςφmax V 1+ς 2ς (t) ≤ -τ γV 1+ς 2ς (t), where τ = 2 1+ς 2ς κ1 min{1, θ-ς -1 } min{1, φ ς -1 max }e -(ςe) -1 . Conse- quently, V (t) ≤ V ς-1 2ς (t0) -γ τ ς (ς -1)(t -t0) 2ς ς-1
for all t ∈ [t0, t0 + T ), and the restriction on γ follows ensuring the desired convergence time T .

Thus, the idea of the algorithm (13) consists in the utilization of a nonlinearity such that the function |φ(t)| |φ(t)| ςφmax becomes separated with zero overcoming the absence of excitation in the system. The price for that is the robustness with respect to a noise W with a welldefined ratio W (t) φ(t) . In particular, if φ(t) ≥ β for all t ∈ [t0, t0 + T ) for an arbitrary small β > 0, then the algorithm stays continuous being independent in the excitation of (1) and robust with respect to the noise W .

Remark 2. One of the most important features of estimation algorithms, after estimation error convergence in the ideal case, is their robustness with respect to measurement noises. In our case, since the regressor φ(t) may converge to zero, the appearance of W (t) = 0 additionally limit the time of convergence, since it is reasonable to use the output Y (t) for estimation with t ∈ [t0, t0+T ] only while

|Y (t)| > |W (t)| + ε for some ε > 0. If Y (t)| ≤ |W (t)| + ε (or |Y (t)| is almost equal to |W (t)
| for a sufficiently small ε), then the measured output mainly contains the measurement noise, and it is ambiguous to ask an algorithm to estimate θ due to a bad ratio between the signal and the noise. In this sense the requirement, that the signal W (t)

φ(t)
is well-defined, is not much restrictive (roughly speaking it just assumes that the ratio between the useful signal and the noise lies in reasonable limits).

If the regressor φ(t) is just asymptotically converging without crossing zero, then all these algorithms, ( 9), [START_REF] Kamalapurkar | Model-based reinforcement learning for approximate optimal regulation[END_REF] and [START_REF] Haddad | Finite-time stability for time-varying nonlinear dynamical systems[END_REF], can be applied for any finite T > 0 and t0 ∈ R. Let us check their performances in examples with converging regressors φ(t). Let α = 0.5 and γ = 5 for the algorithm (9), γ = 5 and η = 1.5 for the algorithm [START_REF] Kamalapurkar | Model-based reinforcement learning for approximate optimal regulation[END_REF], γ = 1, ς = 2 and φmax = 1 + 2β for the algorithm [START_REF] Haddad | Finite-time stability for time-varying nonlinear dynamical systems[END_REF]. For the case W ∞ = 0 the results of simulation are shown in Fig. 1, for W (t) = 0.01 sin(10t) the results are given in Fig. 2, where the blue line represents the linear algorithm (5), the magenta line stays for (9), the green line corresponds to [START_REF] Kamalapurkar | Model-based reinforcement learning for approximate optimal regulation[END_REF] and the red line is for [START_REF] Haddad | Finite-time stability for time-varying nonlinear dynamical systems[END_REF]. As we can conclude from these results, the linear algorithm (5) suffers from the absence of excitation, all the rest converge in a finite time to the ideal value of θ in the absence of noise, however, in the presence of noise the algorithm [START_REF] Kamalapurkar | Model-based reinforcement learning for approximate optimal regulation[END_REF] becomes unstable (since the perturbation W (t) is divided by the time-varying gain approaching 0), while the algorithms ( 9) and (13) generate bounded trajectories, but (13) becomes more sensitive to the noise since it is approaching discontinuity.

For the second scenario, let φ(t) = cos(2t) e -t , then φ(t) crosses zero several times on the interval [0, T ). Thus, the algorithm (11) cannot be used in such a case, while the corresponding results for the algorithms (9), ( 13) and (5) (with the same values of parameters) are given in figures 3 and 4 for noisefree and noisy cases, respectively. All these results confirm the theoretical findings of this note.

VI. CONCLUSIONS

The problem of adaptive estimation of a vector of constant parameters in the linear regressor model is studied without the hypothesis that regressor is PE. For this purpose, the initial estimation problem is transformed to a series of the scalar ones using the DREM approach. Three adaptive estimation algorithms are proposed for the scalar scenario converging in a short-finite-time, their robust abilities with respect to the measurement noise are analyzed using the notion

∂x 2 ≥ 0

 20 for all x ∈ [0, 1]. Therefore, fς (xmin) = e -1 ςe and inf ς>1 fς (xmin) = e -e -1 , then

  θ = 1, t0 = 0, T = 5, and for the first scenario chooseβ = 0.01, φ(t) = 2β + cos 2 (t) e -t .

Figure 1 .Figure 2 .

 12 Figure 1. The results of simulation without noise in case 1

Figure 3 .Figure 4 .

 34 Figure 3. The results of simulation without noise in case 2

  R+ → R+ belongs to the class K if α(0) = 0 and the function is strictly increasing, a function α ∈ K belongs to the class K∞ if it is increasing to infinity. A function β : R+ × R+ → R+ belongs to the class KL if β(•, t) ∈ K for each fixed t ∈ R+ and β(s, •) is decreasing and limt→+∞ β(s, t) = 0 for each fixed s > 0. A function β : R+ × R+ → R+ belongs to the class GKL if β ∈ KL and for each s ∈ R+ there is Ts ∈ R+ such that β(s, t) = 0 for all t ≥ Ts.• The identity matrix of dimension n × n is denoted as In.

• For a Lebesgue measurable and essentially bounded function x : R → R n denote x ∞ = sup t∈R |x(t)|, and define by L∞(R, R n ) the set of all such functions with finite norms • ∞; if +∞ -∞ |x(t)| 2 dt < +∞ then this class of functions is denoted by L2(R, R n ). • A continuous function α : • A sequence of integers 1, 2, ..., n is denoted by 1, n.

This work was partially supported by 111 project No. D17019 (China), by the Government of Russian Federation (Grant 08-08) and by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898.