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Journal of Mathematical Physics, Analysis, Geometry

2018, Vol. 14, No. 2, pp. 169–196

doi: https://doi.org/10.15407/mag14.02.169

Lagrange Stability of Semilinear

Diﬀerential-Algebraic Equations and

Application to Nonlinear Electrical Circuits

Maria S. Filipkovska

A semilinear diﬀerential-algebraic equation (DAE) is studied focusing

on the Lagrange stability (instability). The conditions for the existence and

uniqueness of global solutions (a solution exists on an inﬁnite interval) of

the Cauchy problem, as well as the conditions of the boundedness of the

global solutions, are obtained. Furthermore, the obtained conditions of the

Lagrange stability of the semilinear DAE guarantee that every its solution

is global and bounded and, in contrast to the theorems on the Lyapunov

stability, allow us to prove the existence and uniqueness of global solutions

regardless of the presence and the number of equilibrium points. We also

obtain the conditions for the existence and uniqueness of solutions with a

ﬁnite escape time (a solution exists on a ﬁnite interval and is unbounded, i.e.,

is Lagrange unstable) for the Cauchy problem. The constraints of the type of

global Lipschitz condition are not used which allows to apply eﬃciently the

work results for solving practical problems. The mathematical model of a

radio engineering ﬁlter with nonlinear elements is studied as an application.

The numerical analysis of the model veriﬁes theoretical studies.

Key words: diﬀerential-algebraic equation, Lagrange stability, instability,

regular pencil, bounded global solution, ﬁnite escape time, nonlinear electri-

cal circuit.

Mathematical Subject Classiﬁcation 2010: 34A09, 34D23, 65L07.

1. Introduction

Diﬀerential-algebraic equations (DAEs), which are also called descriptor,

algebraic-differential and degenerate diﬀerential equations, have a wide range

of practical applications. Certain classes of mathematical models in radioelec-

tronics, control theory, economics, robotics technology, mechanics and chemical

kinetics are described by semilinear DAEs. Semilinear DAEs comprise in par-

ticular semiexplicit DAEs and in turn can be attributed to quasilinear DAEs.

The Lagrange stability of a DAE guarantees that every its solution is global and

bounded. The presence of a global solution of the equation guarantees a suf-

ﬁciently long action term of the corresponding real system. The properties of

c

Maria S. Filipkovska, 2018

170 Maria S. Filipkovska

boundedness and stability of solutions of the equations describing mathematical

models are used in the design and synthesis of the corresponding real systems and

processes. The application of the DAE theory to the study of electrical circuits

can be found in various monographs and papers, [4,5,9,11,13–17] are among

them.

In the present paper, the semilinear diﬀerential-algebraic equation (DAE)

d

dt[Ax] + Bx =f(t, x) (1.1)

with a nonlinear function f: [0,∞)×Rn→Rnand the linear operators A,B:

Rn→Rnis considered. The operator Ais degenerate (noninvertible), the opera-

tor Bmay also be degenerate. Note that the solutions of a semilinear DAE of the

form Ad

dt x+Bx =f(t, x) must be smoother than the solutions of a semilinear

DAE of the form (1.1). The availability of a noninvertible operator (matrix) at

the derivative in the DAE means the presence of algebraic connections, which

inﬂuence the trajectories of solutions and impose restrictions on the initial data.

For the DAE (1.1) with the initial condition

x(t0) = x0,(1.2)

the initial value x0must be chosen so that the initial point (t0, x0) belongs to

the manifold L0={(t, x)∈[0,∞)×Rn|Q2[Bx −f(t, x)] = 0}(which is

also deﬁned in (3.1), where Q2is a spectral projector considered in Section 2).

The initial value x0satisfying the consistency condition (t0, x0)∈L0is called

a consistent initial value. A solution x(t) of the Cauchy problem (1.1), (1.2)

(see Deﬁnition 2.1) is called global if it exists on the whole interval [t0,∞).

The inﬂuence of the linear part d

dt [Ax] + Bx of the DAE (1.1) is determined

by the properties of the pencil λA +B(λis a complex parameter). It is assumed

that λA +Bis a regular pencil of index 1, i.e., there exists the resolvent of the

pencil (λA +B)−1and it is bounded for suﬃciently large |λ|(see Section 2).

This property of the pencil allows one to use the spectral projectors P1,P2,Q1,

Q2, which can be calculated by contour integration and reduce the DAE to the

equivalent system of a purely diﬀerential equation and a purely algebraic equa-

tion (see Section 2). This is one of the reasons why we use the requirement of

index 1 for the characteristic pencil λA +Bof the linear part of the DAE and

not for the DAE, as, for example, in [11,12,22]. Another reason is as follows.

The requirement that the DAE have index 1 does not give us the necessary re-

sult and it is too restrictive for our research (this will be discussed in Section 2).

It is also worth noting that semilinear DAEs of the form (1.1) arise in many

practical problems, examples of which can be found in the books and papers by

R. Riaza, A.G. Rutkas, A. Favini, L.A. Vlasenko, A.D. Myshkis, S.L. Camp-

bell, L.R. Petzold, K.E. Brenan, E. Hairer, G. Wanner, J. Huang, J.F. Zhang,

R.E. Showalter and other authors. However, in present literature these equations

are often written in the form d

dt [Ax] = g(t, x) or in the form of semiexplicit DAE.

The objective of the paper is to ﬁnd the conditions of the Lagrange stability

and instability of the semilinear DAE (see Deﬁnitions 2.4–2.6). A mathematical

Lagrange Stability and Instability of Semilinear DAEs and Applications 171

model of a radio engineering ﬁlter with nonlinear elements is considered as an

application. It should be noticed that if the operator Ain the semilinear DAE is

invertible, then the results obtained in the paper remain valid (in this case, the

semilinear DAE is equivalent to an ordinary diﬀerential equation).

In Section 3, the theorem on the Lagrange stability, which gives suﬃcient

conditions for the existence and uniqueness of global solutions of the Cauchy

problem for the semilinear DAE, as well as conditions of the boundedness of

global solutions, is proved. Furthermore, the theorem gives conditions of the

Lagrange stability of the semilinear DAE, which ensure that each solution of the

DAE starting at the time moment t0∈[0,∞) exists on the whole inﬁnite interval

[t0,∞) (is global) and is bounded. In Section 4, the theorem on the Lagrange

instability, which gives suﬃcient conditions for the existence and uniqueness of

solutions with a ﬁnite escape time for the Cauchy problem, is proved. It is

important that the proved theorems do not contain restrictions of the type of

global Lipschitz condition, including the condition of contractivity, which enable

using them for solving more general classes of applied problems. Theorems on the

unique global solvability of semilinear DAEs that comprise conditions equivalent

to global Lipschitz conditions are known (cf. [23]). Also, the proved theorems do

not contain the requirement that the DAE have index 1 globally (this requirement

is found, for example, in [11, Theorem 6.7]). For comparison, the theorems

from [11,12,22] are considered in Sections 1and 2.

The Lagrange stability of the ordinary diﬀerential equation (ODE) d

dt x=

f(t, x) (t≥0, xis an n-dimensional vector) was studied in [10, Chapter 4]

using the method obtained by extending the direct (second) method of Lyapunov.

The results of [10, Chapter 4] concerning the Lagrange stability are extended to

semilinear DAEs in the present paper. The existence and uniqueness theorem of

a global solution of the Cauchy problem for the semilinear DAE with a singular

pencil λA +Bwas proved in the author’s paper [7]. The results on the Lagrange

stability of the semilinear DAE with the regular pencil, obtained by the author in

[6], have been improved and have been applied for a detailed study of evolutionary

properties of the mathematical model for a radio engineering ﬁlter in the present

paper.

The stability of linear DAEs and descriptor control systems described by

linear DAEs was studied by many authors (see, for example, [5,11,16,21] and

references therein).

In [12], R. M¨arz studied the Lyapunov stability of an equilibrium point of the

autonomous “quasilinear” DAE

Ad

dtx+g(x)=0,(1.3)

where A∈L(Rn) is singular (noninvertible) and g:D→Rn,D⊆Rnopen.

The theorem [12, Theorem 2.1] allows to prove the existence and uniqueness of

global solutions only in some (suﬃciently small) neighborhood of an equilibrium

point x∗of (1.3), i.e., g(x∗) = 0, x∗∈D. If there are the two equilibrium

points x∗

1∈Dand x∗

2∈D,x∗

16=x∗

2, then the theorem [12, Theorem 2.1] can

172 Maria S. Filipkovska

not guarantee the existence of a unique global solution in D. Namely, if the

conditions of the theorem are fulﬁlled for the equilibria x∗

1and x∗

2, then for some

initial time moment t0there exists the unique global solution x=ϕ(t) of (1.3)

(with the initial condition [12, (2.8)]) in some neighborhood of x∗

1and the unique

global solution x=ψ(t) of (1.3) in some neighborhood of x∗

2, but this does not

guarantee the existence of a unique global solution in D.Theorem 3.1 allows to

prove the existence and uniqueness of global solutions for all possible initial points

(as noted in Remark 3.2), that is, regardless of the presence of an equilibrium

point, in the presence of several equilibrium points or the inﬁnite number of

equilibrium points, and for a more general equation than (1.3).

A theorem similar to [12, Theorem 2.1] was proved by C. Tischendorf [22]

for the autonomous nonlinear DAE f(x0(t), x(t)) = 0. The theorem [22, Theo-

rem 3.3] gives conditions for the asymptotic stability (in Lyapunov’s sense) of

a stationary solution x∗, i.e., f(0, x∗) = 0. The deﬁnition of asymptotic sta-

bility from [22, Section 3] is equivalent to the fulﬁllment of conditions (i)–(iii)

from [12, Theorem 2.1], and if we take f(x0(t), x(t)) = Ax0(t) + g(x(t)), then [22,

Theorem 3.3] and [12, Theorem 2.1] are analogous.

For a global solution of a nonautonomous nonlinear DAE, the conditions for

the asymptotic stability (in Lyapunov’s sense) which can also be considered only

locally (in a suﬃciently small neighborhood of this solution) are given in the

theorem [11, Theorem 6.16]. Under the conditions of the theorem, it is assumed

that the regular index-1 DAE has the global solution, and a DAE linearized along

this solution is strongly contractive [11, Deﬁnition 6.5].

It is important to note that the theorem on the Lagrange stability (Theo-

rem 3.1) gives conditions for the existence and uniqueness of global solutions

(as well as conditions of the boundedness) independently of the presence and the

number of equilibrium points. In contrast to Lyapunov stability, Lagrange stabil-

ity can be considered as the stability of the entire system, not just of its equilibria.

From this, in particular, it follows that a globally stable dynamic system can be

not only monostable (as in the case of the global stability in Lyapunov’s sense),

but also multistable (cf. [24, Section I]). In [24], A. Wu and Z. Zeng studied

the Lagrange stability of neural networks, which are described by ODEs with

delay. It is known that neural networks are also described by DAEs (including

semilinear DAEs), therefore the research of the present paper is useful for the

analysis and synthesis of the neural networks. Lagrange stability is also used

for the analysis of ecological stability. The theorem on the Lagrange instability

(Theorem 4.1) can be used, in particular, for the analysis of nonlinear control

systems. For example, the study of the Lagrange instability allows to ﬁnd such

a property as a blow up of the solution for a nonlinear control system on a ﬁnite

time interval.

It is also important to note that even for an ordinary diﬀerential equation

containing a nonlinear part, the Lyapunov stability of a nontrivial solution does

not imply that the solution is bounded, i.e., Lagrange stable. Since the DAE

considered in the paper contains the nonlinear part, the Lyapunov stability of

its solution does not imply the Lagrange stability. Also, in the general case, the

Lagrange Stability and Instability of Semilinear DAEs and Applications 173

Lyapunov instability does not imply the Lagrange instability, but the converse

assertion is true. Therefore, the proved Lagrange instability theorem can also be

regarded as the Lyapunov instability theorem.

Thus, the results obtained on the Lagrange stability of semilinear DAEs are

important for the development of the DAE theory and for applied problems. The

Lagrange stability of various types of ODEs and its applications are considered

in many works, e.g., [1,3,10,24]. However, in [11,12,22] and other cited works,

the Lagrange stability of DAEs was not studied.

In Sections 5and 6, the mathematical model of a radio engineering ﬁlter with

nonlinear elements is studied with the help of the theorems proved in the previ-

ous sections. The restrictions on the initial data and parameters for the electrical

circuit of the ﬁlter, which ensure the existence, uniqueness and boundedness of

global solutions, and the existence and uniqueness of solutions with a ﬁnite es-

cape time for the dynamics equation of the electrical circuit are obtained. Certain

functions and quantities (including nonlinear functions that are not global Lips-

chitz) deﬁning the circuit parameters and satisfying the obtained restrictions are

given. The numerical analysis of the mathematical model is carried out.

The paper has the following structure. The main theoretical results are given

in Sections 3,4. Namely, the theorems on the Lagrange stability and instabil-

ity of the DAE are proved. In Sections 5,6, the mathematical model of the

nonlinear radio engineering ﬁlter is studied with the help of the obtained theo-

rems. The conclusions and explanations of the obtained results from a physical

point of view are given in Subsections 5.1,6.1, and the numerical analysis of the

mathematical model is carried out in Subsections 5.2,6.2. In Section 2, we give

a problem setting, preliminary information and deﬁnitions. Section 7contains

general conclusions.

The following notation will be used in the paper: EXis the identity operator

in the space X;A|Xis the restriction of the operator Ato X;L(X, Y ) is the

space of continuous linear operators from Xto Y,L(X, X) = L(X); the notation

R+∞

cf(t)dt < +∞(R+∞

cf(t)dt =∞) means that the integral converges (does

not converge); xTis the transpose of x. Sometimes the function fis denoted by

the same symbol f(x) as its value at the point xin order to explicitly indicate

that the function depends on the variable x, but from the context it will be clear

what exactly is meant.

2. Problem setting and preliminaries

Consider the Cauchy problem (1.1), (1.2) for the semilinear DAE, where

t, t0≥0, x, x0∈Rn,f: [0,∞)×Rn→Rnis a continuous function, A,B:Rn→

Rnare linear operators to which n×nmatrices A,Bcorrespond. The operator

Ais degenerate (noninvertible), the operator Bmay also be degenerate. The

matrix pencil as well as the corresponding operator pencil λA +Bis regular, i.e.,

det(λA +B)6≡ 0.

Deﬁnition 2.1. A function x(t) is called a solution of the Cauchy prob-

lem (1.1), (1.2) on some interval [t0, t1), t1≤ ∞, if x∈C([t0, t1),Rn),

174 Maria S. Filipkovska

Ax ∈C1([t0, t1),Rn), xsatisﬁes equation (1.1) on [t0, t1) and the initial condi-

tion (1.2).

It is assumed that λA +Bis a regular pencil of index 1, that is, there exist

constants C1,C2>0 such that

(λA +B)−1

≤C1,|λ| ≥ C2.(2.1)

For the pencil λA +Bsatisfying (2.1) there exist the two pairs of mutually

complementary projectors P1,P2and Q1,Q2(i.e., PiPj=δij Pi,P1+P2=ERn,

and QiQj=δijQi,Q1+Q2=ERn,i, j = 1,2, δij is the Kronecker delta) ﬁrst

introduced by A.G. Rutkas [17, Lemma 3.2]. The projectors can be constructively

determined by the formulas similar to [19, (5), (6)] (where X=Y=Rn) or [17,

(3.4)] (for the real operators A,Bthe projectors are real). These projectors

decompose the space Rninto direct sums of subspaces

Rn=X1˙

+X2,Rn=Y1˙

+Y2, Xj=PjRn, Yj=QjRn, j = 1,2,(2.2)

such that the operators A,Bmap Xjinto Yj, and the induced operators

Aj=A|Xj:Xj→Yj,Bj=B|Xj:Xj→Yj,j= 1,2, (X2=Ker A,Y1=

ARn=A1X1) are such that A2= 0, the inverse operators A−1

1∈L(Y1, X1),

B−1

2∈L(Y2, X2) exist (cf. [17, Lemma 3.2], [19, Sections 2,6]), and

APj=QjA, BPj=QjB, j = 1,2.(2.3)

With respect to the decomposition (2.2) any vector x∈Rncan be uniquely

represented as the sum

x=xp1+xp2, xp1=P1x∈X1, xp2=P2x∈X2.(2.4)

This representation will be used further. We will also use the auxiliary operator

G∈L(Rn), (cf. [19, Sections 2, 6])

G=AP1+BP2=A+BP2, GXj=Yj, j = 1,2,

which has the inverse operator G−1∈L(Rn) with the properties G−1AP1=P1,

G−1BP2=P2,AG−1Q1=Q1,BG−1Q2=Q2.

By using the projectors P1,P2,Q1,Q2, the DAE can be reduced to the

equivalent system of a purely diﬀerential equation and a purely algebraic equa-

tion. Applying Q1,Q2to (1.1) and taking into account (2.3), we obtain the

equivalent system

d

dt(AP1x) + BP1x=Q1f(t, x),

Q2f(t, x)−BP2x= 0.

Further, using G−1, we obtain the system, which is equivalent to the DAE (1.1):

d

dt(P1x) = G−1−BP1x+Q1f(t, P1x+P2x),

G−1Q2f(t, P1x+P2x)−P2x= 0.

(2.5)

Lagrange Stability and Instability of Semilinear DAEs and Applications 175

Remark 2.1.We consider various notions of an index of the pencil, an index of

the DAE, a relationship between them and their relationship with the mentioned

notion of the pencil of index 1. In [23, Section 6.2], the maximum length of the

chain of an eigenvector and adjoint vectors of the matrix pencil A+µB at the

point µ= 0 is called the index of the matrix pencil λA +B. Following [23,

Sections 6.2, 2.3.1], the regular pencil λA +Bwith the property (2.1) is called

a regular pencil of index 1. Taking into account the properties of the projectors

Pj,Qjand the induced operators Aj,Bj,j= 1,2, if the condition (2.1) holds,

then the index of the pencil (or the index of nilpotency of the matrix pencil)

(A, B) is 1 in the sense as deﬁned in the works of C.W. Gear, L.R. Petzold, for

example, [8, p. 717–718] (it is easy to verify using [8, Theorem 2.2]). In [11,

Deﬁnition 1.4], the index of nilpotency of the matrix pencil (A, B ) [8] is called

the Kronecker index of the regular matrix pair {A, B}which forms the matrix

pencil λA +B. Also, by the index of the pencil one can determine the index

of the corresponding system of diﬀerential-algebraic equations [8, p. 718]. In

particular, the index of the pencil (A, B ) (the Kronecker index of the regular

matrix pair {A, B}) coincides with the index of the linear DAE (the Kronecker

index of the regular DAE [11, Deﬁnition 1.4]) Ad

dt x+Bx =g(t). This is analogous

to the fact that the pencil λA +Bcorresponds to the linear part d

dt [Ax] + Bx of

the DAE (1.1) and the inﬂuence of the linear part is determined by the properties

of the corresponding pencil. For comparison with the notion of the “tractability

index” from the works of R. M¨arz, C. Tischendorf and R. Lamour [11,12,22],

note that the linear DAE d

dt [Ax] + Bx =q(t) with the regular pencil λA +Bof

index 1 (i.e., (2.1) is fulﬁlled) is regular with tractability index 1 [11, p. 65, 91,

Deﬁnition 2.25].

But if we consider a semilinear DAE Ad

dt x+Bx =f(t, x) (in this case the

solution x(t) must be smoother than the solution of the DAE (1.1)), then for it

to have index 1 for all t≥0, x∈D⊆Rn(to be exact, (t, x)∈L0, where L0is

deﬁned in (3.1)), it is necessary that the pencil A, B −∂

∂x f(t, x)have index 1

for all t≥0, x∈D. This condition is too restrictive for our research and it

does not allow us to prove the existence of a unique global solution since the

uniqueness of the solution can be proved only locally (in [2, Ch. 9], the same is

shown for a “semi-explicit index-1” DAE).

One of the conditions for proving the existence of a unique global solution of

the Cauchy problem (1.1), (1.2) for any consistent initial value x0is the condition

of the basis invertibility of an operator function (Deﬁnition 2.3) which will be

discussed below. To begin, we introduce the deﬁnitions.

Deﬁnition 2.2. A system of one-dimensional projectors {Θk}s

k=1, Θk:Z→

Zsuch that ΘiΘj=δijΘi(δij is the Kronecker delta) and EZ=Ps

k=1 Θkis

called an additive resolution of the identity in an s-dimensional linear space Z.

The additive resolution of the identity generates a direct decomposition of Z

into the sum of sone-dimensional subspaces: Z=Z1˙

+Z2˙

+· · · ˙

+Zs,Zk= ΘkZ.

176 Maria S. Filipkovska

Deﬁnition 2.3. Let W,Zbe s-dimensional linear spaces, D⊂W. An

operator function (a mapping) Φ : D→L(W, Z ) is called basis invertible on the

convex hull conv{ˆw, ˆ

ˆw}of vectors ˆw, ˆ

ˆw∈Dif for any set of vectors {wk}s

k=1,

wk∈conv{ˆw, ˆ

ˆw}, and some additive resolution of the identity {Θk}s

k=1 in the

space Zthe operator

Λ =

s

X

k=1

ΘkΦ(wk)∈L(W, Z)

has the inverse operator Λ−1∈L(Z, W ).

Let us represent the operator Φ(w)∈L(W, Z) as a matrix relative to some

bases in the s-dimensional spaces W,Z:

Φ(w) =

Φ11(w)· · · Φ1s(w)

.

.

..

.

.

Φs1(w)· · · Φss(w)

.

Deﬁnition 2.3 can be stated as follows: the matrix function Φ is basis invertible

on the convex hull conv{ˆw, ˆ

ˆw}of the vectors ˆw, ˆ

ˆw∈Dif for any set of vectors

{wk}s

k=1 ⊂conv{ˆw, ˆ

ˆw}, the matrix

Λ =

Φ11(w1)· · · Φ1s(w1)

.

.

..

.

.

Φs1(ws)· · · Φss(ws)

has the inverse Λ−1.

Note that the property of basis invertibility does not depend on the choice

of a basis or an additive resolution of the identity in Z. This statement follows

directly from Deﬁnitions 2.2,2.3.

Obviously, if the operator function Φ is basis invertible on conv{ˆw, ˆ

ˆw}, then

it is invertible at any point w∗∈conv{ˆw, ˆ

ˆw}(w∗=αˆ

ˆw+ (1 −α) ˆw,α∈[0,1]),

i.e., for each point w∗∈conv{ˆw, ˆ

ˆw}, its image Φ(w∗) under the mapping Φ is

an invertible continuous linear operator from Wto Z. The converse is not true

unless the spaces W,Zare one-dimensional. We give an example.

Example 2.1.Let W=Z=R2,D= conv{ˆw, ˆ

ˆw}, ˆw= (1,−1)T,ˆ

ˆw= (1,1)T,

w= (a, b)T∈D,

Φ(w) = ab 1

−1ab.

For the set of vectors {w1, w2} ⊂ conv{ˆw, ˆ

ˆw},w1= (a1, b1)T,w2= (a2, b2)T,

the operator Λ has the form

Λ = a1b11

−1a2b2.

Since det Φ(w) = a2b2+ 1 6= 0 for any w∈D, then Φ(w) is invertible on D.

However, the operator Λ is not invertible for {w1, w2}={ˆw, ˆ

ˆw}and hence the

operator function Φ is not basis invertible on D. If we take ˆw= (1,0)T, then Φ

is basis invertible on D.

Lagrange Stability and Instability of Semilinear DAEs and Applications 177

Now we will explain why this deﬁnition is needed. As shown above, the DAE

(1.1) is equivalent to the system of a purely diﬀerential equation and a purely

algebraic equation. The algebraic equation deﬁnes one of the components of

a DAE solution as an implicitly given function. With the help of the implicit

function theorem this component can be deﬁned as a (unique) explicitly given

function, but only locally, i.e., in some suﬃciently small neighborhood. But we

need a unique globally deﬁned explicit function for further application of the re-

sults on Lagrange stability to the diﬀerential equation, which will be obtained by

substitution of the found component (function). For this purpose, the condition

of the basis invertibility of an operator function (Deﬁnition 2.3), which was ﬁrst

introduced in [18], is used. Note that this condition does not impose restrictions

of a type of a global Lipschitz condition, including the condition of contractivity,

and does not require the global boundedness of the norm for an inverse function

on the whole domain of deﬁnition (see Remark 3.1).

In the theorem [11, Theorem 6.7], the conditions of global solvability are

given for the nonlinear DAE f(D(t)x)0, x, t= 0 [11, (4.1)]. It is as-

sumed that the DAE [11, (4.1)] is globally regular of index 1, i.e., for all

Dx ∈Rn,x∈Rm,t∈[0,∞) [11, Theorem 6.7]. This condition means

that the pencil λ∂

∂y f(y, x, t)D(t) + ∂

∂x f(y, x, t) is regular with Kronecker in-

dex 1 for all y∈Rn,x∈Rm,t∈[0,∞) [11, p. 318-320]. Therefore, there

must exist the constants C1, C2>0 independent of t,x,yand such that

λ∂

∂y f(y, x, t)D(t) + ∂

∂x f(y, x, t)−1

≤C1for all y∈Rn,x∈Rm,t∈[0,∞),

|λ| ≥ C2, i.e., the norm is globally bounded. Also, the theorem contains the

requirement of the contractivity of the regular index-1 DAE (see [11, Deﬁnition

6.1, 6.5]) which is an additional condition. Taking into account Remark 3.1, in

the case of a semilinear DAE these conditions are more restrictive than those of

global solvability from Theorem 3.1.

Concerning the theorems [12, Theorem 2.1], [22, Theorem 3.3], note that

they are obtained for the autonomous DAE. If we consider the nonautonomous

DAEs, namely, Ax0+g(t, x) = 0 or f(x0, x, t) = 0, where f(x0, x, t) = Ax0+

g(t, x), then, as said above, the requirement that the pencil λA +∂

∂x g(t, x∗) have

index 1 means that there exist the constants C1, C2>0 independent of tand

such that

λA +∂

∂x g(t, x∗)−1

≤C1,|λ| ≥ C2for all t∈[0,∞), i.e., the norm

is globally bounded in t. Hence, this requirement is more restrictive than the

requirement that the operator function Φ is basis invertible and the pencil λA +

Bhas index 1.

Also note that in [12, Theorem 2.1] and [22, Theorem 3.3], the nonlinear

function is required to be twice continuously diﬀerentiable, while in Theorem 3.1

fis required to be continuous and have the continuous ∂

∂x f(t, x).

Deﬁnition 2.4. A solution x(t) of the Cauchy problem (1.1), (1.2) has a

ﬁnite escape time if it exists on some ﬁnite interval [t0, T ) and is unbounded, i.e.,

there exists T < ∞(T > t0) such that limt→T−0kx(t)k= +∞.

If the solution has a ﬁnite escape time, it is called Lagrange unstable.

178 Maria S. Filipkovska

Deﬁnition 2.5. A solution x(t) of the Cauchy problem (1.1), (1.2) is called

Lagrange stable if it is global and bounded, i.e., the solution x(t) exists on [t0,∞)

and supt∈[t0,∞)kx(t)k<+∞.

Deﬁnition 2.6. Equation (1.1)is Lagrange stable if every solution of the

Cauchy problem (1.1), (1.2) is Lagrange stable.

Equation (1.1)is Lagrange unstable if every solution of the Cauchy problem

(1.1), (1.2) is Lagrange unstable.

3. Lagrange stability of the semilinear DAE

The theorem on the Lagrange stability of the DAE (1.1), which gives suﬃcient

conditions for the existence and uniqueness of global solutions of the Cauchy

problem (1.1), (1.2), where the initial points satisfy the consistency condition

(t0, x0)∈L0(the manifold L0is deﬁned in (3.1)), and gives conditions of the

boundedness of the global solutions, is given below.

Theorem 3.1. Let f∈C([0,∞)×Rn,Rn)have the continuous partial deriva-

tive ∂

∂x f(t, x)on [0,∞)×Rn,λA +Bbe a regular pencil of index 1 and

∀t≥0∀xp1∈X1∃xp2∈X2

(t, xp1+xp2)∈L0={(t, x)∈[0,∞)×Rn|Q2[Bx −f(t, x)] = 0},(3.1)

where X1,X2from (2.2). Let for any ˆxp2,ˆ

ˆxp2∈X2such that (t∗, x∗

p1+ ˆxp2),

(t∗, x∗

p1+ˆ

ˆxp2)∈L0the operator function

Φ: X2→L(X2, Y2),Φ(xp2) = ∂

∂x Q2f(t∗, x∗

p1+xp2)−BP2,(3.2)

be basis invertible on the convex hull conv{ˆxp2,ˆ

ˆxp2}. Suppose that for some self-

adjoint positive operator H∈L(X1)and some number R > 0there exist functions

k∈C([0,∞),R),U∈C((0,∞),(0,∞)) such that

Z+∞

c

dv

U(v)= +∞(c > 0),

HP1x, G−1[−BP1x+Q1f(t, x)]

≤k(t)U1

2(HP1x, P1x),(t, x)∈L0,kP1xk ≥ R. (3.3)

Then for each initial point (t0, x0)∈L0, there exists a unique solution x(t)of the

Cauchy problem (1.1),(1.2)on [t0,∞).

If, additionally,

Z+∞

t0

k(t)dt < +∞,

there exists ˜xp2∈X2such that for any ˜

˜xp2∈X2such that (t∗, x∗

p1+˜

˜xp2)∈L0

the operator function (3.2)is basis invertible on conv{˜xp2,˜

˜xp2}\{˜xp2}, and

sup

t∈[0,∞),kxp1k≤M

kQ2f(t, xp1+ ˜xp2)k<+∞, M > 0is a number, (3.4)

then for the initial points (t0, x0)∈L0the equation (1.1)is Lagrange stable.

Lagrange Stability and Instability of Semilinear DAEs and Applications 179

Remark 3.1.Now we explain the restriction which is imposed on Φ (3.2)

(for the existence and uniqueness of global solutions). In the case when the

space X2is one-dimensional (then the basis invertibility is equivalent to the

invertibility), it is required that the continuous linear operator Λ = Φ(x∗

p2),

x∗

p2∈conv{ˆxp2,ˆ

ˆxp2}, have a continuous linear inverse operator for any ﬁxed ˆxp2,

ˆ

ˆxp2,t∗,x∗

p1such that (t∗, x∗

p1+ ˆxp2),(t∗, x∗

p1+ˆ

ˆxp2)∈L0. In the case when

the dimension of X2is greater than 1, the operator Λ ∈L(X2, Y2), which is

constructed from the operator function Φ (as shown in Deﬁnition 2.3) for ﬁxed

ˆxp2,ˆ

ˆxp2,t∗,x∗

p1such that (t∗, x∗

p1+ ˆxp2),(t∗, x∗

p1+ˆ

ˆxp2)∈L0, is required to

be invertible. At the same time, the global boundedness of the norm of the

mapping [Φ]−1on X2and the global boundedness of the norm of the function

∂

∂x Q2f(t, xp1+xp2)P2−BP2−1on [0,∞)×Rnare not required (i.e., the norm

of the function is not required to be bounded by a constant for all t,xp1,xp2).

For comparison, the condition of index 1 for the DAE was discussed above.

Proof. The DAE (1.1) is equivalent to system (2.5) (as shown in Section 2).

Denote dim X1=a, dim X2=d(d=n−a). Any vector x∈Rncan be repre-

sented as x=z

u∈Ra×Rd, where z∈Ra,u∈Rdare column vectors. We

introduce the operators (the method of the construction of the operators is given

in [18, Section 2]) Pa:Ra→X1,Pd:Rd→X2, which have the inverse operators

P−1

a:X1→Ra,P−1

d:X2→Rd. Then z=P−1

aP1x,u=P−1

dP2x,x=Paz+Pdu

(recall that (2.4)), and P−1

aP1Pa=ERa,P−1

dP2Pd=ERd. Multiplying the equa-

tions of system (2.5) by P−1

a,P−1

d, and replacing P1xand P2xby Pazand Pdu,

respectively, we get the equivalent system

d

dtz=P−1

aG−1−BPaz+Q1˜

f(t, z, u),(3.5)

P−1

dG−1Q2˜

f(t, z, u)−u= 0,(3.6)

where ˜

f(t, z, u) = f(t, Paz+Pdu).

Thus, the semilinear DAE (1.1) is equivalent to system (3.5), (3.6).

Further we are going to prove the theorem in two steps.

I (The existence and uniqueness). We prove the ﬁrst part of the theorem,

that is, the existence and uniqueness of global solutions.

Consider the mapping

F(t, z, u) = P−1

dG−1Q2˜

f(t, z, u)−u. (3.7)

It is continuous on [0,∞)×Ra×Rdand has continuous partial derivatives

∂

∂z F(t, z , u) = P−1

dG−1∂

∂x (Q2f(t, x))Pa,

∂

∂u F(t, z, u) = P−1

dG−1∂

∂x (Q2f(t, x)) −P2Pd=P−1

dG−1Φ(Pdu)Pd,

where Φ is the operator function (3.2), Φ(Pdu) = Φ(xp2), xp2=Pdu∈X2.

180 Maria S. Filipkovska

Let us prove that for any ˆu, ˆ

ˆu∈Rdsuch that (t∗, z∗,ˆu), (t∗, z∗,ˆ

ˆu)∈˜

L0, where

˜

L0=n(t, z, u)∈[0,∞)×Ra×Rd|P−1

dG−1Q2˜

f(t, z, u)−u= 0o,(3.8)

the operator function Ψ : Rd→L(Rd), Ψ(u) = ∂

∂u F(t∗, z∗, u), is basis invert-

ible on conv{ˆu, ˆ

ˆu}. Since (3.2) is basis invertible on conv{ˆxp2,ˆ

ˆxp2}for any

ˆxp2,ˆ

ˆxp2∈X2such that (t∗, x∗

p1+ ˆxp2), (t∗, x∗

p1+ˆ

ˆxp2)∈L0, there exists an

additive resolution of the identity {Θk}d

k=1 in Y2such that the operator Λ1=

Pd

k=1 ΘkΦ(xk

p2)∈L(X2, Y2) is invertible for any set of vectors {xk

p2}d

k=1 ⊂

conv{ˆxp2,ˆ

ˆxp2}. With the help of the invertible operator N=P−1

dG−1:Y2→

Rdwe introduce the system of one-dimensional projectors ˆ

Θk=NΘkN−1, which

form the additive resolution of the identity {ˆ

Θk}d

k=1 in Rd. Chose any ˆu, ˆ

ˆu∈Rd

such that (t∗, z∗,ˆu), (t∗, z∗,ˆ

ˆu)∈˜

L0and any uk∈conv{ˆu, ˆ

ˆu},k= 1, d. Taking

into account that (t, z, u)∈˜

L0⇔(t, xp1+xp2)∈L0and for ˆxp2=Pdˆu,ˆ

ˆxp2=

Pdˆ

ˆu,xk

p2=Pduk,x∗

p1=Paz∗the operator Λ1is invertible, the operator

Λ2=

d

X

k=1

ˆ

Θk

∂

∂u F(t∗, z∗, uk) =

d

X

k=1

ˆ

ΘkP−1

dG−1Φ(Pduk)Pd=NΛ1Pd

acting in Rdis also invertible. Hence, Ψ is basis invertible on conv{ˆu, ˆ

ˆu}.

Let (t∗, z∗) be an arbitrary (ﬁxed) point of [0,∞)×Ra. Due to the condition

(3.1), choose u∗∈Rdsuch that (t∗, z∗, u∗)∈˜

L0. From the basis invertibility of Ψ,

it follows that there exists a continuous linear inverse operator ∂

∂u F(t∗, z∗, u∗)−1.

By the implicit function theorems [20], there exist neighborhoods Uδ(t∗, z∗) =

Uδ1(t∗)×Uδ2(z∗) (if t∗= 0, then Uδ1(t∗) = [0, δ1)), Uε(u∗) and a unique function

u=u(t, z)∈C(Uδ(t∗, z∗), Uε(u∗)), which is continuously diﬀerentiable in zsuch

that F(t, z, u(t, z)) = 0, (t, z)∈Uδ(t∗, z∗), and u(t∗, z∗) = u∗. We deﬁne a global

function u=η(t, z) : [0,∞)×Ra→Rdat the point (t∗, z∗) as η(t∗, z∗) = u(t∗, z∗).

Let us prove that

∀(t, z)∈[0,∞)×Ra∃!u∈Rd(t, z, u)∈˜

L0.(3.9)

Consider arbitrary (ﬁxed) points (t∗, z∗,ˆu),(t∗, z∗,ˆ

ˆu)∈˜

L0. Clearly, F(t∗, z∗,ˆu) =

0, F(t∗, z∗,ˆ

ˆu) = 0. The projections Fk(t, z, u) = ˆ

ΘkF(t, z, u), k= 1, d, are

the functions with values in the one-dimensional spaces Rk=ˆ

ΘkRdisomor-

phic to R. According to the formula of ﬁnite increments [20], Fk(t∗, z∗,ˆ

ˆu)−

Fk(t∗, z∗,ˆu) = ∂

∂u Fk(t∗, z∗, uk)(ˆ

ˆu−ˆu) = 0, uk∈conv{ˆu, ˆ

ˆu},k= 1, d.

Hence, ˆ

Θk∂

∂u F(t∗, z∗, uk)(ˆ

ˆu−ˆu) = 0, k= 1, d, from which, by summing

these expressions over k, we obtain that Λ2(ˆ

ˆu−ˆu) = 0, where the operator

Λ2=Pd

k=1 ˆ

Θk∂

∂u F(t∗, z∗, uk) = Pd

k=1 ˆ

ΘkΨ(uk) is invertible by virtue of the ba-

sis invertibility of Ψ (see above). Consequently, ˆ

ˆu= ˆu.

Thus, (3.9) is proved. It is also proved that in some neighborhood of each

point (t∗, z∗)∈[0,∞)×Rathere exists a unique solution u=ν(t, z) of (3.6),

which is continuous in (t, z) and continuously diﬀerentiable in z. So, the function

Lagrange Stability and Instability of Semilinear DAEs and Applications 181

u=η(t, z) coincides with ν(t, z) in this neighborhood and it is a solution of (3.6)

with the corresponding smoothness properties. Let us show that the function

u=η(t, z) is unique on the whole domain of deﬁnition. Indeed, if there exists

a function u=µ(t, z) having the same properties as u=η(t, z) at some point

(t∗, z∗)∈[0,∞)×Ra, then by (3.9), η(t∗, z∗) = µ(t∗, z∗) = u∗. Therefore, η(t, z) =

µ(t, z) on [0,∞)×Ra.

Substituting the function u=η(t, z) into (3.5) and denoting g(t, z) =

Q1˜

f(t, z, η(t, z)), we get

d

dtz=P−1

aG−1[−BPaz+g(t, z)].(3.10)

By the properties of η,Q1˜

f, the function g(t, z) is continuous in (t, z) and

continuously diﬀerentiable in zon [0,∞)×Ra. Hence, for each initial point

(t0, z0) such that (t0, z0, η(t0, z0)) ∈˜

L0there exists a unique solution z(t) of

the Cauchy problem for equation (3.10) on some interval [t0, ε) with the initial

condition z(t0) = z0. Note that if (t0, x0)∈L0and x0=Paz0+Pdη(t0, z0), then

(t0, z0, η(t0, z0)) ∈˜

L0.

Introduce the function

V(P1x) = 1

2(HP1x, P1x) = 1

2(HPaz, Paz) = 1

2(P∗

aHPaz, z) = 1

2(ˆ

Hz, z) = ˆ

V(z),

where ˆ

H=P∗

aHPaand His an operator from (3.3). Then grad ˆ

V(z) = ˆ

Hz,

where grad ˆ

Vis the gradient of the function ˆ

V. Since HPaz, G−1[−BPaz+

Q1f(t, Paz+Pdη(t, z))]=ˆ

H z, P −1

aG−1[−BPaz+g(t, z)], then, by (3.3), there

exists ˆ

R > 0 such that

(ˆ

H z, P −1

aG−1[−BPaz+g(t, z)]) ≤k(t)U(ˆ

V), t ≥0,kzk ≥ ˆ

R, (3.11)

where k∈C([0,∞),R) and U∈C((0,∞),(0,∞)) such that R+∞

c

dv

U(v)= +∞.

Taking into account (3.11), for all t≥0 and all zsuch that kzk ≥ ˆ

R, the

derivative ˙

ˆ

V(3.10)of the function ˆ

Valong the trajectories of (3.10) (see the

deﬁnition in [10, Chapter 2]) satisﬁes the estimate

˙

ˆ

V(3.10)= ( ˆ

H z, P −1

aG−1[−BPaz+g(t, z)]) ≤k(t)U(ˆ

V).

It follows from the properties of the functions k,Uthat the inequality

˙v≤k(t)U(v), t≥0, has no positive solution v(t) with ﬁnite escape time (see [10,

Chapter 4]). Then, by [10, Ch. 4, Theorem XIII], every solution z(t) of (3.10) is

deﬁned in the future (i.e., the solution is deﬁned on [t0,∞)). Thus, the function

x(t) = Paz(t) + Pdη(t, z(t)) is a solution of the Cauchy problem (1.1), (1.2) on

[t0,∞).

Let us verify the uniqueness of the global solution. It follows from what

has been proved that the global solution x(t) is unique on some interval [t0, ε).

Assume that the solution is not unique on [t0,∞). Then there exists t∗≥εand

182 Maria S. Filipkovska

two diﬀerent global solutions x(t), ˜x(t) with the common value x∗=x(t∗) =

˜x(t∗). Let us take the point (t∗, x∗) as the initial point. Then there must be a

unique solution of (1.1) on some interval [t∗, ε1) with the initial value x(t∗) = x∗,

which contradicts the assumption.

II (Boundedness). We prove the second part of the theorem, that is, the

Lagrange stability of the DAE. Suppose that the additional conditions of the

theorem are satisﬁed.

Since R+∞

t0k(t)dt < +∞, the inequality ˙v≤k(t)U(v), t≥0, has no un-

bounded positive solution for t≥0 [10, Ch. 4]. Then by [10, Chapter 4, The-

orem XV], equation (3.10) is Lagrange stable. Hence, supt∈[t0,∞)kz(t)k<+∞,

i.e.,

∃M∗∈(0,∞)∀t∈[t0,∞)kz(t)k ≤ M∗.(3.12)

Taking into account the properties of Φ (3.2) and the connection between

Φ and the operator function Ψ : Rd→L(Rd) introduced in part I of the proof,

we get that there exists a point ˜u∈Rd(˜u=P−1

d˜xp2) such that for any ˜

˜u∈

Rd, satisfying (t∗, z∗,˜

˜u)∈˜

L0, the operator function Ψ is basis invertible on

conv{˜u, ˜

˜u}\{˜u}. Let (t∗, z∗,˜

˜u)∈˜

L0be an arbitrary (ﬁxed) point and ˜ube a

point with the property imposed above. Then using the formula of ﬁnite incre-

ments for Fk(t∗, z∗,˜

˜u) and Fk(t∗, z∗,˜u), where Fk(t, z, u) = ˆ

ΘkF(t, z, u), Fis the

mapping (3.7) and {ˆ

Θk}d

k=1 is an additive resolution of the identity in Rd, and

summing the obtained equalities over k, we get that F(t∗, z∗,˜

˜u)−F(t∗, z∗,˜u) =

Λ2(˜

˜u−˜u), where Λ2=Pd

k=1 ˆ

ΘkΨ(uk), Ψ(uk) = ∂

∂u F(t∗, z∗, uk), uk∈conv{˜u, ˜

˜u}\

{˜u}, i.e., uk=α˜

˜u+ (1 −α)˜u,α∈(0,1], k= 1, d. It follows from the ba-

sis invertibility of Ψ on conv{˜u, ˜

˜u} \ {˜u}that there exists the inverse operator

Λ−1

2∈L(Rd). The mentioned above and the fact that F(t∗, z∗,˜

˜u) = 0, give us

˜

˜u= ˜u−Λ−1

2[P−1

dG−1Q2˜

f(t∗, z∗,˜u)−˜u], which is fulﬁlled for an arbitrary point

(t∗, z∗,˜

˜u)∈˜

L0. Consequently, for each t∗∈[t0,∞), the equality η(t∗, z(t∗)) =

˜u−Λ−1

2[P−1

dG−1Q2˜

f(t∗, z(t∗),˜u)−˜u], where z(t) and η(t, z (t)) are components of

the global solution x(t) = Paz(t)+Pdη(t, z(t)) of the Cauchy problem (1.1), (1.2),

holds. Denote ˜

M=k˜uk. Taking into account that Λ−1

2is a bounded linear opera-

tor (since Λ−1

2∈L(Rd)), there exists a constant N > 0 such that kη(t∗, z(t∗))k ≤

(1 + N)˜

M+NkP−1

dG−1k kQ2˜

f(t∗, z(t∗),˜u)kfor each t∗∈[t0,∞). Then it follows

from (3.12), (3.4) that there exists a constant C > 0 such that kη(t∗, z(t∗))k ≤ C

for each t∗∈[t0,∞).

Since the estimate kx(t)k=kPaz(t) + Pdη(t, z(t))k≤kPakM∗+kPdkCis

fulﬁlled for all t∈[t0,∞), the solution x(t) of the Cauchy problem (1.1), (1.2)

is Lagrange stable, which holds for each initial point (t0, x0)∈L0. Hence, for

the initial points (t0, x0)∈L0, equation (1.1) is Lagrange stable. The theorem is

proven.

Remark 3.2. The consistency condition (t0, x0)∈L0for the initial point

(t0, x0) is one of the necessary conditions for the existence of a solution of the

Cauchy problem (1.1), (1.2).

Remark 3.3.If Φ (3.2) is basis invertible on conv{ˆxp2,ˆ

ˆxp2}for any ˆxp2,ˆ

ˆxp2∈

X2,t∗∈[0,∞), x∗

p1∈X1, then obviously it is basis invertible on conv{ˆxp2,ˆ

ˆxp2}

Lagrange Stability and Instability of Semilinear DAEs and Applications 183

for any ˆxp2,ˆ

ˆxp2such that (t∗, x∗

p1+ˆxp2),(t∗, x∗

p1+ˆ

ˆxp2)∈L0and on conv{˜xp2,˜

˜xp2}\

{˜xp2}for any ˜xp2and any ˜

˜xp2such that (t∗, x∗

p1+˜

˜xp2)∈L0. The veriﬁcation

of the basis invertibility of Φ on conv{ˆxp2,ˆ

ˆxp2}for any ˆxp2,ˆ

ˆxp2,t∗,x∗

p1may be

more convenient for applications.

4. Lagrange instability of the semilinear DAE

Below is the theorem on the Lagrange instability of the DAE (1.1), which

gives suﬃcient conditions for the existence and uniqueness of solutions with a

ﬁnite escape time for the Cauchy problem (1.1), (1.2), where the initial points

(t0, x0) satisfy the consistency condition (t0, x0)∈L0and the corresponding

components P1x0belong to a certain region Ω.

Theorem 4.1. Let f∈C([0,∞)×Rn,Rn)have a continuous partial deriva-

tive ∂

∂x f(t, x)on [0,∞)×Rn,λA +Bbe a regular pencil of index 1 and (3.1)be

fulﬁlled. Let for any ˆxp2,ˆ

ˆxp2∈X2such that (t∗, x∗

p1+ ˆxp2),(t∗, x∗

p1+ˆ

ˆxp2)∈L0

the operator function (3.2)be basis invertible on conv{ˆxp2,ˆ

ˆxp2}. Further, let there

exist a region Ω⊂X1such that P1x= 0 6∈ Ωand the component P1x(t)of each

existing solution x(t)with the initial point (t0, x0)∈L0, where P1x0∈Ω, remains

all the time in Ω. Suppose for some self-adjoint positive operator H∈L(X1)there

exist the functions k∈C([0,∞),R),U∈C((0,∞),(0,∞)) such that

Z+∞

c

dv

U(v)<+∞(c > 0),Z+∞

t0

k(t)dt =∞,

(HP1x, G−1[−BP1x+Q1f(t, x)])

≥k(t)U1

2(HP1x, P1x),(t, x)∈L0, P1x∈Ω.(4.1)

Then for each initial point (t0, x0)∈L0, where P1x0∈Ω, there exists a unique

solution of the Cauchy problem (1.1),(1.2)and this solution has a ﬁnite escape

time.

Proof. The beginning of the proof of Theorem 4.1 coincides with the proof of

Theorem 3.1 up to the following statement. For each initial point (t0, z0) such that

(t0, z0, η(t0, z0)) ∈˜

L0, there exists a unique solution z(t) of the Cauchy problem

for equation (3.10) on some interval [t0, ε) with the initial condition z(t0) = z0.

Hence, for each initial point (t0, x0)∈L0, where x0=Paz0+Pdη(t0, z0), there

exists a unique solution x(t) = Paz(t) + Pdη(t, z(t)) of the Cauchy problem (1.1),

(1.2) on [t0, ε).

Further, the proof takes the form.

By the condition of Theorem 4.1, there exists a region Ω ⊂X1such that

P1x= 0 6∈ Ω and the component P1x(t) of each solution x(t) with the initial point

(t0, x0)∈L0, where P1x0∈Ω, remains all the time in Ω. Taking into account

that P1x=Paz, each solution z(t) of equation (3.10) starting in the region ˆ

Ω =

{z∈Ra|Paz∈Ω}=P−1

aΩ remains all the time in it, and z= 0 6∈ ˆ

Ω. Introduce

the function ˆ

V(z) = 1

2(ˆ

Hz, z), where ˆ

H=P∗

aHPaand His an operator from

184 Maria S. Filipkovska

(4.1). Clearly, the function ˆ

V(z) is positive for all z∈ˆ

Ω. It follows from (4.1)

that

(ˆ

H z, P −1

aG−1[−BPaz+g(t, z)]) ≥k(t)U(ˆ

V), t ≥0, z ∈ˆ

Ω,(4.2)

where k∈C([0,∞),R), U∈C((0,∞),(0,∞)) such that R+∞

c

dv

U(v)<+∞,

R+∞

t0k(t)dt =∞.

By (4.2), for all t≥0 and all z∈ˆ

Ω, the derivative of ˆ

Valong the trajectories

of (3.10) satisﬁes the estimate

˙

ˆ

V(3.10)= ( ˆ

H z, P −1

aG−1[−BPaz+g(t, z)]) ≥k(t)U(ˆ

V).

It follows from the properties of the functions k,Uthat the inequality

˙v≥k(t)U(v), t≥0, has no positive solution deﬁned in the future (see [10, Chap-

ter 4]). By [10, Chapter 4, Theorem XIV], each solution z(t) of (3.10) satisfying

the condition z(t0) = z0, where z0∈ˆ

Ω and (t0, z0, η(t0, z0)) ∈˜

L0, has a ﬁnite

escape time, i.e., it exists on some ﬁnite interval [t0, T ) and limt→T−0kz(t)k=

+∞. Then each function x(t) = Paz(t) + Pdη(t, z(t)) with the corresponding

initial values (t0, x0), where x0=Paz0+Pdη(t0, z0), is a solution of the Cauchy

problem (1.1), (1.2) with a ﬁnite escape time, i.e., the solution x(t) is deﬁned on

the corresponding ﬁnite interval [t0, T ) and limt→T−0kx(t)k= +∞.

Let us verify the uniqueness of the solution x(t), t∈[t0, T ). It was proved

that the solution x(t) is unique on some interval [t0, ε). Assume that the solution

is not unique on [t0, T ). Then there exists t∗∈[ε, T ) and two diﬀerent solutions

x(t), ˜x(t) with the common value x∗=x(t∗) = ˜x(t∗) such that (t∗, x∗)∈L0and

P1x∗∈Ω. Let us take the point (t∗, x∗) as the initial point. Then there must be

a unique solution of (1.1) on some interval [t∗, ε1)⊂[t0, T ) with the initial value

x(t∗) = x∗, which contradicts the assumption. The theorem is proven.

5. Lagrange stability of the mathematical model of a radio

engineering ﬁlter

Let us consider the electrical circuit of a radio engineering ﬁlter given in

Fig. 5.1. A voltage source e, nonlinear resistances ϕ,ϕ0,ψ, a nonlinear con-

ductance h, a linear resistance r, a linear conductance g, an inductance Land a

capacitance Care given.

The currents and voltages in the circuit satisfy the Kirchhoﬀ equations, as

well as the constraint equations which describe operation modes of the electric

circuit elements:

IL=I+Iψ, Uψ=Uϕ+Ur+UC, e =Uϕ0+UL+Uψ,

UL=d(LIL)

dt , I =d(CUC)

dt +gUC+h(UC),

Ur=rI, Uϕ=ϕ(I), Uϕ0=ϕ0(IL), Uψ=ψ(Iψ).

Lagrange Stability and Instability of Semilinear DAEs and Applications 185

Fig. 5.1: The electric circuit diagram of the radio engineering ﬁlter.

From these equations we obtain the system with the variables x1=IL,x2=

UC,x3=I:

Ld

dtx1+x2+r x3=e(t)−ϕ0(x1)−ϕ(x3),(5.1)

Cd

dtx2+gx2−x3=−h(x2),(5.2)

x2+rx3=ψ(x1−x3)−ϕ(x3).(5.3)

The system describes a transient process in the electrical circuit (i.e., the

process of transition from one operation mode of the electric circuit to another).

It is assumed that the linear parameters L,C,r,gare positive and real,

ϕ0∈C1(R), ϕ∈C1(R), ψ∈C1(R), h∈C1(R) and e∈C([0,∞),R).

The vector form of system (5.1)–(5.3) is the semilinear DAE

d

dt[Ax] + Bx =f(t, x),(5.4)

where x= (x1, x2, x3)T= (IL, UC, I)T∈R3,

f(t, x) =

e(t)−ϕ0(x1)−ϕ(x3)

−h(x2)

ψ(x1−x3)−ϕ(x3)

, A =

L0 0

0C0

0 0 0

, B =

0 1 r

0g−1

0 1 r

.

It is easy to verify that λA +Bis a regular pencil of index 1.

The projection matrices Pi,Qiand the matrix G−1have the form

P1=

1 0 0

0 1 0

0−r−10

, P2=

0 0 0

0 0 0

0r−11

,

Q1=

1 0 −1

0 1 r−1

0 0 0

, Q2=

0 0 1

0 0 −r−1

0 0 1

,

G−1=

L−10−L−1

0C−1(Cr)−1

0−(Cr)−1(Cr −1)C−1r−2

.

186 Maria S. Filipkovska

The projections of the vector xhave the form

xp1=P1x= (x1, x2,−r−1x2)T= (a, −rb, b)T,

xp2=P2x= (0,0, r−1x2+x3)T= (0,0, u)T,

where a=x1,b=−r−1x2,u=r−1x2+x3∈R.

The equation Q2[Bx −f(t, x)] = 0, determining the manifold L0from (3.1), is

equivalent to equation (5.3). Taking into account the new notation, the condition

(3.1) holds if for any a, b ∈Rthere exists u∈Rsuch that

ru =ψ(a−b−u)−ϕ(b+u).(5.5)

Consider the operator function ˜

Φ: X2→L(R3, Y2),

˜

Φ(xp2) = ∂

∂x Q2f(t∗, x∗

p1+xp2)−BP2

=ψ0(a∗−b∗−u) + ϕ0(b∗+u) + r

0−r−1−1

0r−2r−1

0−r−1−1

,

where ψ0(a−b−u) = dψ(y)

dy y=a−b−u,ϕ0(b+u) = dϕ(y)

dy y=b+u,t∗∈[0,∞),

a∗, b∗∈R,x∗

p1= (a∗,−rb∗, b∗)T. Since the spaces X2,Y2are one-dimensional,

the invertibility of the operator function Φ = ˜

ΦX2

:X2→L(X2, Y2) (i.e., the

operator Φ(xp2)∈L(X2, Y2) is the restriction of the operator ˜

Φ(xp2)∈L(R3, Y2)

to X2) is equivalent to the basis invertibility of Φ. Let for any (ﬁxed) ˆu, ˆ

ˆu, a∗, b∗∈

Rsatisfying (5.5), the condition ψ0(a∗−b∗−u∗) + ϕ0(b∗+u∗)6=−rbe fulﬁlled

for any u∗∈conv{ˆu, ˆ

ˆu}. Then the operator Λ = ˜

ΛX2

∈L(X2, Y2), where ˜

Λ =

˜

Φ(x∗

p2), x∗

p2= (0,0, u∗)T, is invertible since from ˜

Λxp2= 0, xp2∈X2, it follows

that xp2= 0. Hence, for any ˆu, ˆ

ˆu, a∗, b∗∈Rsatisfying (5.5), the operator function

Φ (3.2) is basis invertible on the convex hull conv{ˆxp2,ˆ

ˆxp2}, where ˆxp2= (0,0,ˆu)T,

ˆ

ˆxp2= (0,0,ˆ

ˆu)T.

Choose

H=

2L0 0

0Cr 0

0 0 Cr3

.

Then

HP1x, G−1[−BP1x+Q1f(t, x)]

= 2−(gr+1)x2

2−x1ϕ0(x1)+(x2−x1)ψ(x1−x3)−rx2h(x2)−x2ϕ(x3)+x1e(t).

Since ϕ, ψ ∈C1(R), there exists a constant Csuch that for any ﬁxed

˜xp2= (0,0,˜u)T, where ˜u∈R, and for all t∈[0,∞), kxp1k ≤ M, where Mis

a number, the estimate

kQ2f(t, xp1+ ˜xp2)k ≤ p2 + r−2max

kxp1k≤M|ψ(a−b−˜u)−ϕ(b+ ˜u)| ≤ C

Lagrange Stability and Instability of Semilinear DAEs and Applications 187

is fulﬁlled. Hence, the condition (3.4) is satisﬁed for any ﬁxed ˜xp2= (0,0,˜u)T

(i.e., any ﬁxed ˜u∈R).

5.1. Conclusions. By Theorem 3.1 for each initial point (t0, x0)∈[0,∞)×

R3(x0= (x0

1, x0

2, x0

3)T) satisfying the consistency condition (the equation (5.3))

x0

2+rx0

3=ψ(x0

1−x0

3)−ϕ(x0

3),(5.6)

there exists a unique solution x(t) of the Cauchy problem for the DAE (5.4) with

the initial condition

x(t0) = x0(5.7)

on the whole interval [t0,∞) if:

1) for any a, b ∈Rthere exists u∈Rsuch that (5.5) is fulﬁlled;

2) for any ˆu, ˆ

ˆu, a∗, b∗∈Rsatisfying (5.5), the condition ψ0(a∗−b∗−u∗)+ ϕ0(b∗+

u∗)6=−ris fulﬁlled for any u∗∈conv{ˆu, ˆ

ˆu};

3) for some number R > 0, there exist the functions k∈C([0,∞),R), U∈

C((0,∞),(0,∞)) such that R+∞

c

dv

U(v)= +∞and

−(gr + 1)x2

2−x1ϕ0(x1)+(x2−x1)ψ(x1−x3)−rx2h(x2)−x2ϕ(x3)+

+x1e(t)≤k(t)ULx2

1+Crx2

2

for any t≥0, x∈R3such that (5.3), kP1xk=px2

1+ (1 + r−2)x2

2≥R.

If, additionally, R+∞

t0k(t)dt < +∞and

4) there exists ˜u∈Rsuch that for any ˜

˜u, a∗, b∗∈Rsatisfying (5.5), the condition

ψ0(a∗−b∗−u∗) + ϕ0(b∗+u∗)6=−ris fulﬁlled for any u∗∈conv{˜u, ˜

˜u}\{˜u}

(i.e., u∗=α˜

˜u+ (1 −α)˜u,α∈(0,1] ),

then for the initial points (t0, x0) equation (5.4) is Lagrange stable.

In terms of physics it means that if the input voltage e(t)∈C([0,∞),R), the

nonlinear resistances ϕ, ϕ0, ψ ∈C1(R) and the nonlinear conductance h∈C1(R)

satisfy the aforementioned conditions 1)–3), then for any initial time moment

t0≥0 and any initial values IL(t0), UC(t0), I(t0) satisfying UC(t0) + rI(t0) =

ψ(IL(t0)−I(t0)) −ϕ(I(t0)), there exist the currents IL(t), I(t) and voltage UC(t)

in the circuit (Fig. 5.1) for all t≥t0, which are uniquely determined by the

initial values. The functions IL(t), UC(t) are continuously diﬀerentiable and the

function I(t) is continuous on [t0,∞). The currents and voltage are bounded for

all t≥t0(Lagrange stability) if, additionally, R+∞

t0k(t)dt < +∞, and condition

4) is satisﬁed. The remaining currents and voltages in the circuit are uniquely

expressed in terms of IL(t), I(t), UC(t).

Let us consider the particular cases:

ϕ0(y) = α1y2k−1, ϕ(y) = α2y2l−1, ψ(y) = α3y2j−1, h(y) = α4y2s−1,(5.8)

ϕ0(y) = α1y2k−1, ϕ(y) = α2sin y, ψ(y) = α3sin y, h(y) = α4sin y, (5.9)

188 Maria S. Filipkovska

where k, l, j, s ∈N,αi>0, i= 1,4, y∈R. Note that the functions of the types

(5.8), (5.9) for nonlinear resistances and conductances are encountered in real

radio engineering devices.

For the functions of the form (5.8) and each initial point (t0, x0) satisfying

(5.6), there exists a unique solution of the Cauchy problem (5.4), (5.7) on [t0,∞)

if j≤k,j≤sand α3is suﬃciently small. For the functions of the form

(5.9) and each initial point (t0, x0) satisfying (5.6), there exists a unique solution

of the Cauchy problem (5.4), (5.7) on [t0,∞) if α2+α3< r. If, additionally,

supt∈[0,∞)|e(t)|<+∞or R+∞

t0|e(t)|dt < +∞, then for the initial points (t0, x0)

the DAE (5.4) is Lagrange stable (in both cases), i.e., every solution of the DAE

is bounded. In particular, these requirements are fulﬁlled for voltages of the form

e(t) = β(t+α)−n, e(t) = βe−αt , e(t) = βe−(t−α)2

σ2, e(t) = βsin(ωt +θ),(5.10)

where α > 0, β, σ, ω ∈R,n∈N,θ∈[0,2π]. For voltage having the form

e(t) = β(t+α)n, α, β ∈R, n ∈N,(5.11)

global solutions exist, but they are not bounded on the whole interval [t0,∞).

5.2. Numerical analysis. We ﬁnd approximate solutions of the DAE

(5.4) (system (5.1)–(5.3)) with the initial condition (5.7) using the numerical

method given in [6].

Choose the parameters L= 500, C= 0.5, r= 2, g= 0.2 and the input

voltage e(t) = 100 e−tsin(5 t). For the nonlinear resistances and conductance of

the form (5.8) with k=l=j=s= 2, αi= 1, i= 1,4, the numerical solution

with the initial values t0= 0, x0= (0,0,0)Tis obtained. The components of the

obtained solution are shown in Fig. 5.2.

The components of the solution for the electrical circuit with the linear pa-

rameters L= 50, C= 1, r= 0.001, g= 1, the nonlinear parameters (5.8), where

k=l=j=s= 2, αi= 1, i= 1,3, α4= 0.01, and the input voltage e(t) = 2 sin t,

and for the initial values t0= 0, x0= (0,0,0)T, are shown in Fig. 5.3.

For the linear parameters L= 300, C= 0.5, r= 2.6, g= 0.2, the nonlinear

resistances and conductance (5.9), where k= 2, α1= 0.5, α2= 1.5, α3= 1,

α4= 3, and the voltage e(t) = 200 sin(0.5t)−0.2, the solution components with

the initial values t0= 0, x0= (π/6,0.5,0)Tare shown in Fig. 5.4.

For the linear parameters L= 1, C= 5, r= 1.51, g= 5, the nonlinear param-

eters (5.9), where k= 2, αi= 1, i= 1,2,4, α3= 0.5, the voltage e(t)=(t+ 30)−2

and the initial values t0= 0, x0= (0,0,0)Tthe solution components are shown

in Fig. 5.5.

The components of the solution for the electrical circuit with the linear pa-

rameters L= 1000, C= 0.5, r= 2, g= 0.3, the nonlinear parameters (5.8) with

k=l=j=s= 2, αi= 1, i= 1,4, the input voltage e(t) = −t2, and for the

initial values t0= 0, x0= (0,0,0)Tare shown in Fig. 5.6.

Lagrange Stability and Instability of Semilinear DAEs and Applications 189

(a) The current IL(t) (b) The voltage UC(t)

(c) The current I(t)

Fig. 5.2: (a)–(c): The components of the numerical solution.

(a) The current IL(t) (b) The voltage UC(t)

(c) The current I(t)

Fig. 5.3: (a)–(c): The components of the numerical solution.

For the linear parameters L= 100, C= 5, r= 3, g= 4, the nonlinear

parameters (5.9), where k= 2, α1= 1, α2= 0.9, α3= 2, α4= 5, the volt-

age e(t) = (t−50)3and the initial values t0= 0, x0= (0,0,0)Tthe solution

components are shown in Fig. 5.7.

190 Maria S. Filipkovska

(a) The current IL(t) (b) The voltage UC(t)

(c) The current I(t)

Fig. 5.4: (a)–(c): The components of the numerical solution.

(a) The current IL(t) (b) The voltage UC(t)

(c) The current I(t)

Fig. 5.5: (a)–(c): The components of the numerical solution.

The numerical solutions shown in Figs. 5.2–5.5 are bounded on the corre-

sponding time intervals. When we increase the time intervals by a factor of 5–10,

the solutions are also bounded. The analysis of these numerical solutions indicates

Lagrange Stability and Instability of Semilinear DAEs and Applications 191

(a) The current IL(t) (b) The voltage UC(t)

(c) The current I(t)

Fig. 5.6: (a)–(c): The components of the numerical solution.

(a) The current IL(t) (b) The voltage UC(t)

(c) The current I(t)

Fig. 5.7: (a)–(c): The components of the numerical solution.

that there exist bounded global solutions of equation (5.4) (system (5.1)–(5.3))

with the input voltage of the form (5.10) and the nonlinear resistances and con-

ductance of the form (5.8), (5.9). The analysis of the numerical solutions shown

192 Maria S. Filipkovska

in Figs. 5.6,5.7 indicates that there exist global solutions, increasing without

bound with increasing time (as t→ ∞), for equation (5.4) (system (5.1)–(5.3))

with the input voltage of the form (5.11) and the nonlinear parameters of the

form (5.8), (5.9). Similar results follow from the application of Theorem 3.1.

Therefore, the conclusions obtained with the help of this theorem are veriﬁed by

a numerical experiment.

6. Lagrange instability of the mathematical model of a radio

engineering ﬁlter

Consider system (5.1)–(5.3) (the DAE (5.4)) with the nonlinear resistances

and conductance

ϕ0(x1) = −x2

1, ϕ(x3) = x3

3, ψ(x1−x3)=(x1−x3)3, h(x2) = x2

2.(6.1)

It is assumed that there exists Me= supt∈[t0,∞)|e(t)|<+∞.

The veriﬁcation of the condition (3.1) and the condition for the operator

function (3.2) is similar to that given in Section 5. Thus, it is easy to verify that

these requirements are fulﬁlled.

Denote z= (x1, x2)T∈R2. Choose

ΩR2=z=x1

x2∈R2|x1> m1,

m1= max (1 + pMe,3

rg+1

r,3C

L,smax L

3rC −r

3,0),

x2<−rx1−x3

1−m2, m2= max g−2Cr

L,0,(6.2)

Ω = {xp1=P1x∈X1|z∈ΩR2}.

Since xp1= (x1, x2,−r−1x2)T, then xp1∈Ω⇔z∈ΩR2. Obviously, xp1= 0 6∈ Ω.

The boundary of the region ΩR2consists of the parts x1=m1and x2+

rx1+x3

1+m2= 0. Since x1≥m1,d

dt x1>0 and x2+rx1+x3

1+m2≤

0, d

dt (x2+rx1+x3

1+m2)<0 for all t≥0, x= (x1, x2, x3)T∈R3satisfying

(5.3) (the condition (t, x)∈L0), where z= (x1, x2)T∈ΩR2(ΩR2is the closure

of ΩR2), the component z(t) = (x1(t), x2(t))Tof each existing solution, which

starts at time t0≥0 in the region ΩR2, cannot leave this region. Consequently,

the component xp1(t) = P1x(t) of each existing solution x(t) with the initial

point (t0, x0)∈[0,∞)×R3(x0= (x0

1, x0

2, x0

3)T) satisfying (5.6), where P1x0∈Ω

((x0

1, x0

2)T∈ΩR2), remains all the time in Ω.

We choose H=

2L0 0

0C0

0 0 C r2

. Then for any x= (x1, x2, x3)Tsatisfying

(5.3) and such that (x1, x2)T∈ΩR2, the condition

HP1x, G−1[−BP1x+Q1f(t, x)]= 2e(t)x1−(g+r−1)x2

2+x3

1+

Lagrange Stability and Instability of Semilinear DAEs and Applications 193

+ (r−1x2−x1)(x1−x3)3−x3

2−r−1x2x3

3>2−(g+r−1)x2

2+x3

2≥α v3/2,

where v=1

2(HP1x, P1x) = Lx2

1+Cx2

2and α > 0 is a certain constant, is fulﬁlled.

Hence, the condition (4.1), where k(t)≡1, U(v) = α v3/2, is fulﬁlled.

Thus, all the conditions of Theorem 4.1 are satisﬁed.

6.1. Conclusions. By Theorem 4.1, for each initial point (t0, x0)∈

[0,∞)×R3satisfying (5.6) and such that (x0

1, x0

2)T∈ΩR2, where ΩR2is the

region (6.2), there exists a unique solution of the Cauchy problem for the DAE

(5.4) with the initial condition (5.7), where the functions ϕ0,ϕ,ψ,hhave the

form (6.1) and supt∈[t0,∞)|e(t)|<+∞, and this solution has a ﬁnite escape time

(the solution exists on some ﬁnite interval and it is unbounded).

In terms of physics it means that if supt∈[t0,∞)|e(t)|<+∞and the nonlinear

resistances and conductance have the form (6.1), then for any initial time moment

t0≥0 and any initial values IL(t0), UC(t0), I(t0) satisfying UC(t0) + rI(t0) =

ψ(IL(t0)−I(t0)) −ϕ(I(t0)) and such that (IL(t0), UC(t0))T∈ΩR2, on some ﬁnite

interval t0≤t < T there exist the currents IL(t), I(t) and the voltage UC(t) in

the circuit in Fig. 5.1, which are uniquely determined by the initial values, and

limt→T−0

(IL(t), UC(t), I(t))T

= +∞.

6.2. Numerical analysis. We ﬁnd approximate solutions for the DAE

(5.4) (system (5.1)–(5.3)) with the functions of nonlinear resistances and conduc-

tance (6.1) and the initial condition (5.7). The initial values t0,x0= (x0

1, x0

2, x0

3)T

are chosen such that (5.6) is satisﬁed and (x0

1, x0

2)T∈ΩR2, where ΩR2is (6.2).

Choose the parameters L= 10, C= 0.5, r= 2, g= 0.2, the input voltage

e(t) = 2 sin tand the initial values t0= 0, x0= (2.45,−20.625125,2.5)T. The

components of the obtained numerical solution are shown in Fig. 6.1.

(a) The current IL(t) (b) The voltage UC(t) (c) The current I(t)

Fig. 6.1: (a)–(c): The components of the numerical solution.

For the electrical circuit with the linear parameters L= 5, C= 0.5, r= 2,

g= 0.5 and the input voltage e(t) = 0, the components of the numerical solution

with the initial values t0= 0, x0= (1.1,−4.129,1.2)Thave the form similar to

that shown in Fig. 6.1.

194 Maria S. Filipkovska

The analysis of the obtained numerical solutions shows that the corresponding

exact solutions have a ﬁnite escape time and veriﬁes the results obtained with

the help of Theorem 4.1.

7. Conclusions

The theorems, enabling to prove the existence and boundedness of global so-

lutions (Lagrange stability) of the semilinear DAE (1.1) or their non-existence

(solutions have a ﬁnite escape time, i.e., they are Lagrange unstable), are ob-

tained. Using these theorems, we have found the restrictions on the initial data

and the parameters of the electrical circuit (Fig. 5.1) of the nonlinear radio engi-

neering ﬁlter under which the mathematical model (the DAE (5.4)) of the circuit

is Lagrange stable, and the conditions under which the mathematical model is

Lagrange unstable. The functions and quantities deﬁning the circuits parameters

(resistances, conductivities and others) and satisfying the obtained conditions

have been given. It has been checked that the mentioned conditions of the La-

grange stability are fulﬁlled for certain classes of nonlinear functions which do

not satisfy the global Lipschitz condition. In particular, it has been proven that

the presence of nonlinear resistances and conductivities of the form (5.8), (5.9)

in electric circuits admits the Lagrange stability of the corresponding mathemat-

ical models. Notice that nonlinear resistances and conductivities of this type are

often encountered in real radio engineering systems.

The results of the study of the mathematical model have shown that the

obtained theorems can be eﬀectively applied in practice. The analysis of the

numerical solutions of the mathematical model veriﬁes the results of theoretical

studies.

Supports. The publication is based on the research provided by grant

support of the State Fund for Fundamental Research of Ukraine (project

F83/45808).

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Received November 4, 2017.

Maria S. Filipkovska,

B. Verkin Institute for Low Temperature Physics and Engineering of the National

Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine,

E-mail: filipkovska@ilt.kharkov.ua

Стiйкiсть за Лагранжем напiвлiнiйних

диференцiально-алгебраїчних рiвнянь та

застосування до нелiнiйних електричних кiл

Maria S. Filipkovska

Проводиться дослiдження напiвлiнiйного диференцiально-алгебра-

їчного рiвняння (ДАР) з акцентом на стiйкiсть (нестiйкiсть) за Ла-

гранжем. Отримано умови iснування та єдиностi глобальних розв’яз-

кiв (розв’язок iснує на нескiнченному iнтервалi) задачi Кошi, а також

умови обмеженостi глобальних розв’язкiв. Бiльш того, отриманi умови

стiйкостi за Лагранжем напiвлiнiйного ДАР гарантують, що кожний йо-

го розв’язок є глобальним i обмеженим, та, на вiдмiну вiд теорем про

стiйкiсть за Ляпуновим, дозволяють довести iснування та єдинiсть гло-

бальних розв’язкiв незалежно вiд наявностi та кiлькостi точок рiвно-

ваги. Також отримано умови iснування та єдиностi розв’язкiв зi скiн-

ченним часом визначення (розв’язок iснує на скiнченному iнтервалi та є

необмеженим, тобто нестiйким за Лагранжем) для задачi Кошi. Не вико-

ристовуються обмеження типу глобальної умови Лiпшиця, що дозволяє

ефективно використовувати результати роботи у практичних застосува-

ннях. В якостi застосування дослiджено математичну модель радiоте-

хнiчного фiльтру з нелiнiйними елементами. Чисельний аналiз моделi

пiдтверджує результати теоретичних дослiджень.

Ключовi слова: диференцiально-алгебраїчне рiвняння, стiйкiсть за

Лагранжем, нестiйкiсть, регулярний жмуток, обмежений глобальний

розв’язок, скiнченний час визначення, нелiнiйне електричне коло.