UDC 531.391.5+517.919

MECHANICS

K. A. ABGARYAN

ON THE STABILITY OF MOTION ON A FINITE TIME INTERVAL

(Presented by Academician V. P. Mishin, 1 IV 1968)

1. We consider systems whose perturbed motion is described by an equation of the form

\[ dx/dt = U(t)x + H(t,x), \tag{1} \]

where \(x\) is a column matrix composed of the deviations \(x_i\) \((i=1,\ldots,n)\) of the system, and \(U\) and \(H\) are matrices of type \(n\times n\) and \(n\times 1\), respectively, possessing on the set \(S=[t_0\leq t\leq t_0+T,\ \|x\|\leq a]\) the properties: a) \(U\) is differentiable \(l\) times \((l=1,2,\ldots)\); b) \(H\) is continuous in \(t\) and \(x\);

\[ \lim_{x\to 0}\frac{H(t,x)}{\|x\|}=0. \tag{2} \]

For such systems there is given below a family of necessary and sufficient conditions for stability and instability of the unperturbed motion (the trivial solution of equation (1)) on a finite time interval, proceeding from the following definition of stability.

Definition. If the equations of perturbed motion are such that, for sufficiently small \(\rho>0\), any solution \(x(t)\) of the equations whose initial value \(x_0=x(t_0)\) satisfies the condition

\[ (G(t_0)x_0,\;G(t_0)x_0)\leq \rho, \tag{3} \]

on some finite interval \([t_0,\ t_0+\Delta t]\) satisfies the condition

\[ (G(t)x,\;G(t)x)\leq \rho, \tag{4} \]

where \(G\) is a given bounded matrix, then the unperturbed motion with respect to the domain (4) is stable on the interval \([t_0,\ t_0+\Delta t]\); otherwise it is unstable, i.e. \(\Delta t=0\).

Unlike the definition of stability given by G. V. Kamenkov \((^1)\), here, as in works \((^2,^3)\) (but with different restrictions), a variable domain of limiting deviations \(x_i\) \((i=1,\ldots,n)\) is introduced.

1. Instead of (1) let us consider the more general equation

\[ dx/dt = U(\tau)x + H(t,x), \tag{5} \]

where \(\tau=\varepsilon t\) is the so-called “slow time,” and \(\varepsilon\) is a parameter. When \(\varepsilon=1\), equations (1) and (5) coincide.

We shall assume that \(U(\varepsilon t)\) on the set \(S\) is still differentiable \(l\) times.

By the transformation

\[ x=K^{(m)}(\tau,\varepsilon)y \tag{6} \]

we bring equation (5) to the form

\[ dy/dt=\Lambda^{(m)}(\tau,\varepsilon)y - K^{(m)-1}(\tau,\varepsilon)N^{(m)}(\tau,\varepsilon) + K^{(m)-1}(\tau,\varepsilon)H(t,K^{(m)}y), \tag{7} \]

where

\[ N^{(m)}(\tau,\varepsilon) = \varepsilon\frac{dK^{(m)}(\tau,\varepsilon)}{d\tau} - U(\tau)K^{(m)}(\tau,\varepsilon) + K^{(m)}(\tau,\varepsilon)\Lambda^{(m)}(\tau,\varepsilon). \tag{8} \]

Using the algorithm indicated in (4), one can construct such a transformation (6) that \(\Lambda^{[m]}\) will have a diagonal or, at least, a quasidiagonal structure, while \(N^{(m)}\), as \(\varepsilon \to 0\), will be a matrix of order not lower than \(\varepsilon^{m+1}\).

We shall restrict ourselves to the case when \(U(\varepsilon t)\) has only simple eigenvalues on \([t_0,T]\). Then \(K^{(m)}\) and \(\Lambda^{(m)}\) can be represented as follows:

\[ K^{(m)}(\tau,\varepsilon)=\sum_{k=0}^{m}\varepsilon^k K^{[k]}(\tau),\qquad \Lambda^{(m)}(\tau,\varepsilon)=\sum_{k=0}^{m}\varepsilon^k \Lambda^{[k]}(\tau), \tag{9} \]

where

\[ K^{[k]}=\bigl(K_1^{[k]}\ldots K_n^{[k]}\bigr),\qquad \Lambda^{[k]}= \begin{pmatrix} \lambda_1^{[k]} & 0\\ & \ddots & \\ 0 & & \lambda_n^{[k]} \end{pmatrix}. \]

By \(\lambda_1,\ldots,\lambda_n\) we denote the eigenvalues of the matrix \(U\), and by \(K_1,\ldots,K_n\) the corresponding normalized eigenvectors of it. Then the column matrices \(K_\sigma^{[k]}\) and the scalar functions \(\lambda_\sigma^{[k]}\) are determined by the recurrence relations

\[ K_\sigma^{[0]}=K_\sigma,\qquad \lambda_\sigma^{[0]}=\lambda_\sigma,\qquad K_\sigma^{[k]}=P_\sigma D_\sigma^{[k-1]}+K_\sigma q_\sigma^{[k]}, \]

\[ \lambda_\sigma^{[k]}=-M_\sigma D_\sigma^{[k-1]} \qquad (k=1,\ldots,m;\ \sigma=1,\ldots,n), \tag{10} \]

where

\[ P_\sigma=\sum_{s\ne\sigma}\frac{K_s M_s}{\lambda_s-\lambda_\sigma},\qquad D_\sigma^{[k-1]}=\sum_{\alpha=1}^{k-1}K_\sigma^{[k-\alpha]}\lambda_\sigma^{[\alpha]}+ \frac{dK_\sigma^{[k-1]}}{dt}, \tag{11} \]

\(M_s\) \((s=1,\ldots,n)\) are the rows of the matrix

\[ M= \begin{pmatrix} M_1\\ \vdots\\ M_n \end{pmatrix} =K^{-1}=(K_1\ldots K_n)^{-1}, \]

and \(q_\sigma^{[k]}\) are arbitrary functions of \(\tau\), differentiable \(m-k+1\) times. With this construction of \(K^{(m)}\) and \(\Lambda^{(m)}\),

\[ N^{(m)}=\varepsilon^{m+1}\sum_{\nu=1}^{m}\sum_{\alpha=\nu}^{m}\varepsilon^{\nu-1}K^{[m-\alpha+\nu]}\Lambda^{[\alpha]} \equiv \varepsilon^{m+1}N_0^{(m)} . \tag{12} \]

The freedom available in the choice of \(q_\sigma^{[k]}\) can be used for “normalizing” the columns of the matrix \(K^{(m)}\). If, for example, one sets

\[ q_\sigma^{[k]}=-K_\sigma^{*}P_\sigma D_\sigma^{[k-1]} -\frac{1}{2}\sum_{\alpha=1}^{k-1}K_\sigma^{[k-\alpha]*}K_\sigma^{[\alpha]} \tag{13} \]

\[ (k=1,\ldots,m;\ \sigma=1,\ldots,n), \]

then the Euclidean norm of the columns of the matrix \(K^{(m)}\), up to quantities of order \(\varepsilon^{m+1}\), will be equal to one. In what follows, unless otherwise specified, we shall assume that the columns of the matrix \(K^{(m)}\) possess the indicated property.

3. We define the domain of limiting deviations by the relation

\[ V(\tau,\varepsilon,x)\equiv \bigl((K^{(m)})^{-1}(\tau,\varepsilon)x,\ K^{(m)-1}(\tau,\varepsilon)x\bigr)\le \rho . \tag{14} \]

Geometrically, the domain (14) is an \(n\)-dimensional ellipsoid bounded by the surface \(V(\tau,\varepsilon,x)=\rho\). Each of the \(2n\) rays \(x=\pm K(\tau,\varepsilon)s\) \((\sigma=1,\ldots,n;\ 0<s\le\infty)\) intersects this surface once, at the value of the parameter \(s=\sqrt{\rho}\). Up to quantities of order \(\varepsilon^{m+1}\), the points of intersection are at a constant distance \(\sqrt{\rho}\) from the origin \((x=0)\).

We shall determine the conditions of stability and instability of the trivial solution of equation (5) with respect to the domain (14).

From (7) we find

\[ \frac{d\|y\|}{dt} = \sum_{\sigma=1}^{n}\operatorname{Re}\lambda_{\sigma}^{(m)} \frac{|y_{\sigma}|^{2}}{\|y\|} + \varepsilon^{m+1}\frac{y^{*}\mathcal{P}^{(m)}y}{\|y\|} + \frac{1}{2\|y\|}\operatorname{Re}\{y^{*}K^{(m)-1}H\}, \tag{15} \]

where

\[ \mathcal{P}^{(m)} = -\frac{1}{2}\left[K^{(m)-1}N_{0}^{(m)} + \left(K^{(m)-1}N_{0}^{(m)}\right)^{*}\right], \]

and \(y_{\sigma}\) \((\sigma=1,\ldots,n)\) are the elements of the column matrix \(y\).

In accordance with expressions (6) and (15), the derivative of the positive definite function \(V\) with respect to \(t\), computed by virtue of equation (5), is equal to

\[ \frac{1}{2}\frac{dV}{dt} \sum_{\sigma=1}^{n}\operatorname{Re}\lambda_{\sigma}^{(m)}|y_{\sigma}|^{2} + \varepsilon^{m+1}y^{*}\mathcal{P}^{(m)}y + \frac{1}{2}\operatorname{Re}\{y^{*}K^{(m)-1}H\}. \tag{16} \]

Let

\[ \mu^{(m)}(\tau,\varepsilon) = \max_{\sigma}\left(\operatorname{Re}\lambda_{\sigma}^{(m)}(\tau,\varepsilon)\right); \]

\(v_{\min}^{(m)}(\tau,\varepsilon)\), \(v_{\max}^{(m)}(\tau,\varepsilon)\) are, respectively, the minimum and maximum eigenvalues of the Hermitian matrix \(\mathcal{P}^{(m)}\).

Theorem 1. If

\[ \mu^{(m)}(\tau_{0},\varepsilon) + \varepsilon^{m+1}v_{\max}^{(m)}(\tau_{0},\varepsilon) <0 \qquad (\tau_{0}=\varepsilon t_{0}), \tag{17} \]

then the trivial solution of equation (5) is stable on the finite interval \([t_{0},\,t_{0}+\Delta t]\).

Proof. From condition (2) and the boundedness of \(K^{(m)}\) it follows that, uniformly in \(t\) on the segment \([t_{0},T]\),

\[ \lim_{y\to 0}\frac{H(t,K^{(m)}y)}{\|y\|}=0. \tag{18} \]

Taking this into account, from (16) we obtain

\[ \frac{1}{2}\frac{dV}{dt} \leq \left(\mu^{(m)}(\tau,\varepsilon) + \varepsilon^{m+1}v_{\max}^{(m)}(\tau,\varepsilon)\right)\|y\|^{2} + o(\|y\|^{2}). \]

It is therefore clear that, if inequality (17) holds, then for sufficiently small \(\|y\|\) at the point \(t=t_{0}\), and by continuity also within some finite interval

\[ [t_{0},\,t_{0}+\Delta t]\subset [t_{0},T], \]

\[ dV/dt<0, \]

which proves the theorem.

Theorem 2. If

\[ \mu^{(m)}(\tau_{0},\varepsilon) + \varepsilon^{m+1}v_{\min}^{(m)}(\tau_{0},\varepsilon) >0, \tag{19} \]

then the trivial solution of equation (5) is not stable on the finite interval \([t_{0},\,t_{0}+\Delta t]\), i.e. \(\Delta t=0\).

Proof. Integrating (15), we obtain

\[ \|y(t,\varepsilon)\| = \|y(t_{0},\varepsilon)\| \exp \left\{ \int_{t_{0}}^{t} \left[ \sum_{\sigma=1}^{n} \operatorname{Re}\lambda_{\sigma}^{(m)} \frac{|y_{\sigma}|^{2}}{\|y\|^{2}} + \varepsilon^{m+1} \frac{y^{*}\mathcal{P}^{(m)}y}{\|y\|^{2}} + O(\|y\|) \right]dt \right\}. \tag{20} \]

If

\[ \varphi(\tau,\varepsilon,y) \equiv \sum_{\sigma=1}^{n} \operatorname{Re}\lambda_{\sigma}^{(m)} \frac{|y_{\sigma}|^{2}}{\|y\|^{2}} + \varepsilon^{m+1} \frac{y^{*}\mathcal{P}^{(m)}y}{\|y\|^{2}} \neq 0, \]

then, for sufficiently small \(\|y\|\), the sign of the integrand coincides with the sign of the function \(\varphi\).

Suppose that \(\mu^{(m)}(\tau_0,\varepsilon)=\operatorname{Re}\lambda_s^{(m)}(\tau_0,\varepsilon)\) and that \(\bar{x}=K^{(m)}\bar{y}\) is a particular solution of equation (5), determined by the initial conditions \(y_s(t_0,\varepsilon)=\sqrt{\rho}\), \(y_\sigma(t_0,\varepsilon)=0\) \((\sigma\ne s)\). For this solution,

\[ \varphi\bigl(\tau_0,\varepsilon,\bar{y}(t_0,\varepsilon)\bigr) \ge \mu^{(m)}(\tau_0,\varepsilon)+\varepsilon^{m+1}\nu_{\min}^{(m)}(\tau_0,\varepsilon)>0. \]

Therefore, at the point \(t_0\), for sufficiently small \(\rho\), the integrand in equality (20) is positive. By continuity it is positive also in some neighborhood of the point \(t_0\). Hence, in this neighborhood,

\[ dV(\tau,\varepsilon,\bar{x})/dt=2\|\bar{y}\|\,d\|\bar{y}\|/dt>0, \]

and therefore, along the indicated solution, condition (4) is not satisfied.

Theorem 3. If

\[ \mu^{(m)}(\tau_0,\varepsilon)+\varepsilon^{m+1}\nu_{\min}^{(m)}(\tau_0,\varepsilon)\le 0 \le \mu^{(m)}(\tau_0,\varepsilon)+\varepsilon^{m+1}\nu_{\max}^{(m)}(\tau_0,\varepsilon), \tag{21} \]

then the trivial solution of equation (5) may fail to possess stability on the finite interval \([t_0,t_0+\Delta t]\).

Proof. The inequalities (21) allow the existence of a particular solution \(\bar{x}=K^{(m)}\bar{y}\) such that

\[ \varphi\bigl(\tau_0,\varepsilon,\bar{y}(t_0,\varepsilon)\bigr)=0,\qquad \|\bar{y}(t_0,\varepsilon)\|=\sqrt{\rho}. \]

For this solution the sign of the integrand in equality (20) is determined by the sign of \(O(\|y\|)\), so that, depending on the properties of the nonlinear terms at \(t=t_0\), and by continuity also within the limits of some neighborhood of the point \(t_0\), the integrand may be positive. Then, in this neighborhood,

\[ dV(\tau,\varepsilon,\bar{x})/dt>0, \]

and, consequently, condition (4) will not be satisfied.

4. Applying the results of item 3 to equation (1), we shall have:

\[ \begin{aligned} \mu^{(m)}(t_0)+\nu_{\max}^{(m)}(t_0)&<0 &&\text{— sufficient condition for stability,}\\ \mu^{(m)}(t_0)+\nu_{\min}^{(m)}(t_0)&\le 0 &&\text{— necessary condition for stability,}\\ \mu^{(m)}(t_0)+\nu_{\min}^{(m)}(t_0)&>0 &&\text{— sufficient condition for instability,}\\ \mu^{(m)}(t_0)+\nu_{\max}^{(m)}(t_0)&\ge 0 &&\text{— necessary condition for instability.} \end{aligned} \tag{22} \]

Here

\[ \mu^{(m)}(t)=\mu^{(m)}(\tau,\varepsilon)\big|_{\varepsilon=1},\qquad \nu_{\min,\max}^{(m)}(t)=\nu_{\min,\max}^{(m)}(\tau,\varepsilon)\big|_{\varepsilon=1}. \]

The inequalities (22) constitute a whole family of necessary and sufficient conditions corresponding to the numbers \(m=0,1,2,\ldots,l-1\).

The criteria (22), for a given \(m\), do not solve the problem in those cases when \(\mu^{(m)}\) lies within the strip

\[ -\nu_{\min}^{(m)}\ge \mu^{(m)}\ge -\nu_{\max}^{(m)}. \tag{23} \]

With increasing \(m\) (at least up to a certain value) one may expect a substantial reduction in the width of the “strip of indeterminacy,” especially when the coefficients of the equations of the first approximation are slowly varying functions of \(t\).

Moscow Aviation Institute
named after S. Ordzhonikidze

Received
29 II 1968

REFERENCES

\({}^{1}\) G. V. Kamenkov, PMM, 17, no. 5 (1953).
\({}^{2}\) A. A. Lebedev, PMM, 18, no. 2 (1954).
\({}^{3}\) A. A. Lebedev, Izv. vyssh. uchebn. zaved., ser. Aviation Technology, no. 1 (1958).
\({}^{4}\) K. A. Abgaryan, DAN, 166, no. 2 (1966).