On a new class of differential variational inequalities and a stability result
Math. Program., Ser. B (2013) 139:205–221 DOI 10.1007/s10107-013-0669-5 FULL LENGTH PAPER
On a new class of differential variational inequalities and a stability result Joachim Gwinner
Received: 14 March 2011 / Accepted: 4 December 2011 / Published online: 30 March 2013 © Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013
Abstract This paper addresses a new class of differential variational inequalities that have recently been introduced and investigated in finite dimensions as a new modeling paradigm of variational analysis to treat many applied problems in engineering, operations research, and physical sciences. This new subclass of general differential inclusions unifies ordinary differential equations with possibly discontinuous righthand sides, differential algebraic systems with constraints, dynamic complementarity systems, and evolutionary variational systems. The purpose of this paper is two-fold. Firstly, we show that these differential variational inequalities, when considering slow solutions and the more general level of a Hilbert space, contain projected dynamical systems, another recent subclass of general differential inclusions. This relation follows from a precise geometric description of the directional derivative of the metric projection in Hilbert space, which is based on the notion of the quasi relative interior. Secondly we are concerned with stability of the solution set to this class of differential variational inequalities. Here we present a novel upper set convergence result with respect to perturbations in the data, including perturbations of the associated set-valued maps and the constraint set. Here we impose weak convergence assumptions on the perturbed set-valued maps, use the monotonicity method of Browder and Minty, and employ Mosco convergence as set convergence. Also as a consequence, we obtain a stability result for linear complementarity systems. Mathematics Subject Classification (2000)
34G25 · 49J53 · 49K40
Dedicated to Jonathan M. Borwein on the occasion of his sixtieth birthday. J. Gwinner (B) Fakultät für Luft- und Raumfahrttechnik, Institut für Mathematik, Universität der Bundeswehr München, 85577 Neubiberg/München, Germany e-mail: [email protected]
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1 Introduction This paper is concerned with differential variational inequalities that have recently been introduced and studied in depth by Pang and Stewart [31] in finite dimensions as a new modeling paradigm of variational analysis. In their seminal paper the authors have already shown that this new subclass of differential inclusions contains ordinary differential equations with possibly discontinuous right-hand sides, differential algebraic systems with constraints, dynamic complementarity systems, and evolutionary variational systems. As one of the issues of the paper, we show that differential variational inequalities also include projected dynamical systems, another new subclass of differential inclusions, first introduced in finite dimensions by Dupuis and Nagurney [14] and treated in the monograph [30], then studied in a general Hilbert space by Cojocaru and Jonker [10]. Projected dynamical systems have been introduced and investigated in finite dimensions to treat various time dependent disequilibrium problems in management science and operations research, particularly in traffic science. To exhibit such a relationsship we have to lift differential variational inequalities to the more general level of a Hilbert space and consider the concept of a “slow” solution (the solution of minimal norm) for this new subclass of differential inclusions. To derive such a relationship in a Hilbert space we use some results from [21], in particular a precise geometric description of the directional derivative of the metric projection, that extends a basic geometric lemma (Lemma 2.1 in [30]), due to Dupuis [13], to infinite dimensions. As a basic tool for the latter result in a Hilbert space of infinite dimension we employ the concept of the quasi interior known in nonlinear functional analysis (see [37]) and used in duality theory of partially finite convex programming by Borwein and Lewis [6]; in this context we also refer to the recent study of Borwein and Goebel [7] on notions of relative interior in Banach spaces. Such an analysis of the relationship between different subclasses of differential inclusions is not entirely new. Already Heemels, Schumacher, and Weiland [24] succeeded to show that, under mild conditions, projected dynamical systems can be written as gradient-type complementarity systems, a subclass of the afore mentioned differential variational inequalities. Their result extends the well-known fact to the dynamic regime that, under a suitable constraint qualification, variational inequalities with explicit inequality constraints can be written as mixed nonlinear complementarity problems. Then, Brogliato, Daniilidis, Lemaréchal, and Acary [8] were able to prove, under appropriate conditions, the equivalence between several known dynamic formalisms: projected dynamical systems, two types of differential inclusions, and a class of complementarity systems. Their analysis uses exclusively tools from standard convex analysis. However, the work of [8,24] is limited to finite dimensions; it does neither need the refined concept of the quasi interior nor an extension of the basic geometric lemma of Dupuis to infinite dimensions. Furthermore in this paper, we are concerned with stability of the solution set to differential variational inequalities. In this connection let us refer to [32], where at first several sensitivity results are established for inititial value problems of ordinary differential equations with nonsmooth right hand sides and then applied to treat differential variational inequalities. This has to be distinguished from asymptotic Lyapunov
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stability that has been investigated in [1,18,19] for solutions of evolution variational inequalities and nonsmooth dynamical systems. Here we present a novel upper set convergence result with respect to perturbations in the data. In particular, we admit perturbations of the associated set-valued maps and of the constraint set. We impose weak convergence assumptions on the perturbed set-valued maps, use the monotonicity method of Browder and Minty (see e.g. [28,38]) for the analysis of nonlinear variational inequalities, and employ Mosco convergence as set convergence. Thus we extend the stability result of [21] (without considering slow solutions here) to this new more general class of differential variational inequalities of [31]. As a straightforward consequence, we also obtain a stability result for linear complementarity systems [25]. It is to be noted that related stability results for more general evolution inclusions by Papageorgiou [33,34] and by Hu and Papageorgiou in the memoir [26] are not applicable here, since these results need more stringent compactness assumptions. To facilitate the reading of the text we stick to the following notation. Scalar-valued functions and maps between the basic Hilbert spaces are denoted by small latin letters. Matrix-valued functions and derived maps between function spaces are given capital roman letters. Similarly, small greek letters are reserved for set-valued maps between the basic Hilbert spaces, while capital greek letters stand for derived set-valued maps between function spaces. The plan of this paper is as follows. The next Sect. 2 provides the setting of the class of differential variational inequalities considered. In Sect. 3 we exhibit the relation of this class of differential inclusions to projected dynamical systems. To this end, in Sect. 3.1 we provide the basic geometric lemma for the directional derivative of the metric projection. Hence in Sect. 3.2 we derive the relationship result. Section 4 presents our stability analysis for this new class of differential variational inequalities involving general set-valued maps. After some preliminaries in particular on Mosco set convergence (Sect. 4.1), we establish our novel stability result in Sect. 4.2. In the concluding Sect. 5 we shortly discuss possible extensions and limitations of our results, thus clarifying that the Hilbert space setting is the appropriate infinite dimensional framework. Moreover we give an outlook to future work and sketch some main streams of potential applications of our infnite dimensional stability result, namely to nonsmooth dynamic systems with randomness, feasible continous paths to equilibria in stochastic programming, and nonsmooth parabolic PDE problems. 2 The setting of differential variational inequalities considered Let X, V be two real, separable Hilbert spaces that are endowed with norms · X , · V respectively and with scalar products denoted by ·, · , (·, ·) respectively. Further let there be given T > 0, a convex closed subset K ⊂ V , a set-valued map φ : [0, T ] × X × V ⇒ X , a map g : [0, T ] × X × V → V , and some fixed x0 ∈ X . Then following Pang and Stewart [31] we consider the following problem: Find an X -valued function x and an V -valued function u both defined on [0, T ] that satisfy for a.a. (almost all) t ∈ [0, T ] (PS-DVI)(φ, g, K ; x0 )
x(t) ˙ ∈ φ(t, x(t), u(t)) u(t) ∈ σ (K , g(t, x(t), ·))
(1)
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complemented by the initial condition x(0) = x0 . Here x(t) ˙ denotes the time derivative of x(t); σ (K , g(t, x(t), ·)) stands for the solution set of the variational inequality defined by K and g(t, x(t), ·), that is, u(t) has to satisfy u(t) ∈ K , g(t, x(t), u(t)), v − u(t) ≥ 0, ∀v ∈ K . Such a differential inclusion is termed a differential variational inequality by Pang and Stewart [31], here denoted by the acronym PS-DVI to distinguish from the differential variational inequalities treated in the monograph [3] by Aubin and Cellina over 20 years before (called a variational inequality of evolution in [31]) and here denoted by the acronym AC-DVI, see (10) below. To give a precise meaning to a PS-DVI we have to introduce appropriate function spaces and impose some hypotheses on the data. The fixed finite time interval [0, T ] gives rise to the Hilbert space L 2 (0, T ; V ) endowed with the scalar product T [u 1 , u 2 ] :=
u 1 (t), u 2 (t) dt, u 1 , u 2 ∈ L 2 (0, T ; V ).
0
As in [31] we consider weak solutions of the differential inclusion in a PS-DVI in the sense of Caratheodory. In particular, the X -valued function x has to be absolutely continuous with derivative x(t) ˙ defined almost everywhere. Moreover to define the initial condition, the trace x(0) is needed. Therefore (see [11, Theorem 1, p. 473]) we are led to the function space W (0, T ; X ) := {x | x ∈ L 2 (0, T ; X ), x˙ ∈ L 2 (0, T ; X )}, a Hilbert space endowed with the scalar product [x1 , x2 ] + [x˙1 , x˙2 ], x1 , x2 ∈ W (0, T ; X ). Note that W (0, T ; X ) is continuously and densely embedded in the space C[0, T ; X ] of X -valued continuous functions on [0, T ], where the latter space is equipped with the norm of uniform convergence. We assume that the map g satisfies the following growth condition: There exist g0 ∈ L ∞ (0, T ) and g 0 ∈ L 2 (0, T ) such that ∀t ∈ (0, T ) , ∀(y, v) ∈ X × V there holds g(t, y, v)V ≤ g0 (t) (y X + vV ) + g 0 (t).
(2)
Hence a map G that acts from L 2 (0, T ; X ) × L 2 (0, T ; V ) to L 2 (0, T ; V ) can be derived from g by G(x, u)(t) := g(t, x(t), u(t)) , t ∈ (0, T ).
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Also we can introduce the closed convex subset K := L 2 (0, T ; K ) := {w ∈ L 2 (0, T ; V ) | w(t) ∈ K , ∀a.a. t ∈ (0, T )}
(3)
and replace the above pointwise formulation of the variational inequality in a PS-DVI by its integrated counterpart, u ∈ K,
G(x, u), w − u ≥ 0, ∀w ∈ K,
(4)
where its solution set is denoted by Σ(K, G(x, ·)). Clearly, it is sufficient to replace K in (4) by any dense subset of K, e.g. the K − valued continuous functions on [0, T ], as in [31], section 2.1. Concerning the set-valued map φ, we assume that the values φ(t, y, v) are nonvoid, convex and closed. Moreover, we assume the following growth condition similar to (2): There exist h 0 ∈ L ∞ (0, T ) and h 0 ∈ L 2 (0, T ) such that ∀t ∈ (0, T ), ∀(y, v) ∈ X × V, ∀z ∈ φ(t, y, v), there holds z X ≤ h 0 (t) (y X + vV ) + h 0 (t).
(5)
This condition follows from the linear growth condition used in [31]: There exists some constant Cφ > 0 such that ∀t ∈ (0, T ), ∀(y, v) ∈ X × V, ∀z ∈ φ(t, y, v), there holds z X ≤ Cφ (1 + y X + vV ).
(6)
Hence Φ(x, u)(t) := φ(t, x(t), u(t)) , t ∈ (0, T ) gives a set-valued map Φ : L 2 (0, T ; X ) × L 2 (0, T ; V ) ⇒ L 2 (0, T ; X ). Using a standard device in dynamical systems (see e.g. [29]), we can introduce the unknown x˜ := (t, x) and write the above PS-DVI as ⎧ d t 1 ⎨ d x˜ ˜ x(t), = ∈ φ( ˜ u(t)) := dτ dτ x φ(x(t), ˜ u(t)) ⎩ u(t) ∈ σ (K , g(x(t), ˜ ·)) , complemented by the initial condition x(0) ˜ = (0, x0 ). Therefore in the following we can consider the autonomous problem without any loss of generality and drop the dependence on t in PS-DVI. In addition to [31], but analogous to the differential variational inequalities in [3], we also consider the slow solution of PS-DVI. To this end use the shorthand Σ(x) := Σ(K, G(x, ·)) and assume that for any x ∈ W (0, T ; X ), the image set Φ(x, Σ(x)) := ∪{Φ(x, u) : u ∈ Σ(x)} is closed and convex. Then we seek the solution x of x˙ = Φ(x, Σ(x)) := proj(0, Φ(x, Σ(x))).
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This becomes instrumental, when we exhibit the relation of PS-DVI to projected dynamical systems. A simple example of closed convex image sets is as follows. Example 1 Let φ decompose as φ(y, v) = ψ(y) + B(y)v, where for any y ∈ X, ψ(y) is nonvoid, compact and convex, B(y) is a linear continuous operator from V to X with some constant c(y) > 0 (depending on y) such that B(y)v X ≥ c(y)vV , ∀v ∈ V . Note that under monotonicity and weak continuity (hemicontinuity) assumptions on the map g(y, ·) (to be introduced later), the solution set Σ(x) is convex and closed.
3 Relationship of projected dynamical systems to differential variational inequalities We exhibit the relationship of a projected dynamical system to differential variational inequalities in two steps. In the first step we rewrite a projected dynamical system as a differential variational inequality problem in the sense of Aubin and Cellina [3] (called a variational inequality of evolution in [31]). Then in the second step we use a device of [31] and rewrite the obtained problem as finding the slow (minmal norm) solution of a differential variational inequality of the type PS-DVI. For the first step we need some results from [21], in particular on the differentiability of the projection onto closed convex subsets, which are collected next.
3.1 On differentiability of the projection onto closed convex subsets Let H be a Hilbert space with norm | · | and scalar product ·, ·. For a closed convex subset Z of H and for any z ∈ Z , the tangent cone (also support cone or contingent cone,
see e.g. [3]) to Z at z, denoted by TZ (z), is the closure of the convex cone λ(Z − z) : λ > 0 . Then TZ (z) is clearly a closed convex cone with vertex 0 and is the smallest cone C whose translate z + C has vertex z and contains Z . Note that taking polars with respect to the scalar product in H, (TZ (z))0 = (TZ (z))− = N Z (z), the normal cone to Z at x. The utility of the tangent cone for projected dynamical systems (to be introduced in the subsequent subsection) stems from the fact that for the metric projection p Z = proj (·, Z ) onto Z there holds for any z ∈ Z , h ∈ H p Z (z + th) = z + t pTZ (z) (h) + o(t), t > 0 in a Hilbert space (see [37] Lemma 4.6, p. 300).
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On the other hand, in [14,30], the ”differential projection” of the vector h at z ∈ Z with respect to Z is defined by the directional derivative p Z (z, h) = lim
t→0
p Z (z + th) − z . t
Thus the differential projection coincides with the metric projection onto the tangent cone TZ (z), p Z (z, h) = pTZ (z) (h).
(7)
Now following [6,7,37] we call the quasi interior of a closed convex set Z (denoted qi Z ) the set of those z ∈ Z for which TZ (z) = H . Further we define the quasi boundary of a closed convex set Z (denoted by qbdr y Z ) as the set Z \qi Z . Then, in virtue of the strong separation theorem, z ∈ qbdr y Z if and only if there exists some nonzero w ∈ (TZ (z))0 . As the key result for our discussion of projected dynamical systems, we take from [21] the following geometric interpretation of p Z that extends [13], [30, Lemma 2.1] to infinite dimensions. Lemma 1 i) If z ∈ qi Z , then p Z (z, ·) = Id. ii) If z ∈ qbdr y Z , then for any v ∈ H \TZ (z) there exists w ∈ (TZ (z))0 such that p Z (z, v) = v − β w, |w| = 1, β = v, w > 0. For the convenience of the reader we also reproduce its short Proof i) If z ∈ qi Z , then TZ (z) = H and p Z (z, ·) is the identity. ii) If z ∈ qbdr y Z , then by (7) we obtain for vˆ := p Z (z, v) = pTZ (z) (v) that v − v, ˆ y − v ˆ ≤ 0 , ∀y ∈ TZ (z). Since TZ (z) is a cone, it follows 0 v − v, ˆ v ˆ = 0, v − vˆ ∈ TZ (z) .
(8)
By assumption v − vˆ = 0 , hence β := |v − v| ˆ > 0, and (8) yields v − vˆ = βw ˆ w = 0 implies with |w| = 1 and w ∈ (TZ (z))0 . Moreover the orthogonality v, β = v, w proving the lemma. From (8) we can simply derive the following characterization: Corollary 1 Let z ∈ Z . Then for any v ∈ H # p Z (z, v) = v − N Z (z) := proj(0, v − N Z (z)). For the convenience of the reader we also reproduce its short proof.
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Proof Let vˆ = p Z (z, v). In the special case z ∈ qi Z , N Z (z) reduces to zero and hence the claim trivially follows. Thus suppose z ∈ qbdr y Z . Then by (8) we obtain that vˆ ∈ v − N Z (z), which holds trivially if vˆ = v ∈ TZ (z). Since by (7) vˆ ∈ TZ (z) and N Z (z) = (TZ (z))0 , it follows from (8) that v, ˆ v − vˆ − y ≥ 0, ∀ y ∈ N Z (z). This means that vˆ = proj (0, v − N Z (z)) and proves the claim.
3.2 From projected dynamical systems to differential variational inequalities Following [30] a projected dynamical system P DS(F, Z ) is an ordinary differential equation of the form x˙ = p Z x, − f (x) ,
(9)
complemented by the initial condition x(0) = x0 , where the convex closed set Z , x0 ∈ Z , and the vector field f : Z → H are given. Here one looks for a function x: [0, T ] → Z that is absolutely continuous and satisfies x(t) ˙ = p Z x(t), − f x(t) except on a set of Lebesgue measure zero. By Corollary 1, (9) is equivalent to # x˙ = p TZ (x) − f (x) = − N Z (x) + f (x) . Thus the initial value problem (9) consists in finding the slow solution (the solution of minimal norm) to the given initial condition and the differential variational inequality [3, chapter 6, section 6] (AC-DVI) x(t) ˙ ∈ − N Z (x(t)) + f (x(t)) .
(10)
Finally we use a device of [31] for the reformulation of a AC-DVI as a PS-DVI. We introduce additional time-dependent variables p and q by p(t) = x(t), q(t) = x(t) ˙ + f (x(t)). Then by definition of the normal cone, AC-DVI writes p(t) ∈ Z , q(t), z − p(t) ≥ 0, ∀ z ∈ Z . Thus we arrive at a differential variational inequality PS-DVI with the settings
v = ( p, q), φ(y, v) = q − f (y), K = Z × H, g(y, v) = (q, y − p).
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4 Stability of differential variational inequalities In this section we study stability of differential variational inequalities formulated as PS-DVI and admit perturbations x0,n of x0 in the initial condition x(0) = x0 , φ n , g n of the maps φ : X × V ⇒ X, g : X × V → V , and K n of the convex closed subset K ⊂ V . Suppose that (x n , u n ) solves (PS-DVI)(φ n , g n , K n ; x0,n ) and assume that (x n , u n ) → (x, u) with respect to an appropriate convergence for X -valued, respectively V -valued functions on [0, T ]. Then we seek conditions on φ n → φ, g n → g, K n → K , x0,n → x0 that guarantee that (x, u) solves the limit problem (PSDVI)(φ, g, K ; x0 ). Such a stability result can be understood as a result of upper set convergence for the solution set of (PS-DVI)(φ, g, K ; x0 ). 4.1 Preliminaries; Mosco convergence of sets As the convergence of choice in variational analysis we employ Mosco set convergence for a sequence {K n } of closed convex subsets which is defined as follows. A sequence {K n } of closed convex subsets of the Hilbert space V is called Mosco convergent to a M
closed convex subset K of V , written K n −→ K , if and only if σ − lim sup K n ⊂ K ⊂ s − lim inf K n . n→∞
n→∞
Here the prefix σ means sequentially weak convergence in contrast to strong convergence denoted by the prefix s; lim sup, respectively lim inf are in the sense of Kuratowski upper, resp. lower limits of sequences of sets (see [2,4] for more information on Mosco convergence). As a preliminary result we next show that Mosco convergence of convex closed sets K n inherits to Mosco convergence of the polars K n0 and to Mosco convergence of the associated sets Kn = L 2 (0, T ; K n ), derived from K n similar to (3). M
M
M
Lemma 2 Let K n −→ K . Then (a) K n0 −→ K 0 ; (b) Kn −→ K in L 2 (0, T ; V ). Proof To show (a) we verify 1. σ − lim supn→∞ K n0 ⊂ K 0 : Let ζ = σ − lim ζn with ζn ∈ K n0 . Choose z ∈ K arbitrarily. Then by assumption, n→∞ there exist (eventually for a subsequence) z n ∈ K n with z = s − limn→∞ z n . By definition of polar, (ζn , z n ) ≤ 1, ∀n, hence in the limit (ζ, z) ≤ 1, ∀z ∈ K which gives ζ ∈ K 0 . 2. K 0 ⊂ s − lim inf n→∞ K n0 : M
By Proposition 3.23 in [2], s(K n ) −→ s(K ), where s(K )(ζ ) := sup (ζ, z), ζ ∈ V z∈K
is the support function of K . This means in particular, that for any ζ ∈ dom s(K ) there exist ζn ∈ dom s(K n ) such that ζ = s − lim ζn and s(K )(ζ ) ≥ lim supn→∞ s(K n )(ζn ).
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Now let ζ ∈ K 0 , then s(K )(ζ ) ≤ 1 and hence ζ ∈ dom s(K ). For the sequences {ζn }, {sn } with sn := s(K n )(ζn ) and ζn as above, we obtain lim supn sn ≤ 1. If for a subsequence sn k ≤ 1 holds, then ζn k ∈ K n0k and the argument is complete. Otherwise, for almost all n we have sn > 1. But then limn sn = 1, ζ˜n := sn−1 ζn ∈ K n0 and s − limn ζ˜n = ζ ; the proof of part (a) is complete. To show (b) we verify 1. σ − lim supn→∞ L 2 (0, T ; K n ) ⊂ L 2 (0, T ; K ): Let w = σ − lim wn with wn ∈ L 2 (0, T ; K n ). By the bipolar theorem (K 00 = K ) n→∞
it is enough to show that ∀ζ ∈ K 0 , for a.a. t ∈ (0, T ) there holds (w(t), ζ ) ≤ 1. Assume the contrapositive. Then there exist ζ˜ ∈ K 0 , A ⊂ (0, 1) with measure 1 χ A ζ˜ (with χ A denoting the |A| > 0 such that (w(t), ζ˜ ) > 1 on A. Define w˜ = |A| characteristic function of A). By part (a), there exist ζ˜n ∈ K n0 such that s−limn ζ˜n = 1 χ A ζ˜n we have s − limn w˜ n = w˜ in L 2 (0, T ; V ). By ζ˜ ; hence for w˜ n = |A| construction, 1 [wn , w˜ n ] = (wn (t), ζ˜n ) dt ≤ 1. |A| A
Thus in the limit we arrive at 1 1 ≥ [w, w] ˜ = |A|
(w(t), ζ˜ ) dt > 1,
A
a contradiction, proving the claim. 2. K ⊂ s − lim inf n→∞ Kn : It is enough to verify the claim for a dense subset of K; this follows from a diagonal sequence argument, see also [20], Lemma 2.6 for a similar reasoning. Here we use the well-known fact from Bochner-Lebesgue integration theory that the set S(0, T ; V ) of simple V -valued functions on (0, T ) is dense in L 2 (0, T ; V ). This extends to density of S(0, T ; V ) ∩ K in K. This can be seen by taking averages or mean value approximations; see [20] for approximations on a multidimensional integration domain instead of the interval (0, T ). Thus let w be a K -valued simple function on (0, T ), i.e. w = j∈J χ A j z j , where J is finite, z j ∈ K , and A j ⊂ (0, T ) are pairwise disjoint with measure M
|A j | > 0 and ∪ j∈J A j = (0, T ). Since K n −→ K , there exist z j,n ∈ K n such that ∀ j ∈ J, z j,n → z j (n → ∞). Hence wn := j∈J χ A j z j,n lies in Kn and w = s − limn→∞ wn in L 2 (0, T ; V ) follows. Let us note that part (b) of the preceding lemma is of intrinsic interest for timedependent variational inequalities. An analogous implication (b) 1. was already shown in [20] in the more general setting of probability spaces instead of the interval (0, T ), however for the restricted class of translated convex closed cones.
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As a further tool in our stability analysis we recall from [21] the following technical result. Lemma 3 Let H be a separable Hilbert space and let T > 0 be fixed. Then for any sequence {z n }n∈N converging to some z in L 1 (0, T ; H ) there exists a subsequence s {z n k }k∈N such that for some set N of zero measure, z n k (t) −→ z(t) for all t ∈ [0, T ]\N . 4.2 The stability result Before stating the result, some remarks are in order. In view of the existence theory of variational inequalities in infinite dimensional spaces (see e.g. [28]) the best one can hope for is weak convergence of the perturbations u n in the general case of nonunique solutions of the underlying variational inequalities in (PS-DVI). Weak convergence can namely be readily derived from a posteriori estimates. However, continuity of a nonlinear map (here g, φ) with respect to weak convergence is a hard requirement. To circumvent these difficulties with weak convergence we apply the monotonicity method of Browder and Minty. Then as we shall see below, a stability condition on the maps g n with respect to the underlying Hilbert space norm suffices. These weak convergence difficulties also affect φ n . Therefore we have to impose a generally strong stability condition on the nonlinear maps φ n . In the situation of Example 1 this condition can be simplified to a stability condition on the linear operators B n with respect to norm convergence. On the other hand, stronger assumptions on g n , like uniform monotonicity imply that the solution sets Σ(K n , G(x n , ·)) are single-valued. Uniform monotonicity with respect to u moreover entails that the sequence u n strongly converges. Then the stability σ assumption for φ n can be relaxed; in the subsequent hypothesis (H1) wn → w can be s replaced by wn → w. Since our stability assumptions pertain to the given maps g n and g (not to the derived maps G n and G), we have a delicate interplay between the pointwise almost everywhere formulation and the integrated formulation of the variational inequality in the perturbed PS-DVI and in the limit PS-DVI. We need the following hypotheses on the convergence of (φ n , g n ) to (φ, g): s
σ
s
(H1) Let z n → z in X and wn → w in V . Moreover, let pn ∈ φ n (z n , wn ) and pn → p in X . Then p ∈ φ(z, w). s s (H2) All maps g n (z, ·) for any z ∈ X are monotone. - If z n → z, vn → v in X , s respectively in V , then g n (z n , vn ) → g(z, v) in V . - g is hemicontinuous in the sense, that for any y ∈ X ; v, w ∈ V the real-valued function r ∈ IR → (g(y, v + r w), r w) is lower semicontinuous. Now we can state the following stability result: Theorem 1 Let (x n , u n ) solve (PS-DVI)(φ n , g n , K n ; x0,n ). Suppose, φ n and φ satisfy (H1), and that g n and g satisfy (H2). Let the convex closed sets K n Moscos s converge to K and let x0,n → x0 . Assume that x n → x in W (0, T ; X ) and that
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u n ∈ L 2 (0, T ; V ) converges weakly to u pointwise in V for a.a. t ∈ (0, T ) with u n (t)V ≤ m(t), ∀ a.a.t ∈ (0, T ) for some m ∈ L 2 (0, T ). Then (x, u) is a solution to (PS-DVI)(φ, g, K ; x0 ). Proof The proof consists of three parts. 1. Feasibility: u ∈ K, x(0) = x0 . First we observe that for any w ∈ L 2 (0, T ; V ), in virtue of Lebesgue’s theorem of dominated convergence, T [u , w] = n
n u (t), w(t) dt → [u, w].
0 σ
Thus u n → u and u ∈ L 2 (0, T ; V ). Moreover directly by Mosco convergence of {K n } or invoking lemma 2 (b), u ∈ K follows. Since by continuous embedding s s x n → x in C[0, T ; X ], we conclude x n (0) = x0,n → x(0) = x0 . 2. u solves the variational inequality in (PS-DVI)(φ, g, K ; x0 ): u(t) ∈ σ (K , g(x(t), ·)), ∀ a.a.t ∈ (0, T ). Fix an arbitrary w ∈ K. Then by lemma 2 (b), there exist w n ∈ Kn such that s w n → w in L 2 (0, T ; V ). Moreover, by extracting eventually a subsequence, we have by Lemma 3 that also w n (t) strongly converges to w(t) for a.a. t ∈ (0, T ). For any measureable set A ⊂ (0, T ) we can define w nA ∈ L 2 (0, T ; V ) by w nA = w n on A, w nA = u n on (0, T )\A. Hence w nA ∈ Kn and by construction,
g n (x n (t), u n (t)), w n (t) − u n (t) dt ≥ 0.
A
Hence a contradiction argument shows that we have pointwise for a.a. t ∈ n n n n n entails (0,n T ),n g (xn (t), u n(t)), w n(t) − u (t) ≥ 0. By (H2), monotonicity g (x (t), w (t)), u (t)−w (t) ≤ 0. Again by (H2), in the limit g(x(t), w(t)), u(t) − w(t) ≤ 0. In virtue of the growth condition (2), we arrive at T [G(x, w), u − w] =
g(x(t), w(t)), u(t) − w(t) dt ≤ 0, ∀w ∈ K.
0
Hence by a well-known argument in monotone operator theory (see e.g. [27,38]) we obtain that u ∈ K satisfies the variational inequality [G(x, u), w − u] ≥ 0, ∀w ∈ K, provided the map G(x, ·) is hemicontinuous. This follows from the hemicontinuity assumption in (H2) for g(y, ·) as follows: To show for any z ∈ L 2 (0, T ; X ); v, w ∈ L 2 (0, T ; V ),
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lim inf [G(z, v + r w), r w] ≥ [G(z, v + r0 w), r0 w], r →r0
subtract lim [G(z, v), r w] = [G(z, v), r0 w],
r →r0
and apply Fatou’s lemma to the nonnegative integrand to conclude T lim inf r →r0
g(z(t), v(t) + r w(t)), r w(t) dt
0
T ≥
lim inf g(z(t), v(t) + r w(t)), r w(t) dt r →r0
0
T ≥
g(z(t), v(t) + r0 w(t)), r0 w(t) dt
0
3. (x, u) solves the limit (PS-DVI)(φ, g, K ; x0 ). By Lemma 3 applied to {x n } and {x˙ n }, we can extract a subsequence such that ˙ strongly in X pointwise for all t ∈ (0, T )\N0 , x n (t) → x(t) and x˙ n (t) → x(t) where N0 is a null set. Fix t ∈ (0, T )\N0 . Then by assumption, for all n ∈ N we have for some pn (t) ∈ φ n (x n (t), u n (t)), x˙ n (t) = pn (t). Then in virtue of (H1), x(t) ˙ ∈ φ(x(t), u(t)) follows and (x, u) solves the (PS-DVI)(φ, g, K ; x0 ). Let us specialize the stability result to linear complementarity systems. Following [25] a linear complementarity system (LCS)(A, B, C, D; K , x0 ) is governed by the simultaneous equations x(t) ˙ = Ax(t) + Bu(t) q(t) = C x(t) + Du(t) 0 ≤ q(t) ⊥ u(t) ≥ 0, with the initial condition x(0) = x0 , where in infinite dimensional setting, A, B, C, and D are given appropriate bounded linear operators in the Hilbert spaces X, V , and the convex closed set K becomes the ordering cone in the complementarity condition. Similarly we consider the perturbed linear complementarity systems (LCS)(An , B n , C n , D n ; K n , x0,n ). Then the above hypotheses simplify to the following conditions: (H˜ 1) Convergence of the operators An → A, B n → B holds in the operator norm topology.
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(H˜ 2) All operators D n are monotone, i.e. for any v ∈ V, (D n v, v) ≥ 0 holds. Convergence of the operators C n → C, D n → D holds in the operator norm topology. Thus we obtain the following stability result for linear complementarity systems: Corollary 2 Let (x n , u n ) solve (LCS)(An , B n , C n , D n ; K n , x0,n ). Suppose, that An , B n , C n , D n satisfy (H˜ 1), (H˜ 2), respectively. Let the convex closed cones K n s s Mosco-converge to K and let x0,n → x0 . Assume that x n → x in W (0, T ; X ) and n 2 that u ∈ L (0, T ; V ) converges weakly to u pointwise in V for a.a. t ∈ (0, T ) with u n (t)V ≤ m(t), ∀ a.a.t ∈ (0, T ) for some m ∈ L 2 (0, T ). Then (x, u) is a solution to (LCS)(A, B, C, D; K , x0 ). 5 Some concluding remarks Let us shortly remark on possible extensions and limitations of the stability result above. The monotonicity method extends easily to set-valued operators. Also, monotonicity, can be replaced by the more general, however more abstract notion of (order-) pseudomonotonicity. Here we discussed differential variational inequalities and projected dynamical systems in a Hilbert space framework. While for projected dynamical systems the metric projection (and its directional derivative) is needed what confines the analysis to Hilbert spaces, differential variational inequalities and their stability using Mosco convergence can be investigated in more general reflexive Banach spaces, but not beyond [5]. Finally let us give an outlook of future work and outline some main streams of potential applications of our infinite dimensional stability result, namely to nonsmooth dynamic systems with randomness, feasible continous paths to equilibria in stochastic programming, and nonsmooth parabolic PDE problems. Firstly, modelling uncertainty by randomness leads to random evolution inclusions and random differential variational inequalities, see e.g. [35] for a general existence theory. For such problems the abstract Hilbert spaces X, V become the standard L 2 (Ω, μ, IRn ) spaces for a given measure space (Ω, A, μ). Then the question arises, how such random problems can be approximated by deterministic surrogate problems written as finite dimensional PS-DVI. Consider for simplicity the linear complementarity system (LCS)( A, B, C, D; K , x0 ). Let now the matrices A, B, C, D be random and let in a more structured case a finite Karhunen-Loève expansion [12] in its simplest form leads to a separation of the random and the deterministic variables such that for any M ∈ {A, B, C, D} we have M = M0 + ρ M1 with fixed (deterministic) matrices M0 , M1 and some random variable ρ. Then we can employ the discretization procedure in [22,23] with respect to the random variable based on averaging and truncation. While boundedness of the approximates and existence of (weakly) convergent subsequences result from coercivity assumptions, the stability of the approximation procedure is the most important step of the proof of norm convergence, for more details see [22] in the case of linear-affine variational inequalities and [23] in the more general case of nonlinear monotone variational inequalities.
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Secondly, a much studied application of infinite dimensional programming is stochastic programming. In [16], Flåm and Seeger propose to solve cone-constrained convex programs by differential inclusions and give an application to multi-stage stochastic programming. They combine penalty techniques and Lagrangian relaxation to treat the cone constraint and prove convergence of a primal-dual differential method to saddle points, In contrast to [16], the formulation of a (PS-DVI) enforces the complementarity constraints for each time instant, thus yielding a feasible continuous path to equilibria. Here too, the question arises in the application to stochastic programming, how approximation of the stochastic variables and time discretization can be combined to a numerically efficient approximation procedure. In this context, a stability result as above can shed some new light on the numerical solution of such problems. Thirdly, the infinite dimensional setting in the present paper allows the treatment of the large class of distributed nonsmooth systems where the state variable is space dependent and the associated differential equation is a parabolic partial differential equation, see e.g. [15]. As a prominent example of this class let us mention the quasistatic viscoelasic contact problem with short memory and given friction, see [15] and more detailed [36]: Find u(t) such that u(0) = u 0 , u (t) ∈ Uad , a 1 (u (t), v − u (t)) + a 0 (u(t), v − u (t)) + j (v) − j (u (t)) ≥ ( f (t), v − u (t)) + FN (t)(v − u (t)) dΓ, ∀v ∈ Uad . ΓF
The material under consideration has a short memory, since the state of stress at time t depends only on the configuration at the instant t and at the immediately preceding instants. Here, a 0 and a 1 are the bilinear forms of elasticity and of viscosity, respectively, which can be clearly rewritten as appropriate linear operators. Moreover, introducing an appropriate Lagrangian multiplier to treat the nonsmoothness of the given friction frictional j, the above problem can be recast as an (infinite dimensional) differential variational inequality problem, since the admissible set Uad can be considered as independent of time, thus as a convex closed set K defined by the unilateral contact with a fixed foundation. When it comes to numerical approximation, note that Galerkin approximation with respect to space discretization leads in the case of higher than piecewise linear approximation and in the case of a not necessarily piecewise linear obstacle in the contact constraint to a nonconforming approximation, that is , the approximating convex set K h is not a subset of the given convex K . In this situation Mosco convergence (and more refined Stummel-Glowinski convergence) is instrumental to arrive at convergence of Galerkin approximation, see [17]. Thus our stability result can be seen as an important step towards convergence of semidiscretization methods that provide finite dimensional differential variational inequalities. Clearly a full space time discretization needs an additional analysis (see e.g. [9]) and is beyond the scope of the present paper.
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J. Gwinner The author is indebted to two anonymous referees for their constructive suggestions.
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28. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1984) 29. Kunze, M.: Non-Smooth Dynamical Systems. In: Lecture Notes in Mathematics, vol. 1744. Springer, Berlin (2000) 30. Nagurney, A., Zhang, D.: Projected Dynamical Systems and Variational Inequalities with Applications. Kluwer Boston, Dordrecht (1996) 31. Pang, J.-S., Stewart, D.E.: Differential variational inequalities. Math. Program. 113, 345–424 (2008) 32. Pang, J.-S., Stewart, D.E.: Solution dependence on initial conditions in differential variational inequalities. Math. Program. 116, 429–460 (2009) 33. Papageorgiou, N.S.: A stability result for differential inclusions in Banach spaces. J. Math. Anal. Appl. 118, 232–246 (1986) 34. Papageorgiou, N.S.: On parametric evolution inclusions of the subdifferential type with applications to optimal control problems. Trans. Am. Math. Soc. 347, 203–231 (1995) 35. Papageorgiou, N.S., Papalini, F.: Random evolution inclusions of the subdifferential type. Stoch. Anal. Appl. 15, 801–821 (1997) 36. Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact. Variational Methods. In: Lecture Notes in Physics, vol. 655. Springer, Berlin (2004) 37. Zarantonello, E.H.: Projections on convex sets in Hilbert space and spectral theory. In: Zarantonello, E.H. (ed.) Contributions to Nonlinear Functional Analysis, pp. 237–424. Madison, USA (1971) 38. Zeidler, E.: Nonlinear Functional Analysis and its Applications, vol. II/B, Nonlinear Monotone Operators. Springer, New York (1990)
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On a new class of differential variational inequalities and a stability result Joachim Gwinner
Received: 14 March 2011 / Accepted: 4 December 2011 / Published online: 30 March 2013 © Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013
Abstract This paper addresses a new class of differential variational inequalities that have recently been introduced and investigated in finite dimensions as a new modeling paradigm of variational analysis to treat many applied problems in engineering, operations research, and physical sciences. This new subclass of general differential inclusions unifies ordinary differential equations with possibly discontinuous righthand sides, differential algebraic systems with constraints, dynamic complementarity systems, and evolutionary variational systems. The purpose of this paper is two-fold. Firstly, we show that these differential variational inequalities, when considering slow solutions and the more general level of a Hilbert space, contain projected dynamical systems, another recent subclass of general differential inclusions. This relation follows from a precise geometric description of the directional derivative of the metric projection in Hilbert space, which is based on the notion of the quasi relative interior. Secondly we are concerned with stability of the solution set to this class of differential variational inequalities. Here we present a novel upper set convergence result with respect to perturbations in the data, including perturbations of the associated set-valued maps and the constraint set. Here we impose weak convergence assumptions on the perturbed set-valued maps, use the monotonicity method of Browder and Minty, and employ Mosco convergence as set convergence. Also as a consequence, we obtain a stability result for linear complementarity systems. Mathematics Subject Classification (2000)
34G25 · 49J53 · 49K40
Dedicated to Jonathan M. Borwein on the occasion of his sixtieth birthday. J. Gwinner (B) Fakultät für Luft- und Raumfahrttechnik, Institut für Mathematik, Universität der Bundeswehr München, 85577 Neubiberg/München, Germany e-mail: [email protected]
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1 Introduction This paper is concerned with differential variational inequalities that have recently been introduced and studied in depth by Pang and Stewart [31] in finite dimensions as a new modeling paradigm of variational analysis. In their seminal paper the authors have already shown that this new subclass of differential inclusions contains ordinary differential equations with possibly discontinuous right-hand sides, differential algebraic systems with constraints, dynamic complementarity systems, and evolutionary variational systems. As one of the issues of the paper, we show that differential variational inequalities also include projected dynamical systems, another new subclass of differential inclusions, first introduced in finite dimensions by Dupuis and Nagurney [14] and treated in the monograph [30], then studied in a general Hilbert space by Cojocaru and Jonker [10]. Projected dynamical systems have been introduced and investigated in finite dimensions to treat various time dependent disequilibrium problems in management science and operations research, particularly in traffic science. To exhibit such a relationsship we have to lift differential variational inequalities to the more general level of a Hilbert space and consider the concept of a “slow” solution (the solution of minimal norm) for this new subclass of differential inclusions. To derive such a relationship in a Hilbert space we use some results from [21], in particular a precise geometric description of the directional derivative of the metric projection, that extends a basic geometric lemma (Lemma 2.1 in [30]), due to Dupuis [13], to infinite dimensions. As a basic tool for the latter result in a Hilbert space of infinite dimension we employ the concept of the quasi interior known in nonlinear functional analysis (see [37]) and used in duality theory of partially finite convex programming by Borwein and Lewis [6]; in this context we also refer to the recent study of Borwein and Goebel [7] on notions of relative interior in Banach spaces. Such an analysis of the relationship between different subclasses of differential inclusions is not entirely new. Already Heemels, Schumacher, and Weiland [24] succeeded to show that, under mild conditions, projected dynamical systems can be written as gradient-type complementarity systems, a subclass of the afore mentioned differential variational inequalities. Their result extends the well-known fact to the dynamic regime that, under a suitable constraint qualification, variational inequalities with explicit inequality constraints can be written as mixed nonlinear complementarity problems. Then, Brogliato, Daniilidis, Lemaréchal, and Acary [8] were able to prove, under appropriate conditions, the equivalence between several known dynamic formalisms: projected dynamical systems, two types of differential inclusions, and a class of complementarity systems. Their analysis uses exclusively tools from standard convex analysis. However, the work of [8,24] is limited to finite dimensions; it does neither need the refined concept of the quasi interior nor an extension of the basic geometric lemma of Dupuis to infinite dimensions. Furthermore in this paper, we are concerned with stability of the solution set to differential variational inequalities. In this connection let us refer to [32], where at first several sensitivity results are established for inititial value problems of ordinary differential equations with nonsmooth right hand sides and then applied to treat differential variational inequalities. This has to be distinguished from asymptotic Lyapunov
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stability that has been investigated in [1,18,19] for solutions of evolution variational inequalities and nonsmooth dynamical systems. Here we present a novel upper set convergence result with respect to perturbations in the data. In particular, we admit perturbations of the associated set-valued maps and of the constraint set. We impose weak convergence assumptions on the perturbed set-valued maps, use the monotonicity method of Browder and Minty (see e.g. [28,38]) for the analysis of nonlinear variational inequalities, and employ Mosco convergence as set convergence. Thus we extend the stability result of [21] (without considering slow solutions here) to this new more general class of differential variational inequalities of [31]. As a straightforward consequence, we also obtain a stability result for linear complementarity systems [25]. It is to be noted that related stability results for more general evolution inclusions by Papageorgiou [33,34] and by Hu and Papageorgiou in the memoir [26] are not applicable here, since these results need more stringent compactness assumptions. To facilitate the reading of the text we stick to the following notation. Scalar-valued functions and maps between the basic Hilbert spaces are denoted by small latin letters. Matrix-valued functions and derived maps between function spaces are given capital roman letters. Similarly, small greek letters are reserved for set-valued maps between the basic Hilbert spaces, while capital greek letters stand for derived set-valued maps between function spaces. The plan of this paper is as follows. The next Sect. 2 provides the setting of the class of differential variational inequalities considered. In Sect. 3 we exhibit the relation of this class of differential inclusions to projected dynamical systems. To this end, in Sect. 3.1 we provide the basic geometric lemma for the directional derivative of the metric projection. Hence in Sect. 3.2 we derive the relationship result. Section 4 presents our stability analysis for this new class of differential variational inequalities involving general set-valued maps. After some preliminaries in particular on Mosco set convergence (Sect. 4.1), we establish our novel stability result in Sect. 4.2. In the concluding Sect. 5 we shortly discuss possible extensions and limitations of our results, thus clarifying that the Hilbert space setting is the appropriate infinite dimensional framework. Moreover we give an outlook to future work and sketch some main streams of potential applications of our infnite dimensional stability result, namely to nonsmooth dynamic systems with randomness, feasible continous paths to equilibria in stochastic programming, and nonsmooth parabolic PDE problems. 2 The setting of differential variational inequalities considered Let X, V be two real, separable Hilbert spaces that are endowed with norms · X , · V respectively and with scalar products denoted by ·, · , (·, ·) respectively. Further let there be given T > 0, a convex closed subset K ⊂ V , a set-valued map φ : [0, T ] × X × V ⇒ X , a map g : [0, T ] × X × V → V , and some fixed x0 ∈ X . Then following Pang and Stewart [31] we consider the following problem: Find an X -valued function x and an V -valued function u both defined on [0, T ] that satisfy for a.a. (almost all) t ∈ [0, T ] (PS-DVI)(φ, g, K ; x0 )
x(t) ˙ ∈ φ(t, x(t), u(t)) u(t) ∈ σ (K , g(t, x(t), ·))
(1)
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complemented by the initial condition x(0) = x0 . Here x(t) ˙ denotes the time derivative of x(t); σ (K , g(t, x(t), ·)) stands for the solution set of the variational inequality defined by K and g(t, x(t), ·), that is, u(t) has to satisfy u(t) ∈ K , g(t, x(t), u(t)), v − u(t) ≥ 0, ∀v ∈ K . Such a differential inclusion is termed a differential variational inequality by Pang and Stewart [31], here denoted by the acronym PS-DVI to distinguish from the differential variational inequalities treated in the monograph [3] by Aubin and Cellina over 20 years before (called a variational inequality of evolution in [31]) and here denoted by the acronym AC-DVI, see (10) below. To give a precise meaning to a PS-DVI we have to introduce appropriate function spaces and impose some hypotheses on the data. The fixed finite time interval [0, T ] gives rise to the Hilbert space L 2 (0, T ; V ) endowed with the scalar product T [u 1 , u 2 ] :=
u 1 (t), u 2 (t) dt, u 1 , u 2 ∈ L 2 (0, T ; V ).
0
As in [31] we consider weak solutions of the differential inclusion in a PS-DVI in the sense of Caratheodory. In particular, the X -valued function x has to be absolutely continuous with derivative x(t) ˙ defined almost everywhere. Moreover to define the initial condition, the trace x(0) is needed. Therefore (see [11, Theorem 1, p. 473]) we are led to the function space W (0, T ; X ) := {x | x ∈ L 2 (0, T ; X ), x˙ ∈ L 2 (0, T ; X )}, a Hilbert space endowed with the scalar product [x1 , x2 ] + [x˙1 , x˙2 ], x1 , x2 ∈ W (0, T ; X ). Note that W (0, T ; X ) is continuously and densely embedded in the space C[0, T ; X ] of X -valued continuous functions on [0, T ], where the latter space is equipped with the norm of uniform convergence. We assume that the map g satisfies the following growth condition: There exist g0 ∈ L ∞ (0, T ) and g 0 ∈ L 2 (0, T ) such that ∀t ∈ (0, T ) , ∀(y, v) ∈ X × V there holds g(t, y, v)V ≤ g0 (t) (y X + vV ) + g 0 (t).
(2)
Hence a map G that acts from L 2 (0, T ; X ) × L 2 (0, T ; V ) to L 2 (0, T ; V ) can be derived from g by G(x, u)(t) := g(t, x(t), u(t)) , t ∈ (0, T ).
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Also we can introduce the closed convex subset K := L 2 (0, T ; K ) := {w ∈ L 2 (0, T ; V ) | w(t) ∈ K , ∀a.a. t ∈ (0, T )}
(3)
and replace the above pointwise formulation of the variational inequality in a PS-DVI by its integrated counterpart, u ∈ K,
G(x, u), w − u ≥ 0, ∀w ∈ K,
(4)
where its solution set is denoted by Σ(K, G(x, ·)). Clearly, it is sufficient to replace K in (4) by any dense subset of K, e.g. the K − valued continuous functions on [0, T ], as in [31], section 2.1. Concerning the set-valued map φ, we assume that the values φ(t, y, v) are nonvoid, convex and closed. Moreover, we assume the following growth condition similar to (2): There exist h 0 ∈ L ∞ (0, T ) and h 0 ∈ L 2 (0, T ) such that ∀t ∈ (0, T ), ∀(y, v) ∈ X × V, ∀z ∈ φ(t, y, v), there holds z X ≤ h 0 (t) (y X + vV ) + h 0 (t).
(5)
This condition follows from the linear growth condition used in [31]: There exists some constant Cφ > 0 such that ∀t ∈ (0, T ), ∀(y, v) ∈ X × V, ∀z ∈ φ(t, y, v), there holds z X ≤ Cφ (1 + y X + vV ).
(6)
Hence Φ(x, u)(t) := φ(t, x(t), u(t)) , t ∈ (0, T ) gives a set-valued map Φ : L 2 (0, T ; X ) × L 2 (0, T ; V ) ⇒ L 2 (0, T ; X ). Using a standard device in dynamical systems (see e.g. [29]), we can introduce the unknown x˜ := (t, x) and write the above PS-DVI as ⎧ d t 1 ⎨ d x˜ ˜ x(t), = ∈ φ( ˜ u(t)) := dτ dτ x φ(x(t), ˜ u(t)) ⎩ u(t) ∈ σ (K , g(x(t), ˜ ·)) , complemented by the initial condition x(0) ˜ = (0, x0 ). Therefore in the following we can consider the autonomous problem without any loss of generality and drop the dependence on t in PS-DVI. In addition to [31], but analogous to the differential variational inequalities in [3], we also consider the slow solution of PS-DVI. To this end use the shorthand Σ(x) := Σ(K, G(x, ·)) and assume that for any x ∈ W (0, T ; X ), the image set Φ(x, Σ(x)) := ∪{Φ(x, u) : u ∈ Σ(x)} is closed and convex. Then we seek the solution x of x˙ = Φ(x, Σ(x)) := proj(0, Φ(x, Σ(x))).
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This becomes instrumental, when we exhibit the relation of PS-DVI to projected dynamical systems. A simple example of closed convex image sets is as follows. Example 1 Let φ decompose as φ(y, v) = ψ(y) + B(y)v, where for any y ∈ X, ψ(y) is nonvoid, compact and convex, B(y) is a linear continuous operator from V to X with some constant c(y) > 0 (depending on y) such that B(y)v X ≥ c(y)vV , ∀v ∈ V . Note that under monotonicity and weak continuity (hemicontinuity) assumptions on the map g(y, ·) (to be introduced later), the solution set Σ(x) is convex and closed.
3 Relationship of projected dynamical systems to differential variational inequalities We exhibit the relationship of a projected dynamical system to differential variational inequalities in two steps. In the first step we rewrite a projected dynamical system as a differential variational inequality problem in the sense of Aubin and Cellina [3] (called a variational inequality of evolution in [31]). Then in the second step we use a device of [31] and rewrite the obtained problem as finding the slow (minmal norm) solution of a differential variational inequality of the type PS-DVI. For the first step we need some results from [21], in particular on the differentiability of the projection onto closed convex subsets, which are collected next.
3.1 On differentiability of the projection onto closed convex subsets Let H be a Hilbert space with norm | · | and scalar product ·, ·. For a closed convex subset Z of H and for any z ∈ Z , the tangent cone (also support cone or contingent cone,
see e.g. [3]) to Z at z, denoted by TZ (z), is the closure of the convex cone λ(Z − z) : λ > 0 . Then TZ (z) is clearly a closed convex cone with vertex 0 and is the smallest cone C whose translate z + C has vertex z and contains Z . Note that taking polars with respect to the scalar product in H, (TZ (z))0 = (TZ (z))− = N Z (z), the normal cone to Z at x. The utility of the tangent cone for projected dynamical systems (to be introduced in the subsequent subsection) stems from the fact that for the metric projection p Z = proj (·, Z ) onto Z there holds for any z ∈ Z , h ∈ H p Z (z + th) = z + t pTZ (z) (h) + o(t), t > 0 in a Hilbert space (see [37] Lemma 4.6, p. 300).
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On the other hand, in [14,30], the ”differential projection” of the vector h at z ∈ Z with respect to Z is defined by the directional derivative p Z (z, h) = lim
t→0
p Z (z + th) − z . t
Thus the differential projection coincides with the metric projection onto the tangent cone TZ (z), p Z (z, h) = pTZ (z) (h).
(7)
Now following [6,7,37] we call the quasi interior of a closed convex set Z (denoted qi Z ) the set of those z ∈ Z for which TZ (z) = H . Further we define the quasi boundary of a closed convex set Z (denoted by qbdr y Z ) as the set Z \qi Z . Then, in virtue of the strong separation theorem, z ∈ qbdr y Z if and only if there exists some nonzero w ∈ (TZ (z))0 . As the key result for our discussion of projected dynamical systems, we take from [21] the following geometric interpretation of p Z that extends [13], [30, Lemma 2.1] to infinite dimensions. Lemma 1 i) If z ∈ qi Z , then p Z (z, ·) = Id. ii) If z ∈ qbdr y Z , then for any v ∈ H \TZ (z) there exists w ∈ (TZ (z))0 such that p Z (z, v) = v − β w, |w| = 1, β = v, w > 0. For the convenience of the reader we also reproduce its short Proof i) If z ∈ qi Z , then TZ (z) = H and p Z (z, ·) is the identity. ii) If z ∈ qbdr y Z , then by (7) we obtain for vˆ := p Z (z, v) = pTZ (z) (v) that v − v, ˆ y − v ˆ ≤ 0 , ∀y ∈ TZ (z). Since TZ (z) is a cone, it follows 0 v − v, ˆ v ˆ = 0, v − vˆ ∈ TZ (z) .
(8)
By assumption v − vˆ = 0 , hence β := |v − v| ˆ > 0, and (8) yields v − vˆ = βw ˆ w = 0 implies with |w| = 1 and w ∈ (TZ (z))0 . Moreover the orthogonality v, β = v, w proving the lemma. From (8) we can simply derive the following characterization: Corollary 1 Let z ∈ Z . Then for any v ∈ H # p Z (z, v) = v − N Z (z) := proj(0, v − N Z (z)). For the convenience of the reader we also reproduce its short proof.
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Proof Let vˆ = p Z (z, v). In the special case z ∈ qi Z , N Z (z) reduces to zero and hence the claim trivially follows. Thus suppose z ∈ qbdr y Z . Then by (8) we obtain that vˆ ∈ v − N Z (z), which holds trivially if vˆ = v ∈ TZ (z). Since by (7) vˆ ∈ TZ (z) and N Z (z) = (TZ (z))0 , it follows from (8) that v, ˆ v − vˆ − y ≥ 0, ∀ y ∈ N Z (z). This means that vˆ = proj (0, v − N Z (z)) and proves the claim.
3.2 From projected dynamical systems to differential variational inequalities Following [30] a projected dynamical system P DS(F, Z ) is an ordinary differential equation of the form x˙ = p Z x, − f (x) ,
(9)
complemented by the initial condition x(0) = x0 , where the convex closed set Z , x0 ∈ Z , and the vector field f : Z → H are given. Here one looks for a function x: [0, T ] → Z that is absolutely continuous and satisfies x(t) ˙ = p Z x(t), − f x(t) except on a set of Lebesgue measure zero. By Corollary 1, (9) is equivalent to # x˙ = p TZ (x) − f (x) = − N Z (x) + f (x) . Thus the initial value problem (9) consists in finding the slow solution (the solution of minimal norm) to the given initial condition and the differential variational inequality [3, chapter 6, section 6] (AC-DVI) x(t) ˙ ∈ − N Z (x(t)) + f (x(t)) .
(10)
Finally we use a device of [31] for the reformulation of a AC-DVI as a PS-DVI. We introduce additional time-dependent variables p and q by p(t) = x(t), q(t) = x(t) ˙ + f (x(t)). Then by definition of the normal cone, AC-DVI writes p(t) ∈ Z , q(t), z − p(t) ≥ 0, ∀ z ∈ Z . Thus we arrive at a differential variational inequality PS-DVI with the settings
v = ( p, q), φ(y, v) = q − f (y), K = Z × H, g(y, v) = (q, y − p).
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4 Stability of differential variational inequalities In this section we study stability of differential variational inequalities formulated as PS-DVI and admit perturbations x0,n of x0 in the initial condition x(0) = x0 , φ n , g n of the maps φ : X × V ⇒ X, g : X × V → V , and K n of the convex closed subset K ⊂ V . Suppose that (x n , u n ) solves (PS-DVI)(φ n , g n , K n ; x0,n ) and assume that (x n , u n ) → (x, u) with respect to an appropriate convergence for X -valued, respectively V -valued functions on [0, T ]. Then we seek conditions on φ n → φ, g n → g, K n → K , x0,n → x0 that guarantee that (x, u) solves the limit problem (PSDVI)(φ, g, K ; x0 ). Such a stability result can be understood as a result of upper set convergence for the solution set of (PS-DVI)(φ, g, K ; x0 ). 4.1 Preliminaries; Mosco convergence of sets As the convergence of choice in variational analysis we employ Mosco set convergence for a sequence {K n } of closed convex subsets which is defined as follows. A sequence {K n } of closed convex subsets of the Hilbert space V is called Mosco convergent to a M
closed convex subset K of V , written K n −→ K , if and only if σ − lim sup K n ⊂ K ⊂ s − lim inf K n . n→∞
n→∞
Here the prefix σ means sequentially weak convergence in contrast to strong convergence denoted by the prefix s; lim sup, respectively lim inf are in the sense of Kuratowski upper, resp. lower limits of sequences of sets (see [2,4] for more information on Mosco convergence). As a preliminary result we next show that Mosco convergence of convex closed sets K n inherits to Mosco convergence of the polars K n0 and to Mosco convergence of the associated sets Kn = L 2 (0, T ; K n ), derived from K n similar to (3). M
M
M
Lemma 2 Let K n −→ K . Then (a) K n0 −→ K 0 ; (b) Kn −→ K in L 2 (0, T ; V ). Proof To show (a) we verify 1. σ − lim supn→∞ K n0 ⊂ K 0 : Let ζ = σ − lim ζn with ζn ∈ K n0 . Choose z ∈ K arbitrarily. Then by assumption, n→∞ there exist (eventually for a subsequence) z n ∈ K n with z = s − limn→∞ z n . By definition of polar, (ζn , z n ) ≤ 1, ∀n, hence in the limit (ζ, z) ≤ 1, ∀z ∈ K which gives ζ ∈ K 0 . 2. K 0 ⊂ s − lim inf n→∞ K n0 : M
By Proposition 3.23 in [2], s(K n ) −→ s(K ), where s(K )(ζ ) := sup (ζ, z), ζ ∈ V z∈K
is the support function of K . This means in particular, that for any ζ ∈ dom s(K ) there exist ζn ∈ dom s(K n ) such that ζ = s − lim ζn and s(K )(ζ ) ≥ lim supn→∞ s(K n )(ζn ).
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Now let ζ ∈ K 0 , then s(K )(ζ ) ≤ 1 and hence ζ ∈ dom s(K ). For the sequences {ζn }, {sn } with sn := s(K n )(ζn ) and ζn as above, we obtain lim supn sn ≤ 1. If for a subsequence sn k ≤ 1 holds, then ζn k ∈ K n0k and the argument is complete. Otherwise, for almost all n we have sn > 1. But then limn sn = 1, ζ˜n := sn−1 ζn ∈ K n0 and s − limn ζ˜n = ζ ; the proof of part (a) is complete. To show (b) we verify 1. σ − lim supn→∞ L 2 (0, T ; K n ) ⊂ L 2 (0, T ; K ): Let w = σ − lim wn with wn ∈ L 2 (0, T ; K n ). By the bipolar theorem (K 00 = K ) n→∞
it is enough to show that ∀ζ ∈ K 0 , for a.a. t ∈ (0, T ) there holds (w(t), ζ ) ≤ 1. Assume the contrapositive. Then there exist ζ˜ ∈ K 0 , A ⊂ (0, 1) with measure 1 χ A ζ˜ (with χ A denoting the |A| > 0 such that (w(t), ζ˜ ) > 1 on A. Define w˜ = |A| characteristic function of A). By part (a), there exist ζ˜n ∈ K n0 such that s−limn ζ˜n = 1 χ A ζ˜n we have s − limn w˜ n = w˜ in L 2 (0, T ; V ). By ζ˜ ; hence for w˜ n = |A| construction, 1 [wn , w˜ n ] = (wn (t), ζ˜n ) dt ≤ 1. |A| A
Thus in the limit we arrive at 1 1 ≥ [w, w] ˜ = |A|
(w(t), ζ˜ ) dt > 1,
A
a contradiction, proving the claim. 2. K ⊂ s − lim inf n→∞ Kn : It is enough to verify the claim for a dense subset of K; this follows from a diagonal sequence argument, see also [20], Lemma 2.6 for a similar reasoning. Here we use the well-known fact from Bochner-Lebesgue integration theory that the set S(0, T ; V ) of simple V -valued functions on (0, T ) is dense in L 2 (0, T ; V ). This extends to density of S(0, T ; V ) ∩ K in K. This can be seen by taking averages or mean value approximations; see [20] for approximations on a multidimensional integration domain instead of the interval (0, T ). Thus let w be a K -valued simple function on (0, T ), i.e. w = j∈J χ A j z j , where J is finite, z j ∈ K , and A j ⊂ (0, T ) are pairwise disjoint with measure M
|A j | > 0 and ∪ j∈J A j = (0, T ). Since K n −→ K , there exist z j,n ∈ K n such that ∀ j ∈ J, z j,n → z j (n → ∞). Hence wn := j∈J χ A j z j,n lies in Kn and w = s − limn→∞ wn in L 2 (0, T ; V ) follows. Let us note that part (b) of the preceding lemma is of intrinsic interest for timedependent variational inequalities. An analogous implication (b) 1. was already shown in [20] in the more general setting of probability spaces instead of the interval (0, T ), however for the restricted class of translated convex closed cones.
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As a further tool in our stability analysis we recall from [21] the following technical result. Lemma 3 Let H be a separable Hilbert space and let T > 0 be fixed. Then for any sequence {z n }n∈N converging to some z in L 1 (0, T ; H ) there exists a subsequence s {z n k }k∈N such that for some set N of zero measure, z n k (t) −→ z(t) for all t ∈ [0, T ]\N . 4.2 The stability result Before stating the result, some remarks are in order. In view of the existence theory of variational inequalities in infinite dimensional spaces (see e.g. [28]) the best one can hope for is weak convergence of the perturbations u n in the general case of nonunique solutions of the underlying variational inequalities in (PS-DVI). Weak convergence can namely be readily derived from a posteriori estimates. However, continuity of a nonlinear map (here g, φ) with respect to weak convergence is a hard requirement. To circumvent these difficulties with weak convergence we apply the monotonicity method of Browder and Minty. Then as we shall see below, a stability condition on the maps g n with respect to the underlying Hilbert space norm suffices. These weak convergence difficulties also affect φ n . Therefore we have to impose a generally strong stability condition on the nonlinear maps φ n . In the situation of Example 1 this condition can be simplified to a stability condition on the linear operators B n with respect to norm convergence. On the other hand, stronger assumptions on g n , like uniform monotonicity imply that the solution sets Σ(K n , G(x n , ·)) are single-valued. Uniform monotonicity with respect to u moreover entails that the sequence u n strongly converges. Then the stability σ assumption for φ n can be relaxed; in the subsequent hypothesis (H1) wn → w can be s replaced by wn → w. Since our stability assumptions pertain to the given maps g n and g (not to the derived maps G n and G), we have a delicate interplay between the pointwise almost everywhere formulation and the integrated formulation of the variational inequality in the perturbed PS-DVI and in the limit PS-DVI. We need the following hypotheses on the convergence of (φ n , g n ) to (φ, g): s
σ
s
(H1) Let z n → z in X and wn → w in V . Moreover, let pn ∈ φ n (z n , wn ) and pn → p in X . Then p ∈ φ(z, w). s s (H2) All maps g n (z, ·) for any z ∈ X are monotone. - If z n → z, vn → v in X , s respectively in V , then g n (z n , vn ) → g(z, v) in V . - g is hemicontinuous in the sense, that for any y ∈ X ; v, w ∈ V the real-valued function r ∈ IR → (g(y, v + r w), r w) is lower semicontinuous. Now we can state the following stability result: Theorem 1 Let (x n , u n ) solve (PS-DVI)(φ n , g n , K n ; x0,n ). Suppose, φ n and φ satisfy (H1), and that g n and g satisfy (H2). Let the convex closed sets K n Moscos s converge to K and let x0,n → x0 . Assume that x n → x in W (0, T ; X ) and that
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u n ∈ L 2 (0, T ; V ) converges weakly to u pointwise in V for a.a. t ∈ (0, T ) with u n (t)V ≤ m(t), ∀ a.a.t ∈ (0, T ) for some m ∈ L 2 (0, T ). Then (x, u) is a solution to (PS-DVI)(φ, g, K ; x0 ). Proof The proof consists of three parts. 1. Feasibility: u ∈ K, x(0) = x0 . First we observe that for any w ∈ L 2 (0, T ; V ), in virtue of Lebesgue’s theorem of dominated convergence, T [u , w] = n
n u (t), w(t) dt → [u, w].
0 σ
Thus u n → u and u ∈ L 2 (0, T ; V ). Moreover directly by Mosco convergence of {K n } or invoking lemma 2 (b), u ∈ K follows. Since by continuous embedding s s x n → x in C[0, T ; X ], we conclude x n (0) = x0,n → x(0) = x0 . 2. u solves the variational inequality in (PS-DVI)(φ, g, K ; x0 ): u(t) ∈ σ (K , g(x(t), ·)), ∀ a.a.t ∈ (0, T ). Fix an arbitrary w ∈ K. Then by lemma 2 (b), there exist w n ∈ Kn such that s w n → w in L 2 (0, T ; V ). Moreover, by extracting eventually a subsequence, we have by Lemma 3 that also w n (t) strongly converges to w(t) for a.a. t ∈ (0, T ). For any measureable set A ⊂ (0, T ) we can define w nA ∈ L 2 (0, T ; V ) by w nA = w n on A, w nA = u n on (0, T )\A. Hence w nA ∈ Kn and by construction,
g n (x n (t), u n (t)), w n (t) − u n (t) dt ≥ 0.
A
Hence a contradiction argument shows that we have pointwise for a.a. t ∈ n n n n n entails (0,n T ),n g (xn (t), u n(t)), w n(t) − u (t) ≥ 0. By (H2), monotonicity g (x (t), w (t)), u (t)−w (t) ≤ 0. Again by (H2), in the limit g(x(t), w(t)), u(t) − w(t) ≤ 0. In virtue of the growth condition (2), we arrive at T [G(x, w), u − w] =
g(x(t), w(t)), u(t) − w(t) dt ≤ 0, ∀w ∈ K.
0
Hence by a well-known argument in monotone operator theory (see e.g. [27,38]) we obtain that u ∈ K satisfies the variational inequality [G(x, u), w − u] ≥ 0, ∀w ∈ K, provided the map G(x, ·) is hemicontinuous. This follows from the hemicontinuity assumption in (H2) for g(y, ·) as follows: To show for any z ∈ L 2 (0, T ; X ); v, w ∈ L 2 (0, T ; V ),
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lim inf [G(z, v + r w), r w] ≥ [G(z, v + r0 w), r0 w], r →r0
subtract lim [G(z, v), r w] = [G(z, v), r0 w],
r →r0
and apply Fatou’s lemma to the nonnegative integrand to conclude T lim inf r →r0
g(z(t), v(t) + r w(t)), r w(t) dt
0
T ≥
lim inf g(z(t), v(t) + r w(t)), r w(t) dt r →r0
0
T ≥
g(z(t), v(t) + r0 w(t)), r0 w(t) dt
0
3. (x, u) solves the limit (PS-DVI)(φ, g, K ; x0 ). By Lemma 3 applied to {x n } and {x˙ n }, we can extract a subsequence such that ˙ strongly in X pointwise for all t ∈ (0, T )\N0 , x n (t) → x(t) and x˙ n (t) → x(t) where N0 is a null set. Fix t ∈ (0, T )\N0 . Then by assumption, for all n ∈ N we have for some pn (t) ∈ φ n (x n (t), u n (t)), x˙ n (t) = pn (t). Then in virtue of (H1), x(t) ˙ ∈ φ(x(t), u(t)) follows and (x, u) solves the (PS-DVI)(φ, g, K ; x0 ). Let us specialize the stability result to linear complementarity systems. Following [25] a linear complementarity system (LCS)(A, B, C, D; K , x0 ) is governed by the simultaneous equations x(t) ˙ = Ax(t) + Bu(t) q(t) = C x(t) + Du(t) 0 ≤ q(t) ⊥ u(t) ≥ 0, with the initial condition x(0) = x0 , where in infinite dimensional setting, A, B, C, and D are given appropriate bounded linear operators in the Hilbert spaces X, V , and the convex closed set K becomes the ordering cone in the complementarity condition. Similarly we consider the perturbed linear complementarity systems (LCS)(An , B n , C n , D n ; K n , x0,n ). Then the above hypotheses simplify to the following conditions: (H˜ 1) Convergence of the operators An → A, B n → B holds in the operator norm topology.
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(H˜ 2) All operators D n are monotone, i.e. for any v ∈ V, (D n v, v) ≥ 0 holds. Convergence of the operators C n → C, D n → D holds in the operator norm topology. Thus we obtain the following stability result for linear complementarity systems: Corollary 2 Let (x n , u n ) solve (LCS)(An , B n , C n , D n ; K n , x0,n ). Suppose, that An , B n , C n , D n satisfy (H˜ 1), (H˜ 2), respectively. Let the convex closed cones K n s s Mosco-converge to K and let x0,n → x0 . Assume that x n → x in W (0, T ; X ) and n 2 that u ∈ L (0, T ; V ) converges weakly to u pointwise in V for a.a. t ∈ (0, T ) with u n (t)V ≤ m(t), ∀ a.a.t ∈ (0, T ) for some m ∈ L 2 (0, T ). Then (x, u) is a solution to (LCS)(A, B, C, D; K , x0 ). 5 Some concluding remarks Let us shortly remark on possible extensions and limitations of the stability result above. The monotonicity method extends easily to set-valued operators. Also, monotonicity, can be replaced by the more general, however more abstract notion of (order-) pseudomonotonicity. Here we discussed differential variational inequalities and projected dynamical systems in a Hilbert space framework. While for projected dynamical systems the metric projection (and its directional derivative) is needed what confines the analysis to Hilbert spaces, differential variational inequalities and their stability using Mosco convergence can be investigated in more general reflexive Banach spaces, but not beyond [5]. Finally let us give an outlook of future work and outline some main streams of potential applications of our infinite dimensional stability result, namely to nonsmooth dynamic systems with randomness, feasible continous paths to equilibria in stochastic programming, and nonsmooth parabolic PDE problems. Firstly, modelling uncertainty by randomness leads to random evolution inclusions and random differential variational inequalities, see e.g. [35] for a general existence theory. For such problems the abstract Hilbert spaces X, V become the standard L 2 (Ω, μ, IRn ) spaces for a given measure space (Ω, A, μ). Then the question arises, how such random problems can be approximated by deterministic surrogate problems written as finite dimensional PS-DVI. Consider for simplicity the linear complementarity system (LCS)( A, B, C, D; K , x0 ). Let now the matrices A, B, C, D be random and let in a more structured case a finite Karhunen-Loève expansion [12] in its simplest form leads to a separation of the random and the deterministic variables such that for any M ∈ {A, B, C, D} we have M = M0 + ρ M1 with fixed (deterministic) matrices M0 , M1 and some random variable ρ. Then we can employ the discretization procedure in [22,23] with respect to the random variable based on averaging and truncation. While boundedness of the approximates and existence of (weakly) convergent subsequences result from coercivity assumptions, the stability of the approximation procedure is the most important step of the proof of norm convergence, for more details see [22] in the case of linear-affine variational inequalities and [23] in the more general case of nonlinear monotone variational inequalities.
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Secondly, a much studied application of infinite dimensional programming is stochastic programming. In [16], Flåm and Seeger propose to solve cone-constrained convex programs by differential inclusions and give an application to multi-stage stochastic programming. They combine penalty techniques and Lagrangian relaxation to treat the cone constraint and prove convergence of a primal-dual differential method to saddle points, In contrast to [16], the formulation of a (PS-DVI) enforces the complementarity constraints for each time instant, thus yielding a feasible continuous path to equilibria. Here too, the question arises in the application to stochastic programming, how approximation of the stochastic variables and time discretization can be combined to a numerically efficient approximation procedure. In this context, a stability result as above can shed some new light on the numerical solution of such problems. Thirdly, the infinite dimensional setting in the present paper allows the treatment of the large class of distributed nonsmooth systems where the state variable is space dependent and the associated differential equation is a parabolic partial differential equation, see e.g. [15]. As a prominent example of this class let us mention the quasistatic viscoelasic contact problem with short memory and given friction, see [15] and more detailed [36]: Find u(t) such that u(0) = u 0 , u (t) ∈ Uad , a 1 (u (t), v − u (t)) + a 0 (u(t), v − u (t)) + j (v) − j (u (t)) ≥ ( f (t), v − u (t)) + FN (t)(v − u (t)) dΓ, ∀v ∈ Uad . ΓF
The material under consideration has a short memory, since the state of stress at time t depends only on the configuration at the instant t and at the immediately preceding instants. Here, a 0 and a 1 are the bilinear forms of elasticity and of viscosity, respectively, which can be clearly rewritten as appropriate linear operators. Moreover, introducing an appropriate Lagrangian multiplier to treat the nonsmoothness of the given friction frictional j, the above problem can be recast as an (infinite dimensional) differential variational inequality problem, since the admissible set Uad can be considered as independent of time, thus as a convex closed set K defined by the unilateral contact with a fixed foundation. When it comes to numerical approximation, note that Galerkin approximation with respect to space discretization leads in the case of higher than piecewise linear approximation and in the case of a not necessarily piecewise linear obstacle in the contact constraint to a nonconforming approximation, that is , the approximating convex set K h is not a subset of the given convex K . In this situation Mosco convergence (and more refined Stummel-Glowinski convergence) is instrumental to arrive at convergence of Galerkin approximation, see [17]. Thus our stability result can be seen as an important step towards convergence of semidiscretization methods that provide finite dimensional differential variational inequalities. Clearly a full space time discretization needs an additional analysis (see e.g. [9]) and is beyond the scope of the present paper.
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220 Acknowledgments
J. Gwinner The author is indebted to two anonymous referees for their constructive suggestions.
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