Abstract
Full Text
MATHEMATICS
N. N. KRASOVSKII
ON THE STABILITY OF QUASILINEAR SYSTEMS WITH AFTEREFFECT
(Presented by Academician I. G. Petrovskii on 22 VII 1957)
Known stability theorems for ordinary equations \((^{1-4})\) are transferred in this note to systems with aftereffect
\[ \frac{dx}{dt} = X[x(t+\vartheta),t]+R[x(t+\vartheta),t] \qquad (-h\leq \vartheta\leq 0), \tag{1} \]
where \(x\) is an element of the Banach space \(B\); \(X, R\) are operators on continuous curves \(x(\vartheta)\) \((-h\leq \vartheta\leq 0)\), mapping these curves into \(B\), \(t\geq 0\). The space of continuous curves \(x(\vartheta)\) with norm \(\|x(\cdot)\|=\sup\|x(\vartheta)\|\) for \(-h\leq \vartheta\leq 0\) will be denoted by \(B(\cdot)\), and elements of \(B(\cdot)\) by \(x(\cdot)\).
The stability criteria given below are based on the spectral properties of the linear (bounded) operator \(X\) and include certain known results \((^{5-9})\). The linear equation
\[ \frac{dx}{dt} = X[x(t+\vartheta)] \qquad (-h\leq \vartheta\leq 0,\ x\in B,\ \|X\|=L) \tag{2} \]
is equivalent to the “ordinary” equation
\[ \left.\frac{dx(t,\cdot)}{dt}\right|_{dt=+0} = Ax(t,\cdot) \qquad (x(t,\cdot)\in B(\cdot)) \tag{3} \]
with the unbounded operator
\[ Ax(\cdot)=y(\cdot)= \begin{cases} y(\vartheta)=dx/d\vartheta & \text{for } -h\leq \vartheta<0,\\ y(0)=X[x(\vartheta)]. \end{cases} \tag{4} \]
The investigation of equation (3) is based on results from semigroup theory \((^{10})\). The stability of the quasilinear equation (1) is studied by the Lyapunov method, whose extension to systems with aftereffect is described in papers \((^{12,13})\).
Theorem. If the spectrum \(\{\lambda_\sigma\}\) of the operator \(A\) (4) satisfies the condition
\[ \operatorname{Re}\lambda_\sigma \leq -\gamma \qquad (\gamma>0), \tag{5} \]
then there exists a functional \(v[x(\cdot)]\) satisfying the estimates
\[ c_1\|x(\cdot)\|^k \leq v[x(\cdot)] \leq c_2\|x(\cdot)\|^k, \]
\[ \limsup_{\Delta t\to +0} \bigl(\{v[x(t+\Delta t,\cdot)]-v[x(t,\cdot)]\}/\Delta t\bigr) \leq -c_3\|x(\cdot)\|^k, \]
\[ \bigl|v[x''(\cdot)]-v[x'(\cdot)]\bigr| \leq c_4\|x''(\cdot)-x'(\cdot)\| \sup\bigl(\|x''(\cdot)\|^{k-1},\|x'(\cdot)\|^{k-1}\bigr), \tag{6} \]
where \(k>0\) is any preassigned integer, and \(c_i>0\) are constants.
Remark. If \(B=E_n\), then equation (2) has the form \((^{5,11})\)
\[ \frac{dx_i}{dt} = \sum_{j=1}^{n}\int_{-h}^{0} x_j(t+\vartheta)\,d\eta_{ij}(\vartheta) \qquad (i=1,\ldots,n) \tag{7} \]
and the spectrum of the operator \(A\) is determined by the roots of equation \((^5)\)
\[ \left| \int_{-h}^{0} e^{\lambda\vartheta}\,d\eta_{ij}(\vartheta) - \lambda\delta_{ij} \right|_{1}^{n} =0. \]
A second example of equation (2) is the system of integro-differential equations
\[ \frac{\partial\varphi_i(\xi,t)}{\partial t} = \sum_{j=1}^{n} \left[ \int_{a}^{b} K_{ij}(\xi,s)\varphi_j(s,t)\,ds + \int_{-h}^{0}\varphi_j(\xi,t+\vartheta)\,d\eta_{ij}(\vartheta) \right] \]
\[ (i=1,\ldots,n;\ a\leq \xi\leq b;\ t\geq 0;\ x=\{\varphi_1(\xi),\ldots,\varphi_n(\xi)\}). \]
The proof of the theorem is based on a lemma.
Lemma. The solutions of equation (3) are asymptotically stable and satisfy the inequality
\[ \|x(x_0(\cdot),t,\cdot)\| \leq N\|x_0(\cdot)\|\exp(-q\gamma t) \qquad (t\geq 0), \tag{8} \]
where \(q\) is any chosen number, \(0<q<1\), for all initial data \(x_0(\cdot)\in B(\cdot)\), if condition (5) is satisfied.
A similar criterion for linear equations is given in \((^4)\); here, however, the theorem from \((^4)\) is not directly applicable, since in \((^4)\) the operator \(A\) was assumed to be bounded.
We shall briefly describe the proof of the lemma. Consider the operator
\[ A_0x(\cdot)=y(\cdot)= \begin{cases} y(\vartheta)=dx/d\vartheta, & \text{for } -h\leq \vartheta<0,\\ y(0)=X[e^{\gamma\vartheta}x(\vartheta)]+\gamma x(0), \end{cases} \]
whose spectrum \(\{\lambda_\sigma^0\}\), by virtue of (5), satisfies the inequality
\[ \operatorname{Re}\lambda_\sigma^0\leq 0. \]
Denote by \(D[A_0^2]\) the set of elements of \(B(\cdot)\) on which the operator \(A_0^2\) is defined. Writing out the resolvent \(R(\lambda,A_0)\) of the operator \(A_0\) and repeating, with minor changes, the reasoning of \((^{10})\) (pp. 276–282, 190–194), one can verify that the operator
\[ T_0(t)z(\cdot) = \lim_{\omega\to\infty} \frac{1}{2\pi i} \int_{\gamma_1-i\omega}^{\gamma_1+i\omega} e^{\lambda t}R(\lambda,A_0)z(\cdot)\,d\lambda \qquad (\gamma_1>0,\ t\geq 0) \tag{9} \]
is defined for \(z(\cdot)\in D[A_0^2]\) and coincides with the semigroup operator \(T_0(t)\) (on \(D[A_0^2]\)), for which \(A_0\) is an infinitesimal generating operator \((^{10})\).
In other words, \(y(z_0(\cdot),t,\cdot)=T(t)z_0(\cdot)\) for \(z_0(\cdot)\in D[A_0^2]\), where \(y(z_0(\cdot),t,\cdot)\) is a solution of the “equation”
\[ \frac{dy(t,\cdot)}{dt}=A_0y(t,\cdot) \qquad (t\geq 0). \tag{10} \]
Moreover, it is verified that the operator \(T_0(t)\) satisfies the inequality
\[ \left\|T_0(t)z_0(\cdot)\right\|\leq \left[N_0\left\|z_0(\cdot)\right\|+ N_1\left\|A_0z_0(\cdot)\right\|+ N_2\left\|A_0^2z_0(\cdot)\right\|\right]e^{(\gamma_1+\alpha)t} \tag{11} \]
\[ (t\geq 0;\ z_0(\cdot)\in D[A_0^2];\ N_i=\mathrm{const};\ \alpha,\gamma_1 \text{ are positive constants which can be chosen arbitrarily small}). \]
For \(t=3h\), the segments of the curves \(y(y_0(\cdot),t+\vartheta)\) \((-h\leq \vartheta\leq 0)\) of the solutions (10) are twice continuously differentiable with respect to \(\vartheta\), and, moreover,
\[ \frac{d^2y}{dt^2} = X\left[e^{\gamma\vartheta}\frac{dy(t+\vartheta)}{d\vartheta}\right] +\gamma\left(\frac{dy}{d\vartheta}\right)_{\vartheta=0}; \tag{12} \]
\[ \left\|y(y_0(\cdot),t,\cdot)\right\|\leq P_0\left\|y_0(\cdot)\right\| \quad \text{for } 0\leq t\leq 3h; \tag{13} \]
\[ \left\|y(y_0(\cdot),3h,\cdot)\right\|\leq P_1\left\|y_0(\cdot)\right\|; \qquad \left\|A_0^2y(y_0(\cdot),3h,\cdot)\right\|\leq P_2\left\|y_0(\cdot)\right\|. \tag{14} \]
for all \(y_0(\cdot)\in B(\cdot)\). Therefore \(y(y_0(\cdot),3h,\cdot)\in D[A_0^2]\), and the solutions (10) for \(y_0(\cdot)\in B(\cdot)\) satisfy the inequality
\[ \left\|y(y_0(\cdot),t,\cdot)\right\|\leq P\left\|y_0(\cdot)\right\|\exp(\gamma_1+\alpha)t \quad (P=\mathrm{const},\ t\geq 0). \tag{15} \]
The solutions of equation (2) and of the equation
\[ \frac{dy}{dt}=X\left[e^{\gamma\vartheta}y(t+\vartheta)\right]+\gamma y(t), \]
corresponding to the operators \(A\) and \(A_0\), are connected, evidently, by the relation
\[ x(x_0(\cdot),t)=y(y_0(\cdot),t)\exp(-\gamma t) \quad \text{for } t\geq 0, \]
\[ x_0(\vartheta)=y_0(\vartheta)e^{-\gamma\vartheta}, \]
whence, in consequence of (15), we draw the conclusion that the lemma is valid.
Now, for the proof of the theorem it is sufficient to consider the functional
\[ v[x_0(\cdot)]= \int_0^Q \left\|x(x_0(\cdot),\xi,\cdot)\right\|^k\,d\xi + \sup\left[\left\|x(x_0(\cdot),\xi,\cdot)\right\|^k \quad \text{for } 0\leq \xi\leq Q\right] \]
\[ \left(Q=\frac{4}{\gamma}\ln 2N \quad \text{for } q=\frac12 \text{ in formula (8)}\right). \]
The results obtained make it possible to transfer to equations (1) the stability criteria for ordinary quasilinear systems. Assuming that the operator \(R\) in (1) satisfies the conditions for the existence of solutions in some neighborhood of zero, \(\vartheta(\cdot)\in B(\cdot)\), we give, as examples, two such results.
- If condition (5) is fulfilled, then one can indicate a constant \(a>0\) such that the solution \(x(t,\cdot)\equiv \vartheta(\cdot)\) of the equation
\[ \frac{dx}{dt}=X[x(t+\vartheta)]+R[x(t+\vartheta),t] \quad (-h\leq \vartheta\leq 0) \]
is asymptotically stable, provided only that the inequality
\[ \left\|R[x(\vartheta),t]\right\|\leq a\left\|x(\cdot)\right\|. \tag{16} \]
is satisfied.
This assertion remains valid also in the case when the operator \(R\) is defined on curves \(x(\vartheta)\) for \(\vartheta\in[-h_1,0]\), where \(h_1>h\). In this case, in inequality (16), on the right-hand side one should write \(\left\|x(\cdot)\right\|=\sup |x(\vartheta)|\) for \(-h_1\leq \vartheta\leq 0\).
- If the spectrum of the operator \(A_\infty\)
\[ A_\infty x(\cdot)= \begin{cases} dx/d\vartheta, & \text{for } -h \leqslant \vartheta < 0,\\ X_\infty[x(\cdot)], & \text{for } \vartheta=0 \end{cases} \]
satisfies condition (5), and \(\|R[x(\cdot),t]\| \leqslant D\|x(\cdot)\|^{1+\beta}\) \((D,\ \beta>0\text{—const})\),
\[
\lim_{t\to\infty}\|X[x(\cdot),t]-X_\infty[x(\cdot)]\|=0
\]
uniformly for \(\|x(\cdot)\|=1\) as \(t\to\infty\), then the solution \(x(t,\cdot)=\theta(\cdot)\) of equation (1) is asymptotically stable, and the characteristic numbers of the solutions
\[ \rho=-\varlimsup_{t\to\infty}\left(\frac1t\ln\|x(x(\cdot),t,\cdot)\|\right) \]
satisfy the inequality \(\rho \geqslant \gamma\).
Ural Polytechnic Institute
named after S. M. Kirov
Received
8 IV 1957
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