Stabilization of quasistatic evolution of elastoplastic systems subject to

Noname manuscript No. (will be inserted by the editor)

Stabilization of quasistatic evolution of elastoplastic systems subject to periodic loading

arXiv:1708.03084v1 [math.OC] 10 Aug 2017

Ivan Gudoshnikov · Oleg Makarenkov

Received: date / Accepted: date

Abstract This paper develops an analytic framework to design both stress and stretching/compressing T -periodic loadings which make the quasi-static evolution of a one-dimensional network of elastoplastic springs converging to a unique periodic regime. The solution of such an evolution problem is a function t 7→ (e(t), p(t)), where ei (t) and pi (t) are the elastic and plastic deformations of spring i, defined on [t0 , ∞) by the initial condition (e(t0 ), p(t0 )). After we rigorously convert the problem into a Moreau sweeping process with a moving polyhedron C(t) in a vector space E of dimension d, it becomes natural to expect (based on a result by Krejci) that the solution t 7→ (e(t), p(t)) always converges to a T -periodic function. The achievement of this paper is in spotting a class of sweeping processes and closed-form estimates on eligible loadings where the Krejci’s limit doesn’t depend on the initial condition (e(t0 ), p(t0 )) and so all the trajectories approach the same T -periodic solution. The proposed class of sweeping processes is the one for which the normals of any d different facets of the moving polyhedron C(t) are linearely independent. We further link this geometric condition to mechanical properties of the given network of springs. We discover that the normals of any n different facets of the moving polyhedron C(t) are linearely independent, if the number of stretching/compressing constraints is two less the number of nodes of the given network of springs and when the magnitude of the stress loading is sufficiently large (but admissible). In other words, we offer an analogue of the high-gain control method for elastoplastic systems, which can be used to I. Gudoshnikov University of Texas at Dallas, 800 West Campbell Road, Richardson, TX 75080 E-mail: [email protected] O. Makarenkov University of Texas at Dallas, 800 West Campbell Road, Richardson, TX 75080 Tel.: +1-972-883-4617 E-mail: [email protected]

2

Ivan Gudoshnikov, Oleg Makarenkov

design the properties of rheological models of materials (e.g. in creating smart materials). The theoretical results are accompanied by analytic computations for instructive examples. In particular, we convert specific one-dimensional networks of elastoplastic springs into sweeping processes which have never been explicitly addressed in the literature so far. Keywords Elastoplastic springs · Moreau sweeping process · Quasistatic evolution · Periodic loading · Stabilization

Contents 1 2 3

4

5

6 7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The laws of quasistatic evolution for one-dimensional networks of elastoplastic springs Casting the variatonal system as a sweeping process . . . . . . . . . . . . . . . . . 3.1 Derivation of the sweeping process . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Solvability of the sweeping process . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Sweeping processes of particular elastoplastic systems . . . . . . . . . . . . . 3.4 Bounds on the stress loading to satisfy the safe load condition . . . . . . . . . 3.5 Condition on the stretching/compressing loading to eliminate constant solutions 3.6 A computational formula for the moving polyhedron of sweeping processes of elastoplastic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Further discussion of the objective of the paper . . . . . . . . . . . . . . . . . Convergence to a periodic attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Convergence in the case of a moving constraint given by an intersection of translationally moving convex sets . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Strengthening of the conclusion of section 4.1 in the case of a moving constraint given by a polyhedron with translationally moving facets . . . . . . . 4.3 Application: an analytic condition for the convergence of the stresses of elastoplastic systems to an attractor . . . . . . . . . . . . . . . . . . . . . . . . . . Stabilization to a unique non-stationary periodic solution . . . . . . . . . . . . . . 5.1 Stabilization of a general sweeping process with a polyhedral moving set . . . 5.2 Application: an analytic condition for stabilization of elastoplastic systems to a unique periodic regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 5 7 7 12 13 16 19 23 23 25 25 30 31 32 32 36 40 41

1 Introduction No matter how one stretches or compresses an elastic spring, it always returns to the same relaxed length p after the forces are discharged. Elastoplastic springs are different. When the stress s of an elastoplastic spring reaches the boundaries of the interval [c− , c+ ], the respective stretching or compressing changes the relaxed length p permanently, which can be described by the differential inclusion (see Fig. 1)   [0, ∞), if s = c+ , if s ∈ (c− , c+ ), p(t) ˙ ∈ N[c− ,c+ ] (s(t)), N[c− ,c+ ] (s) = {0},  (−∞, 0], if s = c− .

Stabilization of elastoplastic systems

3

A set of connected springs is called a one-dimensional network, if the endpoints of the springs are given by just scalar coordinates (as opposed to vectors). Onedimensional networks of elastoplastic springs are used in rheological modeling of shape memory alloys [12], biomaterials [13, 19, 20, 34], crystalline polymers [16, 32], and other materials. They are also used to model interactions between cells in biomathematics, see [10] and references there in. This paper studies the response of networks of elastoplastic springs to periodic loadings of various nature. Neglecting the inertia (i.e. considering the so-called quasistatic evolution problem), we give sufficient conditions to ensure that the periodic loading stabilizes the network to a single periodic solution, that rules out possible undesirable phenomena of shakedown and ratcheting (see [7, 14] and references therein).

spring stress

t2

t1

c+ t0 c–

stretching the spring compressing the spring

t4

t3 spring length

Fig. 1 The dependence of the stress of an elastoplastic spring on its elongation upon stretching and then compressing. Here c− and c+ are the elasticity limits that the stress is not allowed to exceed.

In his pioneering work [29] Moreau considered an evolving elastoplastic system defined on an abstract configuration space with unilateral and bilateral constraints, and showed that the stresses of springs are described by a sweeping process with a moving convex polyhedron being an intersection of a parallelepiped Π(t) (defined over elastic limits) with a suitable hyperplane V , see Fig. 2. A theory of sweeping processes started to being developed since along with applications in constrained mechanical systems (see e.g. Valadier [36], Monteiro Marques [28], Krejci [23], Ballard [2], Paoli [31], Stewart [35]), while little attention has been paid to explicit connections between particular networks of elastoplastic springs and the respective sweeping process. The best exception is Bastien et al [4] which does offer a method to construct a sweeping process for a given network of springs. However, the modeling in [4] doesn’t take advantage of applied loading in order to reduce the dynamics to a moving constraint. On the contrary, the sweeping process in [4] is a high dimensional sweeping process with an immovable constraint, which hides the dynamics and doesn’t allow to catch convergence of stresses of springs to a unique regime. This paper establishes a rigorous connection between a onedimensional network of elastoplastic springs and a sweeping process with a moving polyhedron.

4


П(t)

П(t)

C(t)

C(t) V

(a)

V (b)

Fig. 2 The moving constraint C(t) = Π(t) ∩ V (dark gray) of a Moreau sweeping process for different locations of the moving parallelepiped Π(t).

As for the mathematical theory of sweeping processes, very little is known about their asymptotic behavior (i.e. the long-term dynamics). The best result is due to Krejci [23], who proved that a sweeping processes with a moving constraint of the form C + c(t), where C is a constant convex set and c(t) is a periodic vector function, admits a periodic attractor. As a matter of fact, the intersection of a moving parallelepiped Π(t) with a fixed hyperplane V (Fig. 2) is not a set of form C + c(t) beginning dimension 2. In agreement with this, a satisfactory theory for asymptotic behavior is currently available for 1-dimensional sweeping processes only (also known as Prandtl-Ishlinskii model, see Brokate [6], Krasnosel’skii-Pokrovskii [22], Visintin [37]. The present paper makes a significant progress in developing a theory of global asymptotic stability of sweeping processes with a moving polyhedron. The paper is organized as follows. The next section formulates the system of laws of quasistatic evolution for a one-dimensional network of m elastoplastic springs on n nodes. Section 3 establishes a correspondence between solutions of this system and the solutions of an associated sweeping process with a moving polyhedron. In particular, we explain how one or another type of periodic loading influences the geometry of the moving polyhedron C(t) and discuss a mechanical interpretation of the dimension dim V if this polyhedron. Furthermore, through specific examples of arrays of elastoplastic springs, Sections 3.3-3.5 provide a guideline for deriving the associated sweeping process in closed form. Section 3.6 rewrites the intersection Π(t) ∩ V through inequalities that are iteratively used throughout the rest of the paper. Building upon the examples of Sections 3.3-3.5, Section 3.7 discusses different possibilities towards the theory of convergence of arrays of elastoplastic springs along with the particular choice this paper takes. In Section 4.1 we consider a general sweeping process with a moving set of a form ∩ki=1 (Ci + ci (t)), where Ci are closed convex sets, and prove (Theorem 2) the convergence of all solutions to a T -periodic attractor X(t). Section 4.2 (Theorem 4) sharpens the conclusion of Theorem 2 for the case when ∩ki=1 (Ci +ci (t)) is the polyhedron Π(t)∩V. Theorem 4 shows that even though X(t) may consist of a family of functions, all those functions exhibit certain similar dynamics. Specifically, we prove that any two function x1 , x2 ∈ X reach


5

(leave) any of the facets of Π(t) ∩ V at the same time. Section 4.3 (Theorem 5) reformulates the conclusion of Theorem 4 in terms of the sweeping process of a network of elastoplastic springs. Section 4.3 concludes by clarifying that the conclusion of Theorem 5 cannot be improved for certain networks of elastoplastic springs. Specifically, Example 2 features families of T -periodic solutions which cannot be destroyed by small perturbations. Section 5 introduces a class of networks of elastoplastic springs whose stresses converge to a unique T -periodic regime regardless of applied T -periodic loadings as long as the magnitudes of those loadings are sufficiently large. We begin Section 5 by addressing a general sweeping process in a vector space E of dimension d with a T -periodic polyhedral moving set with no connection to networks of springs. Theorem 6 of Section 5.1 states that the periodic attractor of such a sweeping process contains at most one non-constant solution, if normals of any d different facets of the moving polyhedron C(t) are linearely independent. Section 5.2 is the main achievement of this paper, where we introduce a class of networks of elastoplastic springs for which the condition of Theorem 6 can be easily expressed in terms of the magnitudes of the periodic loadings. We discovered (Theorem 7) that global stability of a unique periodic regime occurs when both stretching/compressing and stress loadings are large enough.

2 The laws of quasistatic evolution for one-dimensional networks of elastoplastic springs We consider a one-dimensional network of m elastoplastic springs of lengths ek + pk , k ∈ 1, m, where ek and pk are elastic and plastic components respec+ tively. The bounds of the stress of spring k are denoted by [c− k , ck ] and ak stays for the Hooke’s coefficient of this spring. Each spring connects two of n nodes according to ek + pk = ξjk − ξik , where ik and jk are the indexes of the left and right nodes of spring k respectively and ξi is the coordinate of node i. So defined, the one-dimensional network of springs is an oriented graph on n nodes, where the direction from ik to jk is viewed positive through k ∈ 1, m. The paper investigates the evolution of the stresses under the influence of two types of loadings being stretching/compressing loading and stress loading. Stretching/compressing loading locks the distance between nodes Ik and Jk through k ∈ 1, q according to ξJk − ξIk = lk (t). Since we will work with connected graphs of springs only, we assume that each length lk is uniquely determined by the lengths of springs, i.e. for each enforced constraint k ∈ 1, q there exists a chain of springs which connects the left node Ik of the constraint k with its right node Jk . To each enforced constraint k we can, therefore, associate a so-called incidence vector Rk ∈ Rm whose i-th component Rik is −1, 0, or 1 according to whether the spring i increases, not influences, or decreases the displacement when moving from node Ik to Jk along the chain selected, see Fig. 3

6


Rik  1

ei  pi

Ii

Ji

Rik  0 Rik  1

ei  pi

Ii

Ji

Ik

lk (t )

Jk

Ik

lk (t )

Jk

Ik

lk (t )

Jk



0

Ii

ei  pi

Ji

Fig. 3 Illustration of the signs of the components of the incidence vector Rk ∈ Rm . The dotted contour stays for the chain of the springs associated with the vector Rk .

The stress loading fi is a force applied at node i, so that it affects the resultant of forces at node i. We will study a so-called quasi-static evolution problem which further assumes that f (t) can be balanced by the stresses of springs at any time. In other words, we assume that the stresses of springs, the reactions of stretching/compressing constraints, and the applied stress loading compensate one another at each of the n nodes. In particular, for any f (t) = const the system admits at least one equilibrium. With the notations introduced the quasistatic evolution of the stresses sk of springs and reactions rk of enforced constraints can be described by the following variational system (which corresponds to equations (6.1)-(6.6) in the abstract framework by Moreau [29]) Elastic deformation: s = Ae,

(1)

Plastic deformation: p˙ ∈ NC (s),

(2) n

Geometric constraint: e + p ∈ DR ,

(3)

Enforced stretching/compressing: RT (e + p) = l(t), 1

(4)

m

Static balance under stress loading: sk s + . . . + sm s + +r1 r1 + . . . + rq rq + f (t) = 0,

(5)

where s = (s1 , . . . sm )T − stresses of springs, r = (r1 , . . . rq )T − reactions of enforced constraints, e = (e1 , . . . em )T − elastic elongations of springs, p = (p1 , . . . pm )T − plastic elongations of springs, T

l(t) = (l1 (t), . . . , lq (t)) − enforced lengths between nodes Ii and Ji , i ∈ 1, q, T

f (t) = (f1 (t), . . . , fn (t)) − stress loadings at nodes, A = diag(a1 , ..., am ) − matrix of Hooke’s coefficients, − + m NC (s) = ⊗m i=1 N[c− ,c+ ] (s) − Clarke’s normal cone to C = ⊗i=1 [ci , ci ] at s, i

i

m

Dξ = (ξjk − ξik )k=1 − m × n-matrix that defines the graph of springs, R = R1 , . . . , Rq − m × q-matrix of locations of enforced lengths,


7

while the vectors sk = (sk1 , . . . skn )T and rk = (r1k , . . . rnk )T describe the signs of contributions of stresses of spring k and reactions of enforced constraint k into the resultant of forces at nodes 1, ..., n, i.e. (see Fig. 4) – ski = −1, ski = 0, or ski = 1, according to whether the spring k is to the left from node i, not connected to node i, or is to the right from node i, – rik = −1, rik = 0, or rik = 1, according to whether the stretching/compressing loading k is applied to the left from node i, not applied to node i, or applied to the right from node i.

sik  1

sk

s k~

node i

~

sik  1

rkˆ

ˆ

sik  1 sik  1

rk

fi (t )

Fig. 4 Examples of forces applied at node i.

The m×n-matrix D will be termed the kinematic matrix of the one-dimensional network of m springs on n nodes. Note, the matrix −DT will then be the incidence matrix of the associated oriented graph of n nodes and m edges. The static balance law (5) can be written in the equivalent shorter form (25), which is similar to the one used by Moreau [29, formula (3.23)]. We believe, however, that formulation (5) creates a better idea as for why this law does indeed balance the forces. Following Moreau [29], we term system (1)-(5) an elastoplastic system. 3 Casting the variatonal system as a sweeping process 3.1 Derivation of the sweeping process In order for (4) to be solvable in e + p we assume that the enforced constraints {ri (t)}m i=1 are independent in the sense that rank DT R = q. (6) Mechanically, condition (6) ensures that the enforced constraints don’t contradict one another. For example, (6) rules out the situation where two different enforced constraints connect same pair of nodes. Condition (6) implies the existence of an n × q−matrix L such that RT DL = Iq×q .

(7)

Furthermore, as we will show in the proof of Theorem 1, in order for equation (5) to be solvable in s ∈ Rm and r ∈ Rq , the function f (t) must satisfy

8


¯ : R → Rm f (t) ∈ DT Rm . That is why, the existence of a continuous function h such that ¯ f (t) = −DT h(t) (8) is our another assumption. As we clarify in Remark 4, the proof of Theorem 1 implies that assumption (8) is equivalent to f1 (t) + ... + fn (t) = 0.

(9)

Introducing U = x ∈ DRn : RT x = 0 ,

V = A−1 U ⊥ ,

(10)

where U ⊥ = {y ∈ Rm : hx, yi = 0, x ∈ U } , the space V will be the orthogonal complement of the space U in the sense of the scalar product (u, v)A = hu, Avi .

(11)

Therefore, any element x ∈ Rm can be uniquely decomposed as x = x|U + x|V ,

where

x|U ∈ U and x|V ∈ V.

Define g(t) = (DLl(t))|V , ¯ , h(t) = A−1 h(t) U {ξ ∈ Rm : hξ, A(c − x)i ≤ 0, for any c ∈ C} , A NC (x) = ∅, Π(t) = A−1 C + h(t) − g(t),

(12) (13) if x ∈ C, if x ∈ 6 C, (14)

and consider the following differential inclusions A −y˙ ∈ NΠ(t)∩V (y), A z˙ ∈ NΠ(t) (y) + y˙ ∩ U,

(15) (16)

y(0) ∈ Π(t) ∩ V,

(17)

z(0) ∈ U.

(18)

The function g(t) will be termed the effective stretching/compressing loading. Similarly, h(t) is termed the effective stress loading. In what follows we are going to establish an equivalence between systems (1)(5) and (15)-(18). According to Moreau [29, Proposition of §6.d], the problem (16), (18) admits an absolutely continuous (possibly non-unique) solution z on [0, T ] for any absolutely continuous solution y of (15), (17) defined on [0, T ]. The analysis of the dynamics of the elastic deformation e(t) therefore reduces to the analysis of the solution y of the sweeping process (15). In particular, stabilization of (15) will imply stabilization of both elastic deformations e(t) = (e1 (t), ..., em (t))T and stresses s(t) = (s1 (t), ..., sm (t))T of springs.


9

Theorem 1 Let D be the kinematic matrix of a connected network of m elastoplastic springs on n nodes. Let R be a matrix of incidence vectors of q stretching/compressing constraints, which are independent in the sense of (6). Assume that the stress loading doesn’t exceed the safe load bounds, i.e. (C + Ah(t)) ∩ U ⊥ 6= ∅, f or all t ∈ [0, T ],

(19)

holds for C, U and h as defined in (2), (10), and (13). If (s(t), e(t), p(t), r(t)) is a solution of the variational system (1)-(5) on [0, T ], then y(t) = e(t) + h(t) − g(t), z(t) = e(t) + p(t) + h(t) − g(t)

(20)

is a solution of the sweeping process (15)-(18) on [0, T ]. Conversely, if (y(t), z(t)) is a solution of (15)-(18) then (e(t), p(t)) found from (20) is a solution of (1)(5) with s(t) = Ae(t) and with some suitable r(t). Remark 1 Since g(t)|U = 0, condition (19) is equivalent to assuming Π(t) ∩ V 6= ∅, t ∈ [0, T ]. Remark 2 Condition (19) always holds when h(t) ≡ 0 because 0 ∈ C and 0 ∈ U ⊥ . Geometrically, condition (19) means that the parallelepiped Π(t) and the hyperplane V in Fig. 2 do intersect. Mechanically, condition (19) accounts for the fact that the stresses of the elastoplastic springs are bounded and cannot balance arbitrary large stress loadings. Remark 3 The function g(t) and the matrix (DL)|V don’t depend on the choice of matrix L. Indeed, let g˜(t) be the function g(t) obtained by replacing ˜ Then L by L. ˜ = RT ((D(L−L))| ˜ U +(D(L−L))| ˜ V ) = RT (D(L − L))| ˜ V , 0q×q = RT D(L−L) q ˜ ˜ so (D(L − L))|V R ⊂ U . Therefore, (D(L − L))|V = 0m×m and (DL)|V = ˜ . The conclusion about g(t) follows from (12). (DL) V

Proof of Theorem 1. The system of (3) and (4) is equivalent to e(t) + p(t) ∈ U l (t), where U l (t) = x ∈ DRn : RT x = l(t) .

(21)

Applying the both sides of (7) to l(t), we get RT DLl(t) = l(t), which implies DLl(t) ∈ U l (t). Therefore, U l (t) = U + DLl(t) and (21) can be rewritten as e(t) + p(t) ∈ U + DLl(t), or, equivalently, e + p ∈ U + g(t).

(22)

10


–1 j-th spring

  i-th node T D     

       

+1 j-th spring

  i-th node T D      aj

       

aj i-th node

i-th node

Fig. 5 The meaning of the columns and rows of matrix DT . The cell equals +1, if the i-th node is the right endpoint for spring j. Conversely, the cell equals −1, if the i-th node is the left endpoint for spring j.

By the definition of matrix D, the i-th line has +1 (−1) at those nodes which are right (left) endpoints for the j-th spring, see the illustration at fig. 5. Therefore, s1 s1 + . . . + sm sm = −DT s.

(23)

We now claim that r1 r1 + . . . + rq rq = −DT Rr,

where

T

r = (r1 , . . . , rq ) .

(24)

T D is the incidence matrix of the ori−(rk )T ented graph of springs s1 , ..., sm on nodes 1, ..., n supplemented with a virtual spring connecting the nodes ξIk < ξJk . We can now use this virtual spring in order to close the chain of springs given by the incidence vector Rk and obtain a directed cycle where the direction from the node Ik to the node Jk disagree with the direction of the virtual spring. The incidence vector of this cycle is Rk . According to [3, p. 57], we now have −1

Indeed, by Fig. 5, the matrix −

−

D −(rk )T

T

Rk −1

= 0,

k ∈ 1, q,

from which 24) follows. Therefore, taking into account (23), one concludes that (5) can be rewritten as − DT s − DT Rr + f (t) = 0, which has a solution (s(t), r(t)) if and only if ¯ ∈ Ker DT . s(t) + Rr(t) + h(t)

(25)


11

Keeping s(t) fixed, the latter inclusion can be solved for r(t) ∈ Rq if and only if ¯ ∈ Ker DT + R Rq = (DRn )⊥ + x ∈ Rm : RT x = 0 ⊥ = s(t) + h(t) ⊥ = DRn ∩ x ∈ Rm : RT x = 0 = U ⊥ , (26) see e.g. Friedberg et al [11, Exercise 17, p. 367] for the property Ker DT = ⊥ (DRn ) . If s(t) satisfies (26), then by (13) ¯ ¯ ¯ ∈ − A−1 h(t) + A−1 h(t) s(t) + Ah(t) = s + A A−1 h(t) V V U (27) −1 ¯ ⊥ ¯ ∈ s + AA h(t) + AV = s + h(t) + U ∈ U ⊥ . Vice versa, if s(t) satisfies (27) then ¯ ¯ = s + A A−1 h(t) ¯ = + A−1 h(t) s(t) + h(t) V U ¯ ∈ U ⊥, = s(t) + Ah(t) + A A−1 h(t) V which is (26). By applying A−1 to (27), we get e + h(t) ∈ V.

(28)

Since g(t) ∈ V and h(t) ∈ U we can rewrite (22), (28) and (2) as e + p − g(t) + h(t) ∈ U, e + h(t) − g(t) ∈ V, p˙ ∈ NC (Ae). Introducing the change of the variables (20) we have p = z − y and using the substitution e = y − h(t) + g(t) z˙ − y˙ ∈ NAA−1 C (y − h(t) + g(t)), z ∈ U, y ∈ V.

(29)

Let (y, z) be a solution of (29). Since z ∈ U we have −z˙ ∈ −U = U = V ⊥A = NVA (y), where V ⊥A is the orthogonal complement of V in the sense of the scalar product (·, ·)A , and the inclusion (15) computes as follows: −y˙ ∈ NAA−1 C (y − h(t) + g(t)) − z˙ ∈ NAA−1 C+h(t)−g(t) (y) + NVA (y) = A = N(A −1 C+h(t)−g(t))∩V (y), where the last equality holds due to (19) (where both intersecting sets are polyhedral, we use [33, Corollary 23.8.1]). The inclusion (16) follows by combining z˙ = z˙ − y˙ + y˙ ∈ NAA−1 C+h(t)−g(t) (y) + y, ˙

12


with the property z(t) ∈ U observed in (29). Vice versa, if (y, z) is a solution of (15)-(16), then −y ∈ V, z˙ ∈ NAA−1 C+h(t)−g(t) (y) + y, ˙ z˙ ∈ U, which implies (29) when combined with (18).

t u

Remark 4 Having the proof of Theorem 1 behind, we can now clarify why the n equations (5) of static balance in nodes is equivalent to just one equation (9). Indeed, as it follows from the proof of Theorem 1, equation (5) is equivalent to (25) which is in turn equivalent to f (t) ∈ DT Rm .

(30)

It remains to show that (30) is equivalent to (9). Since dim Ker D + Rank D = n, by rank-nullity theorem (see e.g. [11, Theorem 2.3]) and Rank D = n − 1 by Bapat [3, Lemma 2.2] (the dimension of the incidence matrix of a connected graph is one less the number of nodes), one has dim Ker D = 1. On the other hand, D(1, ..., 1)T = 0 by inspection. Therefore, DT Rm = (Ker D)⊥ = {x : (1, ..., 1)x = 0} , i.e. (30) is equivalent to (9). We acknowledge that the ideas of the proof of Theorem 1 are due to Moreau [29], who however worked in abstract configuration spaces and didn’t give details that relate the sweeping process (15) to networks of connected springs (1)-(5). Formulas (12)-(14) establish a connection between mechanical properties of applied loading and geometric properties of the moving constraint Π(t) ∩ V . Specifically, varying the stress loading f (t) moves Π(t) in the direction perpendicular to V in the sense of the scalar product (11). In contrast, varying the stretching/compressing loading l(t) moves Π(t) in the direction parallel V. We also see that the variety of possible perpendicular motions coming from f (t) is limited by the dimension of the space U, which will be computed in section 3.3 (Lemma 1). The dimension of possible directions for the parallel ¯ which motion in V is not always dim V, but is related to the rank of matrix L, we compute in section 3.5, see formula (48).

3.2 Solvability of the sweeping process One of the advantages of associating a part of the dynamics of variational system (1)-(5) to sweeping process (15) is that sweeping processes come with a well developed theory of the existence, data dependence, and regularity of solutions. In particular, for absolutely continuous inputs g(t), h(t) such that the moving set Π(t) ∩ V remains nonempty on an interval [0, T ], the sweeping process (15) with a feasible initial value (17) admits on [0, T ] a unique


13

absolutely continuous solution y(t), in the sense that y(t) is an absolutely continuous function that verifies (64) for a.a. t ∈ [0, T ], see Moreau [29, ch. 5]. Moreover if g, h are Lipschitz-continuous, then such is y, see Kunze and Monteiro Marques [25, sect. 3].

3 2

C(t) 1 _)

Fig. 6 Illustration of a solution of a planar sweeping process (15) whose convex constraint C(t) moves according to a constant vector (bold). The moving constraint doesn’t affect the point (gray cylinder) until the left white boundary reaches the point. The solution of the sweeping process (the coordinate of the point) is, therefore, a constant during phase 1. Beginning phase 2 the constraint catches the point and swipes it towards the corner making the coordinate of the point (i.e. the solution of sweeping process) a piecewise linear function during phase 2-3.

Fig. 6 gives a brief geometric intuition about the dynamics of a planar sweeping process (15). The coordinate of the point on any time interval for an arbitrary absolutely continuous motion of the constraint can be computed using a socalled catch-up algorithm (see Kunze and Monteiro Marques [25]), but we won’t need this computation in our analysis.

3.3 Sweeping processes of particular elastoplastic systems In this section we consider particular networks of elastoplastic springs and offer a guideline that can be used to derive the associated sweeping process (15) in closed form. The following lemma will be used to compute the dimension of U. Lemma 1 If (6) is satisfied, then dim U = n − q − 1.

(31)

Proof. Let E = DRn . Viewing RT as a linear map from E to Rq the ranknullity theorem (see e.g. Friedberg et al [11, Theorem 2.3]) gives dim Ker RT + Rank RT = dim E, where dim Ker RT = dim U by (10), Rank RT = q by (6), and dim E = n − 1 by Bapat [3, Lemma 2.2]. t u

14


Example 1 Consider a one-dimensional network of 3 springs on 4 nodes with the kinematic matrix       ξ ξ2 − ξ1 −1 1 0 0  1  ξ2  Dξ = ξ3 − ξ2  =  0 −1 1 0  (32) ξ3  , ξ4 − ξ3 0 0 −1 1 ξ4 + some 3×3 diagonal matrix A of Hooke’s coefficients and some intervals [c− i , ci ], i ∈ 1, 3, of elasticity bounds. Assume that stretching/compressing loading l(t) ∈ R2 is given by the incidence vectors



 01 (R1 , R2 ) = R =  1 1  , 10

(33)

see Fig. 7. To examine the shapes of the associated moving set Π(t) ∩ V , we l1(t)

f2(t) 1

f1(t)

l2(t) 1

2

f4(t) 3

2

f3(t) 3

4

Fig. 7 A one-dimensional network of 3 springs on 2 nodes with 2 length locking constraints. The circled figures stays for numbers of nodes. The regular figures are the numbers of springs. The figure shows just one possible option for the directions (and magnitudes) of the forces.

find out the eligible values of the function h(t). From (13) we conclude that eligible external forces f (t) lead to h(t) given by h(t) = Ubasis H(t),

(34)

where Ubasis is the m × dim U −matrix of the vectors of a basis of U and H : [0, T ] → Rdim U is any absolutely continuous function. According to (31), there should exist an n × (n − q − 1)−matrix M such that RT DM = 0

and

Rank(DM ) = n − q − 1

(35)

which allows to introduce Ubasis as Ubasis = DM.

(36)

Getting back to the matrices D and R given by (32) and (33) one has dim U = n−q−1 = 4−2−1 = 1. A possible 4×1−matrix that solves (35), the respective


15

Ubasis found from (36), and the respective function h(t) given by (34) are then read as       0 1 1 1      M = (37)  0  , Ubasis = −1 , h(t) = −1 H(t), 1 1 1 where H is an arbitrary absolutely continuous function from [0, T ] to R. Fig. 8 illustrates the shapes of Π(t) ∩ V for different constant values of H(t), where according to (10) we considered V = ((Ubasis )T A)⊥ = (a1 − a2 a3 )⊥ .

(38)

g(t)

h(t) (a)

(b)

(c)

(d)

Fig. 8 Shapes of the moving constraint Π(t) ∩ V for the sweeping process of the net− + − + − work of Fig. 7 with parameters c+ 1 = −c1 = 1, c2 = −c2 = 1.3, c3 = −c3 = 1.6, a1 = a2 = a3 = 1 for different values of stress loading h(t) = (1, −1, 1)T t: a) t = 0, b) t = 0.32, c) t = 0.5, d) t = 0.8. Figure (a) also features the possible directions of the function g(t) (dotted vectors) and the possible direction of the function h(t) (solid vector) that represent stretching/compressing and stress loading respectively.

Example 2 Consider a one-dimensional network of 5 springs on 4 nodes with the kinematic matrix and a single stretching/compressing loading given by      ξ1 −1 1 0 0 0 1  0 −1 1 0 0   ξ2  1          Dξ =  (39)  0 0 −1 1 0   ξ3  , R =  1  ,  0 0 0 −1 1   ξ4  1 0 −1 0 1 0 ξ5 0 + some 5×5 diagonal matrix A of Hooke’s coefficients and some intervals [c− i , ci ], i ∈ 1, 5, of elasticity bounds, see Fig. 9. Our goal is again to examine the shape of C(t) for different values of the external force f (t).

We will follow the lines of Example 1. Formula (31) leads to dim U = 5 − 1 − 1 = 3.

16

Ivan Gudoshnikov, Oleg Makarenkov 1

4

3

2 5

1

2

3

4

5

Fig. 9 A one-dimensional network of 5 springs on 4 nodes with 1 length locking constraint. The circled figures stays for numbers of nodes. The regular figures are the numbers of springs. The arrowed stick is the stretching/compressing loading l1 (t). The stress loadings f1 (t), ..., f5 (t) are applied at nodes, similar to as shown at Fig. 7.

The 5 × 3−matrix M that solves (35), the respective 5 × 3−matrix (36), and the effective external force h(t) given by (34) are found as     1 0 0 000  −1 1 0  1 0 0      , Ubasis =  0 −1 1  , h(t) = Ubasis H(t), 0 1 0 (40) M =      0 0 −1  0 0 1 −1 0 1 000 where H(t) is an arbitrary absolutely continuous function from [0, T ] to R3 . Finally, we use formula (38) and compute V as  ⊥ a1 −a2 0 0 −a5 0  . V =  0 a2 −a3 0 0 0 a3 −a4 a5 The computed formulas for h(t) and V allow us to draw the constraint Π(t)∩V for different values of h(t) as shown at Fig. 10.

3.4 Bounds on the stress loading to satisfy the safe load condition To verify condition (6) it is sufficient to check that stretching/compressing lengths li (t) can be varied independently one from another. Computational algorithms to verify safe load condition (19) for particular systems is a standard topic of computational geometry, see e.g. Bremner et al [5]. In this section we derive analytic conditions which allow to spot classes of elastoplastic systems for which the safe load condition holds. Proposition 1 In order for the safe load condition (19) of Theorem 2 to hold for some t ≥ 0, it is sufficient to assume that − Ah(t) ∈ C.

(41)

Proof. In order to show that (41) implies (19), it is sufficient to observe that 0 ∈ U ⊥ and that (41) yields 0 ∈ C + Ah(t). t u To illustrate Proposition 1, let us consider the elastoplastic system of Fig. 9 (see Example 2).


17

g(t)

n* (a)

(b)

(c)

(e)

(f)

g(t)

(d)

Fig. 10 Shapes of the moving constraint Π(t)∩V for the sweeping process of the network of − Fig. 9 with parameters c+ i = −ci = ai = 1, i ∈ 1, 5 for different values of stress loading h(t) in the view perpendicular to V. Upper row: h(t) = Ubasis (−1/2, −0.8, −1)T t : a) t = 0.62, b) t = 1.06, c) t = 1.3. Lower row: h(t) = Ubasis (1, 0, 0)T t : d) t = 0.55, e) t = 0.8, f) t = 0.92. The bold curves with arrows are solutions of the sweeping process under action of the horizontal displacement g(t). In particular, figures (a)-(c) feature a family of attractive periodic orbits, while (d)-(f) possess a unique globally stable periodic orbit. Furthermore, the figure illustrates that C(t) disappears by shrinking to a line segment when h(t) increases along the vector Ubasis (−1/2, −0.8, −1)T , while C(t) disappears by shrinking to a point when h(t) increases along the vector Ubasis (1, 0, 0)T .

Example 2 (continuation) Let H(t) be as introduced in Example 2 earlier. Based on Proposition 1 we conclude that the safe load condition (19) for the elastoplastic system of Fig. 9 holds at t ≥ 0, if a1 H1 (t) ∈ −a2 H1 (t) + a2 H2 (t) ∈ −a3 H2 (t) + a3 H3 (t) ∈ −a4 H3 (t) ∈ −a5 H1 (t) + a5 H3 (t) ∈

+ [c− 1 , c1 ], − + [c2 , c2 ], + [c− 3 , c3 ], − + [c4 , c4 ], + [c− 5 , c5 ].

(42)

Property (41) is a sufficient condition for the safe load property (19) to hold. Simple necessary and sufficient condition can be offered in the case where dim U = 1. Definition 1 We will say that a spring i is blocked by stretching/compressing loadings, if the family of stretching/compressing loadings {lj }qj=1 contains a chain that connects one end of spring i with its other end. Lemma 2 Assume that the number q of stretching/compressing loadings is 2 less the number of nodes. If none of the springs of the elastoplastic system (1)-(5) is blocked by stretching/compressing loadings, then xi 6= 0 for any i ∈ 1, m, x ∈ U \{0}. Proof. Recall, that Ik and Jk are the left and right endpoints respectively of the stretching/compressing constraint lk (t). Consider the matrix D1 obtained

18


from matrix D by combining the column Ik and the column Jk as follows: 1) add the values of column Jk to the respective values of column Ik , 2) delete the column Jk . Then, D1 ξ : ξ ∈ Rn−1 = Dξ : (Rk )T Dξ = 0, ξ ∈ Rn . Note, D1 is the kinematic matrix for a new elastoplastic system that is obtained from elastoplastic system (1)-(5) by merging the nodes Ik and Jk together and, thus, by reducing the number of nodes by 1. Accordingly, the new elastoplastic system features only q − 1 stretching/compressing loadings and the indexes {Ii , Ji }q−1 i=1 are now from 1, n − 1. Repeating this process through all the incidence vectors {Rk }qk=1 , where q = n − 2 by Lemma 1, we obtain ¯ : ξ ∈ Rn−q = Dξ ¯ : ξ ∈ R2 , U = Dξ : RT Dξ = 0, ξ ∈ Rn = Dξ ¯ is the kinematic matrix of the reduced elastoplastic system that where D is obtained from the original one by merging node Ik with node Jk trough k ∈ 1, q. Since the reduced elastoplastic system has only two nodes (q = n − 2), all the stretching/compressing constraints of the original system split into at most two connected components, which shrink into these two nodes under the proposed reduction process. If spring i is not blocked by stretching/compressing loadings, then the endpoints of spring i belong to different connected components introduced. Therefore, the endpoints of spring i are two different nodes of the reduced elastoplastic system, which implies ui 6= 0, i ∈ 1, m,

¯ such that 0 6= ξ ∈ R2 . for any u = Dξ t u

The proof of the lemma is complete.

Proposition 2 Assume that the conditions of Lemma 2 hold. Let u ¯ be an arbitrary nonzero fixed vector of U (dim U = 1 by Lemma 1) and consider     −sign (¯ u1 ) sign (¯ u1 ) c1 c1     .. .. , , c¯− =  c¯+ =  . .     sign (¯ um )

cm

−sign (¯ um )

cm

+1 where c−1 denotes c− denotes c+ i i and ci i . Then, for each fixed t ≥ 0, the safe load condition (19) holds if and only if

+ − u ¯, c¯ + Ah(t) · u ¯, c¯ + Ah(t) ≤ 0. (43)

Proof. We first show that (43) implies (19). Assume that (19) doesn’t hold for some t ∈ [0, T ]. Therefore, either h¯ u, x + Ah(t)i > 0 for all x ∈ C or h¯ u, x + Ah(t)i < 0 for all x ∈ C. In either case we conclude h¯ u, c¯+ + Ah(t)i · − + − h¯ u, c¯ + Ah(t)i > 0 because c¯ , c¯ ∈ C, which contradicts (43).


19

Let us now show that (19) implies (43). Indeed, since ¯i c¯+ u ¯i c¯− ¯i cji ≤ u i , i ≤u

for any i ∈ 1, m, j ∈ {−1, +1},

we have

+

− u ¯, c¯ + Ah(t) ≤ h¯ u, x + Ah(t)i ≤ u ¯, c¯ + Ah(t) ,

for any x ∈ C.

The latter inequality takes the required form (43) when one plugs x satisfying h¯ u, x + Ah(t)i = 0, which exists because of (19). t u Remark 5 Considering the left-hand-side of (43) as a polynomial P (h¯ u, Ah(t)i) in h¯ u, Ah(t)i, we see that the branches of the polynomial are pointing upwards. Therefore, condition (43) is the requirement for h¯ u, Ah(t)i to stay strictly between the roots of the polynomial. The roots of P (h¯ u, Ah(t)i) are given by h¯ u, Ah(t)i = − h¯ u, c¯− i > 0 and h¯ u, Ah(t)i = − h¯ u, c¯+ i < 0. Therefore, (43) is equivalent to

+

− − u ¯, c¯ ≤ h¯ u, Ah(t)i ≤ − u ¯, c¯ , which highlights that (43) is a restriction on the magnitude of h¯ u, Ah(t)i . Proposition 2 can be e.g. applied to the one-dimensional network of Fig. 7, where dim U = 1 as we noticed in Example 1. Example 1 (continuation) For the elastoplastic system of Example 1 one can consider u ¯ = (1, −1, 1)T and using (37) obtain  −  + c1 c1  , c¯− =  c+  , h¯ u, Ah(t)i = (a1 + a2 + a3 )H(t). (44) c¯+ =  c− 2 2 + − c3 c3 Based on Remark 5 the necessary and sufficient condition for safe load condition (43) to hold is then − + − + − − c+ 1 + c2 − c3 ≤ (a1 + a2 + a3 )H(t) ≤ −c1 + c2 − c3 .

(45)

3.5 Condition on the stretching/compressing loading to eliminate constant solutions Next proposition gives conditions to ensure that any point x which belongs to the moving set Π(t) ∩ V of sweeping process (15) at some initial time t = t1 will lie outside Π(t) ∩ V at time t = t2 . These conditions will, therefore, rule out the existence of constant solutions. Proposition 3 Assume that conditions of Theorem 1 hold. If kA−1 c− − A−1 c+ kA < kg(t1 ) − g(t2 )kA , for some 0 ≤ t1 < t2 , where p − T kxkA = hx, Axi, c− = (c− 1 , ..., cm ) ,

(46)

+ T c+ = (c+ 1 , ..., cm ) ,

then sweeping process (15) doesn’t have any solutions that are constant on [t1 , t2 ].

20


Proof. The claim follows by showing that A−1 C + h(t1 ) − g(t1 ) ∩ A−1 C + h(t2 ) − g(t2 ) = ∅. Since h(t) ∈ U we have A−1 C + h(t) − g(t) ∩ V ⊂ A−1 C V − g(t),

t ∈ [t1 , t2 ],

and it is sufficient to prove that the sets A−1 C V − g(t1 ) and A−1 C V − g(t2 ) don’t intersect. The latter will hold, if the diameter of the set A−1 C V is smaller than the distance between g(t1 ) and g(t2 ), which fact will now be established. Since x|V is the orthogonal projection in the sense of the scalar product (x, y)A = hx, Ayi , we have (see e.g. Conway [9, Theorem 2.7 b)]) kx|V kA ≤ kxkA ,

x ∈ Rm .

Therefore, for any c1 , c2 ∈ C,

−1

A c1 − A−1 c2 ≤ A−1 c1 − A−1 c2 ≤ kA−1 c− − A−1 c+ kA < V A A < kg(t1 ) − g(t2 )kA . The proof of the proposition is complete. Remark 6 Note, the left-hand-side in the squared inequality (46) from the statement of Proposition

3 can be computed as kA−1 c− − A−1 c+ k2A = c− − c+ , A−1 (c− − c+ ) . In what follows we show which kind of computations is required to verify the condition of Proposition 3 in practice. Example 1 (continuation) Given the elastoplastic system of Fig. 7, our goal is to compute the effective stretching/compressing loading g(t) of (12). ¯ such that Observe, that there exists a dim V × q-matrix L ¯ = (DL)|V , Vbasis L

(47)

where Vbasis is the m × dim V −matrix of the vectors of a basis of V . The i-th ¯ is the vector of the coordinates of the respective vector column of matrix L i (DL )V ∈ V in the basis Vbasis , where Li stays for the i-th column on matrix L. Formula (12) can therefore be rewritten as ¯ g(t) = Vbasis Ll(t).

(48)

Computing the effective stretching/compressing loading g(t) has hereby been ¯ turned into computing Vbasis and L. By (31), dim V = m − n + q + 1,

(49)


21

and according to (38), Vbasis is an arbitrary matrix of dim V linearly independent columns that solves (Ubasis )T AVbasis = 0.

(50)

For the particular matrices (32), using the earlier computed Ubasis , see (37), one gets dim V = 2, (Ubasis )T = (1 − 1 1), and a possible solution to (50) is   1/a1 0 Vbasis =  1/a2 1/a2  . 0 1/a3 ¯ we observe that by (7), for any ξ ∈ Rq , we have To find L, (DLξ)|V = DLξ − (DLξ)|U ∈ DRn , as (DLξ)|U ∈ DRn by definition of U. Combining this relation with (7) and ¯ (47) one gets the following equations for L: ¯ = Iq×q , RT Vbasis L ¯ Vbasis L Rq = DRn ,

(51) (52)

¯ can be found. from which L Indeed, by (7) dim U = dim D(Rn ) − q. Taking into account formula (31) for dim U , one gets dim D(Rn ) = n − 1. Thus, for the matrix D given by (32), one has DR4 = R3 and so (52) holds ¯ The matrix L ¯ is therefore a 2 × 2−matrix that solves (51), for any matrix L. which has a unique solution ¯= L

1/a2 1/a2 + 1/a3 1/a1 + 1/a2 1/a2

−1 .

Formula (48), in particular, implies that, for the network of springs of Fig. 7 (where dim V = q = 2), the stretching/compressing constraints are capable to execute any desired motion of C(t) in V. Applying Proposition 3 and Remark 6, we obtain the following condition for non-existence of constant solutions. The elastoplastic system of Fig. 7 doesn’t have constant solutions on [0, T ], if there exist t1 , t2 ∈ [0, T ] such that 4 X

2 1 + 2 ¯

(ci − c− i ) < Vbasis L (l(t1 ) − l(t2 )) A . a i i=1

(53)

Example 2 (continuation) To examine the condition of Proposition 3 for the elastoplastic system of Fig. 9 we will use same formula (48), which will now

22


¯ because (52) will no longer hold idenneed one more step when computing L tically. According to (49) and (50), one gets  1/a1 0  1/a2 −1/a2     =  1/a3 −1/a3   1/a4 0  0 1/a5 

dim V = 5 − 5 + 1 + 1 = 2,

Vbasis

(54)

¯ we express (52) through the equality In order to solve (51)-(52) in L ¯ = 0, (D⊥ )T Vbasis L

(55)

where D⊥ is a column matrix of m − rank D linearly independent vectors that are orthogonal to the columns of D. By (7) dim U = dim DRn − q. Taking into account formula (31) for dim U , one gets dim DRn = n−1, dim D⊥ = m−n+1, and so D⊥ turns out to be an m × (m − n + 1)−matrix that solves the equation (D⊥ )T D = 0(m−n+1)×(m−n+1) .

(56)

Getting back to the particular matrices D given by (39), we compute dim D⊥ = 5 − 5 + 1 = 1 and the 5 × 1-dimensional solution of (56) is  0  1     D⊥ =   1 .  0  −1 

(57)

¯ one Solving the system of (51) and (55) with respect to the 2 × 1−matrix L obtains ¯= L

1111 0 0 1 1 0 −1

Vbasis

−1 1 , 0

¯ g(t) = Vbasis Ll(t),

(58)

where l is an absolutely continuous function from [0,T] to R. Therefore, when the stretching/compressing loading is only capable to move the convex con¯ see Fig. 10. straint C(t) along a single direction Vbasis L, Therefore, Proposition 3 gives the following condition for non-existence of constant solutions. Constant solutions don’t exist on [0, T ], if one can spot t1 , t2 ∈ [0, T ] such that 5 X

2 1 + ¯ 2 · (l(t1 ) − l(t2 ))2 . ci − c− < Vbasis L i A a i=1 i

(59)


23

3.6 A computational formula for the moving polyhedron of sweeping processes of elastoplastic systems To give a deeper look into the possible dynamics of sweeping processes of elastoplastic systems we now rewrite Π(t) ∩ V in a slightly different form which is more suitable for computational analysis. From 1 + 1 − −1 m A C = x ∈ R : ci ≤ xi ≤ ci ai ai we have + A−1 C+h(t)−g(t) = x ∈ Rm : c− i + ai hi (t) ≤ hei , Ax + Ag(t)i ≤ ci + ai hi (t) , where ei ∈ Rm is the vector with 1 in the i-th component and zeros elsewhere. Since g(t) ∈ V, one has hei , Ax + Ag(t)i = h ei |U + ei |V , Ax + Ag(t)i = h ei |V , Ax + Ag(t)i ,

x ∈ V,

and we conclude Π(t) ∩ V =

m \ + x ∈ V : c− i +ai hi (t) ≤ hni , Ax+Ag(t)i ≤ ci +ai hi (t) , (60) i=1

where ni = ei |V . 3.7 Further discussion of the objective of the paper As Examples 1 and 2 illustrate, the stretching/compressing loading can either be capable to move the convex constraint in an arbitrary direction (as in Example 1) or the available directions of motion can be limited (as in Example 2). In principle, the later case may lead to global convergence to a unique periodic solution. Figs. 10(g)-(i) are examples of moving sets whose horizontal displacement back and forth stirs all the trajectories to a globally stable periodic solution. And, at least in the case of one stretching/compressing loading (q = 1), it looks possible to follow the ideas of Adli et al [1] and obtain global asymptotic stability of a periodic solution by assuming that g(t) lies strictly A inside the normal cone NΠ(t)∩V (x) for at least one x and t ∈ [0, T ] (such a point x is the rightmost vertex of the triangle as far as Figs. 10(g)-(i) are concerned). Though simple to formulate, such a condition appears computationally involved in spaces of higher dimension. One may tempt to think that the requirement for g(t) to lie strictly inside A NΠ(t)∩V (x) can be always achieved for at least one pair of t and x and at least for generic networks of elastoplastic systems. In other words, even though A vector g(t) is never in the interior of NΠ(t)∩V (x) for Figs. 10(a)-(f), one may believe that small perturbation will lead to a polyhedron Π(t) ∩ V where g(t)

24


is no longer perpendicular to any of its facets. To out biggest surprise, the latter statement is incorrect, that we now demonstrate. Example 2 (continuation) In what follows we stick to h(t) = Ubasis (−1/2, −0.8, −1)T t, which corresponds to Figures 10(a)-(c). Observe, that some of the edges of C(t) in e.g. 10(a) look simultaneously orthogonal to the direction g(t) of the motion of C(t). In what follows, we prove that those edges of C(t) which look orthogonal to the direction of g(t) are indeed orthogonal to g(t). In other words, our proof of orthogonality of the edges of C(t) to g(t) will use the data from Fig. 10, so it is not completely analytic. This proof is fully convincing as long as one trusts Fig. 10 (in order to plot Fig. 10 and as well as other sets of type (60) we used Matlab together with its standard Optimization Toolbox and external library [18], which implements the algorithm [5]). The complete analytic proof is doable but unreasonably bulky. From formula (60), the normal vectors ni are given by ni = ei |V , which is equivalent to saying that ni = Vbasis n ¯ i , where n ¯ i ∈ Rdim V are found from Vbasis n ¯ i − ei ∈ U. Observe, that x∈U

if and only if

RT (D⊥ )T

x = 0.

Therefore,

RT (D⊥ )T

Vbasis n ¯i =

and so n ¯i =

RT (D⊥ )T

RT (D⊥ )T

−1

Vbasis

RT (D⊥ )T

ei ei .

When a1 = a2 = a3 = a4 = a5 = 1, for R, D⊥ and Vbasis given by formulas (39), (57) and (54), the exact values of vectors n ¯ i compute as 0.375 0.125 0.125 0.375 0.25 (¯ n1 , n ¯2, n ¯3, n ¯4, n ¯5) = , (61) 0.25 −0.25 −0.25 0.25 0.5 while according to formula (58) the function g(t) appears to be 0.375 g(t) = Vbasis l(t). 0.25

(62)

The analysis so far doesn’t say anything as for which of the normals ni = Vbasis n ¯ i , i ∈ 1, 5, is the normal n∗ of the vertical edges of C(t) of Fig. 10(a). However, by observing from the figure that n∗ is almost parallel to g(t), we conclude that the only options for n∗ are n∗ = n1 or n∗ = n4 , because the angle between g(t) and the normals n2 , n3 , n4 exceeds π/4, as can be analytically


25

estimated based on formulas (61)-(62). Since, again using formulas (39) and (57), RT 1111 0 = 0 1 1 0 −1 (D⊥ )T we conclude

RT (D⊥ )T

e1 =

RT (D⊥ )T

1 e4 = . 0

Therefore, n∗ = Vbasis

1111 0 0 1 1 0 −1

Vbasis

−1 1 0

(63)

when a1 = a2 = a3 = a4 = a5 = 1, h(t) = Ubasis (−1/2, −0.8, −1)T t, and when + c− i < ci are arbitrary fixed constants. Vector n∗ is parallel to g(t) by formula (58), so, for the given values of the parameters and provided that (59) is satisfied, the sweeping process (15) of the elastoplastic system of Fig. 9 admits a family of non-constant periodic solutions (see, in particular, Fig. 10(b)). But expression (63) remains parallel to expression (58) also if we slightly perturb the above-mentioned parameters. Therefore, sweeping process (15) of Fig. 9) will keep consisting of a family of non-constant periodic solutions even if we perturb the mechanical parameters of the network of Fig. 9. In other words, the non-existence of a unique attractive periodic solution appears to be a structurally stable property for some elastoplastic systems. To summarize, the objective of this paper is to spot a class of sweeping processes which converge to a single periodic solution for any values of the parameters. Such an objective is well known in control theory for differential equations and is achieved over the method of high-gain feedback, which intuitively corresponds to introducing a ”high amount” of control. As for elastoplastic systems, we will show that such an objective can be achieved by introducing a sufficient number of stretching/compressing loadings.

4 Convergence to a periodic attractor 4.1 Convergence in the case of a moving constraint given by an intersection of translationally moving convex sets In this section we establish convergence properties of a general sweeping process 0 − x˙ ∈ NC(t) (x), y ∈ E, (64) where E is a d-dimensional linear vector space, C(t) ⊂ E is convex closed set for any t, and {ξ ∈ E : (ξ, c − x)0 ≤ 0, for any c ∈ C} , if x ∈ C, NC0 (x) = (65) ∅, if x ∈ 6 C,

26


where (·, ·)0 is some scalar product in E. These convergence properties are then refined in section 4.3 in the context of the particular sweeping process (15). A set-valued function t 7→ C(t) is called globally Lipschitz continuous, if dH (C(t1 ), C(t2 )) ≤ LC |t1 − t2 |, for all t1 , t2 ∈ R, and for some LC > 0, (66) where dH (C1 , C2 ) is the Hausdorff distance between two closed sets C1 , C2 ∈ E defined as dH (C1 , C2 ) = max sup dist(x, C1 ), sup dist(x, C2 ) (67) x∈C2

x∈C1

with dist(x, C) = inf {|x − c| : c ∈ C} . As preliminary discussed in section 3.2, if C(t) is a globally Lipschitz continuous function with nonempty closed convex values from E, then the solution x(t) of sweeping process (64) with any initial condition x(t0 ) = x0 is uniquely defined on [t0 , ∞) in the sense that x(t) is a Lipschitz continuous function that verifies (64) for a.a. t ∈ [t0 , ∞). Let us use t 7→ X(t, x0 ) to denote the solution of sweeping process (64) that takes the value x0 at time 0. In what follows, we consider the set X(t) of T -periodic solutions of (64) [ X(t) = {X(t, x0 )} (68) x0 ∈C(0):X(0,x0 )=X(T,x0 )

and prove that, for T -periodic moving constraint C(t), the set X(t) attracts all the solutions of (64). Note that the condition X(0, x0 ) = X(T, x0 ) implies X(0, x0 ) = X(jT, x0 ), j ∈ N, when t 7→ C(t) is T -periodic. Definition 2 A set-valued function t 7→ Y (t) is a global attractor of sweeping process (64), if dist(x(t), Y (t)) → 0 as t → ∞ for any solution x of sweeping process (64). Finally, we denote by ri(C) the relative interior of a convex set C ⊂ E, see Rockafellar [33, §6]. Theorem 2 Let t 7→ C(t) be a Lipschitz continuous uniformly bounded T periodic set-valued function with nonempty closed convex values from E. Let t 7→ X(t) be the set of T -periodic solutions of sweeping process (64) as defined in (68). Then, X ⊂ C([0, T ], E) is closed and convex. If, in addition, C(t) is an intersection of closed convex sets Ci (some of them, say, first p sets, may be polyhedral) that undergo just translational motions C(t) =

k \

(Ci + ci (t)),

i=1

where ci (t) are single-valued T -periodic Lipschitz functions such that

(69)


p \

(Ci + ci (t)) ∩

k \

27

t ∈ [0, T ],

(70)

x(t) ˙ = y(t), ˙ f or any x, y ∈ X and f or a.a. t ∈ [0, T ], and X(t) is a global attractor of (64).

(71)

i=1

(ri(Ci ) + ci (t)) 6= ∅,

i=p+1

then

The theorem, in particular, implies that X(t) cannot contain non-constant solutions, if it contains at least one constant solution. The proof of theorem 2 is split into 3 lemmas. Lemma 3 establishes the convexity of X (closedness of X(t) follows from the continuous dependence of solutions of (64) on the initial condition, see [25, Corollary 1]). Lemma 4 proves the statement (71). Finally, the global attractivity of X(t) is given by Theorem 3 which is an extension of a result from Krejci [23] for convex sets (69). In what follows, k · k0 is the norm induced by the scalar product in E, i.e. p (72) kxk0 = (x, x)0 . Lemma 3 Let t 7→ C(t) be a Lipschitz continuous set-valued function with nonempty closed convex values from E. Then, both X(t) ⊂ E and X ⊂ C(R, E) are convex. Moreover, kx(t) − y(t)k0 is constant in t,

x, y ∈ X.

(73)

0 Proof. Let x, y ∈ X. Due to monotonicity of NC(t) (x) in x the distance t 7→ kx(t) − y(t)k0 cannot increase (see e.g. [25, Corollary 1]). Notice, that t 7→ kx(t) − y(t)k0 cannot decrease, otherwise it cannot be periodic, so (73) follows.

For any θ ∈ (0, 1) the initial condition θx(0) + (1 − θ)y(0) belongs C(0) by convexity of C. Let xθ be the corresponding solution. Since t 7→ kx(t)−xθ (t)k0 and t 7→ kxθ (t) − y(t)k0 are also non-increasing, then kx(t) − xθ (t)k0 + ky(t) − xθ (t)k0 6 kx(0) − xθ (0)k0 + ky(0) − xθ (0)k0 = = kx(0) − y(0)k0 = kx(t) − y(t)k0 . On the other hand, the triangle inequality yields kx(t) − xθ (t)k0 + ky(t) − xθ (t)k0 > kx(t) − y(t)k0 and so kx(t) − xθ (t)k0 + ky(t) − xθ (t)k0 = kx(t) − y(t)k0 . By strict convexity of inner product space E, the only possibility for xθ is (see e.g. Narici-Beckenstein [30, Th 16.1.4 d)]) xθ (t) = θx(t) + (1 − θ)y(t). This formula, in particular, implies that xθ is T -periodic. The proof of convexity of X is complete.

28


Lemma 4 Let t 7→ C(t) be a set-valued function of the form (69)-(70) with convex closed Ci and Lipschitz-continuous single valued ci (t). Let x and y be two T -periodic solutions of sweeping process (64) defined on [t0 , ∞). Then (71) holds. Proof. The properties (69)-(70) imply (see [33, Corollary 23.8.1]) that 0 NC(t) (x)

=

k X

NC0 i +ci (t) (x),

for all x ∈ C(t) and for a.a. t ∈ [0, T ].

(74)

i=1

Let t ∈ (0, T ) be such that x(t), ˙ y(t), ˙ c˙i (t), i ∈ 1, k, exist and (74) holds. Property (74) allows to spot x˙ ti , y˙ it , i ∈ 1, k, such that x(t) ˙ =

k X

x˙ ti , −x˙ ti ∈ NC0 i +ci (t) (x(t)),

i ∈ 1, k,

y˙ it , −y˙ it ∈ NC0 i +ci (t) (y(t)),

i ∈ 1, k.

i=1

y(t) ˙ =

k X i=1

To show that kx(t) ˙ − y(t)k ˙ 0 = 0, consider 2 kx(t) ˙ − y(t)k ˙ ˙ − y(t), ˙ x(t) ˙ − y(t)) ˙ 0 = (x(t) 0 =

=

k X

x˙ ti , x(t) ˙ − c˙i (t)

+ 0

i=1

−

k X

y˙ it , y(t) ˙ − c˙i (t) 0 −

(75)

i=1 k X

x˙ ti , y(t) ˙ − c˙i (t)

− 0

k X

y˙ it , x(t) ˙ − c˙i (t)

0

. (76)

i=1

i=1

For the value of t ∈ (0, T ) as fixed above, we now prove that each of sums in (75)-(76) vanish. Step 1. Vanishing sums in (75). Fix i ∈ 1, k. By the definition of normal cone, x˙ ti , z + ci (t) − x(t) 0 ≥ 0 and y˙ it , z + ci (t) − y(t) 0 ≥ 0 for all z ∈ Ci . (77) Considering z = x(t + h) − ci (t + h) ∈ Ci , we observe that the function f (h) = x˙ ti , x(t + h) − x(t) − (ci (t + h) − ci (t)) 0 is non-negative in a neighborhood of zero. Since f (0) = 0, we conclude that 0 = f 0 (0) = (x˙ ti , x(t) ˙ − c˙i (t))0 . The relation (y˙ it , y(t) ˙ − c˙i (t))0 = 0 can be proved by analogy using the second inequality of (77). Step 2. Vanishing sums in (76). We claim that k X i=1

k X x˙ ti , zi + ci (t) − y(t) 0 ≥ 0, y˙ it , zi + ci (t) − x(t) 0 ≥ 0, zi ∈ Ci , (78) i=1


29

so that the arguments of Step 1 apply to f (h) =

k X

x˙ ti , y(t + h) − y(t) + ci (t) − ci (t + h) 0

i=1

(similarly for the second sum of (78) with zi = x(t + h) − ci (t + h)) to show that the sums in (76) vanish. To establish (78), we first rewrite it as k X

x˙ ti , zi + ci (t)

− (x(t), ˙ y(t))0 ≥ 0, 0

i=1

k X

y˙ it , zi + ci (t)

0

− (y(t), ˙ x(t))0 ≥ 0,

i=1

and then prove that (x(t), ˙ y(t))0 = (x(t), ˙ x(t))0 and (y(t), ˙ x(t))0 = (y(t), ˙ y(t))0 , t ∈ [0, T ], (79) so that (78) becomes a consequence of (77). To prove (79) we use (73) and observe that d ˙ y(t) − x(t))0 − (y(t), ˙ x(t) − y(t))0 0 = kx(t) − y(t)k20 = − (x(t), dt But x(t), y(t) ∈ C(t) and both these functions are solutions of sweeping process (64). Therefore, (x(t), ˙ y(t) − x(t))0 > 0 and (y(t), ˙ x(t) − y(t))0 > 0, which implies (79). The proof of the lemma is complete.

t u

We acknowledge that the idea of the proof of Step 1 of Lemma 4 has been earlier used by Krejci in the proof of [23, Theorem 3.14], which would suffice for the proof when k = 1. The achievement of Lemma 4 is in considering k > 1, thus the new Step 2. Accordingly, the proof of the next theorem follows the lines of [23, Theorem 3.14] with Lemma 4 used to justify (103), which is the place of the proof that needed further arguments when moving to k > 1. We present a proof for completeness (Section 7) also because [23] employs slightly different notations. The theorem effectively states that any bounded solution of a T -periodic sweeping process is asymptotically T -periodic, which facts is known in differential equations as Massera’s theorem [27]. Theorem 3 (Massera-Krejci theorem for sweeping processes with a moving set of the form C(t) = ∩ki=1 (Ci + ci (t))) Let t 7→ C(t) be a set-valued uniformly bounded function of the form (69)-(70) with convex closed Ci and Lipschitz-continuous single-valued T -periodic ci (t). Then the set X(t) of T -periodic solutions of (64) is a global attractor of (64). The proof of Theorem 3 is given in Section 7. An interested reader can note that sweeping process (64) with k = 1 converts to a perturbed sweeping process −ξ˙ = NC0 1 (ξ) + c˙1 (t) with an immovable constraint by the change of the variables ξ(t) = y(t) − c1 (t), while it is not clear whether or not (64) converts to a perturbed sweeping process with a constant constraint when k > 1. This further highlights the difference between the cases k = 1 and k > 1 as long as potential alternative methods of analysis of the dynamics of (64) are concerned.

30


4.2 Strengthening of the conclusion of section 4.1 in the case of a moving constraint given by a polyhedron with translationally moving facets When applied to a one-dimensional network of elastoplastic springs (1)-(5), the existence of a periodic attractor X(t) for the associated sweeping process (15) follows from Theorem 2. A new geometric property of X(t) that comes with considering the sweeping process (15) is due to the polyhedral shape of the moving constraint Π(t) ∩ V , see Section 3.6. Theorem 4 below states that even if X(t) consists of several periodic solutions, they all exhibit certain identical behavior. As earlier, let E be a finite-dimensional linear vector space equipped with a scalar product (·, ·)0 and let ri(X) be the relative interior of the convex set X ∈ C([0, T ], E). Theorem 4 Assume that a uniformly bounded set-valued function t 7→ C(t) is given by C(t) =

m \

+ x ∈ E : c− i (t) ≤ (ni , x)0 ≤ ci (t) ,

t > 0,

(80)

i=1 + where c− i , ci are single-valued globally Lipschitz continuous functions, ni are given vectors from E. Then the set X(t) of T -periodic solutions of sweeping process (64) is the global attractor of (64). Furthermore, X ⊂ C([0, T ], E) is closed and convex, and all the interior solutions of X follow the same pattern of motion in the sense that

J(t, x(t)) = J(t, y(t)), f or all x, y ∈ ri(X), t ≥ 0,

(81)

where J(t, x) is the active set of the polyhedron C(t) given by + J(t, x) = i ∈ −m, −1 : (n−i , x)0 = c− −i (t) ∪ i ∈ 1, m : (ni , x)0 = ci (t) . Theorem 4 is a corollary of Theorem 2 except for the property (81) which comes from the polyhedral shape of the moving constraint C(t). The property (81) follows from the following general result. Lemma 5 Consider an arbitrary convex set B⊂

k \

{x ∈ E : (yi , x)0 ≤ bi } ,

i=1

where yi ∈ E and bi ∈ R. If x1 , x2 ∈ ri(B), then, for all i ∈ 1, k, (yi , x1 )0 = bi

if and only if

(yi , x2 )0 = bi .

(82)


31

Proof. Consider xθ = θx1 + (1 − θ)x2 . Since x1 , x2 ∈ ri(B), there exists ε > 0 such that x−ε ∈ ri(B) and x1+ε ∈ ri(X). Put x ¯1 = x−ε , x ¯2 = x1+ε . Then there exist θ1 , θ2 ∈ (0, 1), θ1 6= θ2 , such that x1 = θ1 x ¯1 + (1 − θ1 )¯ x2 ,

x2 = θ2 x ¯1 + (1 − θ2 )¯ x2 .

(83)

Assume that (yi , x1 )0 = bi . Then θ1 ((yi , x ¯1 )0 − bi ) = −(1 − θ1 )((yi , x ¯2 )0 − bi )

(84)

by using the first formula of (83). Since x ¯1 , x ¯2 ∈ B, one has (yi , x ¯ 1 ) 0 − bi ≤ 0 and (yi , x ¯2 )0 − bi ≤ 0. Therefore formula (84) can only hold when both (yi , x ¯1 )0 − bi and (yi , x ¯2 )0 − bi vanish. Hence (yi , x2 )0 = θ2 (yi , x ¯1 )0 + (1 − θ2 ) (yi , x ¯2 )0 = θ2 bi + (1 − θ2 )bi = bi . The reverse implication in (82) can be proved by analogy.

t u

4.3 Application: an analytic condition for the convergence of the stresses of elastoplastic systems to an attractor Let JC (x) be the active set of the parallelepiped C, i.e. + . JC (x) = i ∈ −m, −1 : x−i = c− −i ∪ i ∈ 1, m : xi = ci A direct consequence of Theorem 4 is the following result about asymptotic behavior of the stresses of the elastoplastic system (1)-(5). Theorem 5 Let the conditions of Theorem 1 hold and both stretching/compressing and stress loadings are T -periodic. Then, for any initial condition at t = 0, the stresses s1 (t), ..., sm (t) of the springs converge, as t → ∞, to the attractor S(t) = A (X(t) − h(t) + g(t)) , where X(t) is the set of all T -periodic solutions of sweeping process (15), and h(t) and g(t) are the effective loadings given by (8) and (13). The functions of S(t) have equal derivatives for a.a. t ≥ 0 as per (71) and, moreover, JC A−1 s¯i (t) = JC A−1 sˆi (t) , f or all s¯, sˆ ∈ ri(S), t ≥ 0. (85) Proof. We apply Theorem 4 with C(t) = Π(t) ∩ V, where Π(t) and V are those defined in Theorem 1. Since Π(t) is uniformly bounded in t ∈ [0, T ], same holds for C(t). Thus, the conditions of Theorem 4 are satisfied with − + + c− i (t) = ci + ai hi (t) and ci (t) = ci + ai hi (t), and Theorem 4 implies that J(t, A−1 s¯(t)+h(t)−g(t)) = J(t, A−1 sˆ(t)+h(t)−g(t)), for all s¯, sˆ ∈ ri(S), t ≥ 0,

32


which equivalent formulation is (85). Other statements of Theorem 5 follow from Theorem 4 just directly. The proof of the theorem is complete. t u Property (85) says that, for any i ∈ 1, m, the spring i will asymptotically execute a certain pattern of elastoplastic deformation which doesn’t depend on the state of the network at the initial time. We remind the reader that if si , i ∈ 1, m, are the stresses of springs, then the 1 quantities si (t) that appear in (85) are the elastic elongations of the springs. ai Examples 1 and 2 (continuation). For the elastoplastic system (1)-(5) of Example 1 (respectively Example 2) with T -periodic stretching/compressing and stress loadings l(t) and h(t), Theorem 5 implies the convergence of stresses s(t) to a T -periodic attractor S(t) provided that property (45) (respectively (42)) holds. Furthermore, the functions of A−1 S(t) + h(t) − g(t) are all nonconstant, if (53) (respectively (59)) is satisfied. In the next section of the paper we offer a general result which will, in particular, imply that the attractor A−1 S(t) + h(t) − g(t) in Example 1 consists of a single solution. In contrast, computations of Section 3.7 imply that S(t) in Example 2 contains a structurally stable family of T -periodic solutions for open sets of stress loadings h(t). In other words, the conclusion of Theorem 5 cannot be sharpen as far as Example 2 is concern.

5 Stabilization to a unique non-stationary periodic solution In this section we first prove that the periodic attractor X(t) of a general sweeping process (64) in a vector space of dimension d consists of just one non-stationary T -periodic solution, when the normals of any d different facets of the moving polyhedron C(t) are linearely independent. Then we give a sufficient condition for such a requirement to hold for the sweeping process (15) coming from the elastoplastic system (1)-(5).

5.1 Stabilization of a general sweeping process with a polyhedral moving set As earlier, let E be a linear vector space of dimension d and let (·, ·)0 be a scalar product in E. In this subsection it will be convenient to rewrite the set (80) in the following form k \ C(t) = {x ∈ E : (x, ni )0 ≤ ci (t)} , t > 0, (86) i=1

where ci are single-valued functions and ni are given vectors of E. The ad0 vantage of form (86) compared to (80) is that any vector of NC(t) (x) has non-negative coordinates in the basis formed by the normals n1 , ..., nk , as our


33

Lemma 6 shows. Then we establish the following result about global asymptotic stability of sweeping processes. Theorem 6 Let t 7→ C(t) be a uniformly bounded set-valued function given by (86), where the functions ci are globally Lipschitz continuous and k ≥ dim E. Assume that any dim E vectors out of the collection {ni }ki=1 ⊂ E are linearly independent and the cardinality of the set J(t, x) = {i ∈ 1, k : (x, ni )0 = ci (t)} doesn’t exceed dim E for all x ∈ C(t) and t ∈ [0, T ]. Then X(t) contains at most one non-constant T -periodic solution. Note, ni in (86) are, generally speaking, different from ni in (80), but we use same notation as it shouldn’t cause confusion. Accordingly, the active set J(t, x) of Theorem 6 is different from the active set J(t, x) of Theorem 4. Lemma 6 Assume, that for each t ∈ [0, T ] and x ∈ C(t) the collection of vectors {ni : i ∈ J(t, x)} is linearly indepentent. Then for a solution x(t) of sweeping process (64) there is a collection of integrable non-negative λi : [0, T ], i ∈ 1, k, such that − x(t) ˙ =

k X

λi (t) ni ,

for a.a. t ∈ [0, T ].

(87)

i=1

Proof. Recall, that for a fixed t ∈ [0, T ], the normal of form (86) can be equivalently formulated as    {0},    ( ! )   P 0 λi ni : λi ≥ 0 , NC(t) (x) =  i∈J(t,x)       ∅,

cone (65) to the set C(t)

if x ∈ intC(t), if x ∈ ∂C(t), if x 6∈ C(t).

Therefore, for a.a. fixed t ∈ [0, T ], the existence of λ(t) ∈ Rk , λi (t) ≥ 0, i ∈ 1, k, verifying X −x(t) ˙ = λi (t)ni i∈J(t,x(t))

follows from the inclusion (64). We set λi (t) = 0, if i 6∈ J(t, x(t)). The proof of Lebesgue measurability of λ(t) will be split into several steps. Step 1. First we observe that, for any tˆ ∈ [0, T ], the set Ttˆ = t ∈ [0, T ] : J(t, x(t)) = J(tˆ, x(tˆ)) is measurable. This follows from the fact that the set {t ∈ [0, T ] : hx(t), ni (t)i − ci (t) = 0} is measurable for each fixed index i ∈ 1, k and that J(tˆ, x(tˆ)) ⊂ 1, k.

34


Step 2. Now we fix some tˆ ∈ [0, T ] and prove that, for any Borel set B ⊂ Rk , the set Ttˆ(B) = t : [0, T ] : λ(t) ∈ B, J(t, x(t)) = J(tˆ, x(tˆ)) is measurable. If inclusion (64) doesn’t hold at tˆ and mes(Jtˆ) = 0, then Ttˆ(B) is measurable and mes(Ttˆ(B)) = 0. If (64) doesn’t hold at tˆ and mes(Ttˆ) > 0, then we can find t˜ ∈ Ttˆ such that (64) does hold at t˜. Since Ttˆ = Tt˜, we conclude that one won’t restrict generality of the proof, if assume that (64) holds for the initially chosen tˆ ∈ [0, T ]. Let (e1 , ..., ed ) be some canonical basis in E that stay fixed throughout the proof. Let n ¯ 1 , ..., n ¯ d be any other basis in E such that n ¯ i = ni ,

for all i ∈ J(tˆ, x(tˆ)).

The latter basis depends on tˆ. Denote by Stˆ the d × d transition matrix from basis (e1 , ..., ed ) to basis (¯ n1 , ..., n ¯ d ). Then, λ(tˆ) = −Stˆx( ˙ tˆ). Since Stˆ depends on just the indexes of J(tˆ, x(tˆ)), we have λ(t) = −Stˆx(t), ˙

for a.a. t ∈ [0, T ] such that J(t, x(t)) = J(tˆ, x(tˆ)).

Therefore, up to a subset of [0, T ] of zero measure, Ttˆ(B) = t ∈ [0, T ] : −Stˆx(t) ˙ ∈ B, J(t, x(t)) = J(tˆ, x(tˆ)) = (−Stˆx) ˙ −1 (B)∩Ttˆ, and the measurability of Ttˆ(B) follows by combining the absolute continuity of x and the conclusion of Step 1. Step 3. We finally fix a Borel set B ⊂ Rk and prove the measurability of the set λ−1 (B) = {t ∈ [0, T ] : λ(t) ∈ B} . (88) Since J(tˆ, x(tˆ)) can take only a finite number of (set-valued) values when tˆ varies from 0 to T, then there is a finite sequence t1 , ..., tK ∈ [0, T ] such that [ Tti , [0, T ] = i∈1,K

and so we can rewrite (88) as follows [

λ−1 (B) =

Tti (B),

i∈1,K

which is a finite union of measurable sets. The proof of the measurability of λ is complete. The integrability of λ on [0, T ] now follows from its boundedness. Indeed, since, kx(t)k ˙ 0 ≤ M for a.a. t ∈ [0, T ] and some M > 0 [25, p.13], one has |λi (t)| ≤ kλ(t)k = M max kSti k, i∈1,K

for a.a. t ∈ [0, T ].


35

The proof of the lemma is complete.

t u

Proof of Theorem 6. Let x(t) and y(t) be two non-constant distinct T periodic solutions of (15). Theorem 4 implies that we won’t loss generality by assuming that J(t, x(t)) = J(t, y(t)). (89) When applying Theorem 4 we used the fact that the set (86) can be expressed in the form (80) due to the uniform boundedness of C(t). The proof is by reaching a contradiction with the fact that x(t) and y(t) are distinct. By replacing −x(t) ˙ by its representation from Lemma 6, one gets Z

T

Z

T

−x(t)dt ˙ =

0= 0

0

k X

λi (t)ni dt =

i=1

k Z X i=1

T

λi (t)dt ni ,

(90)

0

where λi (t) ≥ 0. Since x(t) is non-constant, the set ) ( Z T λi (t)dt > 0 Jˆ := i ∈ 1, k : 0

is non-empty. The following two cases can take place. ˆ is a linearly independent system. But property (90) yields 1) {ni : i ∈ J} XZ i∈Jˆ

T

λi (t)dt ni = 0,

0

RT that, for linearly independent vectors ni , can happen only when 0 λi (t)dt ≡ 0, i ∈ 1, k. Therefore case 1) cannot take place as x(t) is non-constant. ˆ are linearly dependent. Since, by the assumption 2) The vectors of {ni : i ∈ J} ˆ are linearly independent, one of the theorem, any d vectors from {ni : i ∈ J} ˆ must have |J| > d. Let us show this leads to a contradiction as well. ˆ the function λj (t) is positive on a set of positive measure, Since for each j ∈ J, ˆ where (89) holds along with there are time moments tj , j ∈ J, −x(t ˙ j) =

k X

λi (tj )ni

and λj (tj ) > 0.

i=1

This implies j ∈ J(tj , x(tj ))

and by (89) j ∈ J(tj , y(tj )),

ˆ j ∈ J,

or, equivalently, (x(tj ), nj )0 = cj (tj )

and

(y(tj ), nj )0 = cj (tj ),

ˆ j ∈ J.

36


Therefore, (x(tj ) − y(tj ), nj )0 = 0,

ˆ j ∈ J,

(x(0) − y(0), nj )0 = 0,

ˆ j ∈ J.

and, by Lemma 4,

ˆ > d and so {ni : i ∈ J} ˆ contains d linearly independent vectors, which But |J| d form a basis of R . Therefore, x(0) = y(0), which is a contradiction. t u Theorem 6 can be used for stabilization of general sweeping process with polyhedral moving set such as those considered e.g. in Colombo et al [8] and KrejciVladimirov [24]. A fundamental case where Theorem 6 allows to stabilize an elastoplastic system (1)-(5) to a single periodic solution is when V cut Π(t) along a simplex. Testing the set Π(t) ∩ V for being a simplex can be executed for any given elastoplastic system (1)-(5) using the algorithms of computational geometry (e.g. Bremner et al [5] can be used to compute the vertexes of Π(t) ∩ V whose number needs to equal m + 1). At the same time, establishing analytic criteria for stabilization to occur could be of great use in materials science. A simple (and thus potentially useful for engineers) criterion of this type is offered in the next section of the paper.

5.2 Application: an analytic condition for stabilization of elastoplastic systems to a unique periodic regime Next theorem is the main result of this paper. It can be viewed as an analogue of high gain feedback stabilization in control theory. Indeed, one of the two central assumptions of the theorem is q = n − 2, which means that the elastoplastic system has a sufficient number of control variables to be fully controllable and thus stabilizable. The second central assumption is assuming that the magnitude of the external loading is high enough which literally resembles the high gain requirement of feedback control theory. The idea of Theorem is based on a simple fact that the moving parallelepiped Π(t) intersects the plane V along a simplex, if the the plane V is close to the vertex of the parallelepiped, see Fig. 2(d). At the same time, this geometric statement turned out to hold only if q = n − 2.


37

Theorem 7 In the settings of Proposition 2 assume that the stress loading h(t) is large in the sense that  c¯k1  ..   .   + *   c¯kj−1   −k 

k   ¯, c¯ + Ah(t) ≤ 0, u ¯,  c¯j  + Ah(t) · u  c¯k   j+1   .   ..  c¯km 

j ∈ 1, m, t ∈ [0, T ],

(91)

holds for at least one k ∈ {−1, +1}. Further assume that the stretching/compressing loading g(t) is large in the sense of (46). Then, there exists a T -periodic function s0 (t) such that ks(t) − s0 (t)k → 0 as t → ∞ for the stress component s(t) of any solution of the quasistatic evolution problem (1)-(5). Remark 7 Following the lines of Remark 5, we consider the left-hand-side of (91) as a polynomial P (h¯ u, Ah(t)i) in h¯ u, Ah(t)i, so that the branches of the polynomial are pointing upwards. Therefore, condition (91) is the requirement for h¯ u, Ah(t)i to stay between the roots of

the polynomial. Note, one root of k P (h¯ u, Ah(t)i) is given by h¯ u , Ah(t)i = − u ¯ , c ¯ . By computing the derivative 0 k k P − u ¯, c¯ one concludes that h¯ u, Ah(t)i = − u ¯, c¯ is the smaller or larger root of P (h¯ u, Ah(t)i) according to whether k = +1 or k = −1. Therefore, a sufficient condition for (91) to hold with k = +1 and k = −1 are  c¯+ 1  ..   .   + + *  c¯j−1   − 

 u ¯, −¯ c+ ≤ h¯ u, Ah(t)i ≤ min u ¯, −   c¯+j  j∈1,m  c¯   j+1   .   ..  c¯+ m 

and  c¯− 1  ..   .   − + *  c¯j−1   + 

c¯j  max u ¯, −  ≤ h¯ u, Ah(t)i ≤ u ¯, −¯ c−   j∈1,m  c¯−   j+1   .   ..  c¯− m 

respectively.

38


Proof. We are going to prove that the conditions of Theorem 6 hold for the sweeping process (15) of the elastoplastic system given. Since the set C(t) =

m \

+ x ∈ V : c− i +ai hi (t) ≤ ai xi ≤ ci +ai hi (t)

(92)

i=1

is just a parallel displacement of the polyhedron (60) of sweeping process (15), it is sufficient to prove that conditions of Theorem 6 hold for the set (92). More precisely, we prove that conditions of Theorem 6 hold for the set (92) after it is expressed in the form (86). Fix t ∈ [0, T ] and j ∈ 1, m. Denote by ξ j the solution of the system of m equations

u ¯, Aξ j = 0, (93) ai ξij = c¯ki + ai hi (t),

i 6= j, i ∈ 1, m.

The solution ξ j is unique by Lemma 2. Observe, that h¯ u, A A−1 c¯k + h(t) i = 0 if and only if ξj = c¯k , j ∈ 1, m.

(94)

(95)

We claim that each of the relations of (95) implies that C(t) = {A−1 c¯k +h(t)}. Indeed, assume that there exists x ∈ C(t) such that x 6= A−1 c¯k + h(t). Then x can be expressed as x = A−1 c + h(t) for some c ∈ C. On the other hand, x ∈ V implies h¯ u, Axi = 0. Therefore, u ¯, c¯k − c = 0 and by just expanding the scalar product we get the existence of two indices i1 , i2 ∈ 1, m, such that u ¯i1 (¯ cki1 − ci1 ) > 0,

u ¯i2 (¯ cki2 − ci2 ) < 0,

which is impossible by the construction of c¯k . Therefore C(t) is a singleton, if (95) holds. But if C(t) is a singleton for at least one t ∈ [0, T ], then the statement of the theorem becomes trivial. That is why we now focus on the case where (95) doesn’t hold on [0, T ]. In what follows, we show that the conditions of Theorem 6 hold by proving that the points ξ j , j ∈ 1, m, are vertexes of a simplex that coincides with C(t). Step 1: The vertexes ξ j , j ∈ 1, m, form an m−1-simplex. For a given j ∈ 1, m, we need to show that m − 1 vectors ζ ij = ξ i − ξ j ,

i ∈ 1, m, i 6= j,

are linearly independent. From (93) we have

u ¯, Aζ ij = 0 while from (94) we get ζ ij = (0, ..., 0, ζiij , 0, ..., 0, ζjij , 0, ..., 0)T .

(96)


39

Combining these two properties we conclude that ui ai ζiij + uj aj ζjij = 0. By Lemma 2, we either have ζiij = ζjij = 0 or ζiij ζjij 6= 0. Observe that the former case is impossible. Indeed, if ξ j1 = ξ j2 for some j1 6= j2 , then (94) implies ξ j1 = ξ j2 = A−1 (¯ ck + Ah(t)), which leads to (95) when plugged to (93) which we already excluded. It remains to notice that property ζiij ζjij 6= 0, i 6= j implies that the vectors (96) are linearly independent through i 6= j, i ∈ 1, m. Step 2: It holds ξ j ∈ C(t), j ∈ 1, m. Based on formula (92), we have to show that j + j ∈ 1, m. c− (97) j + aj hj (t) ≤ aj ξj ≤ cj + aj hj (t), Fix j ∈ 1, m and consider the function    c¯k1 a1 h1 (t)   ..   ..   .   .    *   c¯kj−1   aj−1 hj−1 (t) +      .   0 ¯,  b(x) = u   kx  +   c¯   aj+1 hj+1 (t)    j+1     .   ..   ..   . 

c¯km

am hm (t)

By the definition, aj ξjj is the unique root of the equation b(x) = 0. On the other hand, condition (91) implies that b(¯ c−k ckj +aj kj (t)) ≤ 0, so j +aj hj (t))·b(¯ that the unique zero of b(x) must be located between the numbers c¯−k j +aj hj (t) and c¯kj + aj kj (t), which property coincides with (97). Step3: We claim that C(t) = conv ξ j , j ∈ 1, m . From Step 2, C(t) ⊃ conv ξ j , j ∈ 1, m , so it remains to prove that C(t) ⊂ conv ξ j , j ∈ 1, m . Fix j ∈ 1, m. Since by (94) ξji1 = ξji2 ,

i1 , i2 6= j, i1 , i2 ∈ 1, m,

we have aj ξji = c¯kj + aj hj (t),

i 6= j, i ∈ 1, m.

n o In other words, if we fix ˆj ∈ 1, m and consider a facet conv ξ i , i 6= ˆj, of the simplex conv ξ i , i ∈ 1, m then all vertices of the facet share their ˆj-th coordinate. Therefore the whole facet belongs to the plane ˆ

Lj = {x ∈ V : aj xj = c¯ˆkj + aˆj hˆj (t)}.

40


Therefore, m \ conv ξ i , i ∈ 1, m = x ∈ V : pj aj xj − c¯kj − aj hj (t) ≤ 0 , j=1

where pj ∈ {−1, 1} are suitable signs. On the other hand, by (92), C(t) ⊂

m \ x ∈ V : qj aj xj − c¯kj − aj hj (t) ≤ 0 ,

(98)

j=1

where qj ∈ {−1, 1} are suitable signs. Since by Step 2, conv{ξj , j ∈ 1, m} ⊂ C(t), we get pj = qj , j ∈ 1, m. But then (98) takes the form C(t) ⊂ conv{ξj , j ∈ 1, m}. t u

The proof of the theorem is complete.

Example 1 (continuation) Applying Theorem 7 to the elastoplastic system of Example 1 (where we have q = n − 2) we use earlier formulas (37) and (44) together with Remark 7 to obtain the following conclusion: if the T -periodic stretching/compressing loading l(t) satisfies (53) and, for the T -periodic stress loading h(t), one either has −¯ c+ ¯− ¯+ < a1 h1 (t) + a2 h2 (t) + a3 h3 (t) < 1 +c 2 −c 3 − + < min −¯ c1 + c¯− c+ ¯+ ¯+ c+ ¯− ¯− 2 − c3 , −¯ 1 +c 2 −c 3 , −¯ 1 +c 2 −c 3 ,

t ∈ [0, T ],

or + − max −¯ c1 + c¯+ c− ¯− ¯− c− ¯+ ¯+ < 2 − c3 , −¯ 1 +c 2 −c 3 , −¯ 1 +c 2 −c 3 < a1 h1 (t) + a2 h2 (t) + a3 h3 (t) < −¯ c− ¯+ ¯− 1 +c 2 −c 3,

t ∈ [0, T ],

then the stresses of springs of the elastoplastic system of Fig. 7 converge, as t → ∞, to a unique T -periodic regime that depends on l(t) and h(t), and doesn’t depend on the initial state of the system.

6 Conclusions We used Moreau sweeping process framework to analyze the asymptotic properties of quasistatic evolution of one-dimensional networks of elastoplastic springs (elastoplastic systems) under stretching/compressing and stress loadings. This type of elastoplastic systems covers, in particular, rheological models of materials science. We showed that stretching/compressing loading corresponds to parallel displacement of the moving polyhedron C(t) of the respective sweeping process, but doesn’t influence the shape of C(t). We showed that it is the stress loading which is capable to change the shape of C(t). Moreover, we proved that increasing the magnitude of the stress loading always makes C(t) a simplex, if the number q of stretching/compressing constraints is two less the number n of nodes of the network (q = n − 2).


41

The global asymptotic stability result established in this paper ensures convergence of the stresses of springs to a unique periodic solution (output) when the magnitude of the stretching/compressing loading is large enough and when the normals of any d different facets of the moving polyhedron C(t) are linearly independent, e.g. when C(t) is a simplex. Here d is the dimension of the phase space of the polyhedron C(t), given by d = m − n + q + 1, where m is the number of springs, see (49). We documented that the output of an elastoplastic system may no longer converge to a unique periodic regime when q < n − 2, regardless of the magnitude of the stretching/compressing or stress loading. Specifically, we proved that the associated sweeping process may admit families of periodic solutions as an attractor. Moreover, we gave an example where such a situation cannot be destroyed by small perturbations of the parameters of the given elastoplastic system. This reinforces the fact that global convergence to a unique output is not a generic property of elastoplastic systems and requires certain conditions to hold (that the paper offered). Our theory can be viewed as an analogue of the high gain feedback stabilization of the classical control theory, see Isidori [17, §4.7]. The high gain assumption of the control theory corresponds to our condition (91) on the magnitude of stress loading. Our assumption q = n−2 on the network of elastoplastic springs resembles the relative degree in control. The advantage of the proposed restriction dim U = 1 is that it leads to simple analytic conditions (46) and (91) for the convergence of an elastoplastic system, which can be used for the design of elastoplastic systems that converge for the desired set of applied loadings. Extending Theorem 7 to the case where dim U > 1 is a doable task, but the respective inequality (91) transforms into a list of groups of inequalities, where the number of groups equals the number of selections of dim U from m (equation (93) gets replaced by the respective combinations of dim U equations). We don’t see how such a condition can be useful in design of applied loadings, thus we stick to dim U = 1. The results of the paper can be extended to the case of dynamic evolution of elastoplastic systems with small inertia forces along the lines of Martins et al [26]. We like to think that the present paper makes a breakthrough step in introducing the sweeping process framework to the field of elastoplasticity and that it opens a new room of opportunities for researchers interested in applied analysis.

7 Appendix Proof of Theorem 3 (Massera-Krejci Theorem for sweeping processes with a moving set of the form C(t) = ∩ki=1 (Ci + ci (t))).

42


We prove that every solution x of sweeping process (64), that is defined on [0, ∞), satisfies lim kx(t) − x∗ (t)k0 = 0, (99) t→+∞

where x∗ is a T -periodic solution of (64). Notice, that in case of T −periodic input the function t 7→ x(t+T ) coincide with another solution of (64) originating from the point x(T ) at t = 0. Due to mono0 tonicity of NC(t) (x) in x the distance kx(t + T ) − x(t)k0 is non-increasing(see e.g. [25, Corollary 1]) and there exists r = lim kx(t + T ) − x(t)k0 .

(100)

t→+∞

Since x([0, ∞)) is precompact, there is a subsequence {nj }j∈N ⊂ N and a point x∗0 such that lim kx(nj T ) − x∗0 k0 = 0. (101) j→∞

Moreover, since each x(nj T ) ∈ C(nj T ) = C(0) and C(0) is closed we have x∗0 ∈ C(0). Let x∗ be a solution of (64) with the initial condition x∗ (0) = x∗0 . Consider the functions xj (t) = x(t + nj T ),

j ∈ N.

Since C(t) = C(nj T +t), each function xj (t) is the solution of sweeping process (64) with the initial condition xj (0) = x(nj T ). The distance between solutions doesn’t increase, so for any t > 0, 0 6 kxj (t) − x∗ (t)k0 6 kx(nj T ) − x∗0 k0 ,

(102)

and using (101) we obtain (99). Now it remains to prove that x∗ is T -periodic. Combining (102) and (100) we get lim kx(t + nj T + T ) − x(t + nj T )k0 = kx∗ (t + T ) − x∗ (t)k0 = r.

j→∞

Since x∗ and t 7→ x∗ (t+T ) are two solutions of sweeping process (64), lemma 4 yields x˙ ∗ (t) = x˙ ∗ (t + T ), t ≥ 0. (103) Thus, ∗

x (¯ nT ) −

x∗0

Zn¯ T =

∗

ZT

x˙ (t)dt = n ¯ 0

x˙ ∗ (t)dt = n ¯ (x∗ (T ) − x∗0 ),

n ¯ ∈ N,

0

and so kx∗ (¯ nT ) − x∗0 k = n ¯ r, n ¯ ∈ N. Since t 7→ x∗ (t) is bounded, the latter is possible only when r = 0, i.e. when x∗ is T -periodic. The proof of the theorem is complete. t u Similar to Theorem 3 results are obtained in Henriquez [15] (extension to Banach spaces) and in Kamenskii et al [21] (extension to almost periodic solutions). Acknowledgements The work is supported by the National Science Foundation grant CMMI-1436856.


43

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