Abstract
Full Text
Reports of the Academy of Sciences of the USSR
1957. Volume 114, No. 1
MATHEMATICS
V. G. EGOROV
STABILITY OF SOLUTIONS OF PERIODIC SYSTEMS OF EQUATIONS IN TOTAL DIFFERENTIALS
(Presented by Academician I. G. Petrovsky, 28 XI 1956)
Let there be given a system of equations in total differentials of the form
\[ dx=p(u)x\,du+q(v)x\,dv, \tag{1} \]
where \(x\) is a column matrix; \(p(u)\) and \(q(v)\) are square matrices, continuous, bounded, and satisfying the integrability condition
\[ p(u)q(v)=q(v)p(u) \tag{2} \]
for all \(u>0,\ v\geqslant 0\) (\(u, v\) are parameters).
It is known \((^{1})\) that in this case system (1) admits a unique solution corresponding to prescribed initial conditions.
By an integral matrix of system (1) we shall mean a square matrix
\[
X(u,v)=\|x_{ik}(u,v)\|_1^n,
\]
whose columns are \(n\) linearly independent solutions of this system. Clearly, the matrix \(X(u,v)\) satisfies the equation
\[ dX=p(u)X\,du+q(v)X\,dv. \tag{3} \]
It is easy to show that if \(\overline{X}\) is a nonsingular \((|\overline{X}|\neq 0)\) particular solution of equation (3), then the general solution of this equation will be
\[ X(u,v)=\overline{X}(u,v)C, \tag{4} \]
where \(C\) is an arbitrary constant matrix.
If \(X(u,v)\) is the normalized integral matrix of system (1), and \(\overline{X}(u)\) and \(\overline{X}(v)\) are the normalized integral matrices of the systems
\[ dx=p(u)x\,du,\qquad dx=q(v)x\,dv, \]
then
\[ X(u,v)=\overline{X}(u)\overline{X}(v)=\overline{X}(v)\overline{X}(u). \tag{5} \]
Indeed,
\[ d(\overline{X}\overline{X})=d(\overline{X})\overline{X}+\overline{X}d(\overline{X}) =p(u)\overline{X}\overline{X}\,du+\overline{X}q(v)\overline{X}\,dv. \]
But since \((^{2})\)
\[ \overline{X}(u)=E+\int_{u_0}^{u}p(u)\,du+ \int_{u_0}^{u}p(u)\,du\int_{u_0}^{u}p(u)\,du+\ldots, \]
then, taking (2) into account, we conclude that the matrix \(\overline{X}\overline{X}\) (as well as the matrix \(\overline{X}\overline{X}\)) satisfies equation (3). And since
\[
X(u_0,v_0)=\overline{X}(u_0)\overline{X}(v_0)
=\overline{X}(v_0)\overline{X}(u_0)=E,
\]
relation (5) is proved.
It follows from this that every integral matrix of system (1) is nonsingular. Indeed, on the basis of (4) and (5),
\[ |X(u,v)|=|\overline{X}|\,|\overline{\overline{X}}|\,|C| =\exp\left\{\int_{u_0}^{u}\operatorname{Sp}(p)\,du+ \int_{v_0}^{v}\operatorname{Sp}(q)\,dv\right\}|C|. \tag{6} \]
But the determinant \(|X(u,v)|\) cannot be identically equal to zero; hence \(|C|\ne 0\), and the assertion is proved. Thus, all integral matrices of system (1) are obtained by formula (4) for \(|C|\ne 0\).
We shall call system (1) reducible if, by means of the linear transformation
\[ x=L(u,v)y \tag{7} \]
it is reduced to the form
\[ dy=A y\,du+B y\,dv, \tag{8} \]
where \(A\) and \(B\) are constant permutable matrices. With respect to the transformation matrix \(L(u,v)=\|l_{ik}(u,v)\|_1^n\) we shall assume that in the domain \(u\ge 0,\ v\ge 0\): 1) \(L(u,v)\) has continuous derivatives \(\partial L/\partial u,\ \partial L/\partial v,\ \partial^2 L/\partial u\partial v\), of which the first, as well as the matrix \(L(u,v)\) itself, are bounded; 2) \(\bmod |L(u,v)|>m>0\).
The matrix \(L(u,v)\) will be called a Lyapunov matrix.
It is not difficult to establish that if the zero solution of the system
\[ dx=P(u,x)\,du+Q(v,x)\,dv, \tag{9} \]
where the matrices \(P,Q\) satisfy the known conditions (3), is stable or asymptotically stable, then the zero solution of the system
\[ dy=\overline{P}(u,v,y)\,du+\overline{Q}(u,v,y)\,dv, \tag{10} \]
into which system (9) passes after the transformation (7), will possess the same property.
Theorem 1. If \(p(u)\) and \(q(v)\) are periodic matrices, then system (1) is reducible.
In proving this theorem we use a lemma analogous to the lemma of N. P. Erugin \((^5)\).
Lemma. In order that system (1) be reducible to the form (8), it is necessary and sufficient that it have a solution of the form
\[ X(u,v)=L(u,v)e^{Au+Bv}. \tag{11} \]
Corollary 1. In order that the “union”
\[ dx=\left(\varphi(t)p\left(\int_{0}^{t}\varphi\,dt\right)+ \psi(t)q\left(\int_{0}^{t}\psi\,dt\right)\right)x\,dt \]
of the periodic systems \(dx=p(u)x\,du,\ dx=q(v)x\,dv\) be a reducible system, it is sufficient that the matrices \(p(u)\) and \(q(v)\) be permutable, and that the functions \(\varphi(t),\ \psi(t)\) have the form \(\varphi(t)=\alpha+\varphi_1(t),\ \psi(t)=\beta+\psi_1(t)\), where \(\alpha\) and \(\beta\) are constants, and \(\varphi_1(t)\) and \(\psi_1(t)\) are continuous functions for which the integrals \(\int_{0}^{t}\varphi_1(t)\,dt,\ \int_{0}^{t}\psi_1(t)\,dt\) are bounded.
Corollary 2. In order that the zero solution of the system
\[ dx=\left(\varphi(t)p\left(\int_{0}^{t}\varphi\,dt\right)+ \psi(t)q\left(\int_{0}^{t}\psi\,dt\right)\right)x\,dt+ \]
\[ +\left(\varphi(t)P_{1}\left(\int_{0}^{t}\varphi\,dt,x\right)+ \psi(t)Q_{1}\left(\int_{0}^{t}\psi\,dt,x\right)\right)dt, \]
which is the “union” of the systems
\[ dx=p(u)x\,du+P_{1}(u,x)\,du,\qquad dx=q(v)x\,dv+Q_{1}(v,x)\,dv \]
(\(p(u)\) and \(q(v)\) are periodic matrices), stable in the first approximation, be stable in the first approximation, it is sufficient that the matrices \(p(u)\) and \(q(v)\) be permutable, and that the constant \(\mathcal L\) in the inequality
\[ \bmod (P_{1},Q_{1})<\mathcal L\sum_{s=1}^{n}|x_s| \tag{12} \]
be sufficiently small ((3), Theorem 7).
Corollary 2 remains valid when the periodic matrices \(p(u)\) and \(q(v)\) are permutable and the positive numbers \(\alpha\) and \(\beta\) satisfy the inequality
\[ \max \operatorname{Re}(\alpha\lambda^{(a)})+\max \operatorname{Re}(\beta\lambda^{(b)})<0, \]
where \(\lambda^{(a)}\) and \(\lambda^{(b)}\) are the eigenvalues of the matrices \(A\) and \(B\) ((3), Theorem 5).
Let us consider system (9). Suppose it has the form
\[ dx=p(u)x\,du+q(v)x\,dv+P_{1}(u,x)\,du+Q_{1}(v,x)\,dv, \tag{13} \]
where \(p(u)\) and \(q(v)\) are periodic matrices, and the matrices \(P_{1}(u,x)\) and \(Q_{1}(v,x)\) satisfy condition (12).
Theorem 2. The zero solution of system (13) is asymptotically stable if the characteristic numbers of the systems
\[ dx=p(u)x\,du,\qquad dx=q(v)x\,dv \]
are positive, and the constant \(\mathcal L\) in inequality (12) is sufficiently small. In particular, if \(p(u)\), \(q(v)\) are permutable matrices, then the zero solution of system (1) is also asymptotically stable ((3), Theorem 3).
Corollary. The zero solution of the system
\[ dx=\left(\varphi(t)p\left(\int_{0}^{t}\varphi\,dt\right)+ \psi(t)q\left(\int_{0}^{t}\psi\,dt\right)\right)x\,dt+ \]
\[ +\left(\varphi(t)P_{1}\left(\int_{0}^{t}\varphi\,dt,x\right)+ \psi(t)Q_{1}\left(\int_{0}^{t}\psi\,dt,x\right)\right)dt \]
is asymptotically stable if \(\varphi(t)\) and \(\psi(t)\) are continuous, bounded, positive functions for which at least one of the integrals
\[ \int_{0}^{\infty}\varphi(t)\,dt,\qquad \int_{0}^{\infty}\psi(t)\,dt \]
diverges.
Ural Polytechnic Institute
named after S. M. Kirov
Received
31 V 1955
CITED LITERATURE
- T. Y. Thomas, Ann. Math., 35, No. 4, 729 (1934).
- F. R. Gantmacher, Theory of Matrices, Moscow, 1953.
- V. G. Egorov, DAN, 102, No. 4 (1955).
- V. V. Nemytskii, V. V. Stepanov, Qualitative Theory of Differential Equations, Moscow–Leningrad, 1949.
- N. P. Erugin, Proceedings of the Mathematical Institute named after V. A. Steklov, Academy of Sciences of the USSR, 13 (1946).