Observation stability and convergence for neural-type evolutionary variational inequalities Текст научной статьи по специальности «Математика»

r

dx dt

DIFFERENTIAL EQUATIONS AND

CONTROL PROCESSES N. 2, 2021 Electronic Journal, reg. N <S>C77-39410 at 15.04.2010 ISSN 1817-2172

http://dijfjournal.spbu.ru/ e-mail: [email protected]

Control problems in nonlinear systems

Observation stability and convergence for neural-type evolutionary variational inequalities

Stefan Reitmann Freiberg University of Mining and Technology [email protected]

Bernhard Jung Institute of Computer Science , Germany [email protected] Elena V. Kudryashova St. Petersburg State University, Department of Mathematics and Mechanics

[email protected] Volker Reitmann

St. Petersburg State University, Department of Mathematics and Mechanics

[email protected]

Abstract.We derive absolute observation stability and instability results for controlled evolutionary inequalities which are based on frequency-domain characteristics of the linear part of the inequalities.The uncertainty parts of the inequalities (nonlinearities which represent external forces and constitutive laws) are described by certain local and integral quadratic constraints. Other terms in the considered evolutionary inequalities represent contact-type properties of a mechanical system with dry friction. The absolute stability criteria with respect to a class of observation operators (or measurement operators) gives the opportunity to prove the weak convergence of arbitrary solutions of inequalities to their stationary sets. In particular the obtained results can be used for the investigation of continuum type memories in neural networks. In such a way we can introduce a new type of neurons with hys-teretic nonlinearities which are described by evolutionary variational inequalities.

Keywords: Absolute observation stability, evolutionary variational inequalities, frequency-domain conditions,hysteretic neural networks

1 Introduction

In [28] we have presented an approach which brings continuum type memories and neural networks together. In particular hysteretic nonlinearities were used to introduce a boundary control problem for the interaction of acoustic waves or lasers with some control processes on the boundary described by an ODE. In the present paper we investigate similar questions such as stability or convergence to the stationary set for systems given by an evolutionary variational inequality and a distributed control which acts inside and not on the boundary. It is assume that the system nonlinearities are multivalued functions or hysteresis operators. As an input the variational inequality contains a time-dependent function which can be considered as perturbation or measurement function. As an output signal we consider not the whole solution but some transformation of the solution and the nonlinearity. This approach is typical for absolute stability theory and was firstly discussed by V.A. Yakubovich for different classes of systems. With respect to this output stability and optimization problem can be considered. In a forthcoming paper we will investigate adaption and learning principles in the system based on output functionals. For implementation of the proposed methods we consider to resort to both real and synthetic data of acoustic wave and laser based depth sensors, which we virtually simulated in [29]. Let us note that in the present paper we continue the investigation of control systems which are related to [14, 16, 27].

Our plan for the paper is the following. Some basic notations for evolutionary systems in Hilbert spaces is introduced in Section 2. In Section 3 we shortly consider the general theory of evolutionary variational inequalities. Some examples of evolutionary variational inequalities with hysteresis operator and viscoelastic contact part are discussed in Section 4. In Section 5 we investigate the dissipativity of variational inequalities based on the Kalman-Yakubovich-Popov lemma. The main topic of the last Section 6 is the derivation of the efficient observation and frequency-domain conditions for the convergence of the trajectories of a variational inequality to the stationary set. In particular we consider a stationary set wich consists of a continuum of points. It is clear that for neurons which are described such an evolutionary variational inequality this property is closely connected with the construction of a memory by means of the stationary set.

2 Basic notation

Suppose that Fq is a real Hilbert space. We denote by (•, )o and || • ||o the scalar product resp. the norm on Fq. Let A : D(A) ^ Fq be the generator of a Co-semigroup on Fq and define the set Fi := D (A). Here D (A) is the domain of A, which is dense in Fq since A is a generator. We denote with p (A) the resolvent set of A. The spectrum of A, which is the complement of p (A), is denoted by o(A). If we define with an arbitrary but fixed ft e p (A) n R for any y, n e F1, the value

(y, n)i:=((ftI _ A)y, (ftI _ A)n)q , (2.1)

then the set F1 equipped with this scalar product (•, )1 and the corresponding norm || • |1 becomes a Hilbert space (different numbers ft give different but equivalent norms). Denote by F-1 the Hilbert space which is the completion of Fq with respect to the norm

||y|_1 := ||(ftI _ A)-1y||o and which has the corresponding scalar product

(y, n)_1 := ((ftI_A)—1y, (ftI_A)-1 n)q , Vy, n e F_1. (2.2)

Thus, we get the inclusions F1 c Fq c F_1, which are dense with continuous embedding, i.e., Fa c Fa_1, a = 1,0, is dense and ||y||a_1 < c||y||a, Vy e Fa. Sometimes ([1, 2])the introduced triple of spaces (F1,Fq,F_1) is called a Gelfand triple . The pair (F1, F_1) is also called Hilbert rigging of the pivot space Fq, F1 is an interpolation space of Fq, and F_1 is an extrapolation space of Fq ([5, 33]). Since for any y e Fq and u e F1 we have

|(y,u)o| = |(ftI_A)_1y,((ftI_A)u)o| < ||y|_1|u||1, (2.3)

we can extend (•, u )q by continuity onto F_1 obtaining the inequality

|(y,u)o| < ||y||_1||u||1, Vy e F_1,Vu e Fb

Let us denote this extension also by (•, )_1,1 and call it duality product on F_1 x F1. The operator A has a unique extension to an operator in L(Fq, F_1) which we denote by the same symbol. Suppose now that T > 0 is arbitrary and define the norm for Bochner measurable functions in L2(0, T; Fj) (j = 1,0, _1) through

/ T \ 1/2

||y(-)||2,j := yj ||y(t)H2jdt) . (2.4)

Let WT be the space of functions such that y e L2(0, T; Y1) and 3) e L2(0, T; Y-1), where the time derivative 3) is understood in the sense of distributions with values in a Hilbert space. The space WT equipped with the norm

IMIwt := (I|y(0ll2,i + Il3)(^)y2>-i)1/2 (2.5)

is a Hilbert space and will be used for the description of solutions to evolutionary systems.

Remark 2.1 a) Let us denote by CT := C(0, T; Yo) the Banach space of continuous mappings y : [0, T] ^ Yo provided with the norm

Mlkr = sup yy(t)|0.

t e[0,T ]

It is well known ([5, 24, 33]) that WT can be continuously embedded into the space CT, i.e., every function from WT, properly altered by some set of measure zero, is a continuous function y : [0, T] ^ Y0 and ||y(-)||CT < const • ||y(-)||LT. This embedding property shows that the Cauchy problem (3.1), (3.2) in the next section can be considered.

b) Since the infinite-dimensional evolutionary problems which are investigated in this paper with the help of observation operators are assumed to have uncertain parts, the choice of physically motivated phase spaces, control spaces, and observation spaces (see Sect. 3) is crucial for the measurement process. It is easy to show that in some norms the observation operator can be stable, whereas it may be unstable in other norms. In many cases the described Gelfand triple of spaces is not sufficient for this and we need a Hilbert scale {Ya}aeR of spaces. For arbitrary a > 0 we define (with A and 0 as in (2.1) and (2.2)) the space Ya := D((£I - A)a) and the scalar product in Y

a

(y, n)a :=((01 - A)ay, (01 - A)an)0, Vy, n e Ya . (2.6)

When a < 0, the scalar product is also introduced by formula (2.6) and the space Ya is, as Y-1, obtained by the completion of Y0 with respect to their norm. Furthermore,

for any a, 0, y e R with a e (0, Y) the pair (Yy, Yp) is a Hilbert rigging of Ya ([5]).

□

3 Evolutionary variational inequalities

Suppose that T > 0 is arbitrary and consider for a.a. t e [0, T] the observed and controlled evolutionary variational inequality

(y - Ay - Bu - f(t), n - y)-1,1 + W(n) - W(y) > 0, Vn e Y (3.1) y(0) = yo e Yo ,

z(t) = Cy(t), u(t) e 0(t,z(t)), u(0) = u0 e E(y0), (3.2)

u(t)= Dy(t)+ Eu(t). (3.3)

In (3.1) - (3.3) it is supposed that C e L(Y-1,Z), D e L(Y1,T) and E e L(U,T) are linear operators, U, Z and T are real Hilbert spaces, Y1 c Y0 C Y-1 is a real Gelfand triple and A e L(Y0, Y-1), B e L(U, Y-1), 0 : R+ x Z ^ 2U is a set-valued map, W : Y1 ^ R+ and f : R+ ^ Y-1 are given nonlinear maps. The calculation of u(t) in (3.2) shows that this value in general also depends on certain "initial state" u00 of 0 taken from a set E (y0) C U. This situation is typical for hysteresis nonlineari-ties. We call B control operator, C output operator, D and E observation operators, U control space, Z output space, T observation space, 0 (material law) nonlinearity, W (contact) functional and f forcing function (see Fig. 1).

Figure 1: State / linear output / nonlinear output /

_observation diagram_

Electronic Journal. http://diffjournal.spbu.ru/ 130

In the following we denote by || • ||u, || • ||Z and || • ||Y the norm in U,Z resp. Y. Let us now introduce the solution space for the problem (3.1), (3.2).

Definition 3.1 Any pair of functions {y (•), u(-)} with y e WT and u e L^oc(0, U) such that Bu e WT , satisfying (2.1), (2.2) almost everywhere on (0, T), is called solution of the Cauchy problem y (0) = y0, u(0) = u0 defined for (2.1), (2.2).

In order to have an existence property for (2.1) — (2.3) we state the following assumption:

(C1) The Cauchy-problem (2.1), (2.2) has for arbitrary y0 e Y0 and u00 e E (y0) C U at least one solution {y (•), u (•)}.

Assumption (C1) is fulfilled, for example, in the following situation ([26]).

(C2) a) The nonlinearity 0 : R+ x Z ^ U is a function having the property that A(t) := —A — B0 (t, C ) : Y1 ^ Y— is a family of monotone hemicontinuous operators such that the inequality

||A(t)y11—1 < c 11y11 + c2 , Vy e Y1,

is satisfied, where c1 > 0 and c2 e R are constants not depending on t e [0, T]. Furthermore for any y e Y1 and for any bounded set U C Y1 the family of functions {(A(t)n,y)—1,1, n e U} is equicontinuous with respect to t on any compact subin-terval of R+.

b) y is a proper, convex, and semicontinuous from below function on D(y) C Y1.

(C3) f e LfoC(R+; Y-1).

Under the assumptions (C2) — (C3) the following theorem of H. Brezis ([3]) holds:

Theorem 3.1 Suppose that for the family of operators A (t), t e R+, we have

(A(t)y,y)—1,1 > a||y| 1 + 0 , Vy e Y1 , where a > 0 and 0 e R do not depend on t. Then for arbitrary f e L2oc(R+; Y—1)

_yq

and arbitrary y0 e D(y) (i.e., the closure in Y0) there exists a unique weak solution

y e Lioc(R+; Y1) n C(R+; Y0) with y(0) = y0, satisfying the inequality

(n(T) - A(t)y(T) - f (t), n(T) -y(T))-1,1 + W(n(T)) - W(y(T))] dt >

1 2 1 2

2lln(t) -y(t)|2 - 2lln(s) -y(s)^1,

Vs,t: 0 < s < t, Vn e {n e LjJC(R+; Y1): n e L?oc(R+; Y-1)} . Furthermore, the solution satisfies the inequalities

||y||L2(0,T;Y1) < c1 (yf ||L2(0,T;Y_1) , M^) ,

||y||C(0,T;Y0) < c2 (y f ||L2(0,T;Y_1^ W^) ,

where c\ : R+ x R+ ^ R+ , i = 1,2, are continuous and monotonically increasing functions.

(C4) In the sequel we consider only solutions y of (3.1),(3.2) for which y belongs

2 i loc(

to L2oc(R; Y-i).

4 Examples of evolutionary variational inequalities

Note that in the special case when w = 0 in (3.1) the evolutionary variational inequality is equivalent for a.a. t e [0, T] to the equation

y = Ay + Bu + f (t) in Y-1,

y(0) = Y0, z(t) = Cy(t), u(t) e 0(t,z(t)), u(0) e E^0), u (t) = Dy(t)+ Eu(t).

Under the assumption that 0 is a single valued function this class was considered in [1,2, 4, 5, 6, 8, 18, 20,31,36].

A typical frictional contact problem is modeled by the following second-order evolutionary variational inequality (Duvant, Lions [9], Han, Sofonea [12], Jarucek, Eck [13]): Find a displacement function v such that for a.a. t e [0, T]

(v(t), w - i>(t))v_1,r1 + (A v(t), w - v(t ))v-1,v1 + (g(v(t)),w-v(t)) + j(w) - j(v(t)) > 0, V w e V , (4.1)

V / V-1, V1

v(0) = V0 e V, v(0) = w0 e V. (4.2)

Here V C V C V—1 is a Hilbert space rigging structure, A : V ^ V—1 is a linear continuous operator which is called viscosity operator.

The nonlinear map g : V ^ is the elasticity operator and j : V ^ R+ represents the contact functional.

Under a solution v of (4.1), (4.2) on (0, T) we understand a function v( ) e L2(0, T; V) such that v(-) e L2(0, T; V1),V(0 e L2(0, T; V, /0T j(v(T))dT < ~, and (4.1), (4.2) is satisfied for a.a. t e (0, T).

Let us assume that for any (v0, w0) e V x V and any time T > 0 a solution of (4.1), (4.2) exists. In order to rewrite (4.1), (4.2) as a first-order variational inequality (2.1), (2.2) we define the product Hilbert space rigging structure Y1 C Y0 C Y—1 with

Y0 = V x V, Y1 = V x V, Y—1 = V0 x V—1.

Let us introduce the new variables y1 = v, y2 = v and n2 = w. It follows that y1 = y2 and y2 = v. In this notation the variational inequality (4.1) can be rewritten as

(y 2, n 2 — y 2)V—1,V + (A y 2, n 2 — y 2)V—1,V1 + (g(Уl), n 2 — y 2) V—1,V1 +j(n2) — j(y2) > 0, Vn2 e V,

Using the product topology we get for arbitrary y = (y1, y2) e Y—1 = V0 x and n = (n1, n2) e Y1 = x V the representation of the duality pairing on Y—1 x Y1

as

(y, n )—1,1 = (y1, m)Vi + (y2, n2)r_1,V1. (4.3)

It follows from (4.3) that

(y2, n2 — y2)V—1,V1 = C^ n — y) — 1,1 — (У2, n1 — y1)V1 .

A linear bounded operator A : Y1 ^ Y—1 is defined by

(—Ay, n — y)—1,1 = —(y2, n1 — y1)V + (Ay2, n2 — y2)r_1,V ,

Vy = (y 1,y2), n = (n1, n2) e Y1 = V1 x V . (4.4)

It is easy to see that A defined by (4.4) has the representation

A = 0 I 0A

In order to determine the linear operator B : U = V ^ Y_ 1 we use the equation

(_B0 (yi), n - y)-1,1 = (0 (yi), n2 - y2)V-i,Vi,

vy = (yi,y2), n = (ni, n2) e Y = Vi X Vi.

(4.5)

From (4.5) it follows that

B0 (Cy) =

where the linear operator C : Y1 ^ W := V is defined by (y1,y2) ^ y1.

0

-g(yi)

The last remainig element in the inequality (2.1) is the contact functional W: Y1 ^ R+ given by

W(y) := j*(y2), V (y1,y2) e Y1 = V x V.

An important class of uncertainty systems (3.1), (3.2) are connected with hysteresis operators. Let us describe one example of this type which in more details is discussed in [35]. Let us assume that ye and yr are maximal monotone multivalued functions R ^ 2R , such that inf yr(v) < sup ye(v) V v e R, and define the maximal monotone multivalued function 0 as in Fig. 2.

z

0 = R _ 0 =

A = {0} v

= R+

Figure 2: Generalized play operator

The functions ye and yr define a hysteresis which is called generalized play operator. This operator can be set in the form of a differential inclusion as in system

(4.6)

(3.2) by z e 0 (v, z) in [0, T], which is equivalent to the variational inequality ([35])

v e J(z) := [inf Y—1(z), sup7—1(z)], Z(v — w) > 0 Vw e J(z). By coupling this operator with a PDE, we get the problem

dft + u — Av = f in a x (0, T),

§ — u = 0 in a x (0, t) ,

u e 0 (v, z) in a x (0, T),

v(-, 0) = v0 , z(-, 0) = z0 in a . Here it is assumed that a C Rn is smooth, T > 0 ,

f e L2 (0, T;H—1(a)) and v0,z0 e L2(a).

By applying standard results of the theory of nonlinear semigroups it is shown in [35] that under certain additionally conditions the Cauchy-problem (4.6) has one and only one solution in the sense of Benilan. This solution depends continuously and monotonically on the data v0, z0 and f. It is also demonstrated in [35] that the shortly characterized approach can be extended to generalized Prandtl-Ishlinskij operators ofplay type which are used for the description of elasto-plastic material laws.

5 Dissipativity with respect to observations

Our aim is to investigate the properties of the inequality (3.1), (3.2) using only informations (measurements) from the observation operator (3.3). The concrete nonlin-earities (contact laws and material laws) are assumed to be unknown. We consider them as uncertainties in the system. In order to describe the principal parts of such an uncertain dynamic system we use methods from absolute stability theory. The main idea is to characterize a priori information about the considered class of non-linearities by means of suitably chosen quadratic forms. These quadratic forms, together with the description of the linear parts of system (3.1) — (3.3) in form of transfer functions "from an input to an output", give the necessary information for the construction of observers which are stable with respect to initial conditions in the considered class (3.1) — (3.3). The formal definition of the uncertainty parts of (3.1), (3.2) is the following one (see also [4, 18, 19, 20, 21]).

Definition 5.1 a) Suppose F and G are quadratic forms on Y\ x U. The class of nonlinearities N(F, G) defined by F and G consists of all maps 0 : R+ x Z ^ 2U such that for any y(-) £ LOc(0, o Yi) with y(^) £ LOc(0, o Y-i) and any u(-) £ Lj2oc(0, o U) with u(t) £ 0 (t, Cy(t)) for a.e. t > 0, it follows that F (y(t), u(t)) > 0

for a.e. t > 0 and (for any such pair {y, u}) there exists a continuous functional

t

O : Z ^ R such that for any times 0 < s < twe have J G (y(T), u(t ))d t > O(Cy(t)) -

O(Cy(s)) . s

b) The class of functionals M(d) defined by a constant d > 0 consists of all maps w : Yi ^ R+ such that for any y £ L2oc(0, O Y0) with y £ £^(0, 0 Yi) the

00

function t ^ w (y(t)) belongs to Li(0,0; R) satisfying J w (y(t ))dt < d and for any

0

0 £ N(F, G) and any w £ M(d) the Cauchy-problem (3.1) — (3.3) has a solution {y(-), u(^)} on any time interval [0, T].

Remark 5.1 The functional O used in the description of the class N(F, G) can be considered as generalized potential of 0 ([2i]). Let us assume for a moment that Z = U and 0 : Z ^ Z is a continuous nonlinear mapping of gradient or potential type, i.e., there exists a continuous Frechet-differentiable nonlinear functional O : Z ^ R, whose Frechet derivative O (z) £ L(Z, R) at any z £ Z can be represented

in the form O (z)n = (0 (z), n) for any n £ Z. Then we can write for any function

1 2

z £ Z[o'c (0, +0; Z) and any times 0 < s < t the path integral formula

f (z(t ), 0 (z(t ))Zd t = O(z(t)) — O(z(s)).

s

It follows in this case that the properties described in Definition 5.1 a) are satisfied with respect to the quadratic form G if we take (with Z = U) the form

G(y, u) := (CAy + CBu, u)u . (5.1)

□

A basic tool for the derivation of sufficient conditions for absolute stability and instability of observations to variational inequalities is the following version of the Frequency Domain Theorem or Kalman-Yakubovich-Popov lemma (KYP lemma) ([4, 6, 18, 19, 23, 36]). In the infinite dimensional setting certain regularity assumptions are necessary which we formulate at the beginning of this section. In the next part of the paper up to theorem 3.1 it is assumed that all spaces and operators are complex.

(F1) The operator A e L (Y1, Y—1) is regular ([9, 19, 22]), i.e., for any T > 0, y0 e Y1, e Y1 and f e L2(0, T; Y0) the solutions of the direct problem

y = Ay + f(t), y(0) = y0, a.a. t e [0, T]

and of the dual problem

^ = —A*^+f (t), ^(T) = ^t , a.a. t e [0,T]

are strongly continuous in t in the norm of Y1. Here (and in the following) A* e L(Y—1, Y0) denotes the adjoint to A, i.e., (Ay, n )—1,1 = (y, A* n )—1,1, Vy, n e Y1.

Remark 5.2 The assumption (F1) is satisfied ([19]) if the embedding Y1 C Y0 is

completely continuous, i.e., transforms bounded sets from Y1 into compact sets in Y0.

□

(F2) The pair (A, B) is L2-controllable, ([4, 18, 19]) i.e., for arbitrary y0 e Y0 there exists a control u( ) e L2(0, U) such that the problem

y = Ay + Bu, y(0) = y0

is well-posed on the semiaxis [0, +») , i.e., there exists a solution y( ) e L» with

y (0)= y0.

Remark 5.3 It is easy to see that a pair (A, B) is L2-controllable if this pair is exponentially stabilizable, i.e., if an operator K e L(Y0, U) exists such that the solution y( ) of the Cauchy-problem y = (A + BK) y, y(0) = y0, decreases exponentially as t ^ », i.e.,

3 c > 0 3 e > 0: ||y(t)||0 < ce—£i ||y0||0, Vt > 0.

Note that the usual condition of exact controllability (on any finite time interval) is in

general not satisfied for pairs of operators (A, B) arising from PDE problems ([34]).

□

(F3) Let F(y, u) be a Hermitian form on Y1 x U, i.e.,

F (y, u) = (F1y, y)—1,1 + 2 Re (F2y, u)u + (F3 u, u)u,

where

F1 = F1* e L(Y1, Y—1), F2 e L(Y0, U), F3 = F3* e L(U, U). Define the parameter

a := sup (||y||1 + ||u||U)—1F(y,u),

® ,y,u

where the supremum is taken over all triples («, y, u) e R+ x Y1 x U such that io>y = Ay + Bu.

Theorem 5.1 a) (Frequency Theorem for the Nonsingular Case, ([19]) Assume for the linear operators A e L (Y1, Y—), B e L (U, Y—) and the Hermitian form F on Y1 x U that the assumptions (F1), (F2) are satisfied. Then there exist an operator P = P* e L(Y—1, Y0) H L(Y0, Y1) and a number 5 > 0 such that

2Re (Ay + Bu,Py)—1,1 + F (y, u) <—5(||y||? + ||u||U), V(y, u) e Y1 x U, (5.2)

if and only if the frequency-domain condition a < 0 with a from (F3) is satisfied.

b) (Frequency Theorem for the Singular Case, [19]) Let the assumptions in Theorem 5.1a) be satisfied, and, in addition, let B e L (U, Y0). Then, for the existence of an operator P = P* e L (Y—1, Y0) H L (Y0, Y1) such that

Re (Ay + Bu, Py)—1,1 + F (y, u) < 0, V (y, u) e Y1 x U,

it is necessary and sufficient that the following two conditions are fulfilled:

1) a < 0 , where a is from (F3);

(x)

2) The functional J (y (•), u(-)) := J F (y (t ), u(t )) d T is bounded from above on any set My0 := {y(•), 0

u(^) : y = Ay + Buon R+,y(0) = y0, y(-) e L», u(^) e L2(0,»; U)} .

Remark 5.4 a) Let, in addition to the assumptions of Theorem 5.1b), A be the generator of a C0-group on Y0 and the pair (A, —B) be L2-controllable. Then the condition a < 0, where a is from (F3), is sufficient for the assertion of Theorem 5.1b) ([18, 19, 21]). Note that the existence of Q-groups is given for conservative wave equations, plate problems, and other important PDE classes ([10]).

b) Under the assumptions of Theorem 5.1 a) there exist operators

P = P* e L(Y0, Y0),L e L(U, Y0) and K = K* e L(U, U) such that

2Re (Ay + Bu,Py)—1,1 + F (y,u) = —||L*y — Ku||U, V(y,u) e Y1 x U. (5.3)

This equality is equivalent to the system of Lur'e or algebraic Riccati equations

A*P + PA + F1 = —LL*, PB + F2 = LK, F3 = —K*K. (5.4)

□

Let us state the following theorem which is based on the frequency-domain approach and which can be considered as generalization of energy-type equalities for PDE's. With the superscript c we denote the complexification of spaces and operators and the extension of quadratic forms to Hermitian forms.

Theorem 5.2 Consider the evolution problem (3.1) — (3.3) with 0 £ N(F, G) and W £ M (d). Suppose that for the operators Ac, Bc the assumptions (F1) and (F2) are satisfied. Suppose also that there exist an a > 0 such that with the transfer operator

X)(s) = Dc(slc — Ac)—1 Bc + Ec (s £ a(Ac)) (5.5)

the frequency-domain condition

Fc((itolc — Ac)—1Bcu,u)+ Gc((itolc — Ac)—1Bcu,u) < —a||X(u)(i«)u||2c

Vo £ R : io £ a (Ac), Vu £ Uc

(5.6)

is satisfied and the functional

00

J(y(-),u(0) := y [Fc(y(T),u(t)) + Gc(y(T),u(t)) + a||Dcy(T) + Ecu(t)|||c] dT 0

is bounded from above on any set My0 defined in Theorem 5.1b).

Then there exist an (real) operator P = P* £ L (Y—1, Y0) R L(Y0, Y1) and a number 5 > 0 such that with the Lyapunov-functional V (y) := (y, Py)0 (y £ Y0) for an arbitrary solution {y(-), u(^)} of (3.1), (3.2) with 0 £ N(F,G) and O as generalized potential of 0, the inequality

t

V (y(t)) — V (y(s)) + O (Cy(t)) — O (Cy(s))+ i F (y(T), u(T))dT

s

t

(5.7)

+ f [W (y(T)) — w(—Py(T)+ y(T))] dT + 5f ||Dy(T)+ Eu(t)|^ dT

ss < H (f (T ), Py(T )) —1,1d T

is satisfied on any time interval 0 < s < t.

Proof For the Hermitian form Fc + Gc all assumptions of the Frequency Theorem 5.1b) are satisfied. It follows ([19]) that there exist a (real) operator P = p* £ L(Y—1, Y0) R L(Y0, Y1) and a number 5 > 0 such that

(—Ay — Bu,Py)—1,1 > F(y, u)+ G(y, u) + 5||Dy + Eu|2, V(y, u) £ Y1 x U.

(5.8)

t

Consider the inequality (3.1) with the given solution {y( ),u( )}. Take the special test function Pn to see that on an arbitrary time interval [0, T]

(y(t) — Ay(t) — Bu(t) — f (t),Pn — y(t))—1,1 + ¥(Pn) — ¥(y) > 0, ^

V n e Y1 , a.a. t e [0, T]. Substitute now Pn (t) = —Py(t) + y(t). Then (5.9) can be written as

(y(t) — Ay(t) — Bu(t) — f (t), —Py(t))—1,1 + ¥( Py(t) + y(t)) — ¥(y(t)) > 0,

a.a. t e [0, T]

(5.10)

or (y(t),Py(t))—1,1 — (Ay(t) + Bu(t),Py(t))—1,1

— ¥( Py(t) + y(t)) + ¥(y(t)) < (f (t),Py(t))—1,1, a.a. t e [0, T].

(5.11)

Now we use the inequality (5.8) in order to estimate

— (Ay (t)+ Bu(t), Pu(t ))—11

, (5.12)

> F(y(t),u(t)) + G(y(t),u(t)) + 5||Dy(t) + Eu(t)||£, a.a.t e [0,T].

We get from (5.11) and (5.12)

(y(t), Py(t ))—1,1 + F (y(t), u(t))+ G (y(t), u(t))

+ ¥(y(t)) — ¥( Py(t) + y(t)) + 5||Dy(t) + Eu(t)||2 (5.13)

< (f (t),Py(t))—1,1, a.a.t e [0, T].

If we take in the inequality (5.13) the integral on an arbitrary time interval 0 < s < t we receive

t t V (y(t)) — V (y(s)) + | F (y (t ), u(T))dT + J G (y(T), u(r))dr

s s

t t

(5.14)

+ /- ¥(—Py(T) + y(T))]dT + sf ||Dy(T)+ Eu(r)y^dT

s s

< f (f (t ), Py(T))-1,1dT .

s

s

Now it remains to use the properties of G with respect to the solution {y( ),u( )}, i.e.,

t

JG(y(T),u(T))dt > O(Cy(t)) - 0(Cy(s)). (5.15)

s

From (5.14) and (5.15) the assertion of Theorem 5.2) follows immediately. ■

Remark 5.5 Inequality (5.7) can be considered ([5, 9, 19, 21, 31, 32, 37]) as generalized energy balance inequality or dissipation inequality. For a nonnegative operator P the term V(y(t)) in (5.7) is the energy stored in the state y(t). The form

—F which reflects the influence of the constitutive law can be considered as energy

t

supply rate and the integral — /F(y(T),u(t))dT as energy absorbed by the system

s

in the time interval [s, t]. A contact energy term ([9, 12, 13, 24, 30]) in (5.7) is characterized by w. n

The physical interpretation of inequality (5.7) leads to the following definition.

Definition 5.2 The Cauchy problem (3.1), (3.2) is called dissipative in the sense

of Willems ([37]) with respect to the observation u from (3.3) and the classes N (F, G) and M(d) if there exist a 5 > 0 and a bounded self-adjoint operator P = p* g L(Y_i, Y0) n L(Y0, Y\) such that for any solution {y(-), u(-)} of (3.1), (3.2) and any time interval 0 < s < t the inequality (5.7) holds.

In other words we get from Theorem 5.2 the following.

Corollary 5.1 Under the assumptions of Theorem 5.2 the Cauchy problem (3.1),

(3.2) is dissipative in the sense of Willems with respect to the observation u from

(3.3) and the classes N (F, G ) and M (d ).

6 Absolute observation - stability and instability of evolutionary inequalities

We continue the investigation of energy like properties for the observation operators from the inequality problem (3.1), (3.2) with f = 0.

The next definition generalizes the concepts which are introduced in [19, 20, 21, 38] for output operators of evolution equations, namely in extending them to the observation operators of a class of evolutionary variational inequalities. In the

following we denote for a function u ( ) £ L2 (R+; Y) their norm by

0

||u()112,y := / ||u(t)|ydt. 0

Definition 6.1 a) The inequality (3.1), (3.2) is said to be absolutely dichotomic (i.e., in the classes N(F,G),M(d)) with respect to the observation u from (3.3) if for any solution {y(-),u(-)} of (3.1), (3.2) with y(0) = y0, u(0) = u0 e E(y0) the following is true: Either y(-) is unbounded on [0,») in the Y{)-norm or y(-) is bounded in Y0 in this norm and there exist constants c1 and c2 (which depend only on A,B, N(F, G) and M(d)) such that

11Dy(•) + Eu(^)|2,T < c1 (|y0|2 + c2). (6.1)

b) The inequality (3.1), (3.2) is said to be absolutely stable with respect to the observation u from (3.3) if (6.1) holds for any solution {y(-), u(-)} of (3.1), (3.2).

c) The inequality (3.1), (3.2) is said to be absolutely unstable with respect to the observation u from (3.3) if for any constants c1 and c2 in (6.1) there exist initial states y0 e Y0 and u0 e E(y0) such that solutions {y(^), u(-)} of (3.1), (3.2) starting in these points do not satisfy (6.1).

Our first result concerns frequency-domain conditions for absolute observation dichotomy of controlled evolutionary variational inequalities.

Theorem 6.1 Suppose that the assumptions of Theorem 5.2 are satisfied with f = 0. Assume additionally that any potential O from the class N (F, G) is nonnegative. Suppose also that for any potential O there exists a constant c > 0 such that O(Cy) <

c ||y|2 , Vy e Y0 . Then the inequality (3.1), (3.2) is absolutely dichotomic with respect to the observation u from (3.3).

Proof As was shown in the proof of Theorem 5.2 under the assumptions of the present theorem there exist an operator P = P* e L(Y0, Y0) and a number 5 > 0 such that for an arbitrary solution {y( ), u( )} of (3.1), (3.2) the inequality (5.7) holds. Suppose that y( ) is bounded in Y0 on [0,»). Define the function

t

Z(t) : = V(y(t)) + O(Cy(t)) + /F(y(T),u(T))dT

0

+ f [¥(y(T)) — ¥(—Py(T)+ y(T))] dt. 0

From (5.7) it follows that for arbitrary s, t such that 0 < s < t

t

Z(t) — Z(s) <—5 J ||Dy(T)+ Eu(T)||tdt < 0, (6.2)

s

i.e., Z is monotonically decreasing. Since y( ) is bounded in Y\ on [0, the function Z( ) is bounded from below. It follows that there exists the limit lim Z(t) and that

for any t > 0 by (6.2) the inequality

8 ( ||Dy(T)+ Eu(t)||tdt < Z(0) - limZ(t) < c I|y0U0 Jo t ^^

is true. From this we get immediately that u =

Dy + Eu e L2(0, «>; T). Suppose now

t

that Z(t) ^ for t ^ Then from O(z) > 0, J F(y(T), u(t)) dt > 0 and the

0

t

boundedness of J[y(y(T)) - y(-Py(T) + y(T))]dT it follows that V(y(t)) ^ —~ 0

for t ^ From this we conclude that ||y(t) 110 ^ ®

In order to get absolute stability properties of (3.1), (3.2) with respect to the observation u of (3.3) we need an assumption for (3.1) — (3.3) which is called "minimal stability" for the class of evolution equations in [20, 21].

Definition 6.2 The inequality (3.1)—(3.3) with f = 0 is said to be minimally stable if the resulting equation for ^ = 0 is minimally stable, i.e., there exists a bounded linear operator K : Yi ^ U such that the operator A + BK is stable, i.e.,

a (A + BK) c {s e C : Re s < —£ < 0} with F (y, Ky) > 0 , Vy e Y1 , (6.3) t

and [G(y(T), Ky(T))dt > 0 , Vs,t :0 < s < t, Vy e L?oc(R+; Y1). (6.4)

Theorem 6.2 Suppose that the assumptions of Theorem 6.1 are satisfied and the inequality (3.1)-(3.3)withf = 0 is minimally stable, i.e., (6.3) and (6.4) are satisfied with some operator K e L(Y\, U). Suppose also that the pair (A + BK, D + EK) is observable in the sense of Kalman ([4]), i.e., for any solution y(-) of

y = (A + BK )y, y(0) = yo,

with u (t) = (D + EK )y (t) = 0 for a.a. t > 0 it follows that y(0) = y0 = 0.

Then inequality (3.1), (3.2) is absolutely stable with respect to the observation u from (3.3).

Proof Under the assumptions of the given theorem there exist by Theorem 5.1b) a (real) operator P = P* e L (Y-1, Y0) n L(Y0, Y1) and a number 5 > 0 such that the

inequality (5.8) is satisfied. Setting in (5.8) u = Ky from (6.3) with arbitrary y e Y1 we get with (6.4) the inequality

((A + BK)y, Py)-i,i <-8 ||Dy + EKy ||£, V y e Yi. (6.5)

Using the fact that A + BK is a stable operator and the pair (A + BK, D + EK) is observable, it follows ([4, 7, 36]) from (6.5) that P = P* > 0. Suppose now that {y( ),u( )} is an arbitrary solution of (3.1), (3.2) with f = 0. With the Lyapunov-functional

V(y) = (y, Py)o > 0 it follows from (5.7) that for arbitrary t > 0

t

-V(y0) - O(CV0) + yV(y(T)) - ¥(-Py(T)+ y(T))]dt (6.6)

0

t

+ 8 J ||Dy(T)+ Eu(t)|ydT < 0. 0

t

Since by assumption J[¥(y(T) — ¥(—Py(T) + y(T))]dt > —c2 > —^ we get from

0

(6.6) for arbitrary t > 0 the inequality

t

8 J 11 Dy ( T) + Eu(T )|y dT < V (y0)+ O(Cy0)+ C2 < V (y0)+ c ||y0|2 + C2. (6.7) 0

The property (6.7) implies now the estimate (6.1) . ■

In the next theorem we show that under certain assumptions the inequality (3.1) -(3.3) is absolutely unstable with respect to the observation.

Definition 6.3 The inequality (3.1)—(3.3) with f = 0 is said to be minimally unstable if the resulting equation for ¥ = 0 is minimally unstable, i.e., there exists a bounded linear operator K : Yi ^ U such that the operator A + BK is unstable, i.e.,

o (A + BK) D {s e C : Res > £ > 0} = 0 with F(y, Ky) > 0, Vy e Y1, (6.8) t

and J g(y(T), Ky(T))dt > 0 , Vs,t :0 < s < t, Vy e L2oc(R+; Y1). (6.9)

s

Theorem 6.3 Suppose that the assumptions of Theorem 6.1 are satisfied and the inequality (3.1)-(3.3) with f = 0 is minimally unstable, i.e., (6.8) and (6.9) are satisfied with some operator K G L (Y\, U ). Suppose also that for the equation

y = (A + BK)y, y(0) = yo,

in Yo there exists a splitting Yo = Yo+ © Yo- such that for any yo G Yo+ the solution y(-) of this equation satisfies lim y(t ) = o and for any yo G Yo- a unique solution y(-)

t Aœ o

of the last equation exists on (-œ, o) satisfying lim y(t) = o. Assume that the pair

t A-œ

(A + BK,D) is observable on Yo+, i.e., for any solution y(-) with u(t) = Dy(t) = o for a.a. t > o it follows that y(o) = yo = o. Then there exists an operator P = P* G L ( Y-1, Yo ) fl L ( Yo, Yi ), which is non-negative on Yo+ and negative on Y—. Ifyo G Yo-is an arbitrary point satisfying

2 (yo, Pyo)o + O (Cyo) < -d, (6.io)

where d is the parameter from the class N (d ), then for any solution {y(-), u(-)} of (3.1)- (3.3) with y(o) = yo the term ||Dy (oHt2 is unbounded on [o,œ) proposed that D : Yi a T is invertible. It follows that the inequality (3.1)- (3.3) is absolutely unstable with respect to the observation u(•) = Dy(•).

Proof Under the assumptions of the present theorem we get on the basis of Theorem 5.1b) the existence of an operator P = P* G L(Y-1, Yo) R L(Yo, Y1) and a number 8 > o such that inequality (5.8) with E = o is satisfied. Setting again in (5.8) u = Ky from (6.8) with arbitrary y G Y1 we see that inequality (6.5) with E = o is true. Using now the fact that A + BK is an unstable operator, the pair (A + BK, D) is observable and that there exists a splitting Yo = Yo- 0 Yo+ with the above properties, we see ([4, 7, 19]) that the Lyapunov-functional V(y) := (y,Py)o is non-negative on Yo+ and negative on Yo-. Consider now an arbitrary solution {y(-), u(-)} of (3.1), (3.2) with initial state yo G Yo-. It follows that V(yo) < o and, on the basis of (5.7),

t

V(y(t)) - O(Cyo) + J[w(y(T)) - w(-Py(T) + y(T))] dt (6.11)

o

t

+ 8 J ||Dy(T )||t d t < V (yo) < o o

for all t > o . From (6.11) we conclude that for t ^ +œ

Z (t ) := V (y(t )) + T [w (y(T )) - w (-Py(T ) + y(T ))] d t A o. (6.12)

Jo

Indeed, if we suppose that (6.12) does not hold, we get from (6.11) for large t the estimate

O(Cy0) + 8 f ||Dy(T) ||£ dt < 1V (y0) < 0, 02

which is impossible since O(Cy0) > 0 and 8 > 0 . Let us now show that

Z(t) for t ^ . (6.13)

Suppose that (6.13) is not true. This means that there exists a constant c > 0 such that

Z(t) > -c , Vt > 0. (6.14)

From (6.11) and (6.14) it follows now that for all t > 0

t

8 J || Dy(T)||£dT < c + O (Cy0) + V (y0). (6.15)

0

Since D is invertible, the inequality (6.14) shows that y( ) e L2(0,Y1) and, consequently,

V (y(t)) ^ 0 for t ^ . (6.16)

Recall now that for all t 0

t )) - w (-Py(T ) + ))] d t < d. (6.17)

It follows from (6.11), (6.16) and (6.17) that for sufficiently large t

0 < O (Cy0)+ d + 1v (y0). (6.18)

It is clear that (6.18) contradicts (6.10). ■

t

7 Absolute observation convergence of evolutionary inequalities

In this section, we consider the observation stability of the difference of two arbitrary solutions of inequality (3.1), (3.2), i.e., convergence properties with respect to the observation (3.3) of this inequality, in order to get some information about the convergence of arbitrary solutions to the stationary set.

Let us start with the definition of a new class of nonlinearities for the case U = Z.

Definition 7.1 Suppose in (3.1), (3.2) that U = Z and assume that uo > o is an arbitrary number. The class Nc (uo) consists of all maps 0 : R+ x Z a 2z such that

o < (u1 - u2 , Z1 - ll)z < Uo|z1 - Z2|Z ,

(7.1)

Vt > o, V Z1, Z2 G Z, V u1 G 0 (t, Z1 ) , V u2 G 0 (t, Z2) .

Now we are in the position to give a precise interpretation of observation convergence (see also [11, 17, 2o]).

Definition 7.2 The inequality (3.1), (3.2) is said to be absolutely convergent (in the classes Nc(uo) and M(d)) with respect to the observation u from (3.3) if there exist two constants c > o and c2 > o, which depend only on A, B, Nc(uo) and M(d), such that for any two solutions {y(-), u(-)} and {y(-), «(•)} of (3.1), (3.2) we have

||D(y(0 -?(•)) + E (u(^) - u(-))||2,t < c1(|yo -yo|2 + c2). (7.2)

Theorem 7.1 Suppose that the operator A from (3.1) generates a Co-group on Yo,B G L (U, Yo), the pairs (Ac,Bc) and (-Ac,Bc) are L2-controllable, and (3.1), (3.2) with f = o and U = Z is minimally stable, i.e., there exists an operator K G L (Y1, Z) such that A + BK is stable, (Kw, Cw)Z - U7 ||Kw||Z > o, Vw G Y1, and the pair (A + BK, D + EK) is observable. Suppose also that 0 G Nc(uo) and there exists an £ > o such that

Re (u,x(Z)(io)u)zc - u1o||u|Zc < -£||X(u}(iw)u||Zc ,

V œ G R : io G a(Ac), V u G Zc.

Then inequality (3.1), (3.2) is absolutely convergent in the classes Nc(uo) and M(d) with respect to the observation u.

Proof Suppose that {y(-),u(^)} and {y(-),are two arbitrary solutions of (3.1), (3.2). If we consider in (3.1) the solution {y( ), u( )} and take the test function n = y we get

(3) - Ay - Bu - f (t ), y - y ) -1, 1 + w (y) - W (y) > o. (7.4)

Now we consider (3.1) with the solution {y( ), u( )} and take the test function n = y in order to get

( y) - Ay" - Bu - f (t ), y - y)_1,1 + w (y) - w (y) > o. (7.5)

Suppose P = P* e L(Y-1, Y0) fl L(Y0, Y1) is an arbitrary linear operator and write (7.4) with y - y = P(y - y) and (7.5) with y - y = P(y - y) , i.e.,

(3) - Ay - Bu - f (t), P(y - y))-1,1 + ¥ (y - P(y - y)) - ¥ (y) > 0 (7.6)

and

(yy - Ay - BU - f (t)> P (y - y))-1>1 + ¥ (y + P (y - y)) - W(y) > 0. (7.7) If we take the sum of (7.6) and (7.7) we get the inequality

(y - y>P (y - y))-1>1 - (A(y - y) + B(u - U) >P (y -y))-1>1

(7.8)

- ¥ (y - P (y - y)) + ¥(y) - ¥(y + P (y - y;)) + ¥< 0.

Recall now that the frequency-domain inequality (7.3) together with the other assumptions of Theorem 7.1 guarantee on the base of Theorem 5.1b) the existence of a (real) operator P = P* e L (Y-1> Y0) f L (Y0> Y1) and a number 8 > 0 such that

(Aw + BZ >Pw)-1>1 + (Z>Cw)z - ||Z||Z <-8|Dw + EZ>

V w e Y1> VZ e z. .

Using the fact that (3.1), (3.2) is minimally stable with some K e L(Y1>Z) we conclude from (7.9) that ((A + BK)w>Pw)-1>1 < -8||(D + KE)w||£> Vw e Y1. From this inequality it follows that P is non-negative.

For the considered arbitrary solutions {y(-)> u (•)} and |3;(^)> uu(^)} of (3.1), (3.2) the inequalities (7.8) and (7.9) imply that

(y - y)>P (y - .y))-1>1 + (u - u> C(y - 3;))z - 710 ||u - a^z

(7.10)

+ 8||D (y -yy)+ E (u - u)||£ < 0.

The integration of (7.10) on an arbitrary time interval [0> t] (t > 0) gives for the Lyapunov-functional V(y) := (y> Py)0 on the base of (7.1) and the fact that ¥ e M(d) the inequality

t

V(y(t) -y(t)) - V(y(0) -y(0)) + 8 J ||D(y(T) -y"(t)) + E(u(t) - u(tdt < c>

0

(7.11)

where c is a sufficiently large constant. Since V(y(t) -y(t)) > 0 the inequality (7.11) implies

8 J ||D(y(T) -y (t))+ E (u(t) - u(t))|ydT < V(y(0) -y(0))+ c . 0

In the sequel we investigate the special situation of the convergence of an arbitrary solution of (3.1), (3.2) to a stationary solution. In case when such a stationary solution is unique we can use Theorem 7.1. In many applications, however, the inequality (3.1), (3.2) has a continuum of stationary solutions and the approach of Theorem 7.1 is not applicable.

Let us consider the autonomous inequality (3.1), (3.2) with 0 (t, z) = 0 (z) and

f (t) = o, i.e.,

(y - Ay - Bu, n - y)-1,1 + w (n ) - w (y) > o, Vn G Y1 , (7.12) y (o) = yo G Yo ,

z (t ) = Cy(t ), u(t ) G 0 (z(t )), u(o) = uo G E (yo), (7.13)

u (t)= Dy(t)+ Eu(t). (7.14)

Our aim is to show that under certain observation conditions any solution of (7.12), (7.13) converges to the stationary set of the inequality. Let us start with some definitions ([11, 17, 2o]).

Definition 7.3 Any constant solution {y(-), u(-)} of (7.12), (7.13), i.e., y(t) = const, u(t) = const on R+, is called a stationary solution of (7.12), (7.13). The set of all stationary solutions of (7.12), (7.13) is called the stationary set of (7.12), (7.13) and denoted by L x A c Y1 x U. We say that the solution {y(-),u(-)} of (7.12), (7.13) is weakly quasi-convergent ify(t) a L in the weak sense as t a +œ. The inequality (7.12), (7.13) is called weakly quasi-gradient-like if every its solution is weakly quasi-convergent.

It is clear from the definition that a pair {y, u} is a stationary solution of (7.12), (7.13) if and only if

(-Ay - Bu, n - y)-1,1 + w (n ) - w (y) > o, Vn G Y1, (7.15) y G Yo,

Z = Cy, u G 0 (Z) R E (y). (7.16)

With respect to the stationary set described by (7.15), (7.16) we make the following assumptions.

(S1) The stationary set L x A of (7.15), (7.16) is non-empty and consists of isolated points only.

Remark 7.1 Under standard conditions the assumption (S1) is satisfied ([9, 22, 24]). If (7.15) can be written as

a (y> n -y) + c (y> n -y)-1>1 + ¥(n) - ¥(y) > 0 > Vn e Y > (7.17)

where a(y> n) = a (n > y) is a quadratic form on Y1 x Y1 and c e R is a number, the problem (7.15) is equivalent to the minimisation of the functional

1 r 21

J(n) := ^ [a (n > n)+ c ||n ||xj + ¥(n)

on Y1 . □

Let us formulate some simplifications of system (7.12), (7.13). Assume that

(S2) ¥ = 0 and U = Z = R.

It follows that C e L(Y-1> R) > B e L (R > Y-1) > D e L (Y1> R) > E e L (R > R) > and 0 : R — 2R . Let us further assume, as in the case of ODE's with retarded arguments, ([20]) that 0 is piecewise continuous, discontinuous in z = 0 and with the set of isolated discontinuity points {zj}. For any discontinuity point zj the set 0 (zj) C R

is assumed as a closed interval such that 0 (zj) D [liminf 0 (z) > limsup 0 (z)]. Then

z—zj z—yzj

([11, 15]) 0 is an upper-semicontinuous function. The local solution in the sense of Sect. 3 is assumed to exist globally. In order to construct a quadratic constraint F for 0 we assume that 0 satisfies the following property.

(S3) uz > 0 > Vz e R> Vu e 0(z). (7.18)

The last condition implies that 0 (0) = [01> 02] > with 01 < 0 and 02 > 0. Assume 01 < 0 and 02 > 0.

In mechanics many systems with dry friction are described by equation (7.12), (7.13) with nonlinearities satisfying (S1) - (S3). Typical solutions of such systems are sliding solutions. Let us give the formal definition which goes back to [11, 20, 25].

Definition 7.4 A solution {y(-)> u(-)} of (7.12), (7.13) is called sliding solution on the time interval (t1 > t2) if z(t) = Cy(t) = zj on (t1> t2), where zj is a discontinuity point of 0.

Let us investigate observation properties of sliding solutions associated with the discontinuity z = 0 of 0. Consider for s e p (Ac) the transfer operator x(z)(s) = Cc(slc - Ac)-1Bc = Ccx(s) and assume that 0 e p (Ac) and x(z)(0) = 0. It follows

from (7.12), (7.13) that the stationary sliding solutions of the system corresponding to the discontinuity z = 0 of 0 have the form y(t) = y, u(t) = u, where u e 0 (0),y = uA-1 B .

From Theorem 6.1 we derive now sufficient conditions for absolute observation stability. Note, that for the case of ODE's with retarded arguments similar conditions are derived in [20].

Theorem 7.2 Suppose that for the linear part of (7.12), (7.13) the assumptions of Theorem 7.1 are satisfied, (7.12), (7.13) is minimally stable, X(z)(0) = 0 and there exist parameters 0 j > 0 (j = 1,2) such that the following inequalities hold:

Re

(01 + i©02)x(z)(i©) < 0, V© e R\{0}, (7.19) 1

lim —w Re

©A0 ©2

(01 + i© 02) x (z)(i©)

(01 + i© 02) X (z)(i©)

< 0, (7.20)

and

liminf Re (01 + i©02) X(z)(i©) < n < 0. (7.21)

| ©

Then (7.12), (7.13) is absolutely stable with respect to all observations U1 (t) = Dy(t) and u2(t) = Dy(t) = DAy(t) + DBu(t), where D e L (Y1, R) is such that

DA-1B = 0. (7.22)

Proof We consider the quadratic form

F + G = 01F1 + 02G1 := 01Cy u + 02C [Ay + Bu]u. (7.23)

The first part 01F1 comes from the inequality (7.8), the second part 02G1 is introduced through (5.1). For u = 0 we have F(y, 0) + G(y, 0) = 0 and the system is minimally stable. Extend now F + G to a Hermitian form Fc + Gc on C x C by

Fc(y, u) + Gc(y, u) = Re [01Cyu* + 02C(Ay + Bu)u*]. (7.24)

If we consider the observations u1 (t) = Dy(t) together with the quadratic form (7.23) the frequency-domain condition (5.6) becomes

38 > 0 : n(io) < -5|Dx(iffl) |2, V© e R, (7.25)

with n(i©) := Re [(01 + i©02)x(z)(i©)]. In [20] it is shown that (7.25) is true if (7.19)-(7.21) and (7.22) are satisfied. Thus the system (7.12), (7.13) is absolutely stable with respect to all observations u1(t) = Dy(t) with D satisfying (7.22).

If we consider now the observations u2(t) = DAy(t) + DBu(t) together with the quadratic form (7.23) the frequency-domain condition (5.6) becomes

38 > 0 : n(io) < -8|i«Dx(iffl)|2> V« e R. (7.26)

It is easy to check (see [20]) that (7.26) is also satisfied provided that (7.19)-(7.21) and (7.22) are true. This shows that the system (7.12), (7.13) is also absolutely stable with respect to the observation u2 ■

From Theorem 7.2 we get immediately the following.

Corollary 7.1 Suppose that the assumptions of Theorem 7.1 are satisfied. Then every solution {y(-)>u(-)} of (7.12), (7.13) converges weakly to the set

L := {A-1Bu > u e R} >

and, consequently, the system (7.12), (7.13) is weakly quasi-gradient like with respect to the stationary sliding solutions.

Acknowledgments

This paper was prepared with the support by a grant from St.Petersburg State University Event 3 [Pure ID 75207094] (sections 5-7) and by the Leading Scientific Schools of Russia [Project NSh-2624.2020.1] (sections 1-4).

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