On the Stable Solution of the Problem of Compensating Nonsmooth Additive Disturbances with the Help of Feedback Laws

Abstract

The problem of the feedback control of a system of ordinary differential equations, nonlinear in phase variables, subjected to the effect of an unknown nonsmooth disturbance is discussed. The problem consists of constructing a control action formation law that guarantees compensation for a nonsmooth disturbance; i.e., it guarantees that the phase trajectory (as well as the rate of its change) of the given system follows the prescribed phase trajectory (as well as the rate of its change) for any admissible realization of the disturbance. Two cases are considered. In the first case, admissible disturbances are constrained by instantaneous restrictions, and in the second case, any function that is an element of the space of Lebesgue measurable functions summable with the square of the Euclidean norm can be an admissible disturbance. The problem is solved under conditions of inaccurate measurement at discrete times of the phase states of both systems. In the presence of instantaneous restrictions on disturbances, the problem is also solved by measuring some of the phase states. Algorithms for solving this problem, oriented towards computer implementation, are designed that are resistant to information interference and computational errors. Estimates of the rate of convergence of the algorithms are given.

APPENDIX

Proof of Lemma 1. We consider the change in the value ε(t), $t \in T$. For a.a. $t \in {{\delta }_{i}} = [{{\tau }_{i}},{{\tau }_{{i + 1}}})$ and all $i \in \overline {0,m - 1}$, m = m_h, the inequality

$$\begin{gathered} d\varepsilon (t){\text{/}}dt = 2({{x}^{h}}(t) - y(t),f(t,{{x}^{h}}(t)) - f(t,y(t)) + u_{i}^{h} - v(t) + 2L(\psi _{i}^{h} - \xi _{i}^{h}{{))}_{N}} \\ \, \leqslant 2L\varepsilon (t) + 2({{x}^{h}}(t) - y(t),u_{i}^{h} - v(t) + 2L(\psi _{i}^{h} - \xi _{i}^{h}{{))}_{N}} \leqslant 2L\varepsilon (t) + \sum\limits_{j = 1}^5 {{I}_{{ji}}}(t), \\ \end{gathered} $$

(A.1)

where

$${{I}_{{1i}}}(t) = 2(\xi _{i}^{h} - \psi _{i}^{h},u_{i}^{h} - v(t{{))}_{N}},$$

$${{I}_{{2i}}}(t) = 4h\{ {\text{|}}u_{i}^{h}{{{\text{|}}}_{N}}\; + \;{\text{|}}v(t){{{\text{|}}}_{N}}\} ,$$

$${{I}_{{3i}}}(t) = 2\{ {\text{|}}u_{i}^{h}{{{\text{|}}}_{N}}\; + \;{\text{|}}{v}(t){{{\text{|}}}_{N}}\} \int\limits_{{{\tau }_{i}}}^{{{\tau }_{{i + 1}}}} {\text{|}}{{\dot {x}}^{h}}(s) - \dot {y}(s){{{\text{|}}}_{N}}ds,$$

$${{I}_{{4i}}}(t) = 8Lh{\text{|}}{{x}^{h}}(t) - y(t){{{\text{|}}}_{N}},$$

(A.2)

$${{I}_{{5i}}}(t) = - 8L{{({{x}^{h}}(t) - y(t),{{x}^{h}}({{\tau }_{i}}) - y({{\tau }_{i}}))}_{N}}$$

(A.3)

is valid. The inequality

$$\varepsilon ({{\tau }_{{i + 1}}}) \leqslant \varepsilon ({{\tau }_{i}}) + 2L\int\limits_{{{\tau }_{i}}}^{{{\tau }_{{i + 1}}}} \varepsilon (s)ds + \int\limits_{{{\tau }_{i}}}^{{{\tau }_{{i + 1}}}} \sum\limits_{j = 1}^5 {{I}_{{ji}}}(s)ds$$

(A.4)

follows from (A.1). Next, using Young’s inequality ${{(2}^{{ - 1/2}}}a{{)(2}^{{1/2}}}b) \leqslant {{b}^{2}} + {{a}^{2}}{\text{/}}4$ (a > 0, b > 0), we conclude that for $t \in {{\delta }_{i}}$ the following inequalities are valid:

$${{I}_{{4i}}}(t) \leqslant 8L\varepsilon (t) + 8L{{h}^{2}},$$

(A.5)

$${{I}_{{5i}}}(t) \leqslant - 8L\varepsilon (t) + 8L\varepsilon {{(t)}^{{1/2}}}\int\limits_{{{\tau }_{i}}}^t \{ {\text{|}}{{\dot {x}}^{h}}(s){{{\text{|}}}_{N}}\; + \;{\text{|}}\dot {y}(s){{{\text{|}}}_{N}}\} ds \leqslant - 4L\varepsilon (t) + {{c}^{{(1)}}}{{\delta }^{2}}.$$

(A.6)

In turn, taking into account the rule for determining $u_{i}^{h}$ (see (1.1)), we conclude that the following inequality is valid:

$$\int\limits_{{{\tau }_{i}}}^{{{\tau }_{{i + 1}}}} [{{I}_{{1i}}}(t) + \alpha \{ {\text{|}}u_{i}^{h}{\text{|}}_{N}^{2}\; - \;{\text{|}}{v}(t){\text{|}}_{N}^{2}]dt \leqslant 0.$$

(A.7)

It is easy to see that

$$\int\limits_{{{\tau }_{i}}}^{{{\tau }_{{i + 1}}}} {{I}_{{2i}}}(t)dt \leqslant {{c}^{{(2)}}}h\delta .$$

(A.8)

Next, we have

$$\int\limits_{{{\tau }_{i}}}^{{{\tau }_{{i + 1}}}} {{I}_{{3i}}}(t)dt \leqslant {{c}^{{(3)}}}{{\delta }^{2}}.$$

(A.9)

From (A.4), taking into account (A.5)–(A.9), we obtain

$$\varepsilon ({{\tau }_{{i + 1}}}) + \alpha \int\limits_0^{{{\tau }_{{i + 1}}}} \{ {\text{|}}{{u}^{h}}(s){\text{|}}_{N}^{2}\; - \;{\text{|}}{v}(s){\text{|}}_{N}^{2}\} ds \leqslant \varepsilon (0) + {{c}^{{(4)}}}(h + \delta ),\quad i \in \overline {0,m - 1} .$$

(A.10)

Inequalities (1.4) and (1.5) follow from (A.10). The lemma is proven.

Proof of Lemma 3. It is easy to see that for a.a. $t \in {{\delta }_{i}}$ the inequality

$$\dot {\varepsilon }(t) \leqslant 2L\varepsilon (t) + {{I}_{{4i}}}(t) + {{I}_{{5i}}}(t) + {{I}_{{6i}}}(t),$$

(A.11)

where ${{I}_{{4i}}}(t)$ and ${{I}_{{5i}}}(t)$ are defined according to (A.2) and (A.3), respectively:

$${{I}_{{6i}}}(t) = 2({{x}^{h}}(t) - y(t),u_{i}^{h} - {v}(t{{))}_{N}}.$$

Note that when $t \in {{\delta }_{i}}$

$${{\left| {\int\limits_{{{\tau }_{i}}}^t {{I}_{{6i}}}(s)ds} \right|}_{N}} \leqslant \int\limits_{{{\tau }_{i}}}^t \varepsilon (s)ds + 2{{\tilde {I}}_{{2i}}},$$

(A.12)

where

$${{\tilde {I}}_{{2i}}} = \int\limits_{{{\tau }_{i}}}^{{{\tau }_{{i + 1}}}} \{ {\text{|}}u_{i}^{h}{\text{|}}_{N}^{2}\; + \;{\text{|}}{v}(s){\text{|}}_{N}^{2}\} ds.$$

In addition, at $t \in {{\delta }_{i}}$,

$${{\left| {\int\limits_{{{\tau }_{i}}}^t \{ {{I}_{{4i}}}(s) + {{I}_{{5i}}}(s)\} ds} \right|}_{N}} \leqslant \varepsilon ({{\tau }_{i}}) + 8{{L}^{2}}\int\limits_{{{\tau }_{i}}}^t \varepsilon (s)ds + {{h}^{2}} + {{k}^{{(0)}}}{{\delta }^{2}}.$$

(A.13)

From (A.11), due to (A.12) and (A.13), we obtain that for all $t \in {{\delta }_{i}}$, the following estimate is valid:

$$\varepsilon (t) \leqslant 2\varepsilon ({{\tau }_{i}}) + (1 + 2L + 8{{L}^{2}})\int\limits_{{{\tau }_{i}}}^t \varepsilon (s)ds + {{h}^{2}} + 2{{\tilde {I}}_{{2i}}} + {{k}^{{(0)}}}{{\delta }^{2}}.$$

Using the Gronwall lemma, we derive the inequality

$$\varepsilon (t) \leqslant (2\varepsilon ({{\tau }_{i}}) + {{h}^{2}} + 2{{\tilde {I}}_{{2i}}})\exp \{ (1 + 2L + 8{{L}^{2}})(t - {{\tau }_{i}})\} ,\quad t \in {{\delta }_{i}}.$$

(A.14)

Note that

$${{\tilde {I}}_{{2i}}} \leqslant {{k}^{{(1)}}}\delta .$$

(A.15)

From (A.14), taking into account Lemma 1 (see (1.4)) and inequality (A.15), we obtain that for all $t \in T$ the following inequality is valid:

$$\varepsilon (t) \leqslant {{k}^{{(2)}}}(\alpha + \delta + h).$$

(A.16)

Therefore, due to the conditions of the present lemma, (A.16) implies the following chain of inequalities:

$$\begin{gathered} {{\left| {\int\limits_0^t \{ {{u}^{h}}(s) - v(s)\} ds} \right|}_{N}} \leqslant {{k}^{{(3)}}}{{\left| {\int\limits_0^t \{ {{{\dot {x}}}^{h}}(s) - \dot {y}(s) - f(s,{{x}^{h}}(s)) + f(s,y(s)) - 2L({{\xi }^{h}}(s) - {{\psi }^{h}}(s))\} ds} \right|}_{N}} \\ \, \leqslant {{k}^{{(3)}}}\left\{ {{{\varepsilon }^{{1/2}}}(t) + {{\varepsilon }^{{1/2}}}(0) + L\int\limits_0^t {{\varepsilon }^{{1/2}}}(s)ds + 4tLh + 2tL\mathop {\max }\limits_{i \in \overline {0,m - 1} } {{\varepsilon }^{{1/2}}}({{\tau }_{i}})} \right\} \leqslant {{k}^{{(4)}}}{{\{ \alpha + \delta + h\} }^{{1/2}}},\quad t \in T, \\ \end{gathered} $$

(A.17)

where ${{\xi }^{h}}(s) = \xi _{i}^{h}$ and ${{\psi }^{h}}(s) = \psi _{i}^{h}$ at $s \in {{\delta }_{i}}$, $i \in \overline {0,m - 1}$. Further, by Lemma 1 (see (1.5)), we have the estimate

$$\int\limits_0^\vartheta {\text{|}}{{u}^{h}}(s) - v(s){\text{|}}_{N}^{2}ds \leqslant 2\int\limits_0^\vartheta {{(v(s) - {{u}^{h}}(s),v(s))}_{N}}ds + \int\limits_0^\vartheta {\text{|}}v(s){\text{|}}_{N}^{2}ds + {{d}_{2}}(h + \delta ){{\alpha }^{{ - 1}}}.$$

(A.18)

In turn, by Lemma 2, (A.17) implies the inequality

$$\mathop {\sup }\limits_{t \in T} \left| {\int\limits_0^t {{{({{u}^{h}}(s) - v(s),v(s))}}_{N}}ds} \right| \leqslant {{k}^{{(5)}}}{{\{ \alpha + \delta + h\} }^{{1/2}}}.$$

(A.19)

Inequality (1.6) follows from inequalities (A.18), (A.19), and

$${\text{|}}{{x}^{h}}( \cdot ) - y( \cdot ){{{\text{|}}}_{{W(T;{{\mathbb{R}}^{N}})}}} \leqslant {{k}^{{(6)}}}\{ {\text{|}}{{u}^{h}}( \cdot ) - {v}( \cdot ){{{\text{|}}}_{{{{L}_{2}}(T;{{\mathbb{R}}^{N}})}}}\; + \;{\text{|}}{{\tilde {u}}^{h}}( \cdot ){{{\text{|}}}_{{{{L}_{2}}(T;{{\mathbb{R}}^{N}})}}}\} ,$$

$${\text{|}}{{\tilde {u}}^{h}}( \cdot ){{{\text{|}}}_{{{{L}_{2}}(T;{{\mathbb{R}}^{N}})}}} \leqslant {{k}^{{(7)}}}\{ \alpha + \delta + h\} .$$

The lemma is proven.

Proof of Theorem 2. The result of [17, Lemma 2] implies the validity of the inequalities

$${\text{|}}q_{1}^{h}( \cdot ) - x_{2}^{h}( \cdot ){{{\text{|}}}_{{C(T;{{\mathbb{R}}^{{{{n}_{2}}}}})}}} \leqslant {{c}_{1}}(\alpha + (h + \delta ){{\alpha }^{{ - 1}}}),$$

$${\text{|}}q_{2}^{h}( \cdot ) - {{y}_{2}}( \cdot ){{{\text{|}}}_{{C(T;{{\mathbb{R}}^{{{{n}_{2}}}}})}}} \leqslant {{c}_{2}}(\alpha + (h + \delta ){{\alpha }^{{ - 1}}}).$$

Now, repeating the proof of Lemma 1, in which we replace h by $\alpha + (h + \delta ){{\alpha }^{{ - 1}}}$, we obtain the inequalities

$$\mathop {\max }\limits_{i \in \overline {0,{{m}_{h}}} } \varepsilon ({{\tau }_{i}}) \leqslant {{c}_{3}}({{\alpha }_{1}} + \alpha + (h + \delta ){{\alpha }^{{ - 1}}} + \delta ),$$

$$\int\limits_0^\vartheta {\text{|}}{{u}^{h}}(\tau ){\text{|}}_{n}^{2}d\tau \leqslant \int\limits_0^\vartheta {\text{|}}v(\tau ){\text{|}}_{n}^{2}d\tau + {{c}_{4}}[\alpha + (h + \delta ){{\alpha }^{{ - 1}}} + \delta ]\alpha _{1}^{{ - 1}}.$$

Convergence (1.13) is proved similarly to Theorem 2 in [16]. Estimate (1.14) is established similarly to estimate (1.6) by replacing h by $\alpha + (h + \delta ){{\alpha }^{{ - 1}}}$ and α by α₁, respectively. The theorem has been proven.

Proof of Lemma 5. We consider the change in the value ε(t) at $t \in T$. For $t \in {{\delta }_{i}} = [{{\tau }_{i}},{{\tau }_{{i + 1}}}),$ $i \in \overline {0,m - 1}$, inequality (A.1) holds, which in turn implies inequality (A.4). Further, using Young’s inequality, we conclude that for $t \in {{\delta }_{i}}$ the inequalities

$${{I}_{{4i}}}(t) \leqslant \varepsilon (t) + 32L{{h}^{2}},$$

(A.20)

$${{I}_{{5i}}}(t) \leqslant - 4L\varepsilon (t) + 4L{{\varepsilon }^{{1/2}}}(t)\int\limits_{{{\tau }_{i}}}^t \{ {\text{|}}{{\dot {y}}^{h}}(s){{{\text{|}}}_{N}}\; + \;{\text{|}}\dot {y}(s){{{\text{|}}}_{N}}\} ds \leqslant - 3L\varepsilon (t) + 16L{{\tilde {I}}_{{1i}}},$$

(A.21)

where

$${{\tilde {I}}_{{1i}}} = \delta \int\limits_{{{\tau }_{i}}}^{{{\tau }_{{i + 1}}}} \{ {\text{|}}{{\dot {x}}^{h}}(s){\text{|}}_{N}^{2}\; + \;{\text{|}}\dot {y}(s){\text{|}}_{N}^{2}\} ds,$$

are valid. Given the rule of definition $u_{i}^{h}$ (see (2.1)), we conclude that the following inequality is valid:

$$\int\limits_{{{\tau }_{i}}}^{{{\tau }_{{i + 1}}}} [{{I}_{{1i}}}(t) + \alpha \{ {\text{|}}u_{i}^{h}{\text{|}}_{N}^{2}\; - \;{\text{|}}{v}(t){\text{|}}_{N}^{2}]dt \leqslant 0.$$

(A.22)

It is easy to see that

$$\int\limits_{{{\tau }_{i}}}^{{{\tau }_{{i + 1}}}} {{I}_{{2i}}}(t)dt \leqslant {{c}_{0}}{{h}^{2}} + {{\tilde {I}}_{{2i}}},$$

(A.23)

where

$${{\tilde {I}}_{{2i}}} = \delta \int\limits_{{{\tau }_{i}}}^{{{\tau }_{{i + 1}}}} \{ {\text{|}}u_{i}^{h}{\text{|}}_{N}^{2}\; + \;{\text{|}}{v}(t){\text{|}}_{N}^{2}\} dt.$$

(A.24)

In turn, due to (0.3) and (2.1), the inequality

$${\text{|}}u_{i}^{h}{{{\text{|}}}_{N}} \leqslant {{\alpha }^{{ - 1}}}{{c}_{1}}(h + {{\varepsilon }^{{1/2}}}({{\tau }_{i}}))$$

(A.25)

is valid; therefore,

$$\delta \int\limits_{{{\tau }_{i}}}^{{{\tau }_{{i + 1}}}} {\text{|}}{{u}^{h}}(s){\text{|}}_{N}^{2}ds \leqslant 2{{\delta }^{2}}{{\alpha }^{{ - 2}}}c_{1}^{2}({{h}^{2}} + \varepsilon ({{\tau }_{i}})).$$

(A.26)

Due to (A.26), the following estimate is valid:

$$\delta \int\limits_0^{{{\tau }_{{i + 1}}}} {\text{|}}{{u}^{h}}(s){\text{|}}_{N}^{2}ds \leqslant 2{{\delta }^{2}}{{\alpha }^{{ - 2}}}c_{1}^{2}\left( {\sum\limits_{j = 0}^i \varepsilon ({{\tau }_{j}}) + \vartheta {{h}^{2}}{{\delta }^{{ - 1}}}} \right).$$

(A.27)

Taking (A.27) into account, we obtain

$$\sum\limits_{j = 0}^i {{\tilde {I}}_{{2j}}} \leqslant \delta \int\limits_0^{{{\tau }_{{i + 1}}}} {\text{|}}{v}(s){\text{|}}_{N}^{2}ds + 2\vartheta c_{1}^{2}\delta {{h}^{2}}{{\alpha }^{{ - 2}}} + 2c_{1}^{2}{{\delta }^{2}}{{\alpha }^{{ - 2}}}\sum\limits_{j = 0}^i \varepsilon ({{\tau }_{j}}).$$

(A.28)

Then we have

$$\int\limits_{{{\tau }_{i}}}^{{{\tau }_{{i + 1}}}} {{I}_{{3i}}}(t)dt \leqslant 2{{\tilde {I}}_{{1i}}} + 2{{\tilde {I}}_{{2i}}}.$$

(A.29)

By Lemma 4

$$\int\limits_0^{{{\tau }_{i}}} \{ {\text{|}}{{\dot {x}}^{h}}(s){\text{|}}_{N}^{2}\; + \;{\text{|}}\dot {y}(s){\text{|}}_{N}^{2}\} ds \leqslant {{c}_{2}}\left( {1 + \int\limits_0^{{{\tau }_{i}}} \{ {\text{|}}{{u}^{h}}(s){\text{|}}_{N}^{2}\; + \;{\text{|}}{v}(s){\text{|}}_{N}^{2}\} ds} \right).$$

(A.30)

Therefore,

$$\sum\limits_{j = 0}^i {{\tilde {I}}_{{1j}}} \leqslant {{c}_{3}}\left( {\delta + \sum\limits_{j = 0}^i {{{\tilde {I}}}_{{2j}}}} \right).$$

In this case, the following inequality follows from (A.29) and (A.30):

$$\sum\limits_{j = 0}^i \int\limits_{{{\tau }_{j}}}^{{{\tau }_{{j + 1}}}} {{I}_{{3j}}}(s)ds \leqslant {{c}_{4}}\delta + {{c}_{5}}\sum\limits_{j = 0}^i {{\tilde {I}}_{{2j}}}.$$

(A.31)

Thus, from (A.20), (A.21), (A.23), (A.28), and (A.31), we obtain

$$\sum\limits_{j = 0}^i \sum\limits_{k = 2}^5 \int\limits_{{{\tau }_{j}}}^{{{\tau }_{{j + 1}}}} {{I}_{{kj}}}(t)dt \leqslant - 2L\varepsilon (t) + {{c}_{6}}{{h}^{2}}{{\delta }^{{ - 1}}} + {{c}_{7}}\delta + {{c}_{8}}\left( {{{\delta }^{2}}{{\alpha }^{{ - 2}}}\sum\limits_{j = 0}^i \varepsilon ({{\tau }_{j}}) + \delta {{h}^{2}}{{\alpha }^{{ - 2}}}} \right).$$

(A.32)

In turn, from (A.4), using (A.22) and (A.32), we deduce the estimate

$$\begin{gathered} \varepsilon ({{\tau }_{{i + 1}}}) + \alpha \int\limits_0^{{{\tau }_{{i + 1}}}} \{ {\text{|}}{{u}^{h}}(s){\text{|}}_{N}^{2}\; - \;{\text{|}}{v}(s){\text{|}}_{N}^{2}\} ds \leqslant \varepsilon (0) \\ \, + {{c}_{6}}{{h}^{2}}{{\delta }^{{ - 1}}} + {{c}_{7}}\delta + {{c}_{8}}\delta {{h}^{2}}{{\alpha }^{{ - 2}}} + {{c}_{8}}{{\delta }^{2}}{{\alpha }^{{ - 2}}}\sum\limits_{j = 0}^i \varepsilon ({{\tau }_{j}}),\quad i \in \overline {0,m - 1} . \\ \end{gathered} $$

(A.33)

By the discrete Gronwall inequality (see [18, p. 312]), from (A.33) we obtain

$$\begin{gathered} \varepsilon ({{\tau }_{{i + 1}}}) + \alpha \int\limits_0^{{{\tau }_{{i + 1}}}} {\text{|}}{{u}^{h}}(s){\text{|}}_{N}^{2}ds \\ \, \leqslant \left\{ {\varepsilon (0) + {{c}_{6}}{{h}^{2}}{{\delta }^{{ - 1}}} + {{c}_{7}}\delta + {{c}_{8}}\delta {{h}^{2}}{{\alpha }^{{ - 2}}} + \alpha \int\limits_0^{{{\tau }_{{i + 1}}}} {\text{|}}{v}(s){\text{|}}_{N}^{2}ds} \right\}\exp \{ {{c}_{8}}(i + 1){{\delta }^{2}}{{\alpha }^{{ - 2}}}\} . \\ \end{gathered} $$

(A.34)

Note that

$$\varepsilon (0) \leqslant {{h}^{2}},\quad \exp \{ {{c}_{8}}(i + 1){{\delta }^{2}}{{\alpha }^{{ - 2}}}\} \leqslant \exp \{ {{c}_{8}}\vartheta \delta {{\alpha }^{{ - 2}}}\} .$$

In addition, if $\delta (h){{\alpha }^{{ - 2}}}(h) \to 0$ at $h \to 0$, then at $h \in (0,{{h}_{2}})$ and ${{h}_{2}} \in (0,1)$, there are inequalities

$$\exp \{ {{c}_{8}}\vartheta \delta {{\alpha }^{{ - 2}}}\} \leqslant 1 + {{c}_{9}}\delta {{\alpha }^{{ - 2}}},\quad \delta {{\alpha }^{{ - 2}}} \leqslant {{c}_{{10}}},$$

(A.35)

where ${{c}_{9}} = {{c}_{9}}({{h}_{2}}) > 0$ and ${{c}_{{10}}} = {{c}_{{10}}}({{h}_{2}}) > 0$. Hence, due to (A.34) and (A.35) for $h \in (0,{{h}_{2}})$ the following inequality is valid:

$$\varepsilon ({{\tau }_{{i + 1}}}) + \alpha \int\limits_0^{{{\tau }_{{i + 1}}}} {\text{|}}{{u}^{h}}(s){\text{|}}_{N}^{2}ds \leqslant \alpha (1 + {{c}_{{11}}}\delta {{\alpha }^{{ - 2}}})\int\limits_0^{{{\tau }_{{i + 1}}}} {\text{|}}{v}(s){\text{|}}_{N}^{2}ds + {{c}_{{12}}}\{ {{h}^{2}}{{\delta }^{{ - 1}}} + \delta \} .$$

Inequalities (2.2) and (2.3) follow from the last inequality. The lemma is proven.

Proof of Lemma 6. It is easy to see for a.a. $t \in {{\delta }_{i}}$ the validity of the inequality

$$\dot {\varepsilon }(t) \leqslant 2L\varepsilon (t) + {{I}_{{4i}}}(t) + {{I}_{{5i}}}(t) + {{I}_{{6i}}}(t),$$

(A.36)

where ${{I}_{{4i}}}(t)$ and ${{I}_{{5i}}}(t)$ are defined and according to (A.2) and (A.3), respectively,

$${{I}_{{6i}}}(t) = 2({{x}^{h}}(t) - y(t),u_{i}^{h} - {v}(t{{))}_{N}}.$$

Note that when $t \in {{\delta }_{i}}$,

$${{\left| {\int\limits_{{{\tau }_{i}}}^t {{I}_{{6i}}}(s)ds} \right|}_{N}} \leqslant \int\limits_{{{\tau }_{i}}}^t \varepsilon (s)ds + 2{{\delta }^{{ - 1}}}{{\tilde {I}}_{{2i}}},$$

(A.37)

where ${{\tilde {I}}_{{2i}}}$ are found in (A.24). In addition, at $t \in {{\delta }_{i}}$

$$\left| {\int\limits_{{{\tau }_{i}}}^t {{I}_{{4i}}}(s)ds} \right| \leqslant {{L}^{2}}\int\limits_{{{\tau }_{i}}}^t \varepsilon (s)ds + 16{{h}^{2}},$$

$$\left| {\int\limits_{{{\tau }_{i}}}^t {{I}_{{5i}}}(s)ds} \right| \leqslant \varepsilon ({{\tau }_{i}}) + 16{{L}^{2}}\int\limits_{{{\tau }_{i}}}^t \varepsilon (s)ds.$$

Therefore

$${{\left| {\int\limits_{{{\tau }_{i}}}^t \{ {{I}_{{4i}}}(s) + {{I}_{{5i}}}(s)\} ds} \right|}_{N}} \leqslant \varepsilon ({{\tau }_{i}}) + 17{{L}^{2}}\int\limits_{{{\tau }_{i}}}^t \varepsilon (s)ds + 16{{h}^{2}}.$$

(A.38)

From (A.36), due to (A.37) and (A.38), we obtain that for all $t \in {{\delta }_{i}}$ the following estimate is valid:

$$\varepsilon (t) \leqslant 2\varepsilon ({{\tau }_{i}}) + (1 + 2L + 17{{L}^{2}})\int\limits_{{{\tau }_{i}}}^t \varepsilon (s)ds + 16{{h}^{2}} + 2{{\delta }^{{ - 1}}}{{\tilde {I}}_{{2i}}}.$$

Using the Gronwall lemma, we derive the inequality

$$\varepsilon (t) \leqslant (2\varepsilon ({{\tau }_{i}}) + 16{{h}^{2}} + 2{{\delta }^{{ - 1}}}{{\tilde {I}}_{{2i}}})\exp \{ (1 + 2L + 17{{L}^{2}})(t - {{\tau }_{i}})\} ,\quad t \in {{\delta }_{i}}.$$

(A.39)

At $h \in (0,{{h}_{2}})$, due to limitation of function $v( \cdot )$ and inequalities (A.25), the following relations are valid:

$${{\delta }^{{ - 1}}}{{\tilde {I}}_{{2i}}} \leqslant {{k}_{2}}\delta + \delta {\text{|}}u_{i}^{h}{\text{|}}_{N}^{2} \leqslant {{k}_{2}}\delta + {{k}_{3}}\delta {{\alpha }^{{ - 2}}}({{h}^{2}} + \varepsilon ({{\tau }_{i}})) \leqslant {{k}_{4}}\delta {{\alpha }^{{ - 2}}}.$$

(A.40)

From (A.39), taking into account Lemma 5 (see (2.2)) and inequality (A.40), we obtain that for all $t \in T$ the following inequality is valid:

$$\varepsilon (t) \leqslant {{k}_{1}}(\alpha + \delta {{\alpha }^{{ - 2}}} + {{h}^{2}}{{\delta }^{{ - 1}}}).$$

(A.41)

Therefore, due to the conditions of the present lemma, the following chain of inequalities follows from (A.41):

$$\begin{gathered} {{\left| {\int\limits_0^t \{ {{u}^{h}}(s) - v(s)\} ds} \right|}_{N}} \leqslant {{k}_{5}}{{\left| {\int\limits_0^t \{ {{{\dot {x}}}^{h}}(s) - \dot {y}(s) - f(s,{{x}^{h}}(s)) + f(s,y(s))\} ds - 2L({{\xi }^{h}}(s) - {{\psi }^{h}}(s))} \right|}_{N}} \\ \, \leqslant {{k}_{5}}\left\{ {{{\varepsilon }^{{1/2}}}(t) + {{\varepsilon }^{{1/2}}}(0) + L\int\limits_0^t {{\varepsilon }^{{1/2}}}(s)ds + 4tLh + 42tL\mathop {\max }\limits_{i \in \overline {0,m - 1} } {{\varepsilon }^{{1/2}}}({{\tau }_{i}})} \right\} \\ \, \leqslant {{k}_{6}}{{\{ \alpha + \delta {{\alpha }^{{ - 2}}} + {{h}^{2}}{{\delta }^{{ - 1}}}\} }^{{1/2}}},\quad t \in T, \\ \end{gathered} $$

(A.42)

where ${{\xi }^{h}}(s) = \xi _{i}^{h}$ and ${{\psi }^{h}}(s) = \psi _{i}^{h}$ at $s \in {{\delta }_{i}},$ $i \in \overline {0,m - 1}$. Further, by Lemma 5 (see (2.3)) the following estimate is valid:

$$\int\limits_0^\vartheta {\text{|}}{{u}^{h}}(s) - {v}(s){\text{|}}_{N}^{2}ds \leqslant 2\int\limits_0^\vartheta {{({v}(s) - {{u}^{h}}(s),{v}(s))}_{N}}ds + {{d}_{4}}\delta {{\alpha }^{{ - 2}}}\int\limits_0^\vartheta {\text{|}}{v}(s){\text{|}}_{N}^{2}ds + {{d}_{5}}\{ {{h}^{2}}{{(\alpha \delta )}^{{ - 1}}} + \delta {{\alpha }^{{ - 1}}}\} .$$

(A.43)

In turn, Lemma 2 implies the following inequality from (A.42):

$$\mathop {\sup }\limits_{t \in T} \left| {\int\limits_0^t {{{({{u}^{h}}(s) - {v}(s),{v}(s))}}_{N}}ds} \right| \leqslant {{k}_{7}}{{\{ \alpha + \delta {{\alpha }^{{ - 2}}} + {{h}^{2}}{{\delta }^{{ - 1}}}\} }^{{1/2}}}.$$

(A.44)

Inequality (2.4) and the inequality $\delta {{\alpha }^{{ - 1}}} \leqslant {{\delta }^{{1/2}}}{{\alpha }^{{ - 1}}}$ follow from inequalities (A.43) and (A.44). The lemma is proven.