projected dynamical systems on hilbert sps

8/12/2019 Projected Dynamical Systems on Hilbert Sps

http://slidepdf.com/reader/full/projected-dynamical-systems-on-hilbert-sps 1/13

PROJECTED DYNAMICAL SYSTEMS ON HILBERT SPACES

THESIS ABSTRACT

MONICA-GABRIELA COJOCARU

1. Introduction

The notion of projected dynamical systems is relatively new, being introduced in themathematical literature in the last decade. In this thesis we present the generalization of

this notion from Euclidean space to Hilbert spaces of arbitrary dimensions.

These systems are non-smooth dynamical systems and are dened as the solutions to a

class of ordinary differential equations with a discontinuous right-hand side. We refer to

this class of equations as projected differential equations . The word projected indicates

the use of a specic operator in dening a projected differential equation. This operator

restrains the whole Hilbert space X onto a non-empty, closed and convex subset K X .

The motivation for studying a projected dynamical system is that it can be used in thestudy of dynamics of perturbed steady states of problems arising from Economic Theory,

Physics and Engineering.

This study can be conducted via the theory of variational inequalities, a theory closely

related to calculus of variations and free boundary problems. It is the case that the crit-

ical points of a projected differential equation coincide with the solutions to a variational

inequality problem. The solvability of variational inequalities is a question that has been

answered positively in many contexts (see for example [B-C], [C-Y], [Is1], [Is2], [Is3], [Is4],

[Is5], [Is-C2], [Is-C3], [K-S], [Lu], [M2], [H-P], [Is7] and the references therein). In the thesis

we use the notion of exceptional family of elements to assert the solvability of variational

inequality problems on closed, convex and unbounded subsets of a Hilbert space.

Through a projected dynamics we can show, under suitable conditions, whether or not

equilibria have stability and/or attracting properties. In short, we can show if and how the

equilibria will be reached.1



2 MONICA-GABRIELA COJOCARU

The main results of this thesis consist of proving the existence of projected dynamical

systems on Hilbert spaces of any dimension (that is, proving that the discontinuous differ-

ential equation has unique solutions) and proving results about the dynamics of perturbed

equilibria (critical points of a projected differential equation). We will present below the

main results and outline the ideas and approaches for their proofs.

2. Projected differential equations and projected dynamical systems

Let X be a Hilbert space of arbitrary dimension and K X a non-empty, closed and

convex subset. Let · , · be the inner product on X . For any z X , there exists an unique

element in K , denoted by P K (z), such that ||P K (z) − z || = inf y K

|| y − z || . P K is called the

projection operator of the space X onto the subset K . In [K-S] Chapter I, Theorems2.3 and Corollary 2.4 it is shown that for a Hilbert space X and K X a non-empty closed

convex subset, P K is Lipschitz continuous with constant 1.

DEFINITION 2.1. For x K , the set T K (x ) = h> 0

1h (K − x ) is called the tangent cone

to K at the point x. The cone T K (x ) is a closed convex cone. The normal cone to the set

K at the point x is the polar cone of T K (x ) and it is given by N K (x ) := { p X | p, x − x ≥

0, x K }. The normal cone is also a closed, convex cone. According to [A-C], Chapter

5, Section 1, Proposition 2 we have that N K (x ) = T K (x )− and also T K (x ) = N K (x )− .

Now we introduce the directional derivative of the projection operator P K .

PROPOSITION 2.1. For any x K and any element v X the limit

ΠK (x, v ) := limδ→ 0+

P K (x + δv) − xδ

exists and ΠK (x, v ) = P T K (x ) (v).

A proof of this result can be found in [Z] Lemma 4.6 or [S].

REMARK 2.1. Let ΠK : K × X → X be the operator given by ( x, v ) → ΠK (x, v ). Thenwhenever v T K (x ), ΠK (x, v ) = v. The operator Π K is discontinuous on the boundary of

the set. The following result can be found in [I-R], Theorem 2.23.

THEOREM 2.1 (Moreau) . If C X is a closed convex cone, C − its polar cone and

x ,y,z X , then the following statements are equivalent:



PROJECTED DYNAMICAL SYSTEMS ON HILBERT SPACES THESIS ABSTRACT 3

1). z = x + y, x C , y C − and x, y = 0 ;

2). x = P C (z) and y = P C − (z).

The operator Π K plays a crucial role in the context of this thesis. Π K can be characterized

by the following theorem, which is our generalization of a similar result in [Du]:

THEOREM 2.2. Let X be an innite dimensional Hilbert space and K X be a non-

empty closed, convex subset. Then

(i) ΠK (x, v ) = v, if x intK .

(ii) ΠK (x, v ) = v, if x ∂K and supn N K (x )

v, n ≤ 0.

(iii) ΠK (x, v ) = v − v, n n , if x ∂K , supn N K (x )

v, n > 0 and n achieves that supremum.

In the thesis we use a simpler characterization of Π K .

PROPOSITION 2.2. Let X be an arbitrary Hilbert space and K X a closed and convex

subset. Then there exists n N K (x ) such that

ΠK (x, v ) = v − n, for each x K and v X,

and ||ΠK (x, v )|| ≤ || v|| .

Proof. The proof is immediate by Moreau’s theorem.

DEFINITION 2.2. Let X be a Hilbert space of arbitrary dimension and K X be a

non-empty, closed and convex subset. Let F : K → X be a vector eld. Then the ordinary

differential equation

(1) dx(t )

dt = Π K (x (t ), − F (x (t )))

is called the projected differential equation associated with F and K .

The projected differential equation is present in [A-C], [CO] and [He] if one considers the

case of single-valued mappings from K to X . However, the rst formulation of equation(1) in the form above occurred in 1990 in the paper [D-I] and then in [D-N] in Euclidean

space, for convex polyhedral subsets K R n .

DEFINITION 2.3. A point x K is called a critical point or an equilibrium for the

equation (1) if ΠK (x , − F (x )) = 0 .




There is a fundamental relation between a projected dynamical system and a variational

inequality problem.

THEOREM 2.3. Let X be a Hilbert space and K X be a non-empty, closed and convex

subset. Let F : K → X be a vector eld. Consider the variational inequality problem:

(2) nd x K such that F (x ), y − x ≥ 0, y K.

Then the solutions to problem (2) coincide with the critical points of the projected differential

equation (1).

Proof. The proof is simple, using Moreau’s theorem.

REMARK 2.2. A rst proof of this result can be found in [D-I] Section 5.3 for the case of X a Euclidean space and K a convex polyhedral subset.

Consider the initial value problem

(3) dx (t )

dt = Π K (x (t ), − F (x (t ))) , x(0) = x 0 K.

An absolutely continuous function x : [0, l ] R → X such that x(t ) K for all t [0, l]

and dx(t )

dt = Π K (x (t ), − F (x (t ))), for almost all t [0, l ], is called a solution to the initial

value problem (3). If solutions to problem (3) exist on R+ and are unique starting at each

point x0, then we can associate to this initial value problem a dynamical system:

DEFINITION 2.4. A projected dynamical system is given by a mapping

Φ : K × R + → K which solves the initial value problem:

Φx (t ) = Π K (Φx (t ), − F (Φx (t ))) , Φx (0) = x.

We make the convention Φ x (·) := x(·).

3. Existence of projected dynamical systems on H-spaces

In the paper Existence of solutions to projected differential equations on Hilbert spaces

[C-J], we show that problem (3) has solutions in the general context of a Hilbert space of

innite dimension, and for any non-empty, closed, convex subset K X and any Lipschitz

continuous vector eld F : K → X . The paper is attached .

We give here the statement of the main theorem followed by the theorem about the

extension of solutions from an interval in R to R+ .




THEOREM 3.1. Let X be a Hilbert space of arbitrary dimension and K X a non-

empty, closed and convex subset. Let F : K → X be a Lipschitz continuous vector eld

with Lipschitz constant b. Let x0 K and L > 0 so that ||x 0 || ≤ L . Then the initial value

problem dx (t )

dt = Π K (x (t ), − F (x (t )) , x(0) = x 0 has a unique solution on the interval [0, l ],

where l := L

|| F (x 0)|| + bL.

THEOREM 3.2. Consider the initial value problem

(4) dx (t )

dt = Π K (x (t ), − F (x (t ))) , x(0) = x 0

in the hypothesis of Theorem 3.1. Then the solutions can be extended to [0, ∞ ).

Proof. From Theorem 6.1 we can assert the existence of a solution to problem (6) on an

interval [0 , l ], where l = L

|| F (x 0)|| + bL, with b > 0 xed and L > 0 arbitrary.

We note that we can choose L such that l ≥ 11 + b

in the following way: if ||F (x 0) || = 0

we let L = 1 and if ||F (x 0)|| = 0, then we let L ≥ || F (x 0) || . In both cases we obtain

l ≥ 11 + b

. Therefore beginning at each initial point x0 K problem (6) has a solution on

an interval of length at least [0 , 11 + b

]. Now if we consider problem (6) with x(0) = x ( 11 + b

)

and applying again Theorem 6.1, we obtain an extension of the solution on an interval of

length at least 1

1 + b. By continuing this solution we obtain a solution on [0 , ∞ ).

4. Dynamics of perturbed steady-states

In this section we exploit the fundamental relation between a variational inequality prob-

lem dened by the vector eld F on the subset K , and a projected dynamical system,

involving the vector eld − F (Theorem 2.2). Namely, we show that under supplementary

conditions on F , the equilibria of this system, hence the solutions to a variational inequal-

ity, display stability and attracting properties. These supplementary conditions consist of

pseudo- and quasi-monotonicity.

The notion of monotonicity is the natural generalization of the notion of an increasing

real valued function of one variable. There exist generalizations of this notion to various

larger classes of mappings (pseudo-monotone and quasi-monotone), (see [Ka]). We will limit

ourselves in this abstract to a few illustrations.

Let X be a Hilbert space and K X a non-empty subset. A mapping F : K → X is called




monotone if for all x, y K , F (y) − F (x ), y − x ≥ 0. Monotone mappings are used in

the theory of variational inequalities, in Optimization, in Nonlinear Analysis, in the theory

of nonlinear differential equations as can be seen in [A-C], [B-C], [Hk], [K-S], [K-Z], [La],

[M2], [Na6], [P], [R-W], [Z],[Da2].

DEFINITION 4.1. (i) A map F is called pseudo-monotone on K if, for every pair of

distinct points x, y , we have

y − x, F (x ) ≥ 0 = y − x, F (y) ≥ 0.

(iii) A map F is strongly pseudo-monotone on K if, there exists η > 0 such that, for

every pair of distinct points x, y , we have

y − x, F (x ) ≥ 0 = y − x, F (y) ≥ η|| y − x || 2.

(iv) A map F is quasi-monotone on K if for every distinct pair x, y we have

y − x, F (x ) > 0 = y − x, F (y) ≥ 0.

These classes of mappings are distinct (see [Ka]). We now present the dynamics results for

equilibria of projected systems under pseudo- and quasi-monotonicity conditions. First we

use the notion of monotone attractor. This notion is different than the one of an attractor,

as used in the classical theory of dynamical systems. We recall that a compact set D X iscalled an attractor for the projected dynamical system (3) if there exists a neighbourhood

V of the set D and T > 0 such that x (T ) V , for each x V and x(t ) → D , as t → ∞ for

each x V .

DEFINITION 4.2. An equilibrium x of (3) is a monotone attractor if δ > 0 such

that, x B (x , δ ) and x(t ) the unique solution of (3) starting at the point x, the function

d(x, t ) := ||x (t ) − x || is non-increasing as a function of t. The point x is a global

monotone attractor if all the above is true for any x K .

REMARK 4.1. Denition 4.2 is used in [Is-C1], [Is-C3], [Na6], [Na7] and [P] as well. We

note that an equilibrium x may be a monotone attractor, but not necessarily an attractor.

The rst result characterizes the behaviour of equilibria of (3), located both in the interior

of the set and on the boundary. It is easy to see that a strict equilibrium (see the denition




below) can only exist on ∂K , but it is interesting to notice that under different conditions

they display the same behaviour:

THEOREM 4.1. Let x be an equilibrium of (3).

(i) If F is pseudo-monotone on K, then x is a global monotone attractor.

(ii) If F is quasi-monotone on K and x is a strict equilibrium (i.e. F (x , y − x > 0) ,

then x is a global monotone attractor.

This is proved by studying the function D (t ) := 12

|| x (t ) − x || 2, with x(0) K .

Another formulation for a variational inequality problem (VIP), was introduced by G.

Minty. Given a closed, convex subset K in a Hilbert space X and a mapping F : K → X ,

a variational inequality problem in the sense of Minty (MVIP) is to nd x K such thatF (y), x − y ≤ 0 for all y K . There exists a relation between VIP and MVIP.

THEOREM 4.2. (i) If F is continuous on nite dimensional subspaces and x is a solution

to the MVIP, then it is a solution to the VIP.

(ii) If F is pseudo-monotone and x is a solution to the VIP, then it is a solution to the

MVIP.

Proof. A proof of this result can be found in [K-S] Chapter III, Lemma 1.5.

We can characterize the set of solutions to MVIP.

THEOREM 4.3. The set of global monotone attractors for the projected system (3) coin-

cides with the set of solutions of the MVIP associated to the same system.

Proof. We use the same idea as in the theorem above, i.e. we use the fact that

D (t ) := ddt (

12

|| x (t ) − x || 2) ≤ 0, with x(0) K .

DEFINITION 4.3. Let x K be an equilibrium of the system (3) and let x(t ) denote

the solution with x(0) = x K .

(i) The point x is exponentially stable if there exists δ > 0 and B > 0, µ > 0 constants

such that x B (x , δ ) and t ≥ 0 we have ||x (t ) − x || ≤ B || x − x || e− tµ .

(ii) The point x is a nite-time attractor if there exists δ > 0 such that for any x

B (x , δ ), there exists some moment of time T := T (x ) < ∞ such that x(t ) = x for any

t ≥ T (x ).




(iii) The point x is globally exponentially stable or globally nite-time attractor

if (i) or, respectively (ii), takes place for any x K .

We nally state other two results, their proof being based on the same idea as before.

THEOREM 4.4. Suppose that x is an equilibrium of the system (3). If F is strongly

pseudo-monotone on K , then x is globally exponentially stable and is therefore a global

attractor.

DEFINITION 4.4. A map F is called strongly pseudo-monotone with degree α on

K if, there exists η > 0 such that, for every pair of distinct points x, y , we have

y − x, F (x ) ≥ 0 = y − x, F (y) ≥ η|| y − x || α .

THEOREM 4.5. Suppose that x is an equilibrium of (3). If F is strongly pseudo-

monotone with degree α < 2 on K , then x is a globally nite-time attractor.

5. Summary and future work

We presented here in brief the development of the theory of projected dynamical systems,

from Euclidean space to arbitrary Hilbert spaces. We used elements from various areas of Mathematics (Set-valued Mappings, Differential Inclusions, Functional and Convex Analy-

sis) to obtain results about equilibria of some applied problems. We showed the existence

of equilibria through the theory of variational inequalities, we showed the existence of pro-

jected systems on Hilbert spaces and we studied the dynamics of the problems around these

equilibrium points through the projected dynamical systems theory.

This thesis is intended to be a starting point for future work on projected dynamical

systems. There are a few directions that can be followed and explored, based on the

research developed in this thesis. One such direction could consist of studying applications

to this theory and nding relevant interpretations of the mathematical results.

The theory of projected dynamics was born as a result of applied problems and it has

its roots and motivation in Applied Mathematics. However, it is our opinion that a pro-

jected dynamical system is a very rich topic in Pure Mathematics as well, considering the

interesting blend of mathematical notions involved in proving its existence.




References

[A-C] AUBIN, J. P. and CELLINA, A., Differential Inclusions , Springer-Verlag, Berlin (1984).

[B-C] BAIOCCHI, C. and CAPELO, A., Variational And Quasivariational Inequalities. Appli-

cartions to Free Boundary Problems. , J. Wiley and Sons, (1984).

[B] BAZARAA, M. S. and SHETTY, C. M., Nonlinear Programming , J. Wiley and Sons, (1979).

[C-J] COJOCARU, M. G. and JONKER, L. B., Projected differential equations in Hilbert spaces , to

appear in Proc. Am. Math. Soc (2002).

[CO] CORNET, B., Existence of slow solutions for a class of differential inclusions , J. Math. Anal.

Appl., 96(1983), 130-174.

[C] COTTLE, R., Notes on a fundamental theorem in quadratic programming , SIAM J. Appl. Math.,

Vol. 12(1964), 663-665.

[C-Y] COTTLE, R. W. and YAO, J. C., Pseudo-monotone complementarity problems in Hilbert space ,

J. Opt. Theory Appl., Vol. 75, No. 2.(1992), 281-295.

[Da1] DANIILIDIS, A. and HADJISAVAS, N., On the subdifferentials of quasiconvex and pseudoconvex

functions and cyclic monotonicity , J. Math. Anal. Appl., 237 (1999), 30-42.

[Da2] DANIILIDIS, A. and HADJISAVAS, N., Characterization of nonsmooth semistricly quasiconvex

and stricly quasiconvex functions , J. Opt. Theory Appl., Vol. 102, No. 3(1999), 525-536.

[D-S] DUNFORD, N. and SCHWARTZ, J. T., Linear operators , Wiley Classics Library Ed., New

York: Wiley-Interscience (1988).

[Du] DUPUIS, P., Large deviations analysis of reected diffusions and constrained stochastics approxi-

mation algorithms in convex sets , Stochastics, Vol. 21(1987), 63-96.

[D-I] DUPUIS, P. and ISHII, H., On Lipschitz continuity of the solution mapping to the Skorokhod

problem, with applications , Stochastics and Stochastics Reports, Vol. 35(1990), 31-62.




[D-N] DUPUIS, P. and NAGURNEY, A., Dynamical systems and variational inequalities , Annals of

Operations Research 44, (1993), 9-42.

[H-S] HADJISAVAS, N. and SCHAIBLE, S., Quasimonotone variational inequalities in Banach spaces ,

J. Opt. Theory Appl., Vol. 90, No. 1(1996), 95-111.

[H-P] HARKER, P. and PANG, J., Finite-dimensional variational inequality and nonlinear comple-

mentarity problems: a survey of theory, algorithms and aplications , Mathematical Programming,

48(1990), 161-220.

[Hk] HEIKKILAA, S., Monotonone Iterative Techniques for Discontinuous Nonlinear Dif-

ferential Equations , Monographs and Textbooks in Pure and Applied Mathematics, 181, M.

Dekker (1994).

[He] HENRY, C., An existence theorem for a class of diffferential equations with multivalued right hand

sides , J. Math. Anal. Appl. 41 (1973), 179-186.

[Hi] HIPFEL, D., The Nonlinear Differential Complementarity Problem , Ph. D. Thesis, Rensselaer

Polytechnic Institute, (1993).

[I-R] HYERS, D. H., ISAC G. and RASSIAS, TH. M., Topics in Nonlinear Analysis and Appli-

cations , World Scientic (1997).

[Is1] ISAC, G., Complementarity Problems , Lecture Notes in Math. Springer-Verlag No.1528 (1992).

[Is2] ISAC, G., Topological Methods in Complementarity Theory , Kluwer Academic Publishers,

(2000).

[Is3] ISAC, G., A generalization of Karamardian’s condition in complementarity theory , Nonlinear

Analysis Forum, 4 (1999), 49-63

[Is4] ISAC, G., On the solvability of multi-valued complementarity problem: A topological method , in:

Fourth European Workshop on Fuzzy Decision Analysis and Recognition Technology(EFDAN’99) , June 14-15(1999), Germany (Ed. R. Felix), 51-66.




[Is5] ISAC, G., Leray-Schauder type alternatives and the solvability of complementarity problems , to

appear in: Topological Methods in Nonlinear Analisys

[Is7] ISAC, G., Exceptional family of elements, feasability and complementarity , J. Opt. Theory Appl.,

104, No. 3(2000), 577-588

[Is-C1] ISAC, G. and COJOCARU, M. G., The projection operator in a Hilbert space and its directional

derivative. Consequences for the theory of projected dynamical systems , Preprint (2002).

[Is-C2] ISAC, G. and COJOCARU, M. G., Functions without exceptional family of elements and the

solvability of variational inequalities on unbounded sets , To appear in Topological Methods in

Nonlinear Analysis, (2002).

[Is-C3] ISAC, G. and COJOCARU, M. G., Variational inequalities, complementarity problems and

pseudo-monotonicity. Dynamical aspects , Seminar on xed point theory Cluj-Napoca, Vol. 3,

(2002).

[Ka] KARAMARDIAN, S. and SCHAIBLE, S., Seven kinds of monotone maps , J. Opt. Theory Appl.,

Vol. 66, No. 1(1990), 37-46.

[K-S] KINDERLEHRER, D and STAMPACCHIA, G., An Introduction to Variational Inequalities

and Their Applications , Academic Press (1980).

[K-Z] KRASNOSELSKII, M. A. and ZABREIKO, P. P., Geometrical Methods of Nonlinear

Analysis , Springer-Verlag, A Series of Comprehensive Studies in Mathematics, Vol. 263(1984).

[La] LADAS, G. E. and LAKSHMIKANTHAM, V., Differential Equations in Abstract Spaces ,

Academic Press, (1972).

[Ll] LLOYD, N. G., Degree Theory , Cambridge University Press, (1978).

[Lu] LUC, D. T., Existence results for densely pseudomonotone variational inequalities , J. Math. Anal.

Appl. 254(2001), 291-308.

[M1] MINTY, G. J., Monotone (non-linear) operators in Hilbert space , Duke Math. J., 29(1962), 341-346.




[M2] MINTY, G. J., On variational inequalities for monotone operators , Advances in Mathematics

30(1978), 1-7.

[Na1] NAGURNEY, A., Network Economics. A Variational Inequality Approach , Kluwer

Academic Publishers, (1993).

[Na2] NAGURNEY, A., DUPUIS P. and ZHANG, D, A dynamical systems approach for network

oligopolies and variational inequalities , Annals of Regional Science 28, (1994), 263-283.

[Na3] NAGURNEY, A. and SIOKOS, S., Financial Networks: Statics and Dynamics , Springer-

Verlag (1997)

[Na4] NAGURNEY, A., PAN, J. and ZHAO, L, Human migration network , European Journal of Operational Research, Vol. 59, No. 3(1992), 262-274.

[Na5] NAGURNEY, A., TAKAYAMA, T. and ZHANG, D., Projected dynamical systems modelling and

computation of spatial networks equilibria , Networks, Vol. 26(1995), 69-85.

[Na6] NAGURNEY, A. and ZHANG, D., Projected Dynamical Systems and Variational In-

equalities with Applications , Kluwer Academic Publishers (1996).

[Na7] NAGURNEY, A. and ZHANG, D., Stability of spatial price equilibrium modeled as a projected

dynamical system , Journal of Economic Dynamics and Control 20, (1996), 43-63.

[P] PAPPALARDO, M. and PASSACANTANDO, M., Stability for equilibrium problems: from

variational inequalities to dynamical systems , To appear in J. Opt. Theory and Appl. (2002).

[R-W] ROCKAFELLAR, R. T. and WETS, R. J-B., Variational Analysis , Springer-Verlag, A Series

of Comprehensive Studies in Mathematics, Vol. 317(1998).

[S] SHAPIRO, A. S., Existence and differentiability of metric projections in Hilbert spaces , SIAM J.

Optimization 4(1994), 130-141.

[Sw] SWARTZ, C., An Introduction To Functional Analysis , Pure and Applied Mathematics

Series, 154, M. Dekker (1992).




[W] WEIBULL, J. W., Evolutionary Game Theory , The MIT Press, Cambridge, Massachusets

(1997).

[Y] YAO, J.C., Variational inequalities with generalized monotone operators , Mathematics of Opera-

tions Research, No. 3(1994), 691-705.

[Z] ZARANTONELLO, E., Projections on convex sets in Hilbert space and spectral theory , Contri-

butions to Nonlinear Functional Analysis, Publ. No. 27, Math. Res. Center, Univ. Wisconsin,

Academic Press (1971), 237-424.

Department of Mathematics and Statistics, Jeffery Hall, room 229, Queen’s University,

Kingston, Ontario, Canada

E-mail address : [email protected]

projected dynamical systems on hilbert sps

Documents