Stability Analysis of Mechanical Systems with Highly Nonlinear Positional Forces under Distributed Delay

Abstract

This paper considers mechanical systems with linear velocity forces and highly non-linear positional forces containing distributed-delay terms. Asymptotic stability conditions of system equilibria are proved using Lyapunov’s direct method and the decomposition method. The developed approaches are applied to the monoaxial stabilization of a solid body. The theoretical outcomes are confirmed by computer simulation results.

APPENDIX

Proof of Theorem 1. Using the approaches from [20–22, 38], we construct the Lyapunov–Krasovskii functional

$$\begin{gathered} V({{q}_{t}}) = \frac{1}{2}\lambda {{{\dot {q}}}^{ \top }}(t)A\dot {q}(t) + \frac{1}{2}{{q}^{ \top }}(t)Bq(t) + {{q}^{ \top }}(t)A\dot {q}(t) - {{q}^{ \top }}(t)\int\limits_{t - \tau }^t {(\xi - t + \tau )\frac{{\partial \tilde {\Pi }(q(\xi ))}}{{\partial q}}} {\kern 1pt} d\xi \\ + \;\int\limits_{t - \tau }^t {(\alpha + \beta (\xi - t + \tau ))\left\| {q(\xi )} \right\|{{{\kern 1pt} }^{{\mu + 1}}}d\xi } , \\ \end{gathered} $$

(A.1)

where λ, α, and β are positive parameters. Differentiating it along the trajectories of system (3) yields

$$\dot {V} = - \lambda {{\dot {q}}^{ \top }}(t)B\dot {q}(t) + {{\dot {q}}^{ \top }}(t)A\dot {q}(t) - \lambda {{\dot {q}}^{ \top }}(t)\left( {\frac{{\partial \Pi (q(t))}}{{\partial q}} + \int\limits_{t - \tau }^t {\frac{{\partial \tilde {\Pi }(q(\xi ))}}{{\partial q}}d\xi } + P(q(t))q(t)} \right)$$

$$ - \;{{q}^{ \top }}(t)\left( {\frac{{\partial \Pi (q(t))}}{{\partial q}} + \tau \frac{{\partial \tilde {\Pi }(q(t))}}{{\partial q}}} \right) - {{\dot {q}}^{ \top }}(t)\int\limits_{t - \tau }^t {(\xi - t + \tau )\frac{{\partial \tilde {\Pi }(q(\xi ))}}{{\partial q}}d\xi } $$

$$ - \;\beta \int\limits_{t - \tau }^t {{{{\left\| {q(\xi )} \right\|}}^{{\mu + 1}}}d\xi + (\alpha + \beta \tau ){{{\left\| {q(t)} \right\|}}^{{\mu + 1}}} - \alpha {{{\left\| {q(t - \tau )} \right\|}}^{{\mu + 1}}}} .$$

Due to the properties of homogeneous functions [32], we obtain the upper bounds

$$\begin{gathered} \lambda {{c}_{1}}{{\left\| {\dot {q}(t)} \right\|}^{2}} + {{c}_{2}}{{\left\| {q(t)} \right\|}^{2}} - {{c}_{3}}\left\| {q(t)} \right\|\left\| {\dot {q}(t)} \right\| - {{c}_{4}}\tau \left\| {q(t)} \right\|\int\limits_{t - \tau }^t {{{{\left\| {q(\xi )} \right\|}}^{\mu }}d\xi } + \alpha \int\limits_{t - \tau }^t {{{{\left\| {q(\xi )} \right\|}}^{{\mu + 1}}}d\xi } \;\leqslant \;V({{q}_{t}}) \\ \leqslant \;\lambda {{c}_{5}}{{\left\| {\dot {q}(t)} \right\|}^{2}} + {{c}_{6}}{{\left\| {q(t)} \right\|}^{2}} + {{c}_{3}}\left\| {q(t)} \right\|\left\| {\dot {q}(t)} \right\| + {{c}_{4}}\tau \left\| {q(t)} \right\|\int\limits_{t - \tau }^t {{{{\left\| {q(\xi )} \right\|}}^{\mu }}d\xi } + (\alpha + \beta \tau )\int\limits_{t - \tau }^t {{{{\left\| {q(\xi )} \right\|}}^{{\mu + 1}}}d\xi } , \\ \end{gathered} $$

$$\begin{gathered} \dot {V}\;\leqslant \; - {\kern 1pt} (\lambda {{c}_{7}} - {{c}_{8}}){{\left\| {\dot {q}(t)} \right\|}^{2}} + \lambda \left\| {\dot {q}(t)} \right\|\left( {{{c}_{9}}{{{\left\| {q(t)} \right\|}}^{\mu }} + {{c}_{{10}}}\int\limits_{t - \tau }^t {{{{\left\| {q(\xi )} \right\|}}^{\mu }}d\xi } + {{p}_{0}}{{{\left\| {q(t)} \right\|}}^{\sigma }}} \right) - {{c}_{{11}}}{{\left\| {q(t)} \right\|}^{{\mu + 1}}} \\ + \;{{c}_{{12}}}\tau \left\| {\dot {q}(t)} \right\|\int\limits_{t - \tau }^t {{{{\left\| {q(\xi )} \right\|}}^{\mu }}d\xi } - \beta \int\limits_{t - \tau }^t {{{{\left\| {q(\xi )} \right\|}}^{{\mu + 1}}}d\xi } + (\alpha + \beta \tau ){{\left\| {q(t)} \right\|}^{{\mu + 1}}} - \alpha {{\left\| {q(t - \tau )} \right\|}^{{\mu + 1}}}. \\ \end{gathered} $$

Here, c_k are positive constants, k = 1, …, 12.

By Young’s inequality [7], for ${{\left\| {{{q}_{t}}} \right\|}_{\tau }} < \delta$, the positive numbers λ, α, β, and δ can be chosen so that

$$\begin{gathered} \frac{1}{2}\left( {\lambda {{c}_{1}}{{{\left\| {\dot {q}(t)} \right\|}}^{2}} + {{c}_{2}}{{{\left\| {q(t)} \right\|}}^{2}} + \alpha \int\limits_{t - \tau }^t {{{{\left\| {q(\xi )} \right\|}}^{{\mu + 1}}}d\xi } } \right)\;\leqslant \;V({{q}_{t}}) \\ \leqslant \;2\left( {\lambda {{c}_{5}}{{{\left\| {\dot {q}(t)} \right\|}}^{2}} + {{c}_{6}}{{{\left\| {q(t)} \right\|}}^{2}} + (\alpha + \beta \tau )\int\limits_{t - \tau }^t {{{{\left\| {q(\xi )} \right\|}}^{{\mu + 1}}}d\xi } } \right), \\ \end{gathered} $$

(A.2)

$$\dot {V}\;\leqslant \; - {\kern 1pt} {\kern 1pt} \frac{1}{2}\left( {\lambda {{c}_{7}}{{{\left\| {\dot {q}(t)} \right\|}}^{2}} + {{c}_{{11}}}{{{\left\| {q(t)} \right\|}}^{{\mu + 1}}} + \beta \int\limits_{t - \tau }^t {{{{\left\| {q(\xi )} \right\|}}^{{\mu + 1}}}d\xi } } \right).$$

(A.3)

Hence, (A.1) is a Lyapunov–Krasovskii functional of the full type that satisfies the conditions of the asymptotic stability theorem [6, p. 22].

The proof of Theorem 1 is complete.

Proof of Theorem 2. Passing to the new variables x(t) = $\dot {q}(t)$, y(t) = q(t) + ${{B}^{{ - 1}}}A\dot {q}(t)$, we transform system (1) to

$$\begin{gathered} A\dot {x}(t) = - Bx(t) - Q(y(t) - {{B}^{{ - 1}}}Ax(t)) - \int\limits_{t - \tau }^t {D(y(\xi ) - {{B}^{{ - 1}}}Ax(\xi ))d\xi } , \\ B\dot {y}(t) = - Q(y(t) - {{B}^{{ - 1}}}Ax(t)) - \int\limits_{t - \tau }^t {D(y(\xi ) - {{B}^{{ - 1}}}Ax(\xi ))d\xi } . \\ \end{gathered} $$

(A.4)

The trivial solutions of the isolated subsystems (7), (8) are asymptotically stable. Therefore, see [32, 39], for any numbers ${{\nu }_{1}}\; \geqslant \;2$ and ${{\nu }_{2}}\; \geqslant \;2$ there exist twice continuously differentiable Lyapunov functions V₁(x) and V₂(y) with homogeneity orders ν₁ and ν₂, respectively, such that for all $x,y \in {{\mathbb{R}}^{n}}$,

$${{m}_{{11}}}{{\left\| x \right\|}^{{{{\nu }_{1}}}}}\;\leqslant \;{{V}_{1}}(x)\;\leqslant \;{{m}_{{12}}}{{\left\| x \right\|}^{{{{\nu }_{1}}}}},\quad {{m}_{{21}}}{{\left\| y \right\|}^{{{{\nu }_{2}}}}}\;\leqslant \;{{V}_{2}}(y)\;\leqslant \;{{m}_{{22}}}{{\left\| y \right\|}^{{{{\nu }_{2}}}}},$$

$$\left\| {\frac{{\partial {{V}_{1}}(x)}}{{\partial x}}} \right\|\;\leqslant \;{{m}_{{13}}}{{\left\| x \right\|}^{{{{\nu }_{1}} - 1}}},\quad \left\| {\frac{{\partial {{V}_{2}}(y)}}{{\partial y}}} \right\|\;\leqslant \;{{m}_{{23}}}{{\left\| y \right\|}^{{{{\nu }_{2}} - 1}}},$$

$${{\left( {\frac{{\partial {{V}_{1}}(x)}}{{\partial x}}} \right)}^{ \top }}{{A}^{{ - 1}}}Bx(t)\; \geqslant \;{{m}_{{14}}}{{\left\| x \right\|}^{{{{\nu }_{1}}}}},\quad {{\left( {\frac{{\partial {{V}_{2}}(y)}}{{\partial y}}} \right)}^{ \top }}{{B}^{{ - 1}}}(Q(y) + \tau D(y))\; \geqslant \;{{m}_{{24}}}{{\left\| y \right\|}^{{{{\nu }_{1}} + \mu - 1}}}.$$

Here, m_kj are positive constants, k = 1, 2,  j = 1, 2, 3, 4.

Consider the Lyapunov function

$$\tilde {V}(x,y) = {{V}_{1}}(x) + {{V}_{2}}(y).$$

(A.5)

Calculating its derivative along the trajectories of system (A.4) and using the properties of homogeneous functions, we obtain the upper bound

$$\dot {\tilde {V}}\;\leqslant \; - {\kern 1pt} {{m}_{{14}}}{{\left\| {x(t)} \right\|}^{{{{\nu }_{1}}}}} + {{c}_{1}}{{\left\| {x(t)} \right\|}^{{{{\nu }_{1}} - 1}}}\left( {{{{\left\| {x(t)} \right\|}}^{{{{\mu }_{1}}}}} + {{{\left\| {y(t)} \right\|}}^{{{{\mu }_{1}}}}}} \right)$$

$$ + \;{{c}_{2}}{{\left\| {x(t)} \right\|}^{{{{\nu }_{1}} - 1}}}\int\limits_{t - \tau }^t {\left( {{{{\left\| {x(\xi )} \right\|}}^{\mu }} + {{{\left\| {y(\xi )} \right\|}}^{\mu }}} \right)d\xi } - {{\left( {\frac{{\partial {{V}_{2}}(y(t))}}{{\partial y}}} \right)}^{ \top }}{{B}^{{ - 1}}}\int\limits_{t - \tau }^t {D(y(\xi ))d\xi } $$

$$ - \;{{\left( {\frac{{\partial {{V}_{2}}(y(t))}}{{\partial y}}} \right)}^{ \top }}{{B}^{{ - 1}}}Q(y(t)) + {{c}_{3}}{{\left\| {y(t)} \right\|}^{{{{\nu }_{2}} - 1}}}\left\| {Q(y(t)) - Q(y(t) - {{B}^{{ - 1}}}Ax(t))} \right\|$$

$$ + \;{{c}_{4}}{{\left\| {y(t)} \right\|}^{{{{\nu }_{2}} - 1}}}\int\limits_{t - \tau }^t {\left\| {D(y(\xi )) - D(y(\xi ) - {{B}^{{ - 1}}}Ax(\xi ))} \right\|d\xi } ,$$

where c₁, c₂, c₃, and c₄ are positive constants.

Note that for any numbers ε₁ > 0 and ε₂ > 0, it is possible to indicate h₁ > 0 and h₂ > 0 such that

$$\left\| {Q(y) - Q(y - {{B}^{{ - 1}}}Ax)} \right\|\;\leqslant \;{{\varepsilon }_{1}}{{\left\| y \right\|}^{\mu }} + {{h}_{1}}{{\left\| x \right\|}^{\mu }},$$

$$\left\| {D(y) - D(y - {{B}^{{ - 1}}}Ax)} \right\|\;\leqslant \;{{\varepsilon }_{2}}{{\left\| y \right\|}^{\mu }} + {{h}_{2}}{{\left\| x \right\|}^{\mu }}$$

for all $x,y \in {{\mathbb{R}}^{n}}$.

Now, we choose the Lyapunov–Krasovskii functional

$$\begin{gathered} V({{x}_{t}},{{y}_{t}}) = \tilde {V}(x(t),y(t)) - {{\left( {\frac{{\partial {{V}_{2}}(y(t))}}{{\partial y}}} \right)}^{ \top }}{{B}^{{ - 1}}}\int\limits_{t - \tau }^t {(\xi + \tau - t)D(y(\xi ))d\xi } \\ + \;\int\limits_{t - \tau }^t {({{\alpha }_{1}} + {{\beta }_{1}}(\xi + \tau - t)){{{\left\| {x(\xi )} \right\|}}^{{{{\nu }_{1}}}}}d\xi } + \int\limits_{t - \tau }^t {({{\alpha }_{2}} + {{\beta }_{2}}(\xi + \tau - t)){{{\left\| {y(\xi )} \right\|}}^{{{{\nu }_{2}} + \mu - 1}}}d\xi {\kern 1pt} } , \\ \end{gathered} $$

where $\tilde {V}(x,y)$ is the Lyapunov function given by (A.5) and α₁, β₁, α₂, and β₂ are positive parameters. As a result, we have

$${{c}_{5}}{{\left\| {x(t)} \right\|}^{{{{\nu }_{1}}}}} + {{c}_{6}}{{\left\| {y(t)} \right\|}^{{{{\nu }_{2}}}}} - {{c}_{7}}\tau {{\left\| {y(t)} \right\|}^{{{{\nu }_{2}} - 1}}}\int\limits_{t - \tau }^t {{{{\left\| {y(\xi )} \right\|}}^{\mu }}d\xi } $$

$$ + \;{{\alpha }_{1}}\int\limits_{t - \tau }^t {{{{\left\| {x(\xi )} \right\|}}^{{{{\nu }_{1}}}}}d\xi } + {{\alpha }_{2}}\int\limits_{t - \tau }^t {{{{\left\| {y(\xi )} \right\|}}^{{{{\nu }_{2}} + \mu - 1}}}d\xi } \;\leqslant \;V({{x}_{t}},{{y}_{t}})$$

$$\leqslant \;{{c}_{8}}{{\left\| {x(t)} \right\|}^{{{{\nu }_{1}}}}} + {{c}_{9}}{{\left\| {y(t)} \right\|}^{{{{\nu }_{2}}}}} + {{c}_{7}}\tau {{\left\| {y(t)} \right\|}^{{{{\nu }_{2}} - 1}}}\int\limits_{t - \tau }^t {{{{\left\| {y(\xi )} \right\|}}^{\mu }}d\xi } $$

$$ + \;({{\alpha }_{1}} + {{\beta }_{1}}\tau )\int\limits_{t - \tau }^t {{{{\left\| {x(\xi )} \right\|}}^{{{{\nu }_{1}}}}}d\xi } + ({{\alpha }_{2}} + {{\beta }_{2}}\tau )\int\limits_{t - \tau }^t {{{{\left\| {y(\xi )} \right\|}}^{{{{\nu }_{2}} + \mu - 1}}}d\xi } ,$$

$$\begin{gathered} \dot {V}\;\leqslant \; - {\kern 1pt} {{m}_{{14}}}{{\left\| {x(t)} \right\|}^{{{{\nu }_{1}}}}} - {{m}_{{24}}}{{\left\| {y(t)} \right\|}^{{{{\nu }_{2}} + \mu - 1}}} + {{c}_{1}}{{\left\| {x(t)} \right\|}^{{{{\nu }_{1}} - 1}}}({{\left\| {x(t)} \right\|}^{\mu }} + {{\left\| {y(t)} \right\|}^{\mu }}) \\ + \;{{c}_{2}}{{\left\| {x(t)} \right\|}^{{{{\nu }_{1}} - 1}}}\int\limits_{t - \tau }^t {({{{\left\| {x(\xi )} \right\|}}^{\mu }} + {{{\left\| {y(\xi )} \right\|}}^{\mu }})d\xi } + {{c}_{3}}{{\left\| {y(t)} \right\|}^{{{{\nu }_{2}} - 1}}}({{\varepsilon }_{1}}{{\left\| {y(t)} \right\|}^{\mu }} + {{h}_{1}}{{\left\| {x(t)} \right\|}^{\mu }}) \\ \end{gathered} $$

$$\begin{gathered} + \;\tau {{c}_{{10}}}{{\left\| {y(t)} \right\|}^{{{{\nu }_{2}} - 2}}}\int\limits_{t - \tau }^t {{{{\left\| {y(\xi )} \right\|}}^{\mu }}d\xi \left( {{{{\left\| {x(t)} \right\|}}^{\mu }} + {{{\left\| {y(t)} \right\|}}^{\mu }} + \int\limits_{t - \tau }^t {({{{\left\| {x(\xi )} \right\|}}^{\mu }} + {{{\left\| {y(\xi )} \right\|}}^{\mu }})d\xi } } \right)} \\ + \;{{\varepsilon }_{2}}{{c}_{4}}{{\left\| {y(t)} \right\|}^{{{{\nu }_{2}} - 1}}}\int\limits_{t - \tau }^t {{{{\left\| {y(\xi )} \right\|}}^{\mu }}d\xi } + {{h}_{2}}{{c}_{4}}{{\left\| {y(t)} \right\|}^{{{{\nu }_{2}} - 1}}}\int\limits_{t - \tau }^t {{{{\left\| {x(\xi )} \right\|}}^{\mu }}d\xi } \\ \end{gathered} $$

$$\begin{gathered} - \;{{\beta }_{1}}\int\limits_{t - \tau }^t {{{{\left\| {x(\xi )} \right\|}}^{{{{\nu }_{1}}}}}d\xi } - {{\beta }_{2}}\int\limits_{t - \tau }^t {{{{\left\| {y(\xi )} \right\|}}^{{{{\nu }_{2}} + \mu - 1}}}d\xi } + ({{\alpha }_{1}} + {{\beta }_{1}}\tau ){{\left\| {x(t)} \right\|}^{{{{\nu }_{1}}}}} \\ - \;{{\alpha }_{1}}{{\left\| {x(t - \tau )} \right\|}^{{{{\nu }_{1}}}}} + ({{\alpha }_{2}} + {{\beta }_{2}}\tau ){{\left\| {y(t)} \right\|}^{{{{\nu }_{2}} + \mu - 1}}} - {{\alpha }_{2}}{{\left\| {y(t - \tau )} \right\|}^{{{{\nu }_{2}} + \mu - 1}}}. \\ \end{gathered} $$

Here, c_k > 0, k = 5, …, 10.

By Young’s inequality [7], if the homogeneity orders of the functions V₁(x) and V₂(y) satisfy the condition 1 < (ν₂ + μ – 1)/ν₁ < μ and the values ε₁, ε₂, α₁, β₁, α₂, β₂, and δ are sufficiently small, we arrive at the relations

$$\begin{gathered} \frac{1}{2}\left( {{{c}_{6}}{{{\left\| {y(t)} \right\|}}^{{{{\nu }_{2}}}}} + {{\alpha }_{2}}\int\limits_{t - \tau }^t {{{{\left\| {y(\xi )} \right\|}}^{{{{\nu }_{2}} + \mu - 1}}}d\xi } } \right) + {{c}_{5}}{{\left\| {x(t)} \right\|}^{{{{\nu }_{1}}}}} + {{\alpha }_{1}}\int\limits_{t - \tau }^t {{{{\left\| {x(\xi )} \right\|}}^{{{{\nu }_{1}}}}}d\xi } \;\leqslant \;V({{x}_{t}},{{y}_{t}}) \\ \leqslant \;{{c}_{8}}{{\left\| {x(t)} \right\|}^{{{{\nu }_{1}}}}} + ({{\alpha }_{1}} + {{\beta }_{1}}\tau )\int\limits_{t - \tau }^t {{{{\left\| {x(\xi )} \right\|}}^{{{{\nu }_{1}}}}}d\xi } + 2\left( {{{c}_{9}}{{{\left\| {y(t)} \right\|}}^{{{{\nu }_{2}}}}} + ({{\alpha }_{2}} + {{\beta }_{2}}\tau )\int\limits_{t - \tau }^t {{{{\left\| {y(\xi )} \right\|}}^{{{{\nu }_{2}} + \mu - 1}}}d\xi } } \right), \\ \end{gathered} $$

$$\dot {V}\;\leqslant \; - {\kern 1pt} \frac{1}{2}\left( {{{m}_{{14}}}{{{\left\| {x(t)} \right\|}}^{{{{\nu }_{1}}}}} + {{m}_{{24}}}{{{\left\| {y(t)} \right\|}}^{{{{\nu }_{2}} + \mu - 1}}} + {{\beta }_{1}}\int\limits_{t - \tau }^t {{{{\left\| {x(\xi )} \right\|}}^{{{{\nu }_{1}}}}}d\xi } + {{\beta }_{2}}\int\limits_{t - \tau }^t {{{{\left\| {y(\xi )} \right\|}}^{{{{\nu }_{2}} + \mu - 1}}}d\xi } } \right),$$

holding for ${{\left\| {{{x}_{t}}} \right\|}_{\tau }} + {{\left\| {{{y}_{t}}} \right\|}_{\tau }} < \delta$.

The proof of Theorem 2 is complete.

Proof of Theorem 3. We choose the Lyapunov–Krasovskii functional

$$\begin{gathered} V({{s}_{t}},{{\omega }_{t}}) = \frac{1}{2}\lambda {{\omega }^{ \top }}(t)\Theta \omega (t) + \frac{1}{2}{{\left\| {s(t) - r} \right\|}^{2}} + {{(s(t) \times r)}^{ \top }}{{F}^{{ - 1}}}\Theta \omega (t) + b{{(s(t) \times r)}^{ \top }}{{F}^{{ - 1}}} \\ \times \;\int\limits_{t - \tau }^t {(\xi + \tau - t){{{\left\| {s(\xi ) - r} \right\|}}^{{\mu - 1}}}s(\xi ) \times rd\xi } + \int\limits_{t - \tau }^t {(\alpha + \beta (\xi + \tau - t)){{{\left\| {s(\xi ) - r} \right\|}}^{{\mu + 1}}}d\xi } , \\ \end{gathered} $$

where λ, α, and β are positive parameters.

This functional and its derivative along the trajectories of system (10), (12) admit the upper bounds

$${{c}_{1}}\lambda {{\left\| {\omega (t)} \right\|}^{2}} + \frac{1}{2}{{\left\| {s(t) - r} \right\|}^{2}} - {{c}_{2}}\left\| {s(t) - r} \right\|\left\| {\omega (t)} \right\| - {{c}_{3}}\left| b \right|\tau \left\| {s(t) - r} \right\|\int\limits_{t - \tau }^t {{{{\left\| {s(\xi ) - r} \right\|}}^{\mu }}d\xi } $$

$$ + \;\alpha \int\limits_{t - \tau }^t {{{{\left\| {s(\xi ) - r} \right\|}}^{{\mu + 1}}}d\xi } \;\leqslant \;V({{s}_{t}},{{\omega }_{t}})\;\leqslant \;{{c}_{4}}\lambda {{\left\| {\omega (t)} \right\|}^{2}} + \frac{1}{2}{{\left\| {s(t) - r} \right\|}^{2}} + {{c}_{2}}\left\| {s(t) - r} \right\|\left\| {\omega (t)} \right\|$$

$$ + \;{{c}_{3}}\left| b \right|\tau \left\| {s(t) - r} \right\|\int\limits_{t - \tau }^t {{{{\left\| {s(\xi ) - r} \right\|}}^{\mu }}d\xi } + (\alpha + \beta \tau )\int\limits_{t - \tau }^t {{{{\left\| {s(\xi ) - r} \right\|}}^{{\mu + 1}}}d\xi } ,$$

$$\dot {V}\;\leqslant \; - {\kern 1pt} (\lambda {{c}_{5}} - {{c}_{6}}){{\left\| {\omega (t)} \right\|}^{2}} + \lambda a\left\| {\omega (t)} \right\|{{\left\| {s(t) - r} \right\|}^{\mu }} + b(\lambda + {{c}_{7}}\tau )\left\| {\omega (t)} \right\|\int\limits_{t - \tau }^t {{{{\left\| {s(\xi ) - r} \right\|}}^{\mu }}d\xi } $$

$$ + \;{{c}_{8}}{{\left\| {\omega (t)} \right\|}^{2}}\left\| {s(t) - r} \right\| - (a - \tau b){{c}_{9}}{{\left\| {s(t) - r} \right\|}^{{\mu - 1}}}{{\left\| {s(t) \times r} \right\|}^{2}}$$

$$ - \;\beta \int\limits_{t - \tau }^t {{{{\left\| {s(\xi ) - r} \right\|}}^{{\mu + 1}}}d\xi } + (\alpha + \beta \tau ){{\left\| {s(t) - r} \right\|}^{{\mu + 1}}} - \alpha {{\left\| {s(t - \tau ) - r} \right\|}^{{\mu + 1}}}.$$

Here, c_k > 0, k = 1, …, 9.

By Young’s inequality, if the number λ > 0 is sufficiently large and the positive numbers α, β, and δ are sufficiently small, then

$$\begin{gathered} \frac{1}{2}\left( {{{c}_{1}}\lambda {{{\left\| {\omega (t)} \right\|}}^{2}} + \frac{1}{2}{{{\left\| {s(t) - r} \right\|}}^{2}} + \alpha \int\limits_{t - \tau }^t {{{{\left\| {s(\bar {\xi }) - r} \right\|}}^{{\mu + 1}}}d\xi } } \right)\;\leqslant \;V({{s}_{t}},{{\omega }_{t}}) \\ \leqslant \;2\left( {{{c}_{4}}\lambda {{{\left\| {\omega (t)} \right\|}}^{2}} + \frac{1}{2}{{{\left\| {s(t) - r} \right\|}}^{2}} + (\alpha + \beta \tau )\int\limits_{t - \tau }^t {{{{\left\| {s(\bar {\xi }) - r} \right\|}}^{{\mu + 1}}}d\xi } } \right), \\ \end{gathered} $$

$$\dot {V}\;\leqslant \; - {\kern 1pt} {\kern 1pt} \frac{1}{2}\left( {\lambda {{c}_{5}}{{{\left\| {\omega (t)} \right\|}}^{2}} + (a - \tau b){{c}_{9}}{{{\left\| {s(t) - r} \right\|}}^{{\mu + 1}}} + \beta \int\limits_{t - \tau }^t {{{{\left\| {s(\bar {\xi }) - r} \right\|}}^{{\mu + 1}}}d\xi } } \right)$$

under ${{\left\| {{{s}_{t}} - r} \right\|}_{\tau }} + \left\| {\omega (t)} \right\| < \delta$.

Hence, according to [6, p. 22], the equilibrium (11) is asymptotically stable.

The proof of Theorem 3 is complete.