Reports of the Academy of Sciences of the USSR
1966. Vol. 171, No. 3

MATHEMATICS

V. A. YAKUBOVICH

PERIODIC AND ALMOST PERIODIC LIMITING REGIMES OF CONTROL SYSTEMS WITH SEVERAL, GENERALLY SPEAKING, DISCONTINUOUS NONLINEARITIES

(Presented by Academician L. S. Pontryagin, January 24, 1966)

1°. Consider the differential equations of a control system with \(\varkappa\) nonlinear blocks \(\varphi_j=\varphi_j(\sigma_j)\):

\[ dx/dt = Px + q\varphi(\sigma) + f(t), \qquad \sigma = r^*x . \tag{1} \]

Here the matrices \(P,\ q,\ r,\ x,\ f(t),\ \varphi(\sigma)=\|\varphi_j(\sigma_j)\|\) are real and have, respectively, dimensions \(\nu\times\nu,\ \nu\times\varkappa,\ \nu\times\varkappa,\ \nu\times 1,\ \nu\times 1,\ \varkappa\times 1\) (and, consequently, \(\sigma=\|\sigma_j\|\) is a matrix of order \(\varkappa\times 1\)). We shall assume that the \(\varphi_j(\sigma_j)\) are piecewise continuous functions having only discontinuities of the first kind, and that

\[ 0 \leq \Delta\varphi_j/\Delta\sigma_j \leq \mu_j,\qquad j=1,2,\ldots,\varkappa, \tag{2} \]

where \(\Delta\varphi_j=\varphi_j(\sigma_j+\Delta\sigma_j)-\varphi_j(\sigma_j)\), \(-\infty<\sigma_j<+\infty\), \(-\infty<\Delta\sigma_j<+\infty\), and \(\sigma_j,\ \sigma_j+\Delta\sigma_j\) are points of continuity of the function \(\varphi_j(\sigma_j)\). Without loss of generality we shall suppose that \(\mu_j<\infty\) for \(j=1,\ldots,\varkappa_1\); \(\mu_j=\infty\) for \(j=\varkappa_1+1,\ldots,\varkappa_1+\varkappa_2=\varkappa\)*. Suppose that \(f(t)\in L(t_1,t_2)\) for any \(-\infty<t_1<t_2<+\infty\).

We shall call a solution of system (1) any absolutely continuous \(\nu\times 1\) matrix-function \(x(t)\) such that, for some \(\varkappa\times 1\) matrix-function \(\psi(t)=\|\psi_j(t)\|\), summable on every interval and called an augmented function of \(\varphi[\sigma(t)]\), the following holds almost everywhere:
\[ dx/dt=Px+q\psi(t)+f(t), \qquad \varphi_j[\sigma_j(t)-0]\leq \psi_j(t)\leq \varphi_j[\sigma_j(t)+0], \]
where \(\sigma(t)=\|\sigma_j(t)\|=r^*x(t)\). It is easy to prove a local existence theorem, as well as continuous dependence of the solution on the initial conditions in the following sense: if \(x(t,a_k)\) are solutions of system (1) on a finite interval \(\Delta=[t_0,T]\) or \(\Delta=[T,t_0]\) and \(x(t_0,a_k)=a_k\to a\) as \(k\to\infty\), then for some subsequence \(x(t,a_{k_n})\to x(t)\) as \(k_n\to\infty\), \(t\in\Delta\), and \(x(t)\) is a solution of system (1) for \(t\in\Delta\). We shall say that on \((t_1,t_2)\) a solution \(x(t)\) is a sliding regime if \(\sigma(t)=r^*x(t)\) is not a point of continuity of the matrix-function \(\varphi(\sigma)\) for any \(t\in(t_1,t_2)\). For systems (1) of a certain class with
\[ f(t)=\sum_k \operatorname{Re}\bigl(f_k e^{i\omega_k t}\bigr) \]
sliding regimes (on \((-\infty,+\infty)\)) are found simply (see \((^2)\) for the case \(\varkappa=1\)). From Theorem 1 below it follows that in a number of cases the sliding regime \(x^0(t)\) is limiting, i.e., for any solution \(x(t)\) one has \(|x(t)-x^0(t)|\to 0\) as \(t\to\infty\)*.

* It is possible that \(\varkappa_1=0\) or \(\varkappa_2=0\). The number of discontinuous nonlinearities is, obviously, no greater than \(\varkappa_2\). The case in which, instead of (2), \(\mu_j'\leq \Delta\varphi_j/\Delta\sigma_j\leq \mu_j''\) holds reduces to the case under consideration by replacing \(\varphi_j=\varphi_j-\mu_j'\sigma_j\) or \(\varphi_j=\mu_j''\sigma_j-\varphi_j\).

** For \(\varkappa_2=1\) the definition of a solution given here coincides with the definition of A. F. Filippov \((^1)\), but requires, in addition, the specification of the functions \(\psi_j(t)\). For \(\varkappa_2>1\), not every Filippov solution \((^1)\) is a solution in the sense defined here. Analogous definitions and assertions are valid also in the more general case of the system
\[ dx/dt=F(x,\varphi,t), \]
where \(\varphi=\varphi(\sigma)\), \(\sigma=r^*x\), and \(F\) is a continuous function, \(x,\varphi,F(x,\varphi,t)\in L(t_1,t_2)\), \(-\infty<t_1<t_2<+\infty\).

*** Since a sliding regime is a nonclassical solution, this fact justifies the definition of solution given above.

Denote by \(\chi(\lambda)=r^{*}(P-\lambda I)^{-1}q\) the transfer matrix-function of the linear part of the system from the inputs \(\varphi_{1},\ldots,\varphi_{\chi}\) to the outputs \((-\sigma_{1}),\ldots,(-\sigma_{\chi})\), by \(\mu_{\mathrm d}^{-1}\) the diagonal matrix with diagonal elements \(\mu_{1}^{-1},\ldots,\mu_{\chi}^{-1}\) (where \(\mu_i^{-1}=0\) if \(\mu_i=\infty\)), and by \(\tau_{\mathrm d}\) the diagonal \(\chi\times\chi\) matrix with diagonal elements \(\tau_1,\ldots,\tau_\chi\).* Put
\[ \pi(\omega)=\tau_{\mathrm d}\mu_{\mathrm d}^{-1}+\operatorname{Re}\,[\tau_{\mathrm d}\chi(i\omega)]. \]
Represent the matrices \(q,\ \pi(\omega)\) in the form \(q=\|q_1,q_2\|\), \(\pi(\omega)=\|\pi_{jh}\|\) \((j,h=1,2)\), where \(q_j\) are matrices of order \(\nu\times\chi_j\), and \(\pi_{jh}\) are matrices of order \(\chi_j\times\chi_h\) \((j,h=1,2)\).

Theorem 1. Suppose that \(P\) is a Hurwitz matrix, that the rank of the matrix \(q_2\) is \(\chi_2\), and that, for some \(\tau_1>0,\ldots,\tau_\chi>0\), one has \(\pi(\omega)>0\) for \(-\infty<\omega<\infty\),
\[ \lim_{\omega\to\infty}\omega^2(\pi_{22}-\pi_{21}\pi_{11}^{-1}\pi_{12})>0. \]
Then:

a) for (1) exponential convergence holds, i.e., for some numbers \(\gamma>0,\ \varepsilon>0\), for any \(t\geq t_0\) and any solutions \(x_1(t),x_2(t)\),
\[ |x_1(t)-x_2(t)|\leq \gamma \exp[-\varepsilon(t-t_0)]\,|x_1(t_0)-x_2(t_0)|; \tag{3} \]

b) if \(f(t)\) is a matrix-function bounded on \((-\infty,+\infty)\), then there exists a unique solution \(x^0(t)\) bounded on \((-\infty,+\infty)\) (a limiting regime); c) if \(f(t+T)\equiv f(t)\), then \(x^0(t+T)\equiv x^0(t)\); d) if \(f(t)\) is an almost-periodic matrix-function, then \(x^0(t)\) is also an almost-periodic matrix-function.***

Theorem 2. Let the rank of the \(\nu\times\chi\nu\)-matrix \(\|q,Pq,\ldots,P^{\nu-1}q\|\) be equal to \(\nu\); let the roots of the equation \(\det(P-\lambda I_\nu)=0\) lie in the half-plane \(\operatorname{Re}\lambda\leq -\varepsilon<0\), and let \(\pi(-\varepsilon+i\omega)\geq 0\) for some \(\tau_1\geq0,\ldots,\tau_\chi\geq0\) and all \(\omega,\ -\infty<\omega<+\infty\). Then, for some \(\gamma>0\), for any \(t\geq t_0\) and any solutions \(x_1(t),x_2(t)\), (3) holds, and the assertions b), c), d) of Theorem 1 are also valid.

\(2^\circ\). The proof of Theorems 1 and 2 uses the following algebraic propositions, the most essential part of which, for \(\chi>1\), was proved by V. M. Popov \((^5)\).**** For \(\chi=1\) analogous propositions were established in \((^6,^7)\) (see also \((^8,^9)\)). Let \(X=X^*,\ A,\ C=C^*,\ a,\ b,\ \rho=\rho^*\geq0\) be real matrices, respectively of orders \(\nu\times\nu,\ \nu\times\nu,\ \nu\times\nu,\ \nu\times\chi,\ \nu\times\chi,\ \chi\times\chi\). Put
\[ G=C-A^*X-XA,\qquad g=-Xa-b,\qquad Q(X)=\left\|\begin{matrix}G&g\\ g^*&\rho\end{matrix}\right\| \tag{4} \]
and consider the problem of determining conditions for the existence of a matrix \(X=X^*\) satisfying the inequality \(Q(X)\geq0\).

Introduce the notation:
\[ A_\omega=A-i\omega I_\nu,\qquad a_\omega=A_\omega^{-1}a,\qquad \pi_0(\omega)=\rho+2\operatorname{Re}\,b^*a_\omega+a_\omega^*Ca_\omega. \]
We shall call the \(\chi\times\chi\)-matrix \(\pi_0(\omega)\) the characteristic of the matrix function \(Q(X)\).

Theorem 3. a) For the existence of a solution \(X=X^*\) of the inequality \(Q(X)\geq0\), it is necessary that \(\pi_0(\omega)\geq0\) for \(-\infty<\omega<+\infty\); b) if the rank of the \(\nu\times\chi\nu\)-matrix \(\Phi=\|a,Aa,\ldots,A^{\nu-1}a\|\) is equal to \(\nu\), then the preceding condition is also sufficient; c) if \(C\leq0,\ Q(X)\geq0\), then the null space of the matrix \(X\) is contained in the subspace orthogonal

* Here and below the following notation is used. \(I_k\) is the identity \(k\times k\)-matrix. An asterisk denotes Hermitian conjugation, \(\operatorname{Re}N=\frac12(N+N^*)\). The notation \(H>0\) means that \(H\) is a positive definite matrix. \(A+B\left\|\begin{matrix}A&0\\0&B\end{matrix}\right\|\).

** For \(\chi_2=0\) the condition on the matrix \(q_2\) and the last inequality are absent. For \(\chi_1=0\) the last inequality becomes the inequality
\[ \lim_{\omega\to\infty}\omega^2\pi(\omega)>0. \]

*** For \(\chi_2=0\) Theorem 1 was formulated by the author without proof in \((^3)\). For \(\chi=1\) Theorem 1 was proved in \((^2)\), and a closely related assertion in \((^4)\). Note that from (4) right-hand uniqueness follows: if \(x_1(t_0)=x_2(t_0)\), then \(x_1(t)=x_2(t)\) for \(t\geq t_0\). Left-hand uniqueness may fail.

**** Namely assertion b) of Theorem 3, which is given below in a formulation somewhat different from \((^5)\).

by the columns of the matrix \(\Psi=\|b,A^*b,\ldots,A^{*\nu-1}b\|\); if, in particular, \(A\) is a Hurwitz matrix and the rank of \(\Psi\) is \(\nu\), then \(X>0\).

Assume that \(\rho\) has the form \(\rho=\|\rho_{jh}\|\) \((j,h=1,2)\), where the order of \(\rho_{jh}\) is \(\varkappa_j\times\varkappa_h\), \(\varkappa_1+\varkappa_2=\varkappa\), \(\rho_{12}=\rho_{21}^*=0\), \(\rho_{22}=0\)*. Represent the matrices \(\pi_0(\omega)\), \(a\), \(b\), \(g\) in the form \(\pi_0(\omega)=\|\pi_{jh}\|\) \((j,h=1,2)\), \(a=\|a_1,a_2\|\), \(b=\|b_1,b_2\|\), \(g=\|g_1,g_2\|\), where the order of \(\pi_{jh}\) is \(\varkappa_j\times\varkappa_h\), and the orders of \(a_j,b_j,g_j\) are \(\nu\times\varkappa_j\).

Theorem 4. Assume that \(A\) is a Hurwitz matrix and that the rank of \(a_2\) is \(\varkappa_2\). For the existence of a solution of the inequality \(Q(X)\geq0\), where \(Q(X)\) has the (maximally possible) rank \(\nu+\varkappa_1\), it is necessary and sufficient that the following be fulfilled: (I) \(\pi_0(\omega)>0\) for \(-\infty<\omega<+\infty\), (II) \(\pi_\infty=\lim_{\omega\to\infty}\omega^2(\pi_{22}-\pi_{21}\pi_{11}^{-1}\pi_{12})>0\)*.

3°. Proof of Theorem 3. a) We have
\[ a_\omega^*Ga_\omega=2\operatorname{Re}a_\omega^*(g+b)+a_\omega^*Ca_\omega . \]
Putting \(z_\omega^*=\|a_\omega^*,-I_\varkappa\|\), we obtain the identity
\[ z_\omega^*Q(X)z_\omega\equiv \pi_0(\omega). \]
Therefore \(\pi_0(\omega)\geq0\). b) By Lemma 1 (6) (5) there exists a matrix \(X=M=M^*\) such that \(Q(M)\) has the form
\[ Q(M)=\|L,K\|^*\|L,K\|\geq0 . \]
c) Let \(Xn=0\). Then \(n^*Gn=n^*Cn=0\), since \(G\geq0\), \(C\leq0\). Therefore \(Gn=0\), \(Cn=0\), \(XAn=0\), i.e., the null space of the matrix \(X\) is invariant with respect to \(A\). Let \(\xi\) be an arbitrary \(\varkappa\times1\) vector, \(y^*=\|n^*,\xi^*\|\). From the inequality
\[ y^*Q(X)y=2\operatorname{Re}n^*g\xi+\xi^*\rho\xi\geq0 \]
it follows that \(n^*g=0\), \(n^*b=0\). Replacing \(n\) by \(An,A^2n,\ldots\), we obtain that the vector \(n\) is orthogonal to the columns of the matrix \(\Psi\).

4°. Proof of Theorem 4. Necessity. From the relations \(Q(X)\geq0\), \(\rho_{22}=0\), it follows that \(g_2=0\), \(Q(X)=Q_1+\|0\|\), where the \((\nu+\varkappa_1)\times(\nu+\varkappa_1)\)-matrix \(Q_1\) is obtained from \(Q=Q(X)\) (see (4)) by replacing \(g\) by \(g_1\) and \(\rho\) by \(\rho_{11}\). We have (see 3°, a)
\[ \pi_0(\omega)\equiv z_\omega^*Qz_\omega=u^*Q_1u, \]
where \(u^*=\|a_\omega^*,\delta_1^*\|\), \(\delta_1=\|-I_{\varkappa_1},0\|\) (\(\delta_1\) is a matrix of order \(\varkappa_1\times\varkappa\)). Since, by assumption, \(Q_1>0\), it follows that \(\pi_0(\omega)\geq0\). Since the rank of \(\|a^*,\delta_1^*\|\) is \(\varkappa\) and
\[ u^*=\|a^*,\delta_1^*\|S, \]
where \(S=A_\omega^{*-1}+I_{\varkappa_1}\) is a nonsingular matrix, the rank of \(u^*\) is \(\varkappa\) and \(\pi_0(\omega)>0\). Since
\[ \pi_{11}=\rho_{11}+O(\omega^{-1}),\qquad \pi_{12}=g_1^*a_2(i\omega)^{-1}+O(\omega^{-2}),\qquad \pi_{22}=a_2^*Ga_2\cdot\omega^{-2}+O(\omega^{-3}), \]
we have
\[ \pi_\infty=a_2^*(G-g_1\rho_{11}^{-1}g_1^*)a_2 . \]
Since the rank of \(a_2\) is \(\varkappa_2\), \(\rho_{11}>0\), \(Q_1>0\), it follows that \(\pi_\infty>0\).

Sufficiency. Suppose first that the rank of \(\Phi\) is \(\nu\). It is enough to show that, in the preceding notation, the inequality
\[ Q(X)\geq \varepsilon(I_\nu+\rho), \]
where the number \(\varepsilon>0\), has a solution \(X=X^*\). The last inequality is equivalent to the inequality \(Q(X)\geq0\), in which \(C\) and \(\rho\) are replaced respectively by \(C-\varepsilon I_\nu\) and \(\rho-\varepsilon\rho\). According to item b) of Theorem 3, a solution exists if
\[ \alpha(\omega)\equiv \pi_0(\omega)-\varepsilon\rho-\varepsilon a_\omega^*a_\omega\geq0 . \]
The validity of the last inequality for sufficiently small \(\varepsilon>0\) follows from the conditions \(\rho_{11}>0\), \(\pi_0(\omega)>0\), \(\pi_\infty>0\). Here one should use the fact that the inequality
\[ \alpha=\|\alpha_{jh}\|>0 \]
(where \(\alpha_{jh}\) are matrices of order \(\varkappa_j\times\varkappa_h\), \(j,h=1,2\)) is equivalent to the inequalities
\[ \alpha_{11}>0,\qquad \alpha_{22}-\alpha_{21}\alpha_{11}^{-1}\alpha_{12}>0 . \]

Let the rank of \(\Phi\) be \(\nu_1<\nu\). We reduce the problem to an analogous one in which, instead of \(A,a,b\), there will be
\[ A^S=S^{-1}AS=\left\|\begin{array}{cc} A_1 & B\\ 0 & A_2 \end{array}\right\|,\qquad a^S=S^{-1}a=\left\|\begin{array}{c} a'\\ 0 \end{array}\right\|,\qquad b^S=S^*b=\left\|\begin{array}{c} b'\\ b'' \end{array}\right\|, \tag{5} \]
and the rank of the matrix
\[ \Phi_1=\|a',\ldots,A_1^{\nu_1-1}a'\| \]
is \(\nu_1\). (Moreover, the number of rows of the matrices \(A_1,B,a',b'\) is equal to \(\nu_1\).) It is easy to construct, using Theorem 2 \((^{10})\), a nonsingular matrix \(S\) such that \(A^S,a^S\) have the form (5) and the rank of \(\Phi_1\) is \(\nu_1\). We replace the inequality \(Q(X)\geq0\) by the equivalent one
\[ S_0^*Q(X)S_0\geq0, \]
where \(S_0=S+I_\varkappa\). From (4) it follows that
\[ S_0^*Q(X)S_0=Q^S(X^S), \]
where

* This assumption does not restrict generality. Indeed, let the rank of the matrix \(\rho\geq0\) be \(\varkappa_1\). Replacing the inequality \(Q\geq0\) by the equivalent one \(S^*QS\geq0\), where \(S=I_\nu+\delta\) and \(\delta\) is a suitably chosen nonsingular \(\varkappa\times\varkappa\) matrix, we obtain an analogous inequality in which \(\rho\) has the indicated form.

** For \(\varkappa_1=0\) the matrix \(\pi_\infty\) has the form \(\pi_\infty=\lim_{\omega\to\infty}\omega^2\pi_0(\omega)\). For \(\varkappa_2=0\) condition (II) is absent. For \(\varkappa=1\), \(C=0\), Theorem 4 was proved in \((^8)\). The case \(C\ne0\) reduces to the case \(C=0\) (see \((^6)\)).

\(X^S=S^*XS\) and \(Q^S(X^S)\) has the form (4) with the replacement of \(A, C, a, b, X\) by \(A^S, C^S=S^*CS, a^S, b^S, X^S\). From (5) it follows that \(Q^S(X^S)\) and \(Q(X)\) have identical characteristics: \(\pi^S(\omega)\equiv \pi_0(\omega)\). Define the matrix \(X^S\). Let

\[ X^S=\left\| \begin{array}{cc} X_{11} & X_{12}\\ X_{21} & X_{22} \end{array}\right\|,\quad Q^S(X^S)=\left\| \begin{array}{ccc} G_{11} & G_{12} & g^{(1)}\\ G_{21} & G_{22} & g^{(2)}\\ g^{(1)*} & g^{(2)*} & \rho \end{array}\right\|,\quad Q_{11}(X_{11})=\left\| \begin{array}{cc} G_{11} & g^{(1)}\\ g^{(1)*} & \rho \end{array}\right\|, \]

where the matrices \(X_{jh}, G_{jh}, g^{(j)}\) have, respectively, orders \(\nu_j\times \nu_h,\ \nu_j\times \nu_h,\ \nu_j\times \chi,\ \nu_2=\nu-\nu_1\). It is easy to verify that \(Q^S(X^S)\) and \(Q_{11}(X_{11})\) have identical characteristics. Since the rank of \(\Phi_1\) is \(\nu_1\), by what was proved above there exists \(X_{11}=X_{11}^0\) such that \(Q_{11}(X_{11}^0)\ge 0\) and the rank of \(Q_{11}(X_{11}^0)\) is equal to \(\nu_1+\chi_1\). From the form of \(\rho\) it follows that the last (right-hand) \(\chi_2\) columns of \(Q_{11}(X_{11}^0)\) are zero. Since an analogous condition must obviously also be satisfied for the matrix \(Q^S(X^S)\), represent \(g^{(2)}\) in the form \(g^{(2)}=\|g_1^{(2)}, g_2^{(2)}\|\), where \(g_j^{(2)}\) have orders \(\nu_2\times \chi_j\), and require that \(g_2^{(2)}=0\). We have \(g^{(2)}=-X_{21}a_2'-b_2''\), where \(a_2', b_2''\) are matrices composed of the last \(\chi_2\) columns of the matrices \(a', b''\). The rank of \(a_2'\), coinciding with the rank of \(a_2\), is equal to \(\chi_2\). Therefore we take \(X_{21}=X_{21}^0=-b_2''(a_2'^*a_2')^{-1}a_2'^*\). (If \(\chi_2=0\), then \(X_{21}^0\) is an arbitrary matrix.) In the matrix \(Q^S(X^S)\) only \(G_{22}\) has remained undetermined. Let \([Q^S(X^S)]\), \([Q_{11}(X_{11})]\) be the matrices obtained from \(Q^S(X^S), Q_{11}(X_{11})\) by deleting the \(\chi_2\) extreme right (zero) columns and the \(\chi_2\) lower (zero) rows. Since \([Q_{11}(X_{11})]>0\), there exists \(G_{22}^0>0\) such that, after replacing \(G_{22}\) by \(G_{22}^0\), \([Q^S(X^S)]>0\) is fulfilled. Since \(A_2\) is a Hurwitz matrix and \(G_{22}=F-A_2^*X_{22}-X_{22}A_2\), where \(F\) is a matrix depending only on the already found \(X_{11}^0, X_{12}^0\), there exists a solution \(X_{22}=X_{22}^0\) of the equation \(G_{22}=G_{22}^0\). Let \(X^S=\|X_{jh}^0\|\), \(j,h=1,2\). Then \(Q^S(X^S)\ge 0\) and the rank of \(Q^S(X^S)\) is equal to \(\nu+\chi_1\). This implies the assertion of the theorem for \(\nu_1<\nu\).

\(5^\circ\). Proof of Theorem 1. a) Estimate (3) holds if there exists a matrix \(H=H^*>0\) such that, for the function \(V(y)=y^*Hy\), with \(y=x_1(t)-x_2(t)\), one has \(\dot V\le -2\varepsilon V\). The expression \(\dot V\) is transformed to the form \(\dot V=-\Omega_1-\Omega_2\), where

\[ \Omega_1=y^*Gy+2y^*g\psi+\psi^*\tau_{\mathrm d}\mu_{\mathrm d}^{-1}\psi,\quad G=-P^*H-HP,\quad g=-Hq-\tfrac12 r\tau_{\mathrm d}, \]

\[ \Omega_2=(\sigma^{(1)}-\sigma^{(2)})^*\tau_{\mathrm d}\psi-\psi^*\tau_{\mathrm d}\mu_{\mathrm d}^{-1}\psi,\quad \sigma^{(j)}=r^*x_j(t),\quad \psi=\psi^{(1)}-\psi^{(2)}, \]

\(\psi^{(j)}\) is the completed function \(\varphi[\sigma^{(j)}]\). Using condition (2) and the inequalities \(\varphi_n(\sigma_n^{(j)}-0)\le \psi_n^{(j)}\le \varphi_n(\sigma_n^{(j)}+0)\), \(\tau_h>0\), we obtain that \(\Omega_2\ge 0\). By Theorem 4, if the conditions of Theorem 1 are satisfied, there exists a matrix \(H=H^*\) such that \(\Omega_1\) is a nonnegative form in \(y\) and \(\psi\), having rank \(\nu+\chi_1\). Therefore \(G>0,\ H>0,\ \Omega_1\ge 2\varepsilon V\) for sufficiently small \(\varepsilon>0\). Consequently, \(\dot V(y)\le -2\varepsilon V(y)\). b) As above, we obtain that outside a sufficiently large ball \(\dot V(x)<-\varepsilon V(x)\) holds for any solution \(x=x(t)\). Assertion b) follows from Lemmas 1 and 2 \((^2)\), using the property indicated above of continuous dependence of the solution on the initial data. Assertions c), d) are proved in the same way as in paper \((^2)\).

\(6^\circ\). Proof of Theorem 2 repeats the proof of Theorem 1. The only difference is that the existence of a matrix \(H=H^*\), for which \(\dot V(y)\le -2\varepsilon V(y)\) (with prescribed \(\varepsilon>0\)), follows from Theorem 3.

Leningrad State University
named after A. A. Zhdanov

Received
24 I 1966

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