UDC 513.88:517.948.35

MATHEMATICS

E. A. LARIONOV

ON A CRITERION FOR THE STABILITY OF SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH PERIODIC OPERATOR COEFFICIENTS IN A BANACH SPACE

(Presented by Academician I. N. Vekua on 11 III 1968)

Consider in a Banach space (X) the equation

[
dx/dt=A(t)x,\qquad -\infty<t<\infty,
\tag{1}
]

where (A(t)) is a (T)-periodic operator-function whose values are linear operators in the space (X). At present many results are known (see the bibliography in ((^1))) concerning theorems on the existence and uniqueness of an ordinary or generalized solution (x(t)) of equation (1), established under certain conditions imposed on the function (A(t)). In what follows, by (x(t)) we shall everywhere mean an ordinary or generalized solution of equation (1). Denote by (R) the ring of all linear bounded operators acting in the space (X).

An operator-function (U(t)) is called a resolving one if, for any solution (x(t)) of equation (1),

[
x(t)=U(t)x(0).
\tag{2}
]

In relation (2), (U(t)\in R) for every (t\in(-\infty,\infty)). In those cases when an arbitrary element of (X) is taken as the initial value (x(0)) of the solution (x(t)), the operator (U(t)) maps the space (X) one-to-one onto itself and, by the Banach theorem, is continuously invertible.

Equation (1) is called stable if all its solutions (x(t)) are bounded on the axis ((-\infty,\infty)).

By virtue of the uniqueness of the solution (x(t)), it follows from (2) that

[
U(t+T)=U(T)U(t).
\tag{3}
]

The operator (U(T)) is called the monodromy operator of equation (1). Put (r_t=t-nT) for any (t) in ((-\infty,\infty)), where (n=0,\pm1,\pm2,\ldots). Then from (2) and (3) we obtain

[
x(t)=U^n(T)U(r_t)x(0).
\tag{4}
]

The operator-function (U(t)) is bounded on the segment ([0,T]), and therefore from (4) and the Banach–Steinhaus theorem it follows that boundedness of any solution (x(t)) on the entire axis ((-\infty,\infty)) is equivalent to the relation

[
|U^n(T)|\le C,\qquad C=\mathrm{const};\qquad n=0,\pm1,\pm2,\ldots
\tag{5}
]

Definition. An operator (U) from (R) satisfying estimate (5) is called stable. Thus, the stability of equation (1) is equivalent to the stability of the monodromy operator (U(T)) of this equation.

An operator (V) from (R) mapping all of (X) onto (X) and such that

[
|Vx|=|x|,
\tag{6}
]

is called unitary.

In the case when the space (X) is Hilbert, a stable operator (U), by the theorem of B. Sz.-Nagy ((^2)), is similar to a unitary operator (V) in the ordinary sense. The following theorem is a generalization of the theorem of B. Sz.-Nagy.

Theorem 1. A stable operator (U) in the space (X) is similar to a unitary operator (V) acting in this space.

Proof. The group (G={U^n}_{-\infty}^{\infty}) is commutative and bounded by the constant (C), and therefore, by a well-known theorem of A. A. Markov ((^3)), in the space of all bounded numerical functions on (G) there exists a linear functional (M(\varphi)) with the properties:
[
\begin{aligned}
&1)\quad M(\varphi(AB))=M(\varphi(A)),\quad A,B\in G;\
&2)\quad M(\varphi(A))\geq 0\quad \text{if } \varphi(A)\geq 0;\
&3)\quad M(1)=1.
\end{aligned}
]

Let (X^) be the space conjugate to the space (X). By the Hahn—Banach theorem, for any element (x) of (X) there exists a functional (x^) from (X^) such that (x^x=|x|). Thus a mapping (W) from (X) into (X^) is defined, and we can define on the space (X) the form
[
[x,y]=Wyx=y^x.
\tag{7}
]

The form ([x,y]) is linear in the first argument; the quadratic form ([x,x]) is positive definite, and therefore the form ([x,y]) is called a semiscalar product.

The expression (\varphi_{x,y}(A)=[Ax,Ay]), where (A\in G), is a function on the group (G). It is obvious that the function (\varphi_{x,y}(A)) is bounded on (G). In particular, for (A=I) we have
[
\varphi_{x,y}(I)=[x,y].
\tag{8}
]

Take as the new norm (|x|1) of the element (x) of the space (X) the quantity (M([Ax,Ax])). By property 1), for any (B) from (G) we obtain that
[
M([ABx,ABx])=M(\varphi(A))=M([Ax,Ax]).}(AB))=M(\varphi_{x,x
]

Thus, for all elements (U^n) of the group (G) we have the relation
[
|U^n x|_1=|x|_1.
\tag{9}
]

For (n=1), from (9) we obtain
[
|Ux|_1=|x|_1.
\tag{10}
]

The topological equivalence of the norms (|x|) and (|x|_1) is obvious from the definition of the norm (|x|_1). Theorem 1 is proved.

Corollary 1. For stability of equation (1) it is necessary and sufficient that the monodromy operator (U(T)) of this equation be similar to a unitary operator.

We now indicate the class of stable unitary operators in the space (X).

In ((^4)) the concept was introduced of an operator of spectral type (A), decomposed into the sum of an operator of scalar type
[
S=\int_{\sigma(S)} \lambda\,dE_\lambda
]
and a radical operator (N):
[
A=S+N.
\tag{11}
]

It is known ((^5)) that a spectral operator (A) is an operator of scalar type with spectrum lying on the unit circle if and only if estimate (5) is satisfied for the operator (A). Thus, the set (Y(S)) of all scalar-type operators (S) from (R) with spectrum on the unit circle is contained in the set (Y) of all stable operators from (R).

Theorem 2. If (U(T)\in Y(S)), then equation (1) is stable.

If (X) is a Hilbert space, then every operator in (Y) belongs to the set (Y(S)), and consequently in this case

[
Y(S)=Y.
\tag{12}
]

Theorem 3. In a Hilbert space (X), equation (1) is stable if and only if the monodromy operator (U(T)\in Y(S)).

In the general case the relation (12) does not hold; moreover, in ((^6)) examples are given of unitary operators in (X) which are not even operators of spectral type.

Thus, in the general case the condition (U(T)\in Y(S)) is not necessary for the stability of equation (1). All known theorems ((^1)) guaranteeing the existence of a solution (x(t)) of equation (1) contain, along with other conditions, the requirement that the spectrum of the operator (A(t)), for each (t\in(-\infty,\infty)), be contained in a fixed sector (S_z) of the plane, where for all (z\in S_z), (\pi/2<\arg z<3\pi/2). Thus there naturally arises the question of finding simple conditions under which the spectrum (\sigma(A(t))) of the operator (A(t)) will be located in the above-mentioned sector (S_z). We shall now establish one such condition.

In applied problems one often encounters ((^7,^8)) the equation

[
d^2u/dt^2+P(t)u=0,\qquad -\infty<t<\infty,
\tag{13}
]

where (P(t)) is a (T)-periodic operator-function whose values are self-adjoint operators in a Hilbert space (\mathfrak H). Setting (du/dt=u_1,\ x=(u,u_1)), we reduce equation (13) to a system of first-order equations, written in matrix notation in the space (\mathfrak H_0=\mathfrak H\oplus\mathfrak H) in the form (1), where

[
A(t)=
\begin{pmatrix}
0 & I\
-P(t) & 0
\end{pmatrix}.
\tag{14}
]

Denote by (\rho(A(t))) the resolvent set of the operator (A(t)).

Theorem 4. If (\rho(A(t))=\varnothing), and for all (x,y\ne{0}) from the domain of definition (\mathcal D(P(t))) of the operator (P(t)) it follows from (\operatorname{Re}(x,y)=0) that ((P(t)x,y)\ne(y,y)), then the spectrum (\sigma(A(t))) is real.

Proof. Introduce in the space (\mathfrak H_0) an operator (J), whose matrix representation in the decomposition (\mathfrak H_0=\mathfrak H\oplus\mathfrak H) has the form

[
J=
\begin{pmatrix}
0 & I\
I & 0
\end{pmatrix}.
\tag{15}
]

Consider in (\mathfrak H_0) the form

[
[x,y]=(Jx,y),
\tag{16}
]

where ((x,y)=(x_1,y_1)+(x_2,y_2)), (x={x_1,x_2}), (y={y_1,y_2}\in\mathfrak H_0). The quadratic form ([x,x]) on (\mathfrak H_0) assumes the values ([x,x]>0), ([x,x]<0), ([x,x]=0), ({0}\ne x\in\mathfrak H_0).

A Hilbert space (\mathfrak H_0), equipped, in addition to the usual scalar product ((x,y)), with the indefinite scalar product ([x,y]), is called a space with an indefinite metric. The notions of (J)-self-adjoint, (J)-isometric, and (J)-unitary operators in (\mathfrak H_0) with respect to the form ([x,y]) are introduced in the usual way. By definition, an operator (A) in (\mathfrak H_0) with dense domain (D(A)) is (J)-self-adjoint if

[
(JA)^*=JA.
\tag{17}
]

From (14) and (15) we obtain

[
JA(t)=
\begin{pmatrix}
-P(t) & 0\
0 & I
\end{pmatrix};
\tag{18}
]

Thus, for every (t \in (-\infty,\infty)) the operator (A(t)) is a (J)-self-adjoint operator in (\mathfrak H_0). A (J)-self-adjoint operator (A) is said to belong to the class of Pesonen ({}^{(9)}) ((A \in \Pi)) if from ([x,x]=0) and ([Ax,x]=0) ((x \in \mathfrak D(A))) it follows that (x={0}). Denote by (\Pi^+(\Pi^-)) the set of operators in (\Pi) for which from ((x,x)=0,\ {0}\ne x\in \mathfrak D(A)), it follows that ([Ax,x]>0) (( [Ax,x]<0)). It is known ({}^{(10)}) that (\Pi=\Pi^+\cup\Pi^-). Let ({0}=z\in \mathfrak D(A(t))). In view of the decomposition (\mathfrak H_0=\mathfrak H\oplus\mathfrak H), (z=x+y,\ x,y\in\mathfrak H). Obviously,

[
[z,z]=(x,y)+(y,x).
\tag{19}
]

It follows from (19) that ([z,z]=2\operatorname{Re}(x,y)), and therefore ([z,z]=0) if and only if (\operatorname{Re}(x,y)=0).

Next we have

[
A(t)z={y,-P(t)x},
]

[
[A(t)z,z]=(y,y)-(P(t)x,x).
]

By the condition of the theorem, if ([z,z]=0,\ z\ne{0}), then ([A(t)z,z]\ne0). Thus (A(t)\in\Pi). Since (\Pi=\Pi^+\cup\Pi^-), either (A(t)\in\Pi^+), or (A(t)\in\Pi^-). Now from Theorem A.I of ({}^{(11)}) it follows that the spectrum (\sigma(A(t))) of the operator (A(t)) is real. In applications (P(t)), (t\in(-\infty,\infty)), is a positive definite self-adjoint operator in (\mathfrak H).

Theorem 5. If, for all ({0}\ne x,y\in\mathfrak D(P(t))), from (\operatorname{Re}(x,y)=0) it follows that ((P(t)x,x)\ne(y,y)), and, moreover,

[
\operatorname{Re}\bigl[-(P(t)x,y)+\overline{(x,y)}\bigr]\le 0,
]

then the spectrum of the operator (A(t)) lies on the negative half-axis.

Indeed, the condition

[
\operatorname{Re}\bigl[-(P(t)x,y)+\overline{(x,y)}\bigr]\le 0
]

means that (\operatorname{Re}[A(t)z,z]\le0) for all (z\in\mathfrak D(A(t))), and therefore the spectrum (\sigma(A(t))) lies in the left half-plane. Thus, (\rho(A(t))\ne\varnothing), and this, together with the condition that from (\operatorname{Re}(x,y)=0) it follows that ((P(t)x,x)\ne(y,y)), ensures the reality of the spectrum (\sigma(A(t))).

It is easy to see that if the conditions of Theorem 5 are fulfilled, then for every fixed (t) from ((-\infty,\infty)) the operator (A(t)) generates a strongly continuous one-parameter group of contracting operators.

I express my sincere gratitude to Prof. M. A. Naimark for his attention to this work.

Central Economics and Mathematics Institute
Academy of Sciences of the USSR

Received
4 III 1968

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