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Local controllability and the boundary of the attainable set of a control system E. R. Avakova, G. G. Magaril-Il'yaevbcd a V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia b Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia c Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow, Russia d Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of Sciences, Vladikavkaz, Russia
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    § 1. Statement of the problem and general resultsConsider a control system
$$
\begin{equation}
\dot x=\varphi(t,x,u), \qquad u(t)\in U \quad\text{for a.e. } t\in[t_0,t_1],
\end{equation}
\tag{1}
$$
where $\varphi\colon \mathbb R\times\mathbb R^n\times\mathbb R^r\to\mathbb R^n$ is a map of the variables $t$, $x$ and $u$, and $U$ is a nonempty subset of $\mathbb R^r$. Let $g\colon\mathbb R^n\times\mathbb R^n\to \mathbb R^{m_1}$ and $f\colon\mathbb R^n\times\mathbb R^n\to \mathbb R^{m_2}$ be maps of the variables $\zeta_i\in\mathbb R^n$, $i=0,1$. Throughout what follows we assume that $\varphi$ and its partial derivative with respect to $x$ are continuous, while the maps $f$ and $g$ are continuously differentiable. We denote spaces of continuous vector-valued functions on $[t_0,t_1]$ taking values in $\mathbb R^n$, absolutely continuous vector-valued functions taking values in $\mathbb R^n$ and essentially bounded vector-valued functions taking values in $\mathbb R^r$ by $C([t_0,t_1],\mathbb R^n)$, $\operatorname{AC}([t_0,t_1],\mathbb R^n)$ and $L_\infty([t_0,t_1],\mathbb R^r)$, respectively (for $r=1$ the notation is $L_\infty([t_0,t_1])$). A pair $(x(\cdot),u(\cdot))\in \operatorname{AC}([t_0,t_1],\mathbb R^n)\times L_\infty([t_0,t_1],\mathbb R^r)$ is admissible for the control system (1) (we often drop the word ‘control’ in what follows) if it satisfies (1). In this case $x(\cdot)$ is called an admissible trajectory for system (1). We denote the set of admissible trajectories for (1) by $D$. Let $y=(y_1,y_2)\in\mathbb R^{m_1}\times\mathbb R^{m_2}$, and let $D(y)=D(y,g,f)$ denote the subset of $D$ consisting of the trajectories $x(\cdot)$ such that
$$
\begin{equation*}
g(x(t_0), x(t_1))=y_1\quad\text{and} \quad f(x(t_0), x(t_1))\leqslant y_2.
\end{equation*}
\notag
$$
We denote the attainable set for system (1) with respect to the pair $(g, f)$ and an open set $V\subset C([t_0,t_1],\mathbb R^n)$ by
$$
\begin{equation*}
R(V)=R(g, f, t_0,t_1,V)=\{y=(y_1,y_2)\in\mathbb R^{m_1}\times\mathbb R^{m_2} \colon \exists\,x(\cdot)\in D(y)\cap V\}.
\end{equation*}
\notag
$$
If $V$ is the whole space, then we speak about the attainable set for system (1) with respect to $(g, f)$ and denote it by $R=R(g, f, t_0,t_1)$. Usually, by the attainable set for (1) authors mean the set of values at $t_1$ of admissible trajectories for (1) with fixed left endpoint: $x(t_0)=x_0$ (see [1]–[3]). We denote this attainable set by $R_0$. If $R$ is the attainable set defined above (and consisting here of pairs $(x(t_0),x(t_1))$), then clearly
$$
\begin{equation*}
\{y=(y_1,y_2)\in\mathbb R^n\times\mathbb R^n \colon y_1=x_0,\, y_2\in R_0\}\subset R.
\end{equation*}
\notag
$$
In addition, it is easy to show that $\{x_0\}\times \partial R_0\subset\partial R$ (see the text before Proposition 1, which concerns maps $f$ and $g$ of a special form), so necessary conditions for boundary points of $R_0$ follow from ones for boundary points of $R$. Note that in [1] and [2] the control system is defined on a smooth manifold. In [3], where phase constraints are added, the map $\varphi$ is assumed to be Lipschitz in $x$, rather than differentiable. Definition 1. We say that system (1) is locally controllable with respect to maps $g$ and $f$ and a function $\widehat x(\cdot)\in C([t_0,t_1],\mathbb R^n)$ if the following inclusion holds for each neighbourhood $V$ of this function: In the previous definitions of (local) controllability (see [4], [5] and [2]) it was assumed that $\widehat x(\cdot)$ is an admissible trajectory for (1). Definition 1 does no contain this assumption. We need some additional notation and definitions for what follows. Let $N\in\mathbb N$. Set
$$
\begin{equation*}
\begin{aligned} \, \mathcal A_N &=\bigl\{\overline\alpha(\cdot)=(\alpha_1(\cdot),\dots,\alpha_N(\cdot))\in (L_\infty([t_0,t_1]))^N \colon \\ &\qquad\qquad \overline\alpha(t)\in\Sigma^N \ \text{for a.e.}\ t\in[t_0,t_1]\bigr\}, \end{aligned}
\end{equation*}
\notag
$$
where $\Sigma^N=\bigl\{\overline\alpha=(\alpha_1,\dots,\alpha_N)\in \mathbb R_+^N \colon \sum_{i=1}^N\alpha_i=1\bigr\}$. Also consider the set
$$
\begin{equation*}
\mathcal U=\{u(\cdot)\in L_\infty([t_0,t_1],\mathbb R^r) \colon u(t)\in \operatorname{cl}U \text{ for a.e. } t\in[t_0,t_1]\},
\end{equation*}
\notag
$$
where $\operatorname{cl}U$ denotes the closure of $U$. We assign to (1) the following family of control systems:
$$
\begin{equation}
\dot x =\sum_{i=1}^N\alpha_i(t)\varphi(t,x,u_i(t)), \qquad \overline\alpha(\cdot)\in \mathcal A_N, \quad \overline u(\cdot)\in \mathcal U^N, \quad N\in\mathbb N,
\end{equation}
\tag{3}
$$
where $\overline\alpha(\cdot)=(\alpha_1(\cdot),\dots,\alpha_N(\cdot))$ and $\overline u(\cdot)=(u_1(\cdot),\dots,u_N(\cdot))$ are the control variables. This family can be called a convex extension of system (1). Below we call it simply the convex system. We must say that, in contrast to the convex system introduced by Gamkrelidze in [6], which these authors used in their previous works (for instance, see [7] and [8]), here, for the definition of the set $\mathcal U$, in place of $U$ we use its closure $\operatorname{cl}U$. Let $D_c$ denote the sat of admissible triples $(x(\cdot),\overline \alpha(\cdot),\overline u(\cdot))$ for the convex system (3), that is, triples such that $(x(\cdot),\overline \alpha(\cdot),\overline u(\cdot))\in \operatorname{AC}([t_0,t_1],\mathbb R^n)\times\mathcal A_N\times \mathcal U^N$ for some $N\in\mathbb N$ and the differential equation in (3) is satisfied. In this case we call $x(\cdot)$ an admissible trajectory for the convex system (3). Let $y=(y_1,y_2)\in\mathbb R^{m_1}\times\mathbb R^{m_2}$. We denote by $D_c(y)=D_c(y,g,f)$ the subset of $D_c$ consisting of the triples $(x(\cdot),\overline \alpha(\cdot),\overline u(\cdot))$ such that
$$
\begin{equation*}
g(x(t_0), x(t_1))=y_1\quad\text{and} \quad f(x(t_0), x(t_1))\leqslant y_2.
\end{equation*}
\notag
$$
Similarly to above we define the attainable set for the convex system (3) with respect to a pair $(g, f)$:
$$
\begin{equation*}
R_c=R_c(g, f, t_0,t_1)=\{y=(y_1,y_2)\in\mathbb R^{m_1}\times\mathbb R^{m_2} \colon \exists\,(x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))\in D_c(y)\}.
\end{equation*}
\notag
$$
We denote the Euclidean norm in $\mathbb R^n$ by $|\cdot|$. The value of a linear functional $\lambda=(\lambda_1,\dots,\lambda_n)\in(\mathbb R^n)^*$ at an element $x=(x_1,\dots,x_n)^\top\in\mathbb R^n$ ($\top$ denotes transposition) is denoted by $\langle \lambda,x\rangle=\sum_{i=i}^n\lambda_ix_i$. Let $(\mathbb R^n)^*_+$ be the set of functionals on $\mathbb R^n$ that are nonnegative at nonnegative vectors. We denote the boundary of a set $G\subset \mathbb R^k$ by $\partial G$. For a fixed function $\widehat x(\cdot)$ the partial derivatives of $f$ and $g$ with respect to $\zeta_0$ and $\zeta_1$ at the points $(\widehat x(t_0),\widehat x(t_1))$ are denoted for short by $\widehat f_{\zeta_i}$ and $\widehat g_{\zeta_i}$, $i=0,1$, respectively. Let $(x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))\in D_c$, where $\overline\alpha(\cdot)=(\alpha_1(\cdot),\dots,\alpha_N(\cdot))$ and $\overline u(\cdot)=(u_1(\cdot), \dots, u_N(\cdot))$. Then we denote by $\Lambda(x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))$ the set of tuples $(p(\cdot),\lambda_g,\lambda_f)\in \operatorname{AC}([t_0,t_1],(\mathbb R^n)^*)\times(\mathbb R^{m_1})\times(\mathbb R^{m_2})_+^*$ such that
$$
\begin{equation}
\begin{gathered} \, \dot p(t) =-p(t)\sum_{i=1}^N\alpha_i(t)\varphi_x(t,x(t),u_i(t)), \\ p(t_0)=\lambda_f\widehat {f}_{\zeta_0}+\lambda_g\widehat {g}_{\zeta_0}, \qquad p(t_1)=-\lambda_f\widehat{f}_{\zeta_1}-\lambda_g\widehat {g}_{\zeta_1}, \\ \max_{u\in \operatorname{cl}U}\langle p(t),\,\varphi(t,x(t),u)\rangle=\langle p(t),\,\dot {x}(t)\rangle \quad\text{a.e. on } [t_0,t_1]. \end{gathered}
\end{equation}
\tag{4}
$$
Clearly, the zero tuple satisfies these conditions. Note that if $(\widehat x(\cdot),\widehat u(\cdot))$ is an admissible pair for system (1), then taking account of the maximum condition in (4) for $N=1$ (then $\widehat\alpha_1(\cdot)=1$ and we set $\widehat u_1(\cdot)=\widehat u(\cdot)$), for almost all $t\in[t_0,t_1]$ we have
$$
\begin{equation*}
\begin{aligned} \, \max_{u\in U}\langle p(t),\varphi(t,\widehat x(t),u)\rangle &\geqslant \langle p(t),\varphi(t,\widehat x(t),\widehat u(t))\rangle \\ &=\langle p(t),\dot{\widehat x}(t)\rangle =\max_{u\in \operatorname{cl}U}\langle p(t),\varphi(t,\widehat x(t),u)\rangle, \end{aligned}
\end{equation*}
\notag
$$
so that the left- and right-hand sides are actually equal. Hence in this case conditions (4) cover the conditions of Pontryagin’s maximum principle for system (1) with Pontryagin function $H(t,x,u,p)=\langle p,\varphi(t,x,u)\rangle$. We denote these conditions by $\Lambda(x(\cdot),\overline u(\cdot))$ and call them the geometric Pontryagin maximum principle or Pontryagin’s maximum principle in the geometric form. Theorem 1. Let $(\widehat x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))\in D_c$ and $\Lambda(\widehat x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))=\{0\}$. Then system (1) is locally controllable with respect to the maps $g$ and $f$ and function $\widehat x(\cdot)$. Proof. In our previous papers (see [8] and [9]) we introduced the notion of local controllability of system (1) with constraints and obtained sufficient conditions for the local controllability of such a system. This result reads as follows.
The attainable set for system (1), (5) with respect to an open set $V\mkern-2mu\!\subset\!\mkern-1mu C(\mkern-1mu[t_0,\mkern-1mut_1],\mkern-1mu\mathbb R^n\mkern-1.5mu)$ is defined just as $R(V)$ introduced above. We say that system (1), (5) is locally controllable with respect to a function $\widehat x(\cdot)\in C([t_0,t_1],\mathbb R^n)$ if for each neighbourhood $V$ of this function we have With (1), (5) we associate the convex system (3), (5), where we use the set
$$
\begin{equation*}
\mathcal U_0=\{u(\cdot)\in L_\infty([t_0,t_1],\mathbb R^r) \colon u(t)\in U\text{ for a.e. } t\in[t_0,t_1]\}
\end{equation*}
\notag
$$
in place of $\mathcal U$, that is, we replace $\operatorname{cl}U$ by $U$.
If $(\widehat x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))$ is an admissible triple for the convex system (3), (5) (with $\mathcal U_0$ in place of $\mathcal U$), then let $\Lambda_0(\widehat x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))$ denote the set of tuples $(p(\cdot),\lambda_g, \lambda_f)\in \operatorname{AC}([t_0,t_1],(\mathbb R^n)^*)\times(\mathbb R^{m_1})^*\times(\mathbb R^{m_2})_+^*$ satisfying the relations in (4), where $\operatorname{cl}U$ is replaced by $U$ in the maximum condition and, in addition, the condition of complementary slackness $\langle\lambda_f, f(\widehat x(t_0),\widehat x(t_1))\rangle=0$ is satisfied. The result on local controllability in [9] reads as follows. Let $(\widehat x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))$ be an admissible triple for the convex system (3), (5) (with $\mathcal U_0$ in place of $\mathcal U$). If $\Lambda_0(\widehat x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))=\{0\}$, then system (1), (5) is locally controllable with respect to the function $\widehat x(\cdot)\in C([t_0,t_1],\mathbb R^n)$. Let $\overline\Lambda(\widehat x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))$ denote the set of tuples $(p(\cdot),\lambda_g,\lambda_f)\in \operatorname{AC}([t_0,t_1],(\mathbb R^n)^*)\times(\mathbb R^{m_1})^*\times(\mathbb R^{m_2})_+^*$ satisfying the relations in (4) and the condition of complementary slackness $\langle\lambda_f,f(\widehat x(t_0),\widehat x(t_1))\rangle=0$. If in the proof of the results stated above (see [8]) we use the approximation lemma from our § 3, then we obtain a refined version of the local controllability result, which can be formulated as follows. Let $(\widehat x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))$ be an admissible triple for the convex system (3), (5). If $\overline\Lambda(\widehat x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))=\{0\}$, then system (1), (5) is locally controllable with respect to the function $\widehat x(\cdot)\in C([t_0,t_1],\mathbb R^n)$. The refinement consists in the fact that we can select $\widehat x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))$ from a wider set (because the set $U$ is replaced by $\operatorname{cl}U$). The proof remains the same. This result implies already Theorem 1. In fact, let $(\widehat x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))\in D_c$ and $\Lambda(\widehat x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))=\{0\}$. Consider the control system (1) with constraints of the form
$$
\begin{equation}
g_0(x(t_0),x(t_1))=0\quad\text{and} \quad f_0(x(t_0),x(t_1))\leqslant0,
\end{equation}
\tag{7}
$$
where
$$
\begin{equation*}
g_0(x(t_0),x(t_1)) =g(x(t_0),x(t_1))-g(\widehat x(t_0),\widehat x(t_1))
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
f_0(x(t_0),x(t_1)) =f(x(t_0),x(t_1))-f(\widehat x(t_0),\widehat x(t_1).
\end{equation*}
\notag
$$
Since $\Lambda(\widehat x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))=\{0\}$, and the conditions of complementary slackness obviously hold for (1), (7), it follows that $\overline\Lambda(\widehat x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))=\{0\}$. Hence the refined version of the result on local controllability implies the local controllability of system (1), (7), that is, we have inclusion (6), which is equivalent to (2) in the case when (7) holds. This means that system (1) is locally controllable with respect to the maps $g$ and $f$ and function $\widehat x(\cdot)$. If $(\widehat x(\cdot),\widehat u(\cdot))$ is an admissible pair for system (1), (5), then from Theorem 1 we derive the known sufficient controllability conditions (for instance, see [1] and [5]) of the form $\Lambda(\widehat x(\cdot),\widehat u(\cdot))= \varnothing$. However, it is possible that $\Lambda(\widehat x(\cdot),\widehat u(\cdot))\neq \varnothing$, but $\Lambda(\widehat x(\cdot),\widehat{\overline \alpha}(\cdot),\widehat{\overline u}(\cdot))=\varnothing$ for some triple $(\widehat x(\cdot),\widehat{\overline \alpha}(\cdot),\widehat{\overline u}(\cdot))$ (see Examples $1$ and $2$ in [9]). In this case Theorem 1 improves the known controllability conditions stated above (see Example $2$ in [9]). On the other hand, if $\widehat x(\cdot)$ is not an admissible trajectory for (1), (5), then the relations from the geometric Pontryagin maximum principle cannot be used. In this case Theorem 1 gives us some new machinery to approach such problems (see Examples $1$ and $2$ in [9]). Let $(x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))\in D_c$, where $\overline\alpha(\cdot)=(\alpha_1(\cdot),\dots,\alpha_N(\cdot))$ and $\overline u(\cdot)=(u_1(\cdot), \dots, u_N(\cdot))$, and let $y_2\in\mathbb R^{m_2}$. We denote by $\Lambda_1(x(\cdot),\overline\alpha(\cdot),\overline u(\cdot),y_2)$ the set of tuples $(p(\cdot),\lambda_g,\lambda_f)\in \operatorname{AC}([t_0,t_1],(\mathbb R^n)^*)\times(\mathbb R^{m_1})^*\times(\mathbb R^{m_2})_+^*$ satisfying the following relations:
$$
\begin{equation}
\begin{gathered} \, \dot p(t) =-p(t)\sum_{i=1}^N\alpha_i(t)\varphi_x(t,x(t),u_i(t)), \\ p(t_0)=\lambda_f\widehat {f}_{\zeta_0}+\lambda_g\widehat {g}_{\zeta_0}, \qquad p(t_1)=-\lambda_f\widehat{f}_{\zeta_1}-\lambda_g\widehat {g}_{\zeta_1}, \\ \langle \lambda_f, f(x(t_0),x(t_1))-y_2\rangle=0, \\ \max_{u\in \operatorname{cl}U}\langle p(t),\varphi(t,x(t),u)\rangle =\langle p(t), \dot {x}(t)\rangle \quad\text{a.e. on } [t_0,t_1]. \end{gathered}
\end{equation}
\tag{8}
$$
The zero tuple obviously satisfies these relations. Theorem 2. Let $\widehat x(\cdot)\in C([t_0,t_1],\mathbb R^n)$, let $V$ be a neighbourhood of the point $\widehat x(\cdot)$, and let $y=(y_1,y_2)\in\mathbb R^{m_1}\times\mathbb R^{m_2}$ be such that $D_c(y)\ne\varnothing$. If $y\in\partial R(V)$, then $\Lambda_1(\widehat x(\cdot),\overline\alpha(\cdot),\overline u(\cdot),y_2)\ne\{0\}$ for each triple $(\widehat x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))\in D_c(y)$. Proof. Supposing that there exists a triple $(\widehat x(\cdot),\widehat{\overline \alpha}(\cdot),\widehat{\overline u}(\cdot))\in D_c(y)$ such that $\Lambda_1(\widehat x(\cdot),\widehat{\overline \alpha}(\cdot),\widehat{\overline u}(\cdot),y_2)=\{0\}$, we show that then $y=(y_1,y_2)\in \operatorname{int}R(V)$, and therefore $y\notin\partial R(V)$.
The map $F\colon C([t_0,t_1],\mathbb R^n)\to\mathbb R^{m_2}$, acting by the formula is continuously differentiable. Then it follows from the mean value theorem that $F$ is locally Lipschitz continuous, and therefore there exist positive constants $r$ and $C$ such that if $\|x(\cdot)-\widehat x(\cdot)\|_{C([t_0,t_1],\mathbb R^n)}<r$, then
$$
\begin{equation}
|f(x(t_0),x(t_1))-f(\widehat x(t_0),\widehat x(t_1))|\leqslant C\|x(\cdot)-\widehat x(\cdot)\|_{C([t_0,t_1],\mathbb R^n)}.
\end{equation}
\tag{9}
$$
Let $(\widehat x(\cdot),\widehat{\overline \alpha}(\cdot),\widehat{\overline u}(\cdot))\in D_c(y)$ be a triple satisfying (3), and let for definiteness
$$
\begin{equation*}
f_j(\widehat x(t_0),\widehat x(t_1))=y_{2j}, \ j=1,\dots,k,\ \ \text{and} \ \ f_j(\widehat x(t_0),\widehat x(t_1))<y_{2j}, \ j=k+1,\dots,m_2,
\end{equation*}
\notag
$$
where $f=(f_1,\dots,f_{m_2})$ and $y_2=(y_{21},\dots,y_{2m_2})$. Then there clearly exists $\delta$, $0<\delta< 2Cr$, such that $U_{C([t_0,t_1],\mathbb R^n)}(\widehat x(\cdot),\delta/(2C))\subset V$ and
$$
\begin{equation}
f_j(\widehat x(t_0),\widehat x(t_1))<y_{2j}-\delta, \qquad j=k+1,\dots,m_2.
\end{equation}
\tag{10}
$$
Setting $f'=(f_1,\dots,f_k)$ and $y_2'=(y_{21},\dots,y_{2k})$, we let $\Lambda'(\widehat x(\cdot),\widehat{\overline \alpha}(\cdot),\widehat{\overline u}(\cdot),y_2')$ denote the set analogous to $\Lambda_1(\widehat x(\cdot),\widehat{\overline \alpha}(\cdot),\widehat{\overline u}(\cdot),y_2)$ but with $y_2$ replaced by $y'_2$, $f$ by $f'$, and $\lambda_f=(\lambda_{f1},\dots,\lambda_{fm_2})$ by $\lambda'_f=(\lambda_{f1},\dots,\lambda_{fk})$. It follows from the fourth condition in (8) that $\lambda_{fj}=0$, $j=k+1,\dots,m_2$, and since $\Lambda_1(\widehat x(\cdot),\widehat{\overline \alpha}(\cdot),\widehat{\overline u}(\cdot),y_2)=\{0\}$ by assumption, it is obvious that $\Lambda'(\widehat x(\cdot),\widehat{\overline \alpha}(\cdot), \widehat{\overline u}(\cdot))=\{0\}$. Then by Theorem 1 system (1) is locally controllable with respect to the maps $g$ and $f'$ and the function $\widehat x(\cdot)$. Hence, denoting the attainable set for this system with respect to the pair $(g,f')$ by $R'(V)$, we obtain
$$
\begin{equation*}
(y_1,y_2')=(g(\widehat x(t_0),\widehat x(t_1)), f'(\widehat x(t_0),\widehat x(t_1)))\in \operatorname{int}R'(V).
\end{equation*}
\notag
$$
Therefore, there exists $0<\varepsilon<\delta/2$ such that for each pair $z'=(z_1,z'_2)\in\mathbb R^{m_1}\times\mathbb R^k$ ($z'_2=(z_{21},\dots,z_{2k})$) such that $|z'|<\varepsilon$ there exists a function $x(\cdot)\in U_{C([t_0,t_1],\mathbb R^n)}(\widehat x(\cdot),\delta/2C)\subset V$ admissible for (1) and satisfying the conditions
$$
\begin{equation}
g(x(t_0),x(t_1))=y_1+z_1\quad\text{and} \quad f_j(x(t_0),x(t_1))\leqslant y_{2j}+z_{2j}, \quad j=1,\dots,k.
\end{equation}
\tag{11}
$$
Let $(z_{2(k+1)},\dots,z_{2m_2})$ be a vector such that $|z|<\varepsilon$, where $z=(z_1,z_2)$ and $z_2=(z_{21},\dots,z_{2k},z_{2(k+1)},\dots,z_{2m_2})$. Since $\|x(\cdot)-\widehat x(\cdot)\|_{C([t_0,t_1],\mathbb R^n)}<\delta/(2C)<r$, using inequalities (9) and (10), for $j=k+1,\dots,m_2$ we obtain
$$
\begin{equation*}
f_j(x(t_0),x(t_1))\leqslant f_j(\widehat x(t_0),\widehat x(t_1))+\frac{\delta}2< y_{2j}-\frac{\delta}2<y_{2j}-\varepsilon<y_{2j}+z_{2j}.
\end{equation*}
\notag
$$
In combination with (11), this means that $y=(y_1,y_2)\in\operatorname{int}R(U_{C([t_0,t_1],\mathbb R^n)})\subset \operatorname{int}R(V)$. In Theorem 2 we fixed the function $\widehat x(\cdot)$ and its neighbourhood $V$, and the necessary conditions for boundary points of the attainable set depend on these data. This points at certain duality between the statements of Theorems 1 and 2, which is convenient in the derivation from Theorem 2 of necessary conditions for a trajectory of local infimum in an optimal control problem which generalize and refine the classical result of Pontryagin’s maximum principle (see [9]). In the examination of boundary points of the whole set the following necessary conditions can be of use, which are a direct consequence of from Theorem 2. Theorem 3. If $y=(y_1,y_2)\in\partial R$, then $\Lambda_1(x(\cdot),\overline\alpha(\cdot),\overline u(\cdot),y_2)\ne\{0\}$ for each triple $(x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))\in D_c(y)$. Theorem 3 can be expressed as follows:
$$
\begin{equation}
\begin{aligned} \, \notag &\{y=(y_1,y_2)\in \partial R \colon D_c(y)\ne\varnothing\}\subset \{y=(y_1,y_2) \colon \\ &\qquad \Lambda_1(x(\cdot),\overline\alpha(\cdot),\overline u(\cdot), y_2)\ne\{0\} \ \forall\,(x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))\in D_c(y)\}. \end{aligned}
\end{equation}
\tag{12}
$$
If there is no map $f$ in the problem under consideration, $(\widehat x(\cdot),\widehat u(\cdot))$ is an admissible pair for system (1) and $\widehat x(\cdot)\in D(y)$, then it follows from Theorem 3 that $\Lambda(\widehat x(\cdot),\widehat u(\cdot))\ne\{0\}$. Hence the necessary conditions for a boundary point of the attainable set that were proposed in [1]–[3] follow from Theorem 3 under the above assumptions about system (1). At the same time this theorem produces a whole family of relations for boundary points of $R$, improving in this way the necessary conditions just mentioned (see Example 2 in [9]). On the other hand, for example, a boundary point of the attainable set $R_0$ considered above is not necessarily an endpoint of an admissible trajectory for (1). Thus, the geometric Pontryagin maximum principle cannot be used for the investigation of such points. At the same time, Theorem 3 provides meaningful information about such points too (see Examples 1 and 2 in § 2). Consider now a special but important case when the values of some of the functions forming $g$ and $f$ are fixed. Let $g$ and $f$ be maps of the form $g=(g_1,g_2)$, where $g_i\colon \mathbb R^n\times\mathbb R^n\to\mathbb R^{k_i}$, $i=1,2$, and $f=(f_1,f_2)$, where $f_i\colon \mathbb R^n\times\mathbb R^n\to\mathbb R^{n_i}$, $i=1,2$. Fix $z=(z_1,z_2)\in \mathbb R^{k_1}\times\mathbb R^{n_1}$. Let $q=(q_1,q_2)\in\mathbb R^{k_2}\times\mathbb R^{n_2}$. We denote by $D_c((z,q))=D_c((z,q),g,f)$ the subset of $D_c$ consisting of the trajectories $x(\cdot)$ such that
$$
\begin{equation*}
g(x(t_0),x(t_1))=(z_1,q_1)\quad\text{and} \quad f(x(t_0),x(t_1))\leqslant (z_2,q_2).
\end{equation*}
\notag
$$
Consider the following attainable set:
$$
\begin{equation*}
R_z=R_z(g, f, t_0,t_1)=\{q=(q_1,q_2)\in\mathbb R^{k_2}\times\mathbb R^{n_2} \colon \exists\,x(\cdot)\in D_c((z,q))\}.
\end{equation*}
\notag
$$
We claim that $\{z\}\times\partial R_z\subset \partial R$. In fact, let $(z,q)\in \{z\}\times\partial R_z$, let $V$ be a neighbourhood of $(z,q)$ and $V_0$ be the projection of $V$ onto $\mathbb R^{k_2+n_2}$. Clearly, $V_0$ is a neighbourhood of $q$. Since $q\in\partial R_z$, there exist $q'$ and $q''$ in $V_0$ such that $q'\in R_z$ and $q''\notin R_z$. Then it is obvious that $(z,q')$ and $(z,q'')$ are points in $V$, $(z,q')\in R$ and $(z,q'')\notin R$, that is, $(z,q)\in \partial R$. Hence we have the following formula, providing us with necessary conditions for the boundary of $R_z$:
$$
\begin{equation}
\begin{aligned} \, \notag & \{q\in\partial R_z \colon D_c((z,q))\ne\varnothing\}\subset\{q=(q_1,q_2)\in\mathbb R^{k_2}\times\mathbb R^{n_2} \colon \\ &\qquad \Lambda_1(x(\cdot),\overline\alpha(\cdot),\overline u(\cdot), (z_2,q_2))\ne\{0\}\ \forall\,(x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))\in D_c((z,q))\}. \end{aligned}
\end{equation}
\tag{13}
$$
We present a result we will need in an example at the end of this paper. Proposition 1. Let $(x(\cdot),\overline\alpha(\cdot),\overline u(\cdot))\in D_c$. If for fixed $p(\cdot)$ the maximum in (8) is attained at a unique function $\widehat u(\cdot)$, then $\dot x(t)=\varphi(t,x(t),\widehat u(t))$ for almost all $t\in[t_0,t_1]$. Proof. Let $\overline\alpha(\cdot)=(\alpha_1(\cdot),\dots,\alpha_N(\cdot))$ and $\overline u(\cdot)=(u_1(\cdot),\dots,u_N(\cdot))$. Let $T$ denote the set of $t\in[t_0,t_1]$ such that $0\leqslant\alpha_i(t)\leqslant1$, $i=1,\dots,N$, $\sum_{i=1}^N\alpha_i(t)=1$, equality holds in the maximum condition in (8) and $\dot x(t)=\sum_{i=1}^N\alpha_i(t)\varphi(t,x(t),u_i(t))$. Clearly, $T$ is a set of full measure.
Let $t\in T$, and let $J_0(t)$ denote the subset of $J=\{1,\dots,N\}$ such that for $i\in J_0(t)$ we have $\alpha_i(t)>0$ ($J_0(t)\ne\varnothing$ because $\sum_{i=1}^N\alpha_i(t)=1$). We show that
$$
\begin{equation}
\langle p(t),\varphi(t,x(t),u_i(t))\rangle=\max_{u\in\operatorname{cl}U}\langle p(t),\varphi(t,x(t),u)\rangle, \qquad i\in J_0,
\end{equation}
\tag{14}
$$
at this point $t$. Suppose this is not so, and we have strict inequality in (14) for some $i_0\in J_0(t)$. We multiply both sides of this inequality by $\alpha_{i_0}(t)$. Clearly, for all $i\in J$ the left-hand side of (14) does not exceed the right-hand side. Now, for each such $i$ we multiply both sides of the $i$th inequality (the one involving $u_i(\cdot)$) by $\alpha_i(t)$, $i\in J\setminus i_0$, and then, adding these inequalities, we obtain
$$
\begin{equation*}
\biggl\langle p(t),\sum_{i=1}^N\alpha_i(t)\varphi(t,x(t),u_i(t))\biggr\rangle <\max_{u\in\operatorname{cl}U}\langle p(t),\varphi(t,x(t),u)\rangle.
\end{equation*}
\notag
$$
However, the left-hand side is equal to $\langle p(t),\dot x(t)\rangle$ and must coincide with the right-hand side at $t$ by the maximum condition in (8). We have arrived at a contradiction, and therefore (14) holds.
Since the maximum is attained at a unique function $\widehat u(\cdot)$, it follows that $u_i(t)=\widehat u(t)$, $i\in J_0(t)$, and therefore ($\alpha_i(t)=0$ for $i\in J\setminus J_0(t)$)
$$
\begin{equation*}
\dot x(t)=\sum_{i=1}^N\alpha_i(t)\varphi(t,x(t),u_i(t))=\sum_{i\in J_0(t)}\alpha_i(t)\varphi(t,x(t),\widehat u(t))=\varphi(t,x(t),\widehat u(t)).
\end{equation*}
\notag
$$
This holds for each $t\in T$.
The proposition is proved. § 2. Systems linear in the phase variablesIn this section, for a class of control systems linear in the phase variable we find a criterion of a boundary point. Consider the control system
$$
\begin{equation}
\dot x=A(t)x+\psi(t,u), \qquad u(t)\in U \text{ for a.e. } t\in[t_0,t_1],
\end{equation}
\tag{15}
$$
where $A\colon \mathbb R^n\to\mathbb R^n$ is a continuous $n\times n$ matrix-valued function, $\psi\colon \mathbb R\times\mathbb R^r\to\mathbb R^n$ is a continuous map of the variables $t$ and $u$, and $U$ is a nonempty subset of $\mathbb R^r$. Let the maps $g\colon\mathbb R^n\times\mathbb R^n\to \mathbb R^{m_1}$ and $f\colon\mathbb R^n\times\mathbb R^n\to \mathbb R^{m_2}$ act by the formulae
$$
\begin{equation*}
g(\zeta_0,\zeta_1)=B\zeta_0+C\zeta_1\quad\text{and} \quad f(\zeta_0,\zeta_1)=D\zeta_0+E\zeta_1 \quad\forall\,(\zeta_0,\zeta_1)\in\mathbb R^n\times\mathbb R^n,
\end{equation*}
\notag
$$
where $B, C\in\mathcal L(\mathbb R^n, \mathbb R^{m_1})$ and $D, E\in\mathcal L(\mathbb R^n, \mathbb R^{m_2})$ ($\mathcal L(\mathbb R^n, \mathbb R^{m_i})$ is the space of linear maps from $\mathbb R^n$ to $\mathbb R^{m_i}$, $i=1,2$, which we identify with their matrices with respect to the standard bases). As before, we associate with (15) a convex system of the form (3); here it looks as follows:
$$
\begin{equation}
\dot x =A(t)x+\sum_{i=1}^N\alpha_i(t)\psi(t,u_i(t)), \qquad \overline\alpha(\cdot)\in \mathcal A_N, \quad \overline u(\cdot)\in \mathcal U^N, \quad N\in\mathbb N.
\end{equation}
\tag{16}
$$
In this subsection we denote by $D_c$, in place of the set of admissible triples for (16), the set of admissible trajectories for this system. This will be convenient for the results stated below. Let $x(\cdot)\in C([t_0,t_1],\mathbb R^n)$ and $y_2\in \mathbb R^{m_2}$. Let $\Lambda(x(\cdot), y_2)$ denote the set of triples $(p(\cdot),\lambda_g,\lambda_f)\in \operatorname{AC}([t_0,t_1],(\mathbb R^n)^*)\times (\mathbb R^{m_1})\times(\mathbb R^{m_2})_+^*$ such that
$$
\begin{equation}
\begin{gathered} \, \dot p(t) =-p(t)A(t), \\ p(t_0)=\lambda_gB+\lambda_fD, \qquad p(t_1)=-\lambda_gC-\lambda_fE, \\ \langle \lambda_f, Dx(t_0)+Ex(t_1)-y_2\rangle=0, \\ \max_{u\in \operatorname{cl}U}\langle p(t),\psi(t,u)\rangle=\langle p(t),\dot {x}(t)-A(t)x(t)\rangle \quad\text{a.e. on } [t_0,t_1]. \end{gathered}
\end{equation}
\tag{17}
$$
The zero tuple clearly satisfies these relations. Theorem 4. Let $\widehat y=(\widehat y_1,\widehat y_2)\in\mathbb R^{m_1}\times\mathbb R^{m_2}$. Then for $\widehat y$ to belong to $\partial R$ it is necessary that $\Lambda(x(\cdot),\widehat y_2)\ne\{0\}$ for each $x(\cdot)\in D_c(\widehat y)$, and it is sufficient that there exist a trajectory $\widehat x(\cdot) \in D_c(\widehat y)$ such that $\Lambda(\widehat x(\cdot),\widehat y_2) \ne \{0\}$. Proof. Necessity. In our case conditions (17) and (8) are equivalent, so necessity follows directly from Theorem 3.
Sufficiency. Let $\widehat y=(\widehat y_1,\widehat y_2)\in\mathbb R^{m_1}\times\mathbb R^{m_2}$. The inclusion $\widehat x(\cdot)\in D_c(\widehat y)$ means that for some $k\in\mathbb N$ there are tuples $\widehat{\overline \alpha}(\cdot)=(\widehat a_1(\cdot),\dots,\widehat a_k(\cdot))\in \mathcal A_k$ and $\widehat{\overline u}(\cdot)=(\widehat u_1(\cdot),\dots,\widehat u_k(\cdot))\in \mathcal U^k$ such that
$$
\begin{equation}
\dot{\widehat x}(t) =A(t)\widehat x(t)+\sum_{i=1}^k\widehat\alpha_i(t)\psi(t,\widehat u_i(t)),
\end{equation}
\tag{18}
$$
$$
\begin{equation}
B\widehat x(t_0)+C\widehat x(t_1)=\widehat y_1\quad\text{and} \quad D\widehat x(t_0)+E\widehat x(t_1)\leqslant \widehat y_2.
\end{equation}
\tag{19}
$$
First we show that $\widehat y\in \partial R_c$. Let $y=(y_1,y_2)\in R_c$. Then by definition there exists a trajectory $x(\cdot)\in D_c(y)$, that is, there exist tuples $\overline\alpha(\cdot)=(\alpha_1(\cdot),\dots, \alpha_N(\cdot))\in \mathcal A_N$ and $\overline u(\cdot)=(u_1(\cdot),\dots,u_N(\cdot))\in \mathcal U^N$ for some $N$ such that we have (16) and
$$
\begin{equation}
Bx(t_0)+Cx(t_1)=y_1\quad\text{and} \quad Dx(t_0)+Ex(t_1)\leqslant y_2.
\end{equation}
\tag{20}
$$
It follows from the maximum condition in (17) that
$$
\begin{equation*}
\begin{aligned} \, \sum_{i=1}^N\alpha_i(t)\langle p(t),\psi(t,u_i(t))\rangle &\leqslant\max_{u\in \operatorname{cl}U}\langle p(t),\psi(t,u)\rangle \\ &=\langle p(t),\dot {\widehat x}(t)-A(t)\widehat x(t)\rangle=\sum_{i=1}^k\widehat\alpha_i(t)\langle p(t),\psi(t,\widehat u_i(t))\rangle \end{aligned}
\end{equation*}
\notag
$$
for almost all $t\in [t_0,t_1]$.
Now, using in turn (16) and (18), the first equality from (17), and then also the second, third and fourth equalities from there we obtain
$$
\begin{equation*}
\begin{aligned} \, 0&\leqslant\int_{t_0}^{t_1}\biggl(\sum_{i=1}^k\widehat\alpha_i(t)\langle p(t),\psi(t,\widehat u_i(t))\rangle-\sum_{i=1}^N\alpha_i(t)\langle p(t),\psi(t,u_i(t))\rangle\biggr)\,dt \\ &=\int_{t_0}^{t_1}\bigl(\langle p(t),\dot{\widehat x}(t)-A(t)\widehat x(t)\rangle-\langle p(t),\dot x(t)-A(t)x(t)\rangle\bigr)\,dt \\ &=\int_{t_0}^{t_1}\bigl(\langle p(t),\dot{\widehat x}(t)\rangle+\langle \dot p(t),\widehat x(t)\rangle\bigr)\,dt -\int_{t_0}^{t_1}\bigl( \langle p(t),\dot x(t)\rangle+\langle \dot p(t),x(t)\rangle\bigr)\,dt \\ &= \langle p(t_1),\widehat x(t_1)\rangle-\langle p(t_0),\widehat x(t_0)\rangle -\langle p(t_1),x(t_1)\rangle+\langle p(t_0),x(t_0)\rangle \\ &=-\langle\lambda_gC+\lambda_fE,\widehat x(t_1)\rangle-\langle\lambda_gB+ \lambda_fD,\widehat x(t_0)\rangle+\langle\lambda_gC+\lambda_fE,x(t_1)\rangle \\ &\qquad +\langle\lambda_gB+ \lambda_fD,x(t_0)\rangle \\ &=-\langle \lambda_g,B\widehat x(t_0)+C\widehat x(t_1)\rangle-\langle \lambda_f,D\widehat x(t_0)+E\widehat x(t_1)\rangle \\ &\qquad +\langle \lambda_g,Bx(t_0)+C x(t_1)\rangle+\langle \lambda_f,Dx(t_0)+Ex(t_1)\rangle \\ &=-\langle \lambda_g,B\widehat x(t_0)+C\widehat x(t_1)\rangle-\langle \lambda_f,D\widehat x(t_0)+E\widehat x(t_1)-\widehat y_2\rangle-\langle\lambda_f,\widehat y_2\rangle \\ &\qquad +\langle \lambda_g,Bx(t_0)+C x(t_1)\rangle+\langle \lambda_f,Dx(t_0)+Ex(t_1)-y_2\rangle+\langle \lambda_f,y_2\rangle \\ &=-\langle \lambda_g,\widehat y_1\rangle- \langle \lambda_f,\widehat y_2\rangle+\langle \lambda_g, y_1\rangle+\langle \lambda_f, y_2\rangle. \end{aligned}
\end{equation*}
\notag
$$
For $\widehat\lambda=(\lambda_g,\lambda_f)$ the relation just established means that
$$
\begin{equation}
\langle\widehat \lambda, y-\widehat y\rangle\geqslant0 \quad \forall\,y\in R_c.
\end{equation}
\tag{21}
$$
Since $\Lambda(\widehat x(\cdot),\widehat y_2)\ne\{0\}$, we have $\widehat\lambda\ne0$. Let $y_0\in\mathbb R^{m_1}\times\mathbb R^{m_2}$ be such that $\langle\widehat\lambda,y_0\rangle<0$. Then $\langle\widehat\lambda,\alpha y_0\rangle<0$ for each $\alpha>0$, and $\alpha y_0\notin R_c-\widehat y$ by (21), or $\widehat y+\alpha y_0\notin R_c$. Hence $\widehat y\in\partial R_c$ since we clearly have $\widehat y\in R_c$. Now we show that $\widehat y\in\partial R$. As it is clear that $R\subset R_c$, we must verify that each neighbourhood of $\widehat y$ contains elements of $R$. Let $u_s(\widehat{\overline \alpha};\widehat{\overline u})(\cdot)$, $s\in\mathbb N$, be a sequence of controls from the approximation lemma in § 3 (all functions arising here are solutions of differential equations with bounded right-hand sides, and therefore they are Lipschitz continuous). By that lemma the sequence
$$
\begin{equation*}
\beta_s=\int_{t_0}^{t_1}\biggl(\sum_{i=1}^k\widehat\alpha_i(t)\psi(t, \widehat u_i(t))-\psi(t,u_s(\widehat{\overline \alpha};\widehat{\overline u})(t))\biggr)dt
\end{equation*}
\notag
$$
tends to zero as $s\to\infty$.
For each $s\in\mathbb N$ let $x_s(\cdot)$ denote the solution of the Cauchy problem
$$
\begin{equation}
\dot x_s=A(t)x_s+\psi(t,u_s(\widehat{\overline \alpha};\widehat{\overline u})(t)), \qquad x_s(t_0)=\widehat x(t_0).
\end{equation}
\tag{22}
$$
Set
$$
\begin{equation*}
B\widehat x(t_0)+Cx_s(t_1)=y_{s1}\quad\text{and} \quad D\widehat x(t_0)+Ex_s(t_1)\leqslant y_{s2},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
y_{s2}=\max(D\widehat x(t_0)+Ex_s(t_1), \widehat y_2).
\end{equation*}
\notag
$$
Clearly, $y_s=(y_{s1},y_{s2})\in R$.
Set $z(t)=|\widehat x(t)-x_s(t)|$. Then it is easy to see that
$$
\begin{equation*}
z(t)\leqslant\int_{t_0}^t\|A(\tau)\|z(\tau)\,d\tau+|\beta_s|.
\end{equation*}
\notag
$$
Hence by Grönwall’s lemma
$$
\begin{equation*}
|\widehat x(t_1)-x_s(t_1)|\leqslant|\beta_s|\exp\int_{t_0}^{t_1}\|A(t)\|\,dt.
\end{equation*}
\notag
$$
Since $\beta_s\to0$ as $s\to\infty$, it follows that $x_s(t_1)\to \widehat x(t_1)$ as $s\to\infty$, and therefore $y_s\to \widehat y$. Thus, $\widehat y\in \partial R$.
Theorem 4 is proved. We prove the next result for special maps $g$ and $f$, namely, for $g\colon \mathbb R^n\times\mathbb R^n\to\mathbb R^n\times\mathbb R^{m_1}$ and $f\colon \mathbb R^n\times\mathbb R^n\to\mathbb R^{m_2}$ such that
$$
\begin{equation}
g(\zeta_0,\zeta_1)=(\zeta_0, C\zeta_1)\quad\text{and} \quad f(\zeta_0,\zeta_1)=E\zeta_1 \quad\forall\,(\zeta_0,\zeta_1)\in\mathbb R^n\times\mathbb R^n.
\end{equation}
\tag{23}
$$
Let $x(\cdot)\in C([t_0,t_1],\mathbb R^n)$ and $y_2\in \mathbb R^{m_2}$. Here $\Lambda(x(\cdot), y_2)$ is the set of triples $(p(\cdot),\lambda_g,\lambda_f)\in \operatorname{AC}([t_0,t_1],(\mathbb R^n)^*)\times (\mathbb R^{m_1})^*\times(\mathbb R^{m_2})_+^*$ satisfying the relations
$$
\begin{equation}
\begin{gathered} \, \dot p(t) =-p(t)A(t), \\ p(t_1)=-\lambda_gC-\lambda_fE, \\ \langle \lambda_f, Ex(t_1)-y_2\rangle=0, \\ \max_{u\in \operatorname{cl}U}\langle p(t),\psi(t,u)\rangle=\langle p(t),\dot {x}(t)-A(t)x(t)\rangle \quad\text{a.e. on } [t_0,t_1]. \end{gathered}
\end{equation}
\tag{24}
$$
Theorem 5. Let $\widehat y=(\widehat y_1,\widehat y_2)\in\mathbb R^{m_1}\times\mathbb R^{m_2}$, and let $U$ be a bounded set. Then it is necessary for $\widehat y$ to belong to $\partial R$ that $D_c(\widehat y)\ne\varnothing$ and $\Lambda(x(\cdot),y_2)\ne\{0\}$ for all $x(\cdot)\in D_c(\widehat y)$, and it is sufficient that there exist a trajectory $\widehat x(\cdot)\in D_c(\widehat y)$ such that $\Lambda(\widehat x(\cdot),\widehat y_2)\ne\{0\}$. Proof. Necessity. Let $\widehat y\in\partial R$. Then there exist a sequence $y_k=((y_{1k},y'_{1k}),y_{2k})\in(\mathbb R^n\times\mathbb R^{m_1})\times\mathbb R^{m_2}$ in $R$ converging to $\widehat y$ as $k\to\infty$ and a sequence of functions $x_k(\cdot)$ admissible for (15) such that
We show that the sequence $x_k(\cdot)$ is bounded in $C([t_0,t_1],\mathbb R^n)$. Let $\gamma$ be the maximum of $|\psi|$ on $[t_0,t_1]\times \operatorname{cl}U$. Then it follows from (15) that for all $k$ and each $t\in[t_0,t_1]$
$$
\begin{equation*}
|x_k(t)|\leqslant |x_k(t_0)|+\int_{t_0}^{t_1}\|A(t)\||x_k(t)|\,dt+\gamma(t_1-t_0).
\end{equation*}
\notag
$$
By Grönwall’s lemma
$$
\begin{equation*}
|x_k(t)|\leqslant (|x_k(t_0)|+\gamma(t_1-t_0))\exp\int_{t_0}^{t_1}\|A(t)\|\,dt.
\end{equation*}
\notag
$$
The sequence $x_k(t_0)$ is bounded in view of the convergence of $y_k$, hence the sequence $x_k(\cdot)$ is bounded in $C([t_0,t_1],\mathbb R^n)$.
Now we show that the sequence of derivatives $\dot x_k(\cdot)$ is bounded in $L_\infty([t_0,t_1],\mathbb R^r)$. In fact, the map $t\mapsto A(t)$ is continuous, and therefore $\|A(t)\|\leqslant \kappa$ for all $t\in[t_0,t_1]$ for some $\kappa>0$. Then from (15) we obtain
$$
\begin{equation*}
|\dot x_k(t)|\leqslant \kappa|x_k(t)|+\gamma \quad\text{for a.e. } t\in[t_0,t_1],
\end{equation*}
\notag
$$
and since the sequence $x_k(\cdot)$ is bounded in $C([t_0,t_1],\mathbb R^n)$, the sequence $\dot x_k(\cdot)$ is bounded in $L_\infty([t_0,t_1],\mathbb R^r)$.
Since the sequence $\dot x_k(\cdot)$ is bounded, the family $x_k(\cdot)$ is equicontinuous, and, being bounded, it is precompact in $C([t_0,t_1],\mathbb R^n)$ by the Arzelá–Ascoli theorem, so we can extract a subsequence converging to a function $\widehat x(\cdot)\in C([t_0,t_1],\mathbb R^n)$. We can assume that the sequence $x_k(\cdot)$ itself converges to this function. Clearly, the functions $x_k(\cdot)$ satisfy the convex system (16) for $N=1$ and therefore for each $N\in \mathbb N$. Let $N=n+1$. It follows from the proof of Filippov’s theorem (see [10]) that if the set $Q=\Sigma^{n+1}\times (\operatorname{cl}U)^{n+1}$ is compact, the set
$$
\begin{equation*}
G(t,x)=\biggl\{ \sum_{i=1}^{n+1}\alpha_i(A(t)x+\psi(t,u_i))\in\mathbb R^n \colon (\alpha_1,\dots,\alpha_{n+1},u_1,\dots,u_{n+1})\in Q\biggr\}
\end{equation*}
\notag
$$
is convex for any $t\in[t_0,t_1]$ and $x\in\mathbb R^n$, and the sequence of derivatives $\dot x_k(\cdot)$ is bounded in $L_\infty([t_0,t_1],\mathbb R^r)$, then the limit of $x_k(\cdot)$ is an admissible trajectory for system (16) for $N=n+1$.
It is obvious that $Q$ is compact. Let us verify that $G(t,x)$ is convex for all $t\in[t_0,t_1]$ and $x\in\mathbb R^n$. Clearly, $G(t,x)$ belongs to the convex hull of
$$
\begin{equation*}
A(t)x+\psi(t,U)=\{A(t)x+\psi(t,u)\in\mathbb R^n \colon u\in U\}.
\end{equation*}
\notag
$$
On the other hand each element of this convex hull has a representation as a convex combination of at most $n+1$ elements of $A(t)x+\psi(t,U)$ by Carathéodory’s theorem (for instance, see [11]). Thus, $G(t,x)$ is the convex hull of this set, so that it is convex.
That the sequence of derivatives is bounded in $L_\infty([t_0,t_1],\mathbb R^r)$ was proved above. Thus, $\widehat x(\cdot)$ is an admissible trajectory for the convex system (16), and since $y_k\to\widehat y$ as $k\to\infty$, it follows that $\widehat x(\cdot)\in D_c(\widehat y)$, and so $D_c(\widehat y)\ne\varnothing$. The fact that $\Lambda(x(\cdot),\widehat y_2)\ne\{0\}$ for each trajectory $x(\cdot)\in D_c(\widehat y)$ is a direct consequence of the necessary conditions in Theorem 4. This proves necessity. Sufficiency follows from the sufficiency conditions in Theorem 4. Theorem 5 is proved. For maps $g$ and $f$ under consideration and a bounded set $U$ Theorem 5 describes fully the boundary of the attainable set. Now we present yet another criterion of a boundary point in the case of trajectories of system (15) with fixed left endpoint. Let the constraints $g$ and $f$ have the form (23). Fix $z\in\mathbb R^n$. Let $q=(q_1,q_2)\in\mathbb R^{m_1}\times\mathbb R^{m_2}$, and let $D_0(q)=D((x_0,q),g,f)$ denote the subset of $D$ formed by the trajectories $x(\cdot)$ such that
$$
\begin{equation}
x(t_0)=x_0, \qquad Cx(t_1)=q_1\quad\text{and} \quad Ex(t_1))\leqslant q_2.
\end{equation}
\tag{26}
$$
Consider the following attainable set:
$$
\begin{equation*}
R_0=R_0(g, f, t_0,t_1)=\{q=(q_1,q_2)\in\mathbb R^{m_1}\times\mathbb R^{m_2} \colon \exists\,x(\cdot)\in D_0(q)\}.
\end{equation*}
\notag
$$
Let $D_{c0}(q)=D_c((x_0,q),g,f)$ denote the subset of $D_c$ formed by the trajectories $x(\cdot)$ such that (26) holds. Then the corresponding attainable set has the form
$$
\begin{equation*}
R_{c0}=R_{c0}(g, f, t_0,t_1)=\{q=(q_1,q_2)\in\mathbb R^{m_1}\times\mathbb R^{m_2} \colon \exists\,x(\cdot)\in D_{c0}(q)\}.
\end{equation*}
\notag
$$
The following result holds. Theorem 6. Let $\widehat q=(\widehat q_1,\widehat q_2)\in\mathbb R^{m_1}\times\mathbb R^{m_2}$, and let $U$ be a bounded set. Then for $\widehat q$ to belong to $\partial R_0$ it is necessary that $D_{c0}(\widehat q) \ne \varnothing$ and $\Lambda(x(\cdot),\widehat q_2)\ne\{0\}$ for each $x(\cdot)\in D_{c0}(\widehat q)$, and it is sufficient that there exist a trajectory $\widehat x(\cdot)\in D_{c0}(\widehat q)$ such that $\Lambda(\widehat x(\cdot),\widehat q_2)\ne\{0\}$. Proof. Necessity. Let $\widehat q\in\partial R_0$. Then there exists a sequence $q_k=(q_{1k},q_{2k})\in\mathbb R^{m_1}\times\mathbb R^{m_2}$ in $R_0$ converging to $\widehat q$ as $k\to\infty$ and a sequence of trajectories $x_k(\cdot)$ admissible for (15) such that
$$
\begin{equation*}
x_k(t_0)=x_0, \qquad Cx_k(t_1))=q_{1k}\quad\text{and} \quad Ex_k(t_1)\leqslant q_{2k}.
\end{equation*}
\notag
$$
Now repeating the arguments in Theorem 5, in fact, word for word, we obtain $D_{c0}(\widehat q)\ne\varnothing$.
Let $D((\zeta,q))=D((\zeta,q),g,f)$ denote the subset of $D$ formed by the trajectories $x(\cdot)$ such that
$$
\begin{equation*}
x(t_0)=\zeta, \qquad Cx(t_1)=q_1\quad\text{and} \quad Ex(t_1))\leqslant q_2.
\end{equation*}
\notag
$$
Consider the following attainable set:
$$
\begin{equation*}
R=\{(\zeta,q)=(\zeta,q_1,q_2)\in\mathbb R^n\times\mathbb R^{m_1}\times\mathbb R^{m_2} \colon \exists\,x(\cdot)\in D((\zeta,q))\}.
\end{equation*}
\notag
$$
Just as in the remark after Theorem 3, we can show that if $\widehat q\in\partial R_0$, then ${(x_0,\widehat q)\in \partial R}$. Hence Theorem 5 yields $\Lambda(x(\cdot),\widehat q_2)\ne\{0\}$ for each trajectory ${x(\cdot)\in D_{c0}(\widehat q)}$. Sufficiency. The inclusion $\widehat x(\cdot)\in D_{c0}(\widehat q)$ means that for some $k\in\mathbb N$ we can find tuples $\widehat{\overline \alpha}(\cdot)=(\widehat a_1(\cdot),\dots,\widehat a_k(\cdot))\in \mathcal A_k$ and $\widehat{\overline u}(\cdot)=(\widehat u_1(\cdot),\dots,\widehat u_k(\cdot))\in \mathcal U^k$ such that
$$
\begin{equation*}
\begin{gathered} \, \dot{\widehat x}(t) =A(t)\widehat x(t)+\sum_{i=1}^k\widehat\alpha_i(t)\psi(t,\widehat u_i(t)), \\ \widehat x(t_0)=x_0, \qquad C\widehat x(t_1)=\widehat q_1\quad\text{and} \quad E\widehat x(t_1)\leqslant \widehat q_2. \end{gathered}
\end{equation*}
\notag
$$
First we show that $\widehat q\in \partial R_{c0}$. Let $q=(q_1,q_2)\in R_{c0}$. Then by definition there exists a trajectory $x(\cdot)\in D_{c0}(q)$, that is, there exist tuples $\overline\alpha(\cdot)=(\alpha_1(\cdot),\dots,\alpha_N(\cdot))\in \mathcal A_N$ and $\overline u(\cdot)=(u_1(\cdot),\dots,u_N(\cdot))\in \mathcal U^N$ for some value of $N$ such that (16) holds,
$$
\begin{equation}
x(t_0)=x_0, \qquad Cx(t_1)= q_1\quad\text{and} \quad Ex(t_1)\leqslant q_2.
\end{equation}
\tag{27}
$$
Repeating again the arguments after formula (20) in the proof of Theorem 5, and using (27) in place of (20) and (24) in place of (17) we arrive at an analogue of (21):
$$
\begin{equation*}
\langle\widehat \lambda, q-\widehat q\rangle\geqslant0 \quad\forall\,q\in R_{c0},
\end{equation*}
\notag
$$
where $\widehat\lambda=(\lambda_g,\lambda_f)$.
Hence, as before, we obtain $\widehat q\in\partial R_{c0}$. Now we show that $\widehat q\in\partial R_0$. As it is obvious that $R_0\subset R_{c0}$, we must verify that each neighbourhood of $\widehat q$ contains elements of $R_0$. As in Theorem 5, for each $s\in\mathbb N$ let $x_s(\cdot)$ denote the solution of the Cauchy problem (22), and set
$$
\begin{equation*}
Cx_s(t_1)=q_{s1}\quad\text{and} \quad Ex_s(t_1)\leqslant q_{s2},
\end{equation*}
\notag
$$
where Clearly, $q_s=(q_{s1},q_{s2})\in R_0$. Arguing just as in Theorem 4, we see that $q_s\to \widehat q$ as ${s\to\infty}$, and therefore $\widehat q\in \partial R_0$.
Theorem 6 is proved. Now we present two examples showing the potentials of the above result in the construction of attainable sets and their boundaries. Example 1. Consider the control system As above, let $D$ denote the set of admissible trajectories $x(\cdot)=(x_1(\cdot),x_2(\cdot))$ for this system. We look for the boundary of the following set: We will draw on Theorem 6, so we arrange the setting in accordance with the general notation introduced before this theorem. In our example $g(\zeta_0,\zeta_1)=(\zeta_0,\zeta_1)$ and there is no map $f$. Fix $z=0$. Let $q\in \mathbb R^{2}$. Then $D((0,q))$ is the subset of $D$ formed by the trajectories $x(\cdot)$ such that and $R_0$ can be expressed as
$$
\begin{equation*}
R_0=\{q=(q_1,q_2)\in\mathbb R^{2} \colon \exists\,x(\cdot)\in D((0,q))\}.
\end{equation*}
\notag
$$
Here $D_c$ is the set of admissible trajectories for the convex system
$$
\begin{equation}
\dot x_1(t)=\sum_{i=1}^N\alpha_i(t)u_i(t), \qquad \dot x_2(t)=\sum_{i=1}^N\alpha_i(t)u^2_i(t),
\end{equation}
\tag{29}
$$
where $N\in \mathbb N$, $\overline\alpha(\cdot)=(\alpha_1(\cdot),\dots,\alpha_N(\cdot))\in\mathcal A_N$ and $\overline u(\cdot)=(u_1(\cdot),\dots,u_N(\cdot))\in\mathcal U^N$, and $D_c((0,q))$ is the subset of trajectories in $D_c$ satisfying (26). We turn to finding the boundary of $R_0$. In this case $\Lambda(x(\cdot),q_2)$ (below we write $\Lambda(x(\cdot))$ because there is no $f$) is the set of pairs $(p(\cdot),\lambda_g)\in \operatorname{AC}([t_0,t_1],(\mathbb R^2)^*)\times (\mathbb R^{4})^*$, where $\lambda_g=(\lambda_1,\lambda_2)$, that satisfy and ($p=(p_1,p_2)$)
$$
\begin{equation}
\max_{|u|\leqslant1}(p_1u+p_2u^2)=p_1\dot x_1(t)+p_2\dot x_2(t) \quad\text{for a.e. } t\in[0,1].
\end{equation}
\tag{31}
$$
Clearly, $p_1$ and $p_2$ are constants, and the condition $\Lambda(x(\cdot))\ne\{0\}$ is equivalent to conditions (30) and (31) for $p=(p_1, p_2)\ne0$. Now we consider all cases when $p\ne0$ and (31) holds. (1) $p_2=0$. First let $p_1>0$. The trajectory $\widehat x(t)=(t,t)$, $t\in[0,1]$, satisfies (29) for $N=1$ and $u_1=1$, so it belongs to $D_c(0,(1,1))$. Clearly, $\widehat x(\cdot)$ satisfies (31) for $p$ in question. Hence, by the sufficiency conditions in Theorem 6 the point $(1,1)$ lies in $\partial R_0$. For fixed $p$ the boundary of $R_0$ contains no other points. In fact, it follows from Theorem 6 that points ‘suspicious’ for lying on the boundary of $R_0$ (satisfying the necessary conditions) are endpoints of trajectories $x(\cdot)\in D_c$ such that $x(0)=0$ and $\Lambda(x(\cdot))\ne\{0\}$. In our case the maximum in (31) is clearly attained at the unique function $\widehat u(\cdot)=1$, so by Proposition 1 the set of such trajectories consists of the single function $\widehat x(t)=(t,t)$, $t\in[0,1]$. Hence only $(1,1)$ can lie on the boundary of the attainable set $R_0$. In combination with the above, this means that only the point $(1,1)$ belongs to $\partial R_0$. For $p_1<0$ similar arguments show that then $\partial R_0$ contains only the point $(-1,1)$. (2) $p_2<0$ and $p_1\ne0$. Easy calculations show that then the maximum in (31) is attained at the point
$$
\begin{equation*}
u_*=\begin{cases} -\dfrac{p_1}{2p_2}, &|p_1|\leqslant-2p_2, \\ -1, &p_1<-2p_2, \\ 1, & p_1>2p_2. \end{cases}
\end{equation*}
\notag
$$
Let $|p_1|\leqslant-2p_2$. Then for the trajectory $x(\cdot)=(x_1(\cdot),x_2(\cdot))$ defined by
$$
\begin{equation*}
x(t)=\biggl(\frac{-p_1}{2p_2}\,t,\frac{p^2_1}{4p^2_2}\,t\biggr), \qquad t\in[0,1],
\end{equation*}
\notag
$$
we clearly have (31). This function solves (29) for $N=1$ and $u_1=-p_1/(2p_2)$, and therefore $x(\cdot)\in D_c((0,(q_1,q_2))$, where $q_1=-p_1/(2p_2)$ and $q_2=q_1^2$. Hence by Theorem 6 the set of points $x(1)=(x_1(1),x^2_1(1))$, $|x_1(1)|\leqslant1$, lies on the boundary of $R_0$, or
$$
\begin{equation}
\{(q_1,q_2)\in\mathbb R^2 \colon q_2=q_1^2,\, |q_1|\leqslant1\}\subset \partial R_0.
\end{equation}
\tag{32}
$$
If $p_1<2p_2$ or $p_1>-2p_2$, then similarly to the above we see that the points $(-1,1)$ and $(1,1)$ belong to $\partial R_0$. Since for all $p_1\ne0$ and $p_2<0$ the maximum in (31) is attained for a unique function $u_*$, using Proposition 1 we conclude, as in the previous case, that now only the set on the left-hand side of (32) belongs to the boundary of $R_0$. (3) $p_2>0$. If $p_1\ne0$, then it is easy to check that the maximum in (31) is attained at the point $-1$ or at $1$. Now, similarly to the above we obtain that the boundary of $R_0$ contains only the points $(-1,1)$ and $(1,1)$. Let $p_1=0$. Then the maximum is attained at two points, $-1$ and $1$. If in the convex system (29) for $N=2$ we set $\alpha_1(\cdot)=1-\alpha$ and $\alpha_2(\cdot)=\alpha$, where $\alpha\in[0,1]$, $u_1=-1$ and $u_2=1$, then it is easy to see that for each $\alpha\in[0,1]$ the trajectory $x_\alpha(t)=((2\alpha-1)t,t)$ of this system, $t\in[0,1]$, satisfies (31). Therefore, $x_\alpha(\cdot)\in D_c((0,(2\alpha-1,1)))$ for each $\alpha\in[0,1]$, so that the set of endpoints of the trajectories $x_\alpha(1)=(2\alpha-1,1)$, $\alpha\in[0,1]$, namely, the set
$$
\begin{equation}
\{(q_1,q_2)\in\mathbb R^2 \colon |q_1|\leqslant1,\,q_2=1\},
\end{equation}
\tag{33}
$$
lies in $\partial R_0$. We show now that the boundary contains no other points in this case. In fact, each trajectory $x(\cdot)=(x_1(\cdot),x_2(\cdot))\in D_c$, $x(0)=0$, has the form
$$
\begin{equation*}
x(t)=\biggl(\int_{0}^t\biggl(\sum_{i=1}^N\alpha_i(\tau)u_i(\tau)\biggr)d\tau,t\biggr), \qquad t\in[0,1],
\end{equation*}
\notag
$$
for some $N$, where $(\alpha_i(\cdot),\dots,\alpha_N(\cdot)\in \mathcal A_N$ and $u_i(\cdot)\in\{-1,1\}$, $i=1,\dots,N$, for almost all $t\in[0,1]$. Then the integrand clearly has modulus not greater than 1 for almost all $\tau\in[0,1]$. Therefore, the only points ‘suspicious’ for lying on the boundary are endpoints of the trajectories from the set (33), so (as said above) only this set belongs to $\partial R_0$ in our case. Thus we have considered all cases when $p\ne0$ and have shown that
$$
\begin{equation}
\partial R_0 =\{(q_1,q_2)\in\mathbb R^2 \colon q_2=q_1^2,\,|q_1|\leqslant1\}\cup \{(q_1,q_2)\in\mathbb R^2 \colon |q_1|\leqslant1,\,q_2=1\}.
\end{equation}
\tag{34}
$$
Note that, since $|u| < 1$, we cannot use the geometric Pontryagin maximum principle (see [9]) or necessary conditions for a trajectory of geometric local infimum (see [1] and [2]) to examine points in the interval $\{(q_1,q_2)\in\mathbb R^2 \colon |q_1|\leqslant1,\,q_2=1\}$. If we assume that $|u|\leqslant1$ in this example, then the fact that the right-hand side of (34) lies in $\partial R_0$ can be established by using necessary conditions for a trajectory of geometric local infimum, To use the geometric maximum principle for this purpose we must show that all points $q\in\partial R_0$ are endpoints of trajectories in $D((0,q))$. Example 2. Consider the following control system: where $g\colon \mathbb R\to\mathbb R$ is a continuous function. Again, let $D$ be the set of admissible trajectories $x(\cdot)=(x_1(\cdot),x_2(\cdot))$ for this system. Then its attainable set has the form Proof. For any two numbers $u_i\in \mathbb R$ of opposite signs, $i=1,2$, there exist ${\alpha_i \in [0,1]}$, $i=1,2$, $\alpha_1+\alpha_2=1$, such that the triple $(0,0), (\alpha_1,\alpha_2), ((u_1,0), (u_2,0))$ is admissible for the system, $x_1(0)=0$ and $x_1(1)=0$.
In the notation presented before (13) this means that all such triples belong to $D_c((0,0))$. Since $0\in \partial R_0$, by (13), for each such triple there exists a nontrivial vector function $p(\cdot)=(p_1(\cdot), p_2(\cdot))$ such that
$$
\begin{equation}
\dot p_1(t)=-p_2(t)(\alpha_1 g(u_1)+\alpha_2 g(u_2)), \qquad \dot p_2(t)=0,
\end{equation}
\tag{36}
$$
$$
\begin{equation}
\max_{u\in \mathbb R,\,v\geqslant0}(p_1(t)u + p_2(t)v)=\langle p(t),0\rangle=0 \quad\text{for a.e. } t\in[0,1].
\end{equation}
\tag{37}
$$
It follows from (37) that $p_1(\cdot)=0$. Hence the constant $p_2$ is distinct form zero. Then the first relation in (36) turns to the equality which holds for all the $u_i\in \mathbb R$ as indicated and the appropriate $\alpha_i \in [0,1]$, $i=1,2$, $\alpha_1+\alpha_2=1$, satisfying the first relation in (35), which shows that $\alpha_1 = u_2/(u_2-u_1)$ and $\alpha_2= -u_1/(u_2-u_1)$. The second relation in (35) always holds.Substituting the expressions for the $\alpha_i$, $i=1,2$, into (38) we obtain $u_2g(u_1)=u_1g(u_2)$ for any numbers $u_1$ and $u_2$ in $\mathbb R$ of opposite signs, or
$$
\begin{equation*}
\frac{g(u)}{u}=\mathrm{const} \quad \forall\, u\in \mathbb R, \quad u\ne0,
\end{equation*}
\notag
$$
that is, for some $a$ we have $g(u)=au$ for $u\in \mathbb R$, $u\ne0$.
It is obvious that the triple $(0,0), (\alpha_1,\alpha_2), ((0,0), (0,0))$ belongs to $D_c((0,0))$, so it follows from (38) for $u_1=u_2=0$ that $g(0)=0$. Hence we also have $g(u)=au$ for $u=0$. Now elementary calculations show that
$$
\begin{equation*}
x_2(1)=\int_0^1(x_1(t)g(\dot x_1(t))+v(t))\,dt=a\frac{x_1^2(1)}2+\int_0^1v(t)\,dt>a\frac{x_1^2(1)}2.
\end{equation*}
\notag
$$
Proposition 2 is proved. § 3. AppendixLet $\mathcal M$ be a topological space. Let $C(\mathcal M,Z)$ denote the space of continuous bounded maps $F\colon \mathcal M\to Z$ with the norm
$$
\begin{equation*}
\|F\|_{C(\mathcal M, Z)}=\sup_{x\in \mathcal M}\|F(x)\|_Z.
\end{equation*}
\notag
$$
We denote the subset of Lipschitz functions in $C([t_0,t_1],\mathbb R^n)$ with Lipschitz constant $L$ by $Q_L$. Approximation Lemma. Let $M$ be a bounded subset of $ C([t_0, t_1],\mathbb R^n)$, $\Omega$ be a bounded subset of $\mathbb R^n$, and let $N\in\mathbb N$, $\widehat{\overline u}(\cdot)=(\widehat u_1(\cdot),\dots,\widehat u_{N}(\cdot))\in \mathcal U^N$ and $L>0$. Then the map $F\colon C([t_0, t_1],\mathbb R^n)\times\mathbb R^n\times(L_\infty([t_0, t_1]))^N\to C([t_0, t_1],\mathbb R^n)$ defined by for $t\in[t_0, t_1]$, where $\overline\alpha=(\alpha_1(\cdot),\dots,\alpha_N(\cdot))$, belongs to $C((M\cap Q_L)\times\Omega\times\mathcal A_N, C([t_0, t_1],\mathbb R^n))$, and for each $\overline\alpha\in\mathcal A_N$ there exists a sequence of piecewise constant controls $u_s(\overline\alpha,\widehat{\overline u})(\cdot)$, $s\in\mathbb N$, such that $u_s(\overline\alpha,\widehat{\overline u})(t)\in U$ for almost all $t\in[t_0,t_1]$, and there exist maps $F_s\colon C([t_0,t_1],\mathbb R^n)\times\mathbb R^n\times\mathcal A_N\to C([t_0,t_1],\mathbb R^n)$, $s\in\mathbb N$, defined by for $t\in[t_0,t_1]$, that also belong to $C((M\cap Q_L)\times\Omega\times\mathcal A_N,C([t_0, t_1],\mathbb R^n))$ and converge to a map $F$ as $s\to\infty$ in this space.If $\widehat u_i(t)\in U$ for almost all $t\in [t_0,t_1]$, $i=1,\dots,N$ (rather than $\widehat u_i(t)\in \operatorname{cl}U$, as follows from the definition of $\mathcal U$), then this lemma is a particular case of a result in our paper [7]. It turns out that this result also remains valid for $\widehat{\overline u}(\cdot)\in \mathcal U^N$; only quite slight modifications of the proof are required. AcknowledgementThe authors are grateful to Yu. L. Sachkov, who proposed an example which inspired us to write this paper. 
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