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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.