DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS SUPPLEMENT 2007
Website: www.AIMSciences.org pp. 467–476
ON DIFFERENTIAL VARIATIONAL INEQUALITIES AND PROJECTED DYNAMICAL SYSTEMS EQUIVALENCE AND A STABILITY RESULT
J. Gwinner Institut f¨ ur Mathematik Fakult¨ at f¨ ur Luft- und Raumfahrttechnik Universit¨ at der Bundeswehr M¨ unchen 85577 Neubiberg/M¨ unchen, Germany
Abstract. The purpose of this note is two-fold. Firstly we deal with projected dynamical systems that recently have been introduced and investigated in finite dimensions to treat various time dependent (dis)equilibrium and network problems in operations research. Here at the more general level of a Hilbert space, we show that a projected dynamical system is equivalent in finding the “slow” solution (the solution of minimal norm) of a differential variational inequality, a class of evolution inclusions studied much earlier. This equivalence follows easily from a precise geometric description of the directional derivative of the metric projection in Hilbert space. By our approach, we can easily characterize a stationary point of a projected dynamical system as a solution of a related variational inequality. Secondly we are concerned with stability of the solution set to differential variational inequalities. 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 the constraint set, where we impose weak convergence assumptions on the perturbed set-valued maps and employ Mosco convergence as set convergence.
1. Introduction. There is an increasing interest in the study of nonsmooth dynamical systems in applied mathematics, see e.g. [1, 2, 17, 19, 20, 21, 22], because they find a wealth of applications in a diversity of fields, like physics, mechanics, electrical circuits, and economics. They usually take the form of differential variational inequalities or more general evolution inclusions. As a seemingly new class of nonsmooth ordinary differential equations, first in the paper [13] by Dupuis and Nagurney, then in the monograph [24] by Nagurney and Zhang, projected dynamical systems have been introduced and studied in finite dimensions to treat various time dependent (dis)equilibrium and network problems in management science and operations research, particularly in traffic science. In this note at the more general level of a Hilbert space, we show that a projected dynamical system is equivalent in finding the “slow” solution (the solution of minimal norm) of a differential variational inequality [5], a class of differential inclusions that were already analyzed over twenty years ago, see [4, 10]. Such an 2000 Mathematics Subject Classification. Primary: 34G25, 35B30, 35K85; Secondary: 49J53, 49J45. Key words and phrases. Projected variational inequality, tangent cone, quasi interior, slow solution, differential variational inequality, stationarity, parametric stability, Mosco convergence.
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equivalence result is recently shown in finite dimensions in [8]. Here employing the concept of the quasi interior known in nonlinear functional analysis (see [28]) and used in duality theory of partially finite convex programming [7], we can extend a basic geometric lemma (Lemma 2.1 in [24]), due to Dupuis [12] to infinite dimensions. This lemma, which gives a precise geometric description of the directional derivative of the metric projection, is the key to the equivalence result in Hilbert space. Moreover by this lemma, a stationary point of a projected dynamical system can be easily characterized as a solution of a related variational inequality, without any restriction to convex polyhedra. Furthermore in this note, we are concerned with stability of the solution set to differential variational inequalities. For a recent study of Liapunov stability we refer to [15]. 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 map and the constraint set, where we impose weak convergence assumptions on the perturbed set-valued maps and employ Mosco convergence as set convergence. It is to be noted that related stability results for more general evolution inclusions by Denkowski and Staicu [11], respectively by Papageorgiou [25, 26] and by Hu and Papageorgiou in the memoir [27] are not applicable here, since these results are limited to finite dimensions, respectively need more stringent compactness assumptions. The outline of this paper is as follows. Section 1 is devoted to projected dynamical systems. In subsection 2.1 we provide the basic geometric lemma for the directional derivative of the metric projection. Hence in subsection 2.2 we derive the equivalence and stationarity results. Section 2 deals with differential variational inequalities involving general set-valued maps. After a discussion of the stability of the metric projection and of the tangent cone under Mosco set convergence (subsection 3.1) and a preliminary convergence result (subsection 3.2), we establish our novel stability result in subsection 3.3. The paper concludes with a remark (section 4) suggesting a new direction in the applied field of nonsmooth dynamical systems. 2. Projections and Projected Dynamical Systems. 2.1. 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 x ∈ Z, the tangent cone (also support cone or contingent cone,nsee e.g. [5]) to Z oat x, denoted by TZ (x), is the closure of S the convex cone λ(Z − x) : λ > 0 . Then TZ (x) is clearly a closed convex cone with vertex 0 and is the smallest cone C whose translate x + C has vertex x and contains Z. Note that taking polars with respect to the scalar product in H, (TZ (x))0 = (TZ (x))− = NZ (x), 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 PZ = proj (·, Z) onto Z there holds for any x ∈ Z, h ∈ H PZ (x + th) = x + tPTZ (x) h + o(t), t > 0 in a Hilbert space (see [28] Lemma 4.6, p. 300). On the other hand, in [24], also [13], the ”differential projection” of the vector h at x ∈ Z with respect to Z is defined by the directional derivative PZ (x + th) − x . ΠZ (x, h) = lim t→0 t
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Thus ΠZ (x, h) = PTZ (x) h ,
(1)
the metric projection onto the tangent cone TZ (x). Now we call the quasi interior of a closed convex set Z (denoted qi Z) the set of those x ∈ Z for which TZ (x) = H. The refinement used by Zarantonello [28] and by Borwein and Lewis [7] that TZ (x) is a subspace, only, is not needed here. In addition, we define the quasi boundary of a closed convex set Z (denoted by qbdry Z) as the set Z \ qi Z. Then, in virtue of the strong separation theorem, x ∈ qbdry Z, if and only if, there exists some nonzero u ∈ (TZ (x))0 . As is well-known, in infinite dimensions the (topological) interior as well as the relative algebraic interior of a convex set can be void. On the contrary, the quasi interior as well as the quasi boundary are nonempty in important function spaces, as seen in the following Example 1. Let H = L2 (T, µ) on a measure space (T, µ). Consider the closed convex cone Z = {z ∈ H : z(t) ≥ 0 µ - a.e. }. Then e.g. 1 = 1T ∈ qi C. Indeed, by Lebesgue’s theorem of majorized convergence, any x ∈ H can be approximated by the sequence {xn } of truncations, x(t) if x(t) ≥ −n a.e.; xn (t) = −n elsewhere, and clearly xn ∈ n(C − 1). In contrast, 1S ∈ qbdry Z, provided µ(T \S) > 0. As the key result for our discussion of projected dynamical systems, we have the following geometric interpretation of ΠZ that extends [12], [24, Lemma 2.1 ] to infinite dimensions. Lemma 1. i) If x ∈ qi Z, then ΠZ (x, ·) = Id . ii) If x ∈ qbdry Z, then for any v ∈ H \ TZ (x) there exists u ∈ (TZ (x))0 such that |u| = 1, β := hv, ui > 0, ΠZ (x, v) = v − β u . Proof. i) If x ∈ qi Z, then TZ (x) = H and ΠZ (x, ·) is the identity. ii) If x ∈ qbdry Z, then by (1) we obtain for vˆ := ΠZ (x, v) = PTZ (x) v that hv − vˆ, w − vˆi ≤ 0 ,
∀w ∈ TZ (x) .
Since TZ (x) is a cone, it follows hv − vˆ, vˆi = 0,
0 v − vˆ ∈ TZ (x) .
(2)
By assumption v − vˆ 6= 0 , hence β := |v − vˆ| > 0, and (2) yields v − vˆ = βu with |u| = 1 and u ∈ (TZ (x))0 . Moreover the orthogonality hu, vˆi = 0 implies β = hv, ui proving the lemma. Let us note that another extension of Dupuis’ Lemma for Hilbert spaces is due to Isac and Cojocaru [18] that works with the topological interior and boundary and appears somewhat complicated. Here from (2) we simply derive the following characterization. Corollary 1. Let x ∈ Z. Then for any v ∈ H # ΠZ (x, v) = v − NZ (x) := proj (0, v − NZ (x)) .
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Proof. Let vˆ = ΠZ (x, v) . In the special case x ∈ qi Z, NZ (x) reduces to zero and hence the claim trivially follows. Thus suppose x ∈ qbdry Z. Then by (2) we obtain that vˆ ∈ v − NZ (x) , what holds trivially if vˆ = v ∈ TZ (x). Since by (1) vˆ ∈ TZ (x) and NZ (x) = (TZ (x))0 , it follows from (2) moreover that hˆ v , v − vˆ − zi ≥ 0, ∀ z ∈ NZ (x) . This means that vˆ = proj (0, v − NZ (x)) and proves the claim. 2.2. Projected dynamical systems - stationarity and equivalence. Following [24] a projected dynamical system P DS(F, Z) is an ordinary differential equation of the form x˙ = ΠZ x, −F (x) , (3) where the convex closed set Z and the vector field F : Z → H are given. Here one looks for : [0, +∞) → Z that is absolutely continuous and satisfies a function x
x(t) ˙ = ΠZ x(t), −F x(t) except a set of Lebesgue measure zero. Like in the classical theory of ordinary differential equations, (3) has to be complemented by an initial condition x(0) = x0 , (4) where x0 ∈ Z is given. A vector x∗ ∈ Z is termed [24, Definition 2.7] an equilibrium point or stationary point of the projected dynamical system P DS(F, Z), if x∗ satisfies ΠZ x∗ , −F (x∗ ) = 0 . (5) This means that once the projected dynamical system reaches x∗ at some time t ≥ 0, it will remain at x∗ for all future times t ≥ t∗ . Based on Lemma 1 we can easily characterize the stationary points of a projected dynamical system P DS(F, Z) as solutions of a related variational inequality. Thus we can extend [24, Theorem 2.4] by dispensing with its restriction to finite dimension and to convex polyhedra. ∗
Proposition 1. Let Z be closed convex and x∗ ∈ Z. Then x∗ is a stationary point of the projected dynamical system P DS(F, Z), if and only if hF (x∗ ), z − x∗ i ≥ 0, ∀ z ∈ Z .
(6)
Proof. Let x∗ solve (6) which is equivalent to x∗ = PZ x∗ − tF (x∗ ) , ∀ t > 0, whence by definition, ΠZ x∗ , −F (x∗ ) = 0 follows. Conversely let x∗ be a stationary point. Consider the case x∗ ∈ qbdry Z, −F (x∗ ) 6∈ TZ (x∗ ). Then by Lemma 1, there holds F (x∗ ) ∈ −(TZ (xa st))0 . and hence x∗ solves (6). In the remaining cases, we have −F (x∗ ) ∈ TZ (x∗ ), hence F (x∗ ) = 0, and x∗ trivially solves (6).
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Note that by (1) and Corollary 1, (3) is equivalent to # x˙ = P TZ (x) −F (x) ≡ − NZ (x) + F (x) . Thus the initial value problem (3), (4) consists in finding the ”slow” solution (the solution of minimal norm) to the initial condition (4) and the differential variational inequality [5, chapter 6, section 6] x(t) ˙ ∈ − NZ (x(t)) + F (x(t)) (7) what is equivalent [10] to finding the ”slow” solution to the initial condition (4) and the projected variational inequality [5, chapter 6, section 6] x(t) ˙ ∈ PTZ (x(t)) −F (x(t)) . (8) Note that here the operator F is single-valued and thus (8) becomes an equation. By this equvalence earlier existence results ([4, Theorem 2]; see [5], [23] for more results and references) apply to projected dynamical systems in infinite dimensions. Recently existence for a P DS(F, Z) has been directly proved for a Lipschitzian map F by Cojocaru and Jonker [9]. 3. A parametric stability result for differential variational inequalities. In this section we study for some fixed T > 0, differential variational inequalities of the form DVI(F, Z; x0 ) 0 ∈ u(t) ˙ + NZ (u(t)) + F (u(t)) , t ∈ [0, T ] complemented by the initial condition u(0) = x0 with x0 ∈ Z given. Now we admit that F : Z ⇒ H is a multimap. We consider perturbations F n of F , Z n of Z, and x0,n of x0 . Suppose that un solves DVI(F n , Z n ; x0,n ) and assume that un → u in an appropriate function space of H-valued functions on [0, T ]. Then we seek conditions on F n → F, Z n → Z, x0,n → x0 that guarantee that u solves the unperturbed problem DVI(F, Z; x0 ). Such a stability result can be understood as a result of upper set convergence for the solution set of DVI(F, Z; x0 ). As the convergence of choice in variational analysis we employ Mosco set convergence for Z n . Therefore we have first to discuss the stability of metric projection and of tangent, resp. normal cones under Mosco set convergence. 3.1. Projection, tangent cone, and Mosco convergence. A sequence {Zn } of closed convex subsets of the Hilbert space H is called Mosco convergent to a closed M convex subset Z of H, written Zn −→ Z, if and only if σ − lim sup Zn ⊂ Z ⊂ s − lim inf Zn . n→∞
n→∞
Here the prefix σ means sequentially weak convergence in contrast to strong convergence denoted by the prefix s; lim sup, resp. lim inf are in the sense of Kuratowski upper, resp. lower limits of sequences of sets (see [3, 6] for more information on Mosco convergence). For the metric projection we have the following basic result. s
M
Lemma 2. Let yn −→ y, Zn −→ Z. Suppose, d(0, Zn ) ≤ c < +∞. Then σ PZn yn −→ PZ y.
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Proof. Let zˆn = PZn yn . We claim 1. {ˆ zn } is bounded. Indeed, since d(0, Zn ) ≤ c, there exist z˜n ∈ Zn with |˜ zn | ≤ c and hence |ˆ zn |
≤ |ˆ zn − yn | + |yn | ≤ |˜ zn − yn | + |yn | ≤
|˜ zn | + 2|yn | ≤ c˜.
Thus {ˆ zn } possesses weak limit points zˆ. 2. Any weak limit zˆ = σ − lim zˆnk belongs to Z. This is immediate from the k→∞
assumption σ − lim sup Zn ⊂ Z. 3. zˆ = PZ y. Indeed, let z arbitrarily in Z. Then by assumption Z ⊂ s − lim inf Zn , s there exist zn ∈ Zn such that zn −→ z. Thus we can conclude |ˆ z − y| ≤ lim inf |ˆ znk − ynk | ≤ lim |zn − yn | = |z − y|. n→∞
k→∞
Let us comment on this result. Under the natural (and necessary) condition d(0, Zn ) ≤ c we have the boundedness of the projections to Zn . Under Mosco convergence for {Zn } we get both: feasibility (ˆ z ∈ Z) and optimality (ˆ z = minimizer of distance). We would like to apply this lemma to a DVI(F, Z, x0 ), where we can use (7) or M s (8). Thus let Zn −→ Z, xn −→ x (where xn ∈ Zn ). What can be said about the sequences NZn (xn ), resp. TZn (xn )? First, we have σ − lim sup NZn (xn ) ⊂ NZ (x).
(9)
n→∞
σ
Indeed, let wn ∈ NZn (xn ), wn −→ w. To show that w ∈ NZ (x), let z ∈ Z arbitrarily. M s By Zn −→ Z, there exist zn ∈ Zn such that zn −→ z. Thus hwn , zn − xn i ≤ 0, in the limit hw, z − xi ≤ 0, and (9) follows. By duality for convex cones [6, Theorem 1.1.8], (9) implies TZ (x) ⊂ s − lim inf TZn (xn ). n→∞
(10)
However, Lemma 2 is not fully applicable to tangent cones, resp. normal cones. In nonsmooth situations we do not even have NZ (x) ⊂ s − lim NZ (xn ). This comes out in the following simple 1 ,0 → Example 2. Consider the compact convex Z = [0, 1]2 ⊂ R2 . Let xn = n x = (0, 0). Then w = (−1, 0) ∈ NZ (x), but cannot be recovered as the limit of wn ∈ NZ (xn ) = R+ (0, −1). The same situation occurs with the convex cone Z = R2+ . Therefore for our subsequent stability result we have to identify an appropriate topology in the space of absolutely continuous H-valued functions on [0, T ] such that from un → u we can conclude by a direct argument that in the limit there holds u(t) ˙ ∈ TZ (u(t))
a.e. on (0, T ) .
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3.2. A preliminary convergence result. Since the Hilbert space H may be infinite dimensional, we need the following result. Lemma 3. Let H be a separable Hilbert space and let T > 0 be fixed. Then for any sequence {vn }n∈N converging to some v in V := L1 (0, T ; H) there exists a s subsequence {vnk }k∈N such that for some set N of zero measure, vnk (t) −→ v(t) for all t ∈ [0, T ]\N . Proof. By assumption, there exists a denumerable subset M that is dense in H. Let s x ∈ H and {xν }ν∈N in H. To have strong convergence xν −→ x it is enough that |xν | → |x| and for all a ∈ M , hxν , ai → hx, ai for ν → ∞. Indeed, let > 0 fixed. By convergence |xν | → |x|, there exists some C > 0 such that |x| + sup |xν | ≤ C. ν
Since x = lim aµ for some appropriate sequence {aµ } ⊂ M , there exists m ∈ N µ→∞
such that
for µ ≥ m. 2C Now choose µ = m. By assumption, hxν − x, am i → 0 (ν → ∞). Hence there exists ν0 ∈ N, ν0 = ν0 m() such that |x − aµ | ≤
|hxν − x, am i| ≤
for ν ≥ ν0 . 2
Thus for ν ≥ ν0 , |hxν − x, xi|
≤ |hxν − x, am i| + |hxν − x, x − am i| +C · = . ≤ 2 2C
Therefore we can conclude |x − xν |2 = |xν |2 − 2hxν − x, xi − |x|2 → 0. Now let vn → v in V and let M = {aµ }µ∈N . Since |vn | → |v| in L1 (0, T ), by a well-known result in measure theory, there exists a set N0 of Lebesgue measure (0) zero (shortly null set in the following) and a subsequence {vn }n∈N (the index set is (0) again denoted by N) of {vn } such that |vn (t)| → |v(t)| for all t ∈ [0, T ]\N0 . Since (0) (1) hvn , a1 i → hv, a1 i in L1 (0, T ), there exists a null set N1 and a subsequence {vn } (0) (1) of {vn } such that hvn (t), a1 i → hv(t), a1 i (n → ∞) for all t ∈ [0, T ]\N1 . By this (µ) procedure, for any µ ∈ N, we can find a null set Nµ and a subsequence {vn }n∈N (µ−1) (0) (µ) of {vn } ⊂ · · · ⊂ {vn } ⊂ {vn } such that hvn (t), aµ i → hv(t), aµ i (n → ∞) for all t ∈ [0, T ]\Nµ . [ (n) Consider N = Nµ , again a null set, and the diagonal sequence {vn }n∈N ⊂ µ∈N (n)
{vn }. Then by construction, for all t ∈ [0, T ]\N , we have |vn (t)| → |v(t)| and (n) hvn (t), ai → hv(t), ai for all a ∈ M (n → ∞). By the first part of the proof, we s (n) arrive at vn (t) −→ v(t) (n → ∞) for all t ∈ [0, T ]\N . 3.3. The stability result. We need the following hypotheses on the convergence of F n : s
(H1) : Let zn → z. Then for any pn ∈ F n (zn ) there exist p ∈ F (z) and a subseσ quence {pnk } such that pnk → p.
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(H2) : Let zn → z. Then for any p ∈ F (z) there exists a subsequence {pnk } such σ that pnk ∈ F nk (znk ) and pnk → p. With fixed T > 0, we consider the Bochner space W 1,2 (0, T ; H) of H-valued 2 L (0, T ) functions u with time derivative u˙ in L2 (0, T ; H). It is known (see e.g. [14], Section 4.1, Lemma 1.11) that W 1,2 (0, T ; H) is continuously embedded in the space C[0, T ; H] of H-valued continuous functions on [0, T ]. This is the key for the following result. Theorem 1. Let H be a separable Hilbert space. Suppose, Fn satisfy (H1). Let s the convex closed sets Z n Mosco-converge to Z and let x0,n → x0 . Let un solve DVI(F n , Z n ; x0,n ). Assume that un converges to u in W 1,2 (0, T ; H) for n → ∞. Then u is a solution to the DVI(F, Z; x0 ). If moreover Fn satisfy (H2) and un is the slow solution of DVI(F n , Z n ; x0,n ), then u is also the slow solution of the DVI(F, Z; x0 ). Proof. By Mosco convergence, x0 ∈ Z and u(t) ∈ Z for all t ∈ [0, T ], since un → u in C[0, T ; H]. Therefore by the definition of the tangent cone, u(t) ˙ ∈ TZ u(t) a.e. on (0, T ) . (11) s
Moreover un (0) → u(0) and un (0) = x0,n → x0 , hence u(0) = x0 . By assumption, for all n ∈ N we have for some pn (t) ∈ F n (un (t)), u˙ n (t) + pn (t) ∈ −NZ n (un (t)) pointwise for all t ∈ (0, T )\N0 , where N0 is a null set. Fix t ∈ (0, T )\N0 . Then in σ virtue of (H1), with some subsequence {pnk (t)} there holds pnk (t) → p(t) ∈ F (u(t)). Hence by (9), u(t) ˙ + p(t) ∈ −NZ (u(t)) follows and u solves the DVI(F, Z; x0 ). Now let un be the slow solution of DVI(F n , Z n ; x0,n ) for n ∈ N. This means apart from a null set, hu˙ n (t), yn + u˙ n (t)i ≤ 0 , ∀yn ∈ NZ n (un (t)) + F n (un (t)) . Now choose 0 ∈ NZ n (un (t)). Thus hu˙ n (t), pn + u˙ n (t)i ≤ 0 , ∀pn ∈ F n (un (t)) . Then we apply Lemma 3 to u˙ n and use (H2). Thus apart from a null set, we arrive in the limit at hu(t), ˙ p + u(t)i ˙ ≤ 0 , ∀p ∈ F (u(t)) , which in view of (11) proves the theorem. Clearly, in the single-valued case F n : Z → H, instead of (H1) and (H2) it is s σ enough to require that zn → z implies F n (zn ) → F (z). We emphasize that in this case and also in the more general set-valued case we could prove the above stability result without any compactness assumptions. This is important in view of applications to projected differential systems and differential variational inequalities with unilateral unbounded constraints. In contrast, the profound analysis of Papageorgiou [26] and of Hu and Papageorgiou in the memoir [27], dealing with stability of much more general evolution inclusions (including time dependent constraints Z = Z(t)), while introducing refined compactness assumptions (e.g. functionals of “compact type”) targets full Kuratowski set convergence for the solution set.
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4. A concluding remark. In this short note we discussed projected dynamical systems and differential variational inequalities in a Hilbert space framework. Thus we not only lifted this class of ordinary differential equations to a more abstract level, but such an infinite dimensional framework is also important for a functional analytic approach to evolutionary (dis)equilibrium problems with uncertain data, often modelled as random data. Here to suggest a new direction in the applied field of nonsmooth dynamical systems we refer the interested reader to the recent paper [16] dealing with stationary random variational inequalities on random sets. Acknowledgements. The author would like to thank the referees very much for their careful work. REFERENCES [1] K. Addi and D. Goeleven and A. Rodic, Mathematical analysis and numerical simulation of a nonsmooth road-vehicle spatial model, ZAMM Z. Angew. Math. Mech., 86 (2006), 185–209. [2] S. Adly and D. Goeleven, A stability theory for second-order nonsmooth dynamical systems with application to friction problems, J. Math. Pures Appl., IX. S´ er., 83 (2004), 17-51. [3] H. Attouch, “Variational convergence for functions and operators,” Pitman, Boston, London, Melbourne, 1984. [4] H. Attouch and A. Damlamian, Probl` emes d’ ´ evolution dans les Hilberts et applications, J. Math. pures et appl. IX S´ er., 54 (1974), 53–74. [5] J.-P. Aubin and A. Cellina, “Differential Inclusions. Set-valued Maps and Viability Theory,” Springer-Verlag, Berlin, 1984. [6] J.-P. Aubin, and H. Frankowska, “Set-Valued Analysis,” Birkh¨ auser, Boston, Basel, 1990. [7] J. M. Borwein and A. S. Lewis, Partially finite convex programming, part I: quasi relative interiors and duality theory, Math. Programming, 57 (1992), 15–48. [8] B. Brogliato and A. Daniilidis and C. Lemarechal and V. Acary, On the equivalence between complementarity systems, projected systems and differential inclusions, System & Control Letters, 55 (2006), 45–51. [9] M. G. Cojocaru and L. B. Jonker, Existence of solutions to projected differential equations in Hilbert spaces, Proc. Am. Math. Soc., 132 (2004), 183–193. [10] B. Cornet, Existence of slow solutions for a class of differential inclusions, J. Math. Anal. Appl., 96 (1983), 130–147. [11] Z. Denkowski and V. Staicu, Asymptotic behavior of the minima to a class of optimization problems for differential inclusions, J. Optimization Theory Appl., 81 (1994), 21–34. [12] P. Dupuis, Large deviations analysis of reflected diffusions and constrained stochastic approximation algorithms in convex sets, Stochastics, 21 (1987), 63–96. [13] P. Dupuis and A. Nagurney, Dynamical systems and variational inequalities, Ann. Oper. Res., 44 (1993), 9–42. [14] H. Gajewski and K. Gr¨ oger and K. Zacharias, “Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen,” Akademie-Verlag, Berlin, 1974. [15] D. Goeleven and D. Motreanu and V. V. Motreanu, On the stability of stationary solutions of first order evolution variational inequalities, Adv. Nonlinear Var. Inequal., 6 (2003), 1–30. [16] J. Gwinner and F. Raciti, On a class of random variational inequalities on random sets, Numer. Func. Anal. Opt., 27 (2006), 619–636. [17] C. Henry, Differential equations with discontinuous right-hand side for planning procedures, J. Economic Theory, 4 (1972), 545–551. [18] G. Isac and M. G. Cojocaru, Variational inequalities, complementarity problems and pseudomonotonicity. Dynamical aspects, Semin. Fixed Point Theory Cluj-Napoca, 3 (2002), 41–62. [19] P. Krejci, Evolution variational inequalities and multidimensional hysteresis operators, “Nonlinear differential equations,” (Pavel Drabek et al., eds.) Boca Raton, (1990), 47–110. [20] M. D. P. Monteiro Marques, “Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction,” Birkh¨ auser, Basel, Boston, 1993. [21] J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differ. Equations, 26 (1977), 347–374. [22] J. J. Moreau, Numerical aspects of the sweeping process, Comput. Methods Appl. Mech. Engrg., 177 (1999), 329–349.
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Received September 2006; revised December 2006. E-mail address:
[email protected]
