Nonlinear Oscillations, Vol. 10, No. 4, 2007
CLOSEDNESS AND NORMAL SOLVABILITY OF AN OPERATOR GENERATED
BY A DEGENERATE LINEAR DIFFERENTIAL EQUATION WITH VARIABLE
COEFFICIENTS
S. M. Zhuk1
UDC 517.9
n
m
m
For a linear operator D : WF
generated by the differential equation
2 ⊂ L2 → L2 × R
d
F x(t) − C(t)x = f (t),
dt
F x(t0 ) = f0 ,
0
m
m
we prove that its graph is closed and determine the adjoint operator D∗ : WF
→ Ln
2 ⊂ L2 × R
2 . For
F
F0
elements of the linear manifolds W2 and W2 , we propose an analog of the formula of integration by
parts. We establish a criterion for the existence of a pseudosolution of the operator equation Dx(·) =
(f (·), f0 ) and formulate sufficient conditions for the normal solvability of the operator D in terms of
relations for blocks of the matrix C(t). The results obtained are illustrated by examples.
1. Introduction
In the process of solution of various problems in applied fields such as mathematical economics, robotics,
biotechnology, digital image processing, control theory, circuit theory, radiophysics, and chemical and biological
kinetics, researchers encounter systems whose state is described by differential equations with degeneracy:
F (t)ẋ(t) + C(t)x(t) + B(t)f (t) = 0.
(1)
In literature, systems of the form (1) are called degenerate [1], algebraic–differential [2], singular [3–5], descriptor2 [6–9], and implicit [10] or unresolved with respect to the derivative [11].
For autonomous systems (1) in the case where the pencil of matrices λF + C is regular,3 the notion of central
canonical form proposed in [5] proved to be fruitful. Namely, for a smooth vector function f (·), a stationary
regular system can be reduced to the following form by a nondegenerate linear transformation [7, p. 14]:
ẋ1 (t) = Ax1 (t) + Kf (t),
x2 (t) = −Df (t) −
m−1
X
i=1
i
(i)
N Dv (t)),
x1
= Q−1 x(t).
x2
Sufficient conditions for the reducibility of (1) to a central canonical form were proposed in [1]. Note that the
theory of singular matrix pencils was used as early as in [12, p. 348] for the description of the set of solutions of
stationary singular linear differential equations.
1
Shevchenko Kyiv National University, Kyiv.
In the present paper, we use the term “descriptor equation” or “descriptor system” to denote systems of the type (2). The emerging
differences are described in what follows (see Remark 1).
3
A pencil of matrices λF + C is called regular if det(λ0 F + C) 6= 0 for a certain real λ0 .
2
Translated from Neliniini Kolyvannya, Vol. 10, No. 4, pp. 464–480, October–December, 2007. Original article submitted September 18,
2007.
1536–0059/07/01004–0469
c 2007
Springer Science+Business Media, Inc.
469
S. M. Z HUK
470
In the case of variable coefficients, the notion of central canonical form gives information on the structure of a
general solution of the system only in special cases, e.g., [2, p. 8] for
rankF (t) = deg det(λF (t) + C(t)) = const,
t ∈ [t0 , T ].
In [2, 3], the problem of the existence of solutions of initial- and boundary-value problems for Eqs. (1) was studied
on the basis of the notion of left (right) regularizing operator [2]:
Λ∗,r [F (t)ẋ(t) + C(t)x(t) + f (t)] = ẋ(t) + Λ∗,r [C(t)]x(t) + Λ∗,r [f (t)],
where
Λ∗,r =
r
X
Lj (t)
j=0
d
dt
i
.
The existence of a left regularizing operator for autonomous systems [2, p. 11] is equivalent to the existence of
a central canonical form. In the case of variable coefficients, conditions for the existence of a left regularizing
operator [2] are connected with properties of the “extended” system (1), i.e., the system that consists of Eq. (1) and
its total time derivatives up to a certain order.
Attention of some investigators is focused on the study of qualitative problems in the theory of descriptor
systems with operator coefficients in Banach spaces. The behavior of a descriptor system with delay in a Banach
space was investigated in [13] on the basis of analysis of poles of the resolvent of the operator pencil R(t, λ) =
(λF (t) + C(t))−1 .
For detailed information on problems in the theory of descriptor systems such as the existence and uniqueness
of solutions, construction of numerical methods, control and observability, see [6, 8].
2. Statement of the Problem
In most papers presented above, system (1) was studied on the basis of notions of central canonical form
and regularizing operators, i.e., the construction of solutions was based on the reduction of Eq. (1) to a certain
canonical form, which enables one to use methods developed for normal ordinary differential equations. In this
case, the structure of the corresponding descriptor equation is restricted.
In the present paper, we use an alternative approach based on the investigation of the properties of an operator
generated by the linear descriptor equation
d
F x(t) − C(t)x(t) = f (t),
dt
(2)
F x(t0 ) = f0
without assuming the possibility of its reduction to a certain canonical form. Operator methods enable one to use
general properties of a broad class of descriptor systems, including so-called “noncausal” systems (see [6, 10]).
In Eq. (2), let F be an m × n matrix, let t 7→ C(t) be a continuous m × n matrix-valued function, and let
f (·) ∈ L2 ([t0 , T ], Rm ) := Lm
2 ,
−∞ < t0 < T < +∞,
f0 ∈ Rm .
C LOSEDNESS AND N ORMAL S OLVABILITY OF AN O PERATOR
471
It is known that, in the case F = E, a vector function x(·) ∈ Ln2 that satisfies the Volterra integral equation of the
second kind
Zt
x(t) = f0 +
(C(s)x(s) + f (s))ds
t0
is a solution of Eq. (2). An equation of this type has a unique solution x(·) in the class of absolutely continuous
functions, and, furthermore, x(·) satisfies Eq. (2) almost everywhere.
Let us clarify what we mean by a solution of (2) in the case of an arbitrary constant rectangular matrix F =
{Fij }m,n
1,1 . We introduce the set
WF2 := {x(·) ∈ Ln2 : F x(·) ∈ Wm
2 },
m
where Wm
2 denotes the collection of all absolutely continuous vector functions from L2 whose derivatives lie
m
in L2 , and F x is an m-vector function whose ith component is a linear combination of components of x(·)
with coefficients {Fij }nj=1 . The linear set WF2 is everywhere dense in Ln2 because Wn2 ⊂ WF2 . By analogy, we
introduce the set
0
0
n
WF2 := {z(·) ∈ Lm
2 : F z(·) ∈ W2 }.
We define a linear operator D by the relations
Dx(·) =
d
F x(·) − Cx(·), F x(t0 ) ,
dt
x(·) ∈ D(D) := WF2 ,
d
m
F x(·) ∈ Lm
2 is the derivative of the vector function F x(·), Cx(·) ∈ L2 is a vector function t 7→
dt
d
C(t)x(t), and F x(t0 ) is the value4 of t 7→ F x(t) for t = t0 . Note that, for x(·) ∈ WF2 , the equality F x(t) =
dt
F ẋ(t) can be absent.
In the case of an arbitrary constant rectangular matrix F, a solution of (2) is understood as an element of the
set WF2 that satisfies the operator equation
where
Dx(·) = (f (·), f0 ).
The operator D is defined in a similar way as in S. Krein’s work [14] for boundary-value problems for linear
differential equations of order n. It follows from the structure of the introduced operator equation that the vector
function x(·) from the space Ln2 belongs to the set of solutions of (2) for the fixed initial condition f0 and
right-hand side f (·) if the vector function F x(·) is absolutely continuous and has the derivative of the class Lm
2
satisfying the first equality in (2) almost everywhere and, furthermore, the second equality in (2) is true. Descriptor
equations of the form (2) were considered in [9, 11].
The aim of the present paper is to investigate properties of the operator D, namely, its closedness and normal
solvability. In terms of descriptor systems, we investigate the problem of the generalized solvability of Eq. (2),
conditions for the continuous (in the metric of the corresponding Hilbert space) dependence of a pseudosolution on
the right-hand side and initial condition, and the approximation of solutions by elements of a regularizing sequence.
The symbol F x(t) makes sense for any t ∈ [t0 , T ] because, by virtue of the inclusion x(·) ∈ WF
2 , the vector function t 7→ F x(t) is
absolutely continuous, which, in the general case, cannot be stated for x(·).
4
S. M. Z HUK
472
3. Closedness of the Operator D. Form of the Adjoint Operator D ∗
Generally speaking, the introduced operator D is not a Fredholm one (Example 3), and, moreover, it is not
even normally solvable (Example 2) unlike the operator of ordinary differentiation in Ln2 . However, in the general
case, it is densely defined and closed (Theorem 1).
0
Theorem 1. For x(·) ∈ WF2 and z(·) ∈ WF2 , the following analog of the formula of integration by parts is
true:
ZT d 0
d
F x(t), z(t) +
F z(t), x(t)
dt = (F x(T ), F 0+ F 0 z(T )) − (F x(t0 ), F 0+ F 0 z(t0 )).
dt
dt
(3)
t0
The operator D is closed, and its adjoint D∗ is determined by the relations
D∗ (z(·), z0 ) = L(z(·), z0 ) := −
d 0
F z(·) − C 0 z(·),
dt
0
D(D∗ ) = D(L) := {(z(·), F 0+ F 0 z(t0 ) + d) : z(·) ∈ WF2 , F 0 z(T ) = 0, F 0 d = 0}.
0
Proof. We choose x(·) ∈ WF2 and z(·) ∈ WF2 and use the formula of integration by parts for the absolutely
continuous vector functions F + F x(·) and F 0 z(·). We obtain
ZT d +
d
F F x(t), F 0 z(t) + F + F x(t), F 0 z(t)
dt = F + F x(T ), F 0 z(T ) − F + F x(t0 ), F 0 z(t0 ) .
dt
dt
t0
By virtue of the Moore–Penrose theorem, we get F F + F = F. Therefore, almost everywhere, we have
F
d +
d
d
F F x(t) =
F F + F x(t) =
F x(t),
dt
dt
dt
whence
ZT t0
ZT ZT d +
d
d
0
+
F F x(t), F z(t) dt =
F
F F x(t), z(t) dt =
F x(t), z(t) dt
dt
dt
dt
t0
t0
for every absolutely continuous z ∈ Lm
and, hence, for any z ∈ Lm
2
2 .
We prove the theorem in the case5 C(t) ≡ 0. Let us show the closedness of D. To this end, we note that the
d
operator x(·) 7→
F x(·), x(·) ∈ WF2 , is closed.
dt
5
A generalization is carried out by standard reasoning on the closedness of the sum of closed and bounded operators and on the operator
adjoint to this sum.
C LOSEDNESS AND N ORMAL S OLVABILITY OF AN O PERATOR
473
d
F xn (·) → z(·),6 then F xn (·) is absolutely continuous and F xn (·) → F x(·).
dt
Taking into account that the operator of differentiation is closed in Lm
2 , we establish that F x(·) is absolutely
d
continuous and
F x(·) = z(·).
dt
d
F x(·) = z(·), and, according to the
Let Dxn (·) → (z(·), z0 ) and xn (·) → x(·). Then x(·) ∈ WF2 ,
dt
results proved above, it remains to show that z0 = F x(t0 ). Indeed, by assumption, we have F xn (t0 ) → z0 . On
the other hand7
Indeed, if xn (·) → x and
t
Z d
d
→ 0,
kF x(t0 ) − F xn (t0 )k ≤ JkF xn (·) − F x(·)k2 + sup F
x(s)
−
F
x
(s)
ds
n
dt
dt
t∈[t0 ,c] t0
1
where J := (c − t0 )− 2 .
We have shown the closedness of the densely defined operator D, which proves the existence of D∗ [15,
p. 40]. Let us show that the other statements are also true.
Let (z(·), z0 ) ∈ D(L) and x(·) ∈ D(D). Then
ZT hDx(·), (z(·), z0 )i1 =
d
F x(t), z(t) dt + F x(t0 ), F 0+ F 0 z(t0 )
dt
t0
0+
ZT 0
= (F x(T ), F F z(T )) −
d 0
F z(t), x(t) dt = hL(z(·), z0 ), x(·)i2
dt
t0
and, according to (3), L ⊂ D∗ . It remains to prove that D(D∗ ) ⊂ D(L). Indeed, let v(·) = D∗ (z(·), z0 ) for a
certain (z(·), z0 ) ∈ D(D∗ ). Then v(·) ⊥ N (D). Since
+
F (E − F F ) = 0 ⇒
d
+
+
F (E − F F )x(·), F (E − F F )x(t0 ) = (0, 0) ∀x(·) ∈ Wn2 ,
dt
we conclude that v(·) = F + F v(·) almost everywhere. In this case,
ZT t0
ZT
ZT ZT
d
d
F x(t), z(t) dt =
F 0+ v(t), F x(t) dt = F 0+ v(s)ds, F x(t) dt
dt
dt
t0
t0
t
for all x(·) ∈ M := {x(·) ∈ WF2 : F x(t0 ) = 0}, and, hence,
ZT
z(·) −
F 0+ v(s)ds ⊥ R(L0 ),
t
6
7
1/2
Convergence is understood in the sense of the norm k · k2 := h·, ·i2
The quantity k · k is the norm in the space Rn .
of the space Ln
2.
S. M. Z HUK
474
where L0 denotes the operator x(·) 7→
d
F x(·), x(·) ∈ M. If
dt
Zt
g(t) =
F F + ϕ(s)ds,
ϕ(·) ∈ Lm
2 ,
t0
then L0 g(·) = F F + ϕ(·) and, hence, F F + ϕ(·) ∈ R(L0 ) for any ϕ(·) ∈ Lm
2 , which yields
ZT
ZT
z(t) −
t0
F 0+ v(s)ds, F F + ϕ(t) dt = 0 ∀ϕ(·) ∈ Lm
2 .
t
In this case, we have
F 0 z(t) −
ZT
F 0+ v(s)ds = F 0 (E − F 0+ F 0 ) z(t) −
t
ZT
F 0+ v(s)ds = 0,
t
0
i.e., z(·) ∈ WF2 and F 0 z(T ) = 0. Thus, for any x(·) ∈ WF2 , we get
ZT d 0
d
F x(t), z(t) +
F z(t), x(t)
dt = −(F x(t0 ), F 0+ F 0 z(t0 ))
dt
dt
t0
by virtue of (3). On the other hand,
ZT ZT
d
F x(t), z(t) dt + (F x(t0 ), z0 ) = (D∗ (z(·), z0 )(t), x(t))dt
dt
t0
t0
and, hence,
ZT d
∗
(F x(t0 ), z0 − F F z(t0 )) =
D (z(·), z0 )(t) + F x(t), x(t) dt
dt
0+
0
t0
for all x(·) ∈ WF2 and, in particular, for x(·) ∈ M (see the definition of M given above). Taking into account
that cl M = Ln2 , we obtain
(F x(t0 ), z0 − F 0+ F 0 z(t0 )) = 0
for all x(·) ∈ WF2 . It is now clear that z0 = F 0+ F 0 z(t0 ) + d and F 0 d = 0.
The theorem is proved.
C LOSEDNESS AND N ORMAL S OLVABILITY OF AN O PERATOR
475
d
Remark 1. Generally speaking (see Example 1), the operator x(·) 7→ F x(·), x(·) ∈ Wn2 , is not closed.
dt
In particular, this means that
∗
d
d
F0 ⊂
F .
dt
dt
d
On the other hand, the space Wn2 is not always a Hilbert space with respect to the norm of the graph of
F.
dt
d
Indeed, if Wn2 is a Hilbert space, then the restriction of
F to Wn2 is a closed operator [15, p. 26]. However,
dt
d
d
F x(·) = F x(·)
dt
dt
i.e., F
∀x(·) ∈ Wn2 ,
d
d
is the restriction of
F to Wn2 .
dt
dt
Example 1. Let t0 = 0, T = 1, n = 2, and
1 0
F =
.
0 0
d
defined on Wn2 . The Cantor function t 7→ k(t) is continuous and almost everywhere
dt
differentiable on [0, 1], but not absolutely continuous. Therefore, (0, k(·)) 6∈ Wn2 . On the other hand, according
to the Bernstein theorem, the function k(·) can be approximated uniformly in t by the polynomials
Consider the operator F
n
X
i
n
Bn (t) :=
k
ti (1 − t)n−i .
i
n
0
For each n, the polynomial Bn (·) is an absolutely continuous function. Therefore,
xn (·) :=
0
∈ Wn2 .
Bn (·)
d
On the other hand, F xn (·) = (0, 0) for any n, and xn converges to x(·) := (0, k(·)) in Ln2 , which implies
dt
that
xn (·) → x(·)
and F
d
xn (·) → (0, 0)
dt
d
d
is closed, then x(·) ∈ Wn2 by virtue of the closedness of the graph of F
dt
dt
d
[15, p. 17]. However, this inclusion is not true by virtue of the choice of x(·). Consequently, the operator F
is
dt
not closed.
d
On the other hand, x(·) ∈ WF2 . Therefore, for the operator
F defined on WF2 , we get
dt
simultaneously. If the operator F
xn (·) → x(·),
d
F xn (·) → (0, 0) ⇒ x(·) ∈ WF2 ,
dt
d
F x(·) = (0, 0).
dt
S. M. Z HUK
476
4. Conditions for the Normal Solvability of the Operator D
The property of normal solvability of the operator of a system is very important for various applications of
linear differential equations. This property enables one to speak of the continuous dependence of a solution with
the least norm on the right-hand side. The fact that the operator D is closed and densely defined enables one to use
methods of the theory of ill-posed problems [16, 17] for the investigation of the problem of the normal solvability
of D (Theorems 2 and 3).
The theorem presented below gives a criterion for the existence of a pseudosolution8 of Eq. (2) for given
(f (·), f0 ).
Theorem 2. The boundary-value problem
d
F x(t) = C(t)x(t) + z(t) + f (t),
dt
d 0
F z(t) = −C 0 (t)z(t) + ε2 x(t),
dt
F 0 z(T ) = 0,
F x(t0 ) − F 0+ F 0 z(t0 ) − d = f0 ,
(4)
F 0 d = 0,
m
has a unique solution (x(·, ε), z(·, ε), d(ε)) for every ε > 0. For arbitrary (f (·), f0 ) ∈ Lm
2 × R , the descriptor
equation
d
F x(t) = C(t)x(t) + f (t),
dt
F x(t0 ) = f0 ,
(5)
has a pseudosolution x
b(·) if and only if
kx(·, ε)k2 ≤ C
as
ε → 0.
m and show that (4) has a unique solution. We choose ε > 0 and
Proof. We fix g(·) := (f (·), f0 ) ∈ Lm
2 ×R
1
consider the operator D. The problem of the projection
ε
2
1
Dp(·) − g(·) + kp(·)k2 →
2
ε
1
min
p(·)∈D(D)
(6)
1
of the vector (0, g(·)) to the graph of the operator D has a unique solution p̂(·) ∈ D(D) by virtue of the
ε
1
closedness of D because the product of the bounded operator of multiplication by a scalar and a closed operator
ε
is a closed operator. Since9
2
1
Dp(·) − g(·) + kp(·)k22 = kDx(·) − g(·)k21 + ε2 kx(·)k22
ε
1
Here, a pseudosolution is understood as a vector x̂(·) ∈ D(D) with the least norm for which hDx̂(·) − (f (·), f0 ), Dx(·)i = 0 ∀x(·) ∈
D(D).
9
m
The symbol k · k1 denotes the norm in the Hilbert space Lm
2 ×R .
8
C LOSEDNESS AND N ORMAL S OLVABILITY OF AN O PERATOR
477
for p(·) = εx(·), by virtue of the arguments presented above the optimization problem
kDx(·) − g(·)k21 + ε2 kx(·)k22 →
min
x(·)∈D(D)
has a unique solution x̂ε (·) ∈ D(D) that satisfies the system of operator equations
Dx(·) − z(·) = g(·),
−D∗ z(·) − ε2 x(·) = 0.
It is easy to verify that this system is equivalent to (4).
The function
L (ε) :=
min
x(·)∈D(D)
kDx(·) − g(·)k21 + ε2 kx(·)k22 = kDx̂ε (·) − g(·)k21 + ε2 kx̂ε (·)k22
satisfies the relation [16, p. 119] (Lemma 1.25)
L (0+) = L (0) =
min
x(·)∈D(D)
kDx(·) − g(·)k21 .
Thus,
lim kDx̂ε (·) − g(·)k21 =
ε→0
min
x(·)∈D(D)
kDx(·) − g(·)k21 .
(7)
It is known [17, p. 124] (Theorem 18.5) that the convergence of the sequence {x̂ε (·)} as ε → 0 is equivalent
to the existence of x̂(·) ∈ D(D) such that
kDx̂(·) − g(·)k21 = min kDx(·) − g(·)k21 .
x(·)
Let f (·) ∈ cl R(D). Then Dx̂ε (·) → f (·), and the set {Dx̂ε (·)} is bounded. If {x̂ε (·)} is also bounded,
then, by virtue of the weak closedness of D, we have f (·) ∈ R(D). Otherwise, for f (·) ∈
/ R(D), the norms of
the functions {x̂ε (·)} increase unboundedly as ε → 0 by virtue of (7).
In the context of the theorem proved, the following lemma may also be useful:
Lemma 1. For a rectangular m × n matrix F, there exist square matrices L and R such that
F = LΛR, F + = R0 Λ+ L0 , LL0 = Em , RR0 = En ,
1
D2
0r,n−r
,
Λ=
0m−r,r 0m−r,n−r
(8)
where Em is the identity matrix of order m, 0k,s is the zero k × s matrix, and D is the diagonal matrix of order
r formed by the positive eigenvalues λ1 , . . . , λr of the matrix F F 0 .
S. M. Z HUK
478
Proof. Using the theorem on the singular-value decomposition of a rectangular matrix F (see [18, p. 52]),
1
we write F = P D 2 Q, where P and Q are, respectively, m × r and r × n rectangular matrices and P 0 P =
QQ0 = Er .
We decompose the matrix
Pr,r
P =
!
Pm−r,r
into blocks. The columns of the matrix P are orthonormal because P 0 P = Er . Using the Gram–Schmidt orthogonalization, we can complement them by m − r vectors wk to the orthonormal basis of the corresponding
Euclidean space. We represent the matrix T whose columns are the m − r vectors wk in the form
Tr,m−r
T =
Tm−r,m−r
!
.
Then T 0 P = 0 and T 0 T = Em−r by construction. We set
L :=
Pr,r
Tr,m−r
!
.
Pm−r,r Tm−r,m−r
Then L0 L = Em ⇒ LL0 = Em because P 0 P = Er , T 0 P = 0, and T 0 T = Em−r . By analogy, we define
R :=
Qr,r
Qr,n−r
!
,
Wn−r,r Wn−r,n−r
where Q = Qr,r Qr,n−r and W = Wn−r,r Wn−r,n−r is a matrix whose rows are n − r vectors vk that
complement the orthonormal row vectors of the matrix Q to the orthonormal basis of the corresponding Euclidean
1
space. Thus, RR0 = En . One can directly verify that RΛL = P D 2 Q.
Let us show that F + = R0 Λ+ L0 . We determine (LΛR)+ . It is known [18, p. 69] that (AB)+ = B + A+
if and only if R(BB 0 A0 ) ⊆ R(A0 ) and, simultaneously, R(A0 AB) ⊆ R(B) for rectangular matrices A and
B. According to the results proved above, we have L0 L = Em . Therefore, R(ΛRR0 ΛL0 ) ⊂ R(L0 ) and
R(L0 LΛR) = R(ΛR), whence (LΛR)+ = (ΛR)+ L0 because L+ = L−1 = L0 . Since RR0 = En , we
conclude that (ΛR)+ = R0 Λ+ because R(RR0 Λ0 ) = R(Λ0 ) and R(Λ0 ΛR) ⊂ R(R).
The lemma is proved.
In the theorem below, a class of equations of special structure that generate a normally solvable operator is
separated from the singular systems (2).
Theorem 3. Suppose that
0
0
L C(t)R =
C1 C2
C3 C4
!
,
C LOSEDNESS AND N ORMAL S OLVABILITY OF AN O PERATOR
479
where L and R are the same as in Lemma 1 and Ci are stationary rectangular matrices of consistent dimensions.
If 10
sup kQ(ε)C20 kmod < +∞,
1>ε>−1
Q(ε) := (ε2 E + C40 C4 )−1 ,
then the operator D has a closed range of values.
Proof. Let us show that, under the conditions of the theorem, the norms of solutions of the boundary-value
problem (4) remain bounded as ε → 0. To this end, we rewrite (4) in the form
d
dt
F
0
0
F0
x(t)
C(t)
E
x(t)
f (t)
=
+
.
z(t)
ε2 −C 0 (t)
z(t)
0
Using the equality (see Lemma 1)
0
Λ 0
L 0
F
=
0
0 Λ
0 R
0
0
F0
0
R 0
,
0 L
we obtain the equivalent equation
d
dt
0
p(t)
L C(t)R0
E
p(t)
g(t)
Λ 0
=
+
,
q(t)
ε2
−RC 0 (t)L
q(t)
0
0 Λ0
where p(t) := Rx(t), q(t) := L0 z(t), and g(t) := L0 f (t).
The conditions F 0 z(T ) = 0, F x(t0 ) = F 0+ F 0 z(t0 ) + d + f0 , and F 0 d = 0 take the form
Λ0 q(T ) = 0,
+
Λp(t0 ) = Λ0 Λ0 q(t0 ) + L0 d + L0 f0 ,
Λ0 L0 d = 0.
Taking into account the form of Λ, we obtain the system of algebraic–differential equations
1
1
D 2 ṗ1 (t) = C1 p1 (t) + C2 (t)p2 (t) + g1 (t) + D− 2 v1 (t),
0 = C3 p1 (t) + C4 p2 (t) + g2 (t) + q2 (t),
1
v̇1 (t) = ε2 p1 (t) − C10 D− 2 v1 (t) − C30 q2 (t),
1
0 = ε2 p2 (t) − C20 D− 2 v1 (t) − C40 q2 (t),
1
p1 (t0 ) − D−1 v1 (t0 ) = D− 2 g10 ,
10
For a matrix F, we set kF kmod :=
P
i,j
|Fij |.
d2 = g20 ,
v1 (T ) = 0,
(9)
S. M. Z HUK
480
where
1
2
v1 (t) := D q1 (t),
p(t) :=
p1 (t)
,
p2 (t)
0
g
g0 := L f0 = 10 ,
g2
0
q1 (t)
q(t) :=
,
q2 (t)
0
Ld=
.
d2
0
We write the algebraic equations separately:
C4
E
!
ε2 E −C40
!
p2 (t)
q2 (t)
=
−C3 p1 (t) − g2 (t)
1
C20 D− 2 v1 (t)
!
.
(10)
Multiplying (10) from the left by
Q(ε)C40
Q(ε)
!
,
E − C4 Q(ε)C40 −C4 Q(ε)
we get
1
p2 (t) = Q(ε)(−C40 C3 p1 (t) + C20 D− 2 v1 (t) − C40 g2 (t)),
(11)
q2 (t) = (E − C4 Q(ε)C40 )(−C3 p1 (t) − g2 (t)) − C4 Q(ε)C20 D
− 12
v1 (t).
Substituting the obtained relations into (9), we get the following two-point boundary-value problem for a nonnegative-definite Hamiltonian system with parameter:
ṗ1 (t) = A(ε)p1 (t) + R(ε)v1 (t) + g(t, ε),
p1 (t0 ) − D−1 v1 (t0 ) = g0 ,
(12)
v̇1 (t) = −A0 (ε)v1 (t) + S(ε)p1 (t) + `(t, ε),
v1 (T ) = 0,
where
1
g0 := D− 2 g10 ,
`(t, ε) = C30 (E − C4 Q(ε)C40 )g2 (t),
1
A(ε) = D− 2 (C1 − C2 Q(ε)C40 C3 ),
S(ε) = ε2 E + C30 (E − C4 Q(ε)C40 )C3 ,
1
1
R(ε) = D−1 + D− 2 C2 Q(ε)C20 D− 2 ,
1
g(t, ε) = D− 2 (g1 (t) − C2 Q(ε)C40 g2 (t)).
Using the theorem on “lower” and “upper” solutions [19], we can show that the Riccati equation
K̇(t) = A(ε)K(t) + K(t)A0 (ε) − K(t)S(ε)K(t) + R(ε),
K(t0 ) = D−1 ,
(13)
C LOSEDNESS AND N ORMAL S OLVABILITY OF AN O PERATOR
481
has a unique solution t 7→ K(t, ε) defined on [t0 , T ] for every ε > 0, and, furthermore, K(t, ε) is a nonnegativedefinite matrix in its domain of definition. Let q(·, ε) and ϕ(·, ε) be determined as solutions of the Cauchy problem
ϕ̇(t) = (A(ε) − K(t, ε)S(ε))ϕ(t) + g(t, ε) − K(t, ε)`(t, ε),
q̇(t) = (−A0 (ε) + S(ε)K(t, ε))q(t) + S(ε)ϕ(t) + `(t, ε),
ϕ(t0 ) = g0 ,
(14)
q(T ) = 0.
(15)
Performing the substitution, one can verify that the functions
p1 (t, ε) = K(t, ε)q(t, ε) + ϕ(t, ε),
v1 (t, ε) = q(t, ε)
(16)
satisfy (12).
Let us show that, under the conditions of the theorem, the norm of ϕ(·, ε) remains bounded as ε → 0. Indeed,
ϕ(t, ε) =
Φ(t, t0 , ε)f10
Zt
Φ(t, s, ε)(g(s, ε) − K(s, ε)`(s, ε))ds,
+
t0
where Φ(t, s, ε) is the normalized fundamental matrix for (14) for fixed ε > 0. It remains to show the boundedness
of the functions that are elements of the matrices Φ(t, s, ε) and K(t, ε) and vectors g(s, ε) and `(s, ε) as ε → 0.
1
We set g(t) := D− 2 (g1 (t) − C2 C4+ g2 (t)) and `(t) := C30 (E − C4 C4+ )g2 (t). Then, as ε → 0, we get
ZT
(g(t, ε) − g(t), g(t, ε) − g(t)) dt
t0
ZT
=
(D−1 (C2 C4+ − C2 Q(ε)C40 )g2 (t), (C2 C4+ − C2 Q(ε)C40 )g2 (t))dt → 0
t0
because Q(ε)C40 → C4+ as ε → 0. Hence, there exist g, ` > 0 such that
kg(·, ε)k2 ≤ g,
k`(·, ε)k2 ≤ `,
ε → 0.
We introduce a function Kl (·, ε) as a solution of the Bernoulli equation
K̇l (t) = A(ε)Kl (t) + Kl (t)A0 (ε) + R(ε),
Kl (t0 ) = D−1 .
Differentiating the function t 7→ ((Kl (t, ε) − K(t, ε))p, p), we obtain
(Kl (t, ε)p, p) ≥ (K(t, ε)p, p),
t0 ≤ t ≤ T,
ε > 0,
S. M. Z HUK
482
which implies that sp (Kl (t, ε) ≥ sp (K(t, ε)). Let kF ksp = sp(F F 0 ). Then, for a certain U > 0, we have
1
kK(t, ε)kmod ≤ kK(t, ε)ksp ≤ sp(K(t, ε)) ≤ kKl (t, ε)kmod
U
because K(t, ε) is a nonnegative-definite symmetric matrix for t ≥ t0 and ε > 0. Taking into account that
1
kA(ε) − Akmod → 0 as ε → 0 for A := D 2 (C1 − C2 C4+ C3 ) and using the representation
Kl (t, ε) = exp((A(ε) + A0 (ε))(t − t0 )) +
Zt
exp(A(ε)(t − s))R(ε) exp(A0 (ε)(t − s))ds,
t0
one can easily establish that the following relations hold for sufficiently small ε > 0:
kKl (t, ε)kmod ≤ k exp((A + A0 )(t − t0 ))kmod + 1,
Zt
M
(k exp(A(t − s))kmod + 1)(1 + k exp(A0 (t − s))kmod )ds := K 1 (t),
t0
where
kR(ε)kmod ≤ M.
Thus,
kK(t, ε)kmod ≤ U K 1 (t).
Setting
P (t, ε) := A(ε) − K(t, ε)S(ε),
we get
kP (t, ε)kmod ≤ 1 + kAkmod + U K 1 (t)(kC30 (E − C4 C4+ )C3 kmod + 1).
Using this result, one can easily establish (see, e.g., [12, p. 432]) the boundedness of the elements Φ(t, s, ε) as
ε → 0.
By analogy, we prove the boundedness of q(·, ε). Using equalities (10) and (15), we establish the boundedness
of p1 (·, ε) and p2 (·, ε) as ε → 0. Taking into account that x(t, ε) = R0 p(t, ε) and using Theorem 2, we obtain
the required statement.
We illustrate Theorem 2 by an example of a descriptor system of special form that generates an injective
operator D with nonclosed range of values.
Example 2. We set
1 0
F =
,
0 0
1 −1
C(t) ≡
.
1 0
C LOSEDNESS AND N ORMAL S OLVABILITY OF AN O PERATOR
483
It is easy to verify that, in this case, the operator D is injective. Hence, the range of values of D∗ is dense in Ln2
and consists (see Theorem 1) of all vector functions of the form
−ż1 − z1 − z2
1
, z1 ∈ W2 ([t0 , T ]), z1 (T ) = 0, z2 ∈ L2 ([t0 , T ]) .
z1
f1
∈
/ R(D∗ ) if f2 ∈
/ W12 ([t0 , T ]). Thus, the range of values of D∗ and, hence, R(D) are
f2
not closed. Note that the conditions of Theorem 3 are not satisfied because C20 (ε2 E + C40 C4 )−1 = −ε−2 .
Let us show that, in this case, a solution of Eq. (5) can also be approximated with the use of solutions of (12)
for (f (·), f0 ) ∈ R(D). It is easy to verify that the boundary-value problem (12) takes the form
This implies that
ẋ1 (t) = x1 (t) + (1 + ε−2 )z1 (t) + f1 (t),
ż1 (t) = −z1 (t) + (1 + ε2 )x1 (t) + f2 (t),
x1 (t0 ) − z1 (t0 ) = f01 ,
z1 (T ) = 0,
(17)
x2 (t) = −ε2 z1 (t, ε).
The Riccati equation (13) has the form
k̇(t) = 2k(t) + (1 + ε−2 ) − (1 + ε2 )k 2 (t) := U (t, k),
k(t0 ) = 1.
(18)
We set
−
k :=
ε2 −
√
ε2 + 3ε4 + ε6
,
ε2 + ε4
+
k :=
ε2 +
√
ε2 + 3ε4 + ε6
.
ε2 + ε4
Using the Picard–Lindelöf theorem, we establish that, for a certain ε0 > 0, we have
k − < k(t, ε) < k + ,
0 < ε < ε0 ,
t > t0 ,
(19)
which yields k̇(t, ε) > 0, t ≥ t0 , 0 < ε < ε0 , because U (t, k) = (1 + ε2 )(k − k − )(k + − k). Thus, k(t, ε) ≥
k(t0 , ε) > 0 for t ≥ t0 and 0 < ε < ε0 .
Let q(·, ε) denote a solution of the equation
qtt (t) − 2qt (t) + (1 + ε−2 )(1 + ε2 )q(t) = 0,
By direct substitution, we establish that the function t 7→
Rt
q(t, ε) = e
2
t0 (1+ε )k(s,ε)ds
qt (t0 ) = 1 + ε2 ,
q(t0 ) = 1.
qt (t, ε)
satisfies (18). Hence,
(1 + ε2 )q(t, ε)
> 0 ⇒ qt (t, ε) ≥ 0,
t ≥ t0 ,
0 < ε < ε0 .
Carrying out differentiation, one can easily verify that
Zt
q(τ, ε)
q̇(τ, ε)f2 (τ )
ϕ(t, ε) =
f1 (τ ) − τ −t0
dτ
f10 +
τ
−t
0
q(t, ε)
e
e
(1 + ε2 )
et−t0
t0
S. M. Z HUK
484
is a solution of (14) and
q(t, ε)
z(t, ε) = − t
e
ZT
es
f2 (s) + (1 + ε2 )ϕ(s, ε) ds
q(s, ε)
t
is a solution of (15). Therefore, x1 (t, ε) = k(t, ε)z(t, ε) + ϕ(t, ε) and x2 (t, ε) = −ε−2 z(t, ε).
If, in addition, f1 (t) ≡ 0, f2 (t) = −et−t0 , and f10 = 1, then the equation Dx(·) = (f (·), f0 ) has the
unique solution
!
x1 (t)
x(t) =
, x1 (t) = −f2 (t), x2 (t) ≡ 0
x2 (t)
by virtue of the injectivity of D. On the other hand, we have
et−t0
ε2 et−t0
+
,
ϕ(t, ε) =
(1 + ε2 )q(t, ε) 1 + ε2
z(t, ε) =
q(t, ε)
−ε2 t+t0
e
ZT
e2s
ds.
q 2 (s, ε)
t
We show that x1 (·, ε) → x1 and x2 (·, ε) → 0 in Ln2 . Since the function q(·, ε) is increasing, we get
et−t0
et−t0
ε2 et−t0
< ϕ(t, ε) ≤
+
,
2
2
1+ε
1+ε
(1 + ε2 )q(t0 , ε)
q(t, ε)
z(t, ε) ≤ −ε t+t0 2
e
q (t, ε)
2
ZT
e2s ds.
t
Using the equality q(t0 , ε) = 1 and relation (19), we obtain
ZT
2
ZT (ϕ(t, ε) + f2 (t)) dt ≤
t0
ε2 et−t0
et−t0
+
− et−t0
1 + ε2 (1 + ε2 )
2
dt → 0,
ε → 0,
t0
ZT
2
ZT 2
k (t, ε)z (t, ε) dt ≤
t0
2 +e
−ε k
2T
− e2t
2
2et+t0
dt → 0,
ε → 0.
t0
Thus, x1 (·, ε) → −f2 (·).
One can verify that
q(t, ε) =
(ε4
+
√
√
ε2
+
3ε4
ε6 )e
ε2 +
ε2 +3ε4 +ε6
(t−t0 )
ε2
+
√
2 ε2 + 3ε4 + ε6
√
√
ε2 − ε2 +3ε4 +ε6
(t−t0 )
ε2
( ε2 + 3ε4 + ε6 − ε4 )e
√
+
.
2 ε2 + 3ε4 + ε6
Therefore, the norm of q(·, ε) unboundedly increases as ε → 0. On the other hand, we have
−ε−2 z(t, ε) ≤
Therefore, x2 (·, ε) → 0.
e2T − e2t
.
2et+t0 q(t, ε)
C LOSEDNESS AND N ORMAL S OLVABILITY OF AN O PERATOR
485
The next example illustrates the application of sufficient conditions for normal solvability (Theorem 3) to a
descriptor equation that is not decomposed into algebraic and differential components.
Example 3. We set
1
2
F =
1
−
2
1
−
2
,
1
2
1
−
2
C(t) ≡
1
2
1
−
2
.
1
2
The corresponding descriptor system has the form
d
dt
1
1
1
1
x1 − x2 (t) = − x1 (t) − x2 (t) + f1 (t),
2
2
2
2
d
dt
1
1
1
1
− x1 + x2 (t) = x1 (t) + x2 (t) + f2 (t),
2
2
2
2
1
1
1
1
0
x1 − x2 (t0 ) = f1 , − x1 + x2 (t0 ) = f20 .
2
2
2
2
We set
1
√2
T :=
1
−√
2
1
√
2
,
1
√
2
1 0
Λ :=
.
0 0
Then
F = T ΛT
0
and
0
T C(t)T =
0 0
.
1 0
According to Theorem 3, the corresponding operator D is normally solvable because C2 Q(e) ≡ 0. On the other
hand, the closedness of the range of values of the operator
p(·) 7→ D1 p(·) =
d
0
Λp(·) − T C(t)T p(·), Λ p(t0 )
dt
can be verified directly. Indeed, the adjoint operator acts according to the rule
−ż1 (t) − z2 (t)
(z(·), z0 ) 7→
,
0
z2 ∈ L2 (t0 , T ),
z1 ∈ W12 (t0 , T ),
Therefore, R(D1∗ ) = L2 (t0 , T ) × {0} is a closed set together with R(D1 ).
z1 (T ) = 0.
S. M. Z HUK
486
Note that det(λF + C) ≡ 0. The kernel of the operator D is infinite-dimensional because
(F − C(t))T
0
= 0,
f (·)
f (·) ∈ L2 (t0 , T ).
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