A Criterion for the Asymptotic Stability of a Periodic Selector-Linear Differential Inclusion

Abstract

We consider a periodic selector-linear differential inclusion. It is proved that for this inclusion to be uniformly asymptotically stable, it is necessary and sufficient that there exists a time-periodic Lyapunov function of a quasi-quadratic form. We derive estimates for the Lyapunov function that guarantee its positive definiteness and the existence of an infinitesimal upper limit.

APPENDIX

Proof of the Lemma. It follows from the assumptions in the Lemma that $v(t,0)\equiv 0$ for all $t\ge 0$ by virtue of the positive homogeneity of the function $v(t,x)$. Since the function $v(t,x)$ is strictly convex in $x_{0} \in \mathrm {R}^{n}$, it follows that for any $x\ne 0$ and $\mu$ ($0<\mu <1$) one has the inequality $v(t,\mu x+(1-\mu )0)<\mu v(t,x)+(1-\mu )v(t,0)$, or $v(t,\mu x)<\mu v(t,x)$. The last inequality can be rewritten in the form $\mu (1-\mu ^{s-1} )v(t,x)>0$. Consequently, $v(t,x)>0$ for all $x\ne 0$ and $t\ge 0$. Let $t\in [0,T]$. Since the function $v(t,x)$ is continuous, the functions $w_{1} (x)=\min _{t\in [0,T]} v(t,x)$ and $w_{2} (x)=\max _{t\in [0,T]} \nu (t,x)$ are continuous and positive definite in $\mathrm {R}^{n}$, with the inequalities $w_{1} (x)\le v(t,x)\le w_{2} (x)$ holding true for all $x\in \mathrm {R}^{n}$ and $t\in [0,T]$. By virtue of the periodicity of the function $v(t,x)$ in $t$, the last inequalities are satisfied for all $t\ge 0$. Note that one has the equality $w_{1} (x)=w_{1} \left (\left \| x\right \| \frac {x}{\left \| x\right \| } \right )=\left \| x\right \| ^{s} w_{1} \left (\frac {x}{\left \| x\right \| } \right )$, $x\ne 0$. A similar equality is satisfied for the function $w_{2} (x).$ By virtue of the continuity of the functions $w_{1} (x)$ and $w_{2} (x)$ in $\mathrm {R}^{n}$ and the compactness of the unit sphere $S^{n} =\{ x:\left \| x\right \| =1\}$, one has inequality (7), where $\lambda _{1} =\min _{x\in S^{n} } w_{1} (t,x)$, $\lambda _{2} =\max _{x\in S^{n} } w_{2} (t,x)$, and $\lambda _{2} \ge \lambda _{1} >0$. The proof of the Lemma is complete.$\quad \blacksquare$

Proof of the Theorem. Necessity. Consider the system of differential equations

$$ \dot {x}=B(t)x,\quad x_{0}\in \mathrm {R}^{n},\quad B(t)\in \Omega (t),\quad \Omega (t+T)=\Omega (t).$$

(A.1)

Note that each solution $x_{B} (t,t_{0} ,x_{0} )$ of the inclusion (2) can be represented for $x_{B} (t_{0} )=x_{0}$ and $B(t)\in \Omega (t)$, $t\ge t_{0}$, in the form $x_{B} (t,t_{0} ,x_{0} )=\Phi _{B} (t,t_{0} )x_{0}$, where $\Phi _{B} (t,t_{0} )$ is the principal matrix solution of system (A.1).

Consider the function

$$ S(t,t_{0} ,x_{0} )={\mathop {\max }\limits _{B(t)\in \Omega (t)}} \big \| x_{B} (t,t_{0} ,x_{0} )\big \| ^{2} ={\mathop {\max }\limits _{B(t)\in \Omega (t)}} \big \| \Phi _{B} (t,t_{0} )x_{0} )\big \|^{2}. $$

(A.2)

The function $S(t,t_{0} ,x_{0} )$ can be written in the form

$$ S(t,t_{0} ,x_{0} )={\mathop {\max }\limits _{x\in X(t,t_{0} ,x_{0} )}} \left \| x\right \| ^{2} =H^{2}\big (X(t,t_{0} ,x_{0} ),0\big ),$$

(A.3)

where $X(t,t_{0} ,x_{0} )$ is the section at time $t\ge t_{0}$ of the integral funnel of the inclusion (2) issuing from the point $(t_{0} ,x_{0} )$ and $H(X(t,t_{0} ,x_{0} ),0)$ is the Hausdorff distance from the section $X(t,t_{0} ,x_{0} )$ to zero. The function $S(t,t_{0} ,x_{0} )$ is defined for all $x_{0} \in \mathrm {R}^{n}$ and $t\ge t_{0} \ge 0$, because the solution set $\{ x(t,t_{0} ,x_{0} )\}$ of the inclusion (2) is compact [20] on each finite interval $[t_{0} ,t]$. The inequality

$$ \left |\sqrt {S(t,t_{1} ,x_{1} )} -\sqrt {S(t,t_{0} ,x_{0} )} \right |\le H\big (X(t,t_{1} ,x_{1} ),X(t,t_{0} ,x_{0} )\big ),$$

which follows from (A.3) and Theorems 15 and 18 in [20], implies that the function $S(t,t_{0} ,x_{0} )$ is continuous in $(t_{0} ,x_{0} )$, $t\ge 0$, $x_{0} \in \mathrm {R}^{n}$, for any fixed $t\ge t_{0}$ and continuous in $t\ge t_{0}$ for fixed $(t_{0} ,x_{0} )$.

Let us prove that the function $S(t,t_{0} ,x_{0} )$ is strictly convex with respect to $x_{0} \in \mathrm {R}^{n}$. Let $x_{0} \ne y_{0}$ and $\lambda \in (0,1)$. The condition for the strict convexity of the function $S(t,t_{0} ,x_{0} )$ has the form

$$ \eqalign { S\big (\lambda x_{0} +(1-\lambda )y_{0} \big )&={\mathop {\max }\limits _{B(t)\in \Omega (t)}} \big \| \Phi _{B} (t,t_{0} )(\lambda x_{0} +(1-\lambda )y_{0} )\big \| ^{2}\cr &<\lambda {\mathop {\max }\limits _{B(t)\in \Omega (t)}} \big \| \Phi _{B} (t,t_{0} )x_{0} )\big \| ^{2} +(1-\lambda ){\mathop {\max }\limits _{B(t)\in \Omega (t)}} \big \| \Phi _{B} (t,t_{0} )y_{0} )\big \| ^{2}.}$$

(A.4)

Let us show that inequality (A.4) is satisfied. Note that one has the inequality

$$ \big \| \Phi _{B} (t,t_{0} )(\lambda x_{0} +(1-\lambda )y_{0} )\big \| ^{2} \le \Big (\lambda \big \| \Phi _{B} (t,t_{0} )x_{0} \big \| +(1-\lambda )\big \| \Phi _{B} (t,t_{0} )y_{0} \big \| \Big )^{2} . $$

(A.5)

Let us estimate the right-hand side of inequality (A.5). Since $x_{0} \ne y_{0}$, we have the inequality

$$ \left (\left \| \Phi _{B} (t,t_{0} )x_{0} \right \| ^{2} -\left \| \Phi _{B} (t,t_{0} )y_{0} \right \| ^{2} \right )^{2} >0. $$

(A.6)

It follows from the condition $\lambda \in (0,1)$ and inequality (A.6) that

$$ \eqalign { &2\lambda (1-\lambda )\big \| \Phi _{B} (t,t_{0} )x_{0} )\big \| \big \| \Phi _{B} (t,t_{0} )y_{0} \big \| \cr &\qquad \qquad {}-\lambda (1-\lambda )\Big (\big \| \Phi _{B} (t,t_{0} )x_{0} \big \| ^{2} +\big \| \Phi _{B} (t,t_{0} )y_{0} \big \| ^{2} \Big )<0.} $$

(A.7)

By transforming inequality (A.7), we obtain

$$ \eqalign { &\Big (\lambda \big \| \Phi _{B} (t,t_{0} )x_{0} \big \| +(1-\lambda )\big \| \Phi _{B} (t,t_{0} )y_{0} \big \| \Big )^{2} \cr &\qquad \qquad {}<\lambda \big \| \Phi _{B} (t,t_{0} )x_{0} )\big \| ^{2} +(1-\lambda )\big \| \Phi _{B} (t,t_{0} )y_{0} )\big \| ^{2}.} $$

(A.8)

Relations (A.8) and (A.5) imply inequality (A.4); this proves the strict convexity of the function $S(t,t_{0} ,x_{0} )$ with respect to $x_{0}$.

It follows from the generalized Filippov theorem [21] that the function $S(t,t_{0} ,x_{0} )$ is locally Lipschitz with respect to $(t_{0} ,x_{0} )$ uniformly on $t\ge t_{0}$.

As was mentioned above, the uniform absolute stability of the inclusion (2) is equivalent to the uniform exponential absolute stability. Therefore, its solutions $x_{B} (t,t_{0} ,x_{0} )$ satisfy the estimate

$$ \big \| x_{B} (t,t_{0} ,x_{0} )\big \|\le \beta \left \| x_{0} \right \| \exp \big (-\alpha (t-t_{0} )\big ),\quad t\ge t_{0}, $$

(A.9)

where the numbers $\alpha >0$ and $\beta \ge 1$ are independent of $B(t)\in \Omega (t)$, $t\ge t_{0}$, and $x_{0} \in \mathrm {R}^{n}$. It follows from (A.2) and (A.9) that

$$ S(t,t_{0} ,x_{0} )\le \beta ^{2} \left \| x_{0} \right \| ^{2} \exp \big (-2\alpha (\tau -t_{0} )\big ),\quad \tau \ge t_{0}. $$

(A.10)

At each point $(t_{0} ,x_{0} )$, $x_{0} \in \mathrm {R}^{n}$, $t_{0} \ge 0$, we define the Lyapunov function $v(t_{0} ,x_{0})$ by the relation

$$ v(t_{0} ,x_{0} )=\int _{t_{0} }^{t_{0} +T_{1} }S(\tau ,t_{0} ,x_{0} )\,d\tau ,\quad T_{1} >\alpha ^{-1} \ln \beta . $$

(A.11)

The strict convexity of the function $S(t,t_{0} ,x_{0} )$ with respect to $x_{0}$ implies the strict convexity of the function $v(t_{0} ,x_{0} )$ with respect to $x_{0} \in \mathrm {R}^{n}$ for each fixed $t_{0} \ge 0$. By virtue of (A.3), we have $S(\tau ,t_{0} ,0)\equiv 0,$ $0\le t_{0} \le \tau$.

Let $B_{1} (\tau ,t_{0} ,x_{0} )\in \Omega (\tau )$ be a solution of the variational problem (A.3), and let $\Phi _{B_{1} } (\tau ,t_{0} ,x_{0} )$ be the principal matrix solution of system (A.1) for $B(\tau )=B_{1} (\tau ,t_{0} ,x_{0} )$. Then for $x_{0} \ne 0$ we have

$$ \eqalign { S(t,t_{0} ,x_{0} )&=x^{\prime}_{0} \Phi ^{\prime}_{B_{1} } (\tau ,t_{0} ,x_{0} )\Phi _{B_{1} } (\tau ,t_{0} ,x_{0} )x_{0} \cr &=x^{\prime}_{0} P(\tau ,t_{0} ,x_{0} )x_{0} , P^{\prime}(\tau ,t_{0} ,x_{0} )=P(\tau ,t_{0} ,x_{0} ).} $$

(A.12)

It follows from (A.11) and (A.12) that $v(t_{0} ,0)\equiv 0$ for all $t_{0} \ge 0$ and that

$$ v(t_{0} ,x_{0} )=x^{\prime}_{0} \left (\int _{t_{0} }^{{t}_{{0}} +T_{1} }P(\xi ,t_{0} ,x_{0} )\,d\xi \right )x_{0} =x^{\prime}_{0} L(t_{0} ,x_{0} )x_{0},\quad L^{\prime}(t_{0} ,x_{0} )=L(t_{0} ,x_{0} ).$$

(A.13)

Let $X_{B_{1} } (t,t_{0} ,x_{0} )$ be the time $t\ge t_{0}$ section of the integral funnel of the inclusion (2) issuing from the point $(t_{0},x_{0} )$. By virtue of the periodicity of the inclusion (2), we have $X_{B_{1} } (t,t_{0} ,x_{0} )=X_{B_{1} } \times (t+T,t_{0} +T,x_{0} )$ (see [22]). It follows from (A.11)–(A.13) and the last equality that $L(t_{0} ,x_{0} )=L(t_{0} +T,x_{0} )$ for all $x_{0} \in \mathrm {R}^{n}$ and $t_{0} \ge 0$.

Since $S(\tau ,t_{0} ,\mu x_{0} )=\mu ^{2} S(\tau ,t_{0} ,x_{0} )$, we have $v(t_{0} ,\mu x_{0} )=\mu ^{2} v(t_{0} ,x_{0} )$ and

$$ v(t,x)=x^{\prime}L(t,x)x=x^{\prime}L(t,\mu x)x,\quad x\ne 0,\quad \mu \ne 0,\quad t\ge 0.$$

(A.14)

Let $x\ne 0$, $\tau _{1} =\left \| x\right \| ^{-1}$, and $\tau _{2} =-\left \| x\right \| ^{-1}$. Then the function $v(t,x)$ can be represented in the form $v(t,x)=x^{\prime}L_{1} (t,x)x$, where

$$ L_{1} (t,x)=2^{-1} \big [L(t,\tau _{1} x)+L(t,\tau _{2} x)\big ].$$

(A.15)

In this case, the matrix $L_{1} (t,x)$ satisfies the condition $L_{1} (t,\mu x)=L_{1} (t,x)=L^{\prime}_{1} (t,x)$ for all $x\ne 0$, $\mu \ne 0$, and $t\ge 0$.

Consequently, we can assume that the matrix $L(t,x)$ in (A.14) satisfies the relations $L(t,x)=L(t,\mu x)=L^{\prime}(t,x)$ for all $x\ne 0$, $\mu \ne 0$, and $t\ge 0$. Thus, the function $v(t,x)$ defined in accordance with (A.11) is representable in the form (5).

Let us proceed to proving inequality (6). Since the function $S(t,t_{0},x_{0})$ satisfies the local Lipschitz condition with respect to $(t_{0} ,x_{0} )$ uniformly in $t\ge t_{0}$, it follows that the function $v(t,x)$ satisfies the local Lipschitz condition as well. Therefore, for any $x_{0} \in \mathrm {R}^{n}$, $t_{0} \ge 0$, and $y\in F(t_{0} ,x_{0} )$ one has the relation

$$ \eqalign { D_{y}^{+} v(t_{0} ,x_{0} )&={\mathop {\overline {\lim }}\limits _{h\to +0}} h^{-1} \Big [v(t_{0} +h,x_{0} +hy)-v(t_{0} ,x_{0} )\Big ]\cr &={\mathop {\overline {\lim }}\limits _{h\to +0}} h^{-1} \Big [v\big (t_{0} +h,x_{0} +hy+o(h)\big )-v(t_{0} ,x_{0} )\Big ]\cr &={\mathop {\overline {\lim }}\limits _{h\to +0}} h^{-1} \Big [v\big (t_{0} +h,x_{\Lambda } (t_{0} +h,t_{0} ,x_{0} )\big )-v(t_{0} ,x_{0} )\Big ],} $$

(A.16)

where $y\in F(t_{0} ,x_{0} )$, $o(h)$ has been selected based on the condition ${x}_{0} +hy+o(h)=x_{\Lambda } (t_{0} +h,t_{0} ,x_{0} )$, and $x_{\Lambda } (t,t_{0} ,x_{0} )$ is the solution of the inclusion (2) for $B(t)\equiv \Lambda =\mathrm {const}$.

It follows from (A.11) and (A.16) that

$$ \eqalign { &h^{-1} \Big [v\big (t_{0} +h,x_{\Lambda } (t_{0} +h,t_{0} ,x_{0} )\big )-v(t_{0} ,x_{0} )\Big ]\cr &\qquad =h^{-1} \left \{\int _{t_{0} }^{t_{0} +h+T_{1} }S\big (\tau ,t_{0} +h,x_{\Lambda } (t_{0} +h,t_{0} ,x_{0} )\big )\,d\tau -\int _{t_{0} }^{t_{0} +T_{1} }S(\tau ,t_{0} ,x_{0} )\,d\tau \right \}}$$

(A.17)

for $h>0$. Note that the sections $X(t,t_{0} ,x_{0} )$ of the integral funnel of the inclusion (2) satisfy the semigroup property of generalized dynamical systems [23], with this property implying the inclusion

$$ X\big (\tau ,t_{0} +h,x_{\Lambda } (t_{0} +h,t_{0} ,x_{0} )\big )\subset X(\tau ,t_{0} ,x_{0} ).$$

(A.18)

It follows from (A.18) and the definition of $S(\tau ,t_{0} ,x_{0} )$ that one has the inequality

$$ S\big (\tau ,t_{0} +h,x_{\lambda }(t_{0} +h,t_{0} ,x_{0} )\big )\le S(\tau ,t_{0} ,x_{0} ),\quad \tau \le t_{0},\quad h>0.$$

From (A.17) and the last inequality we conclude that

$$ \eqalign { &h^{-1}\Big [v\big (t_{0} +h,x_{\Lambda } (t_{0} +h,t_{0} ,x_{0} )\big )-v(t_{0} ,x_{0} )\Big ]\cr &\qquad \qquad {}\le h^{-1} \left \{\int _{t_{0} +h}^{t_{0} +h+T_{1} }S(\tau ,t_{0} ,x_{0} )d\tau -\int _{t_{0} }^{t_{0} +T_{1} }S(\tau ,t_{0} ,x_{0} )d\tau \right \}\cr &\qquad \qquad {}=h^{-1} \left \{\int _{t_{0} }^{t_{0} +h+T_{1} }S(\tau ,t_{0} ,x_{0} )d\tau -\int _{t_{0} }^{t_{0} +h}S(\tau ,t_{0} ,x_{0} )d\tau \right \}} $$

(A.19)

for $h>0$. From (A.16) and (A.19) we obtain

$$ \eqalign { w(t_{0} ,x_{0} )&={\mathop {\max }\limits _{y\in F(t_{0} ,x_{0} )}} D_{y}^{+} v(t_{0} ,x_{0} )\cr &\le \overline {{\mathop {\lim }\limits _{h\to +0}} }h^{-1} \left \{\int _{t_{0} }^{t_{0} +h+T_{1} }S(\tau ,t_{0} ,x_{0} )d\tau -\int _{t_{0} }^{t_{0} +h}S(\tau ,t_{0} ,x_{0} )d\tau \right \}.}$$

(A.20)

By virtue of the continuity of the function $S(\tau ,t_{0} ,x_{0} )$ with respect to $\tau$ and inequality (A.20), we can pass to the ordinary limit as $h\to +0$. Therefore, from (A.10) and (A.20) we obtain the inequality $w(t,x)\le S(t_{0} +T_{1} ,t_{0} ,x_{0} )-S(t_{0} ,t_{0} ,x_{0} )\le -(1-\beta ^{2} \exp (-2\alpha T_{1} ))\left \| x_{0} \right \| ^{2}$; by virtue of the arbitrariness of $x_{0} \in \mathrm {R}^{n}$ and $t_{0} \ge 0$, we obtain the estimate (6) for $\gamma =1-\beta ^{2} \exp (-2\alpha T_{1} )>0$.

Sufficiency. Consider any absolutely continuous solution $x(t)=x_{B} (t,t_{0} ,x_{0} )$ of the inclusion (2), where $x_{0}$ belongs to the ball of arbitrary radius $\rho (\left \| x_{0} \right \| \le \rho )$ and $0\le t_{0} \le t\le T_{1}$. Since the function $v(t,x)$ satisfies the local Lipschitz condition for all $x\in \mathrm {R}^{n}$, $t\ge 0$, we see that the function $V(t)=v(t,x(t))$ will be locally Lipschitz and hence also absolutely continuous for $t\ge 0$. By virtue of this property of the function $V(t)$, almost everywhere on the segment $[t_{0} ,T_{1} ]$ we have the relation

$$ \eqalign { \frac {dV}{dt} &={\mathop {\lim }\limits _{h\to +0}} h^{-1} \big [V(t+h)-V(t)\big ]= {\mathop {\overline {\lim }}\limits _{h\to +0}} h^{-1} \big [V(t+h)-V(t)\big ]\cr &={\mathop {\overline {\lim }}\limits _{h\to +0}} h^{-1}\Big [v\big (t+h,x(t+h)\big )-v\big (t,x(t)\big )\Big ]\cr &={\mathop {\overline {\lim }}\limits _{h\to +0}} h^{-1} \Big [v\big (t+h,x(t)+h\dot {x}(t)\big )-v\big (t,x(t)\big )\Big ]=D_{\dot {x}(t)}^{+} v\big (t,x(t)\big ),}$$

(A.21)

where $\dot {x}(t)\in F(t,x(t))$ is the derivative of the original function $x(t)$.

From (A.21), with allowance for (3), (4), and (6), we obtain the inequality

$$ \frac {dV}{dt}=D_{\dot {x}(t)}^{+} v\big (t,x(t)\big )\le {\mathop {\max }\limits _{y\in F(t,x(t))}} D_{y}^{+} v\big (t,x(t)\big )=w(t,x)\le -\gamma \left \| x(t)\right \| ^{2},\quad \gamma >0, $$

(A.22)

which holds for almost all $t\ge 0$.

The Lemma implies that, under the assumptions of the Theorem, a function $v(t,x)$ of the form (5) satisfies the inequalities $\lambda _{1} \left \| x\right \| ^{2} \le v(t,x)\le \lambda _{2} \left \| x\right \| ^{2}$ for all $t\ge 0$,$x\in \mathrm {R}^{n}$. These inequalities and formula (A.22) for solutions of inclusion (2) imply the exponential estimate (A.9) for $\alpha =\gamma /2\lambda _{2} >0$ and $\beta =\sqrt {\lambda _{2} /\lambda _{1} } \ge 1$. Consequently, under the assumptions of the Theorem, the inclusion (2) is uniformly asymptotically stable. The proof of the Theorem is complete.$\quad \blacksquare$