---
abstract: 'A semi-process is an analog of the semi-flow for non-autonomous differential equations or inclusions. We prove an abstract result on the existence of measurable semi-processes in the situations where there is no uniqueness. Also, we allow solutions to blow up in finite time and then obtain local semi-processes.'
author:
- 'Jorge E. Cardona and Lev Kapitanski'
title: 'Measurable process selection theorem and non-autonomous inclusions'
---

Introduction
============

Let \[aut\] = f(u) be an archetypical autonomous differential equation. Autonomous refers to the structure of the equation and means that the independent variable, $t$, does not appear explicitly (independently) in the equation. Because of that, is invariant under the time shift (translation) $\theta_\tau: t\mapsto t+\tau$, and if $u(\cdot)$ is a solution of , then $\theta_\tau u(\cdot) = u(\cdot + \tau)$ is a solution as well. Suppose describes evolution/dynamics on some set $X$ (which could be a finite- or infinite-dimensional vector space or manifold). Given an $a\in X$, let $u(t, a)$, $t\in [0, +\infty)$ be a solution of starting at $u(0, a) = a$ (let us assume that global solutions exist forward in time). If $v(\cdot, u(t_1, a))$ is a solution of starting (at $t = 0$) from the point $u(t_1, a)$, we can splice $u$ and $v$ and obtain a (possibly new) solution $w = u \underset{t_1}{\bowtie} v$ starting at $a$: w(t) = u v(t) =

u(t, a), & 0tt\_1,\
v(t - t\_1, u(t\_1, a)), & tt\_1.

If solutions are unique (for every $a\in X$ there is a unique solution $u(t, a)$), then $v(t, u(t_1, a)) = u(t + t_1, a)$. In general, in the case of uniqueness, the solutions of enjoy the semigroup property, i.e., for every $a\in X$, $u(0, a) = a$ and \[sg\] u(t\_2, u(t\_1, a)) = u(t\_1 + t\_2, a),t\_1, t\_2 0. This allows us to define the semigroup $U(t): X\to X$ by the formula $U(t)(a) = u(t, a)$.

For non-autonomous differential equations the situation is similar and different. Consider an archetypical non-autonomous equation \[n-aut\] = g(t, u). Now, in addition to the initial position/state $a\in X$, it is important to specify the initial moment of time, $t_0$. The solution(s) will depend on $a$ and $t_0$; we write $u(t; t_0, a)$ to denote a solution of for $t\ge t_0$ that equals $a$ when $t = t_0$. If we follow the solution $u(t; t_0, a)$ until the moment $t = t_1$ and then follow a solution $v(t; t_1, u(t_1; t_0, a))$ that starts at the point $u(t_1; t_0, a)$ at the moment $t_1$, we obtain a spliced solution of , \[splice\] w(t; t\_0, a) = u v(t) =

u(t; t\_0, a), & t\_0tt\_1,\
v(t; t\_1, u(t\_1;, t\_0, a)), & tt\_1.

If the solutions of equation are unique, we have the following analog of the semigroup property: \[n-sg\] u(t\_0; t\_0, a) = a u(t\_2; t\_1, u(t\_1; t\_0, a)) = u(t\_2; t\_0, a),aXt\_2t\_1t\_00. Also, we can define the *transition* map $U(t_1; t_0)$ that maps $X$ into $X$ by assigning to every $a\in X$ the value $u(t_1; t_0, a)$ of the solution $u(t; t_0, a)$. This transition map has the properties

\[NSG:main\] $$\begin{aligned}
& U(t; t) = \hbox{id}_X\,,\;\forall t\ge 0 \label{NSG:a}\\
& U(t_2; t_1)\circ U(t_1; t_0) = U(t_2; t_0)\,,\;\forall t_2\ge t_1\ge t_0\ge 0\,.\label{NSG:b}\end{aligned}$$

A family of maps $U(t_1; t_0)$ with the properties will be called a *process* (see, e.g., [@Dafermos; @Hale]). To the autonomous case correspond homogeneous processes characterized by time invariance: $U(t_1-\tau; t_0 - \tau) = U(t_1; t_0)$ for all (admissible) $\tau$ (and therefore $U(t) = U(t; 0)$ is the semigroup).

It may be advantageous to think of solutions of the autonomous equation as integral curves (trajectories) in $X$, i.e., view solitons as continuous (infinite, one-sided) paths in $X$. If there is no uniqueness, the solutions/integral curves starting at the point $a$ form an integral funnel, $S(a)$, a subset of all paths starting at $a$. (Analysis of integral funnels for ODEs was initiated by H. Kneser in the 1920s, [@Kneser].) Denote by $\Omega$ the space of all continuous (infinite, one-sided) paths in $X$. Then the map $a\mapsto S(a)$ is a set-valued map (other names: multifunction, correspondence) from $X$ into $2^\Omega$. It has the (already mentioned) properties: if the path $w$ is in $S(a)$, then its shift, $\theta_\tau w$ is in $S(w(\tau))$, and if $u\in S(a)$ and $v\in S(u(t_1))$, then $u \underset{t_1}{\bowtie} v\in S(a)$.

An interesting and important question is whether it is possible to select a solution $u(\cdot, a)$ from every funnel $S(a)$ in such a way, that $u(t, a)$ has the semigroup property . In other words, is it possible to define a semigroup (semiflow) $U(t)$ so that, for every $a\in X$, $u(t, a) = U(t)(a)$, $t\ge 0$, is a path in $S(a)$? In [@C; @C-K], we show that the answer is yes under some very general assumptions. Moreover, we show that the selection $a\mapsto u(\cdot, a)\in S(a)$ is measurable. This is a new type of selection theorems (for measurable selection results see, e.g., [@A-B Section 18.3], [@H-P], and the surveys [@Wa; @Io]). Our results were motivated by the Markov selection theorems, see [@Krylov; @S-V; @F-R; @G-R-Z].

Here, we extend the semiflow selection theorem of [@C; @C-K] to non-autonomous equations and processes. If there is no uniqueness for , but the solutions to the initial-value problem exist forward in time, we have the integral funnels $S(t_0, a)$ formed by all the solution $u(t; t_0, a)$, $t\ge t_0$, such that $u(t_0; t_0, a) = a$. The question we ask is whether there exists a measurable selection $(t_0, a) \mapsto u(\cdot; t_0, a)\in S(t_0, a)$ such that $u(t; t_0, a)$ satisfies . The answer again is yes under the right assumptions. After our previous work [@C; @C-K], this answer is not surprising if we are concerned with solutions of the non-autonomous equation : one could replace with the equivalent autonomous system \[n-aut-2\] = g(t, u), = 1. Then, if we apply our results from [@C; @C-K] and obtain a semigroup ${\tilde U}(s)$, $s\ge 0$, on $X\times [0, +\infty)$ corresponding to , the maps $U(t_1, t_0)(a) = {\tilde U}(t_1)(a, t_0)$ would form a process corresponding to . However, if we are in a more general setting and deal with integral funnels, their *autonomisation* is not clear. Thus, in the next section we present the precise statement and the proof of the existence of a measurable process. After that, in Section \[loc-proc\] we study the existence of *local processes*. These apply to the situations, where in addition to non-uniqueness we allow solutions to blow up in finite time. \[In the 1960–70s there was some interest in abstracting the dynamical and semi-dynamical systems and processes, see, e.g., [@Hajek-1; @Hajek-2; @B-H]. Our approach involves funnels and is different.\] An example on the semi-process selection is presented in Section \[sec:ex\].

A different and important point of view on non-autonomous dynamics involves the skew-product construction, see, e.g., [@S-S; @S]. The skew-product set-up is very convenient for the study of the long term behavior of non-autonomous systems. However, for the measurable processes selection, we believe our direct approach is more natural. We should mention that our abstract results apply not only to non-autonomous ordinary differential equations, but to partial differential equations, to differential and difference inclusions, and to other situations where integral funnels make sense.

Global processes {#glob-proc}
================

Let $X$ be a separable complete metric space with metric $\rho$, which we assume to be bounded: $\rho(x, y)\le 1$ for all $x, y\in X$. Denote by ${\cal B}_X$ the Borel $\sigma$-algebra of $X$. Let $\Omega$ be the space of all continuous infinite one-side paths in $X$ equipped with the compact-open topology. The elements of $\Omega$ are continuous maps $u: [0, +\infty)\to X$, and convergence in $\Omega$ is the uniform convergence on every finite time interval. The space $\Omega$ is Polish; we fix a complete bounded metric on $\Omega$ by setting \[met-Om\] d(u, v) = \_[= 1]{}\^2\^[-]{}\_[t]{}(u(t), v(t))\^[-1]{}. Sometimes it is convenient to view the paths parametrized by $s\in [\tau, +\infty)$. We denote by $\Omega^\tau$ the space of continuous maps from $[\tau, +\infty)$ into $X$. It can be identified with the image of $\Omega$ under the (past erasing) map $\sigma_\tau$: if $u(t)$, $t\ge 0$, is a path in $\Omega$, then the path $\sigma_\tau u : [\tau, +\infty)\to X$ is defined as $\sigma_\tau u(t) = u(t)$ for $t\ge \tau$. On the other hand, $\Omega^\tau$ can be viewed as a subset of $\Omega$ if we extend the paths $v(t)$ in $\Omega^\tau$ to $[0, \tau)$ as staying at $v(\tau)$: $$\Game_\tau : \Omega^\tau \to \Omega\,,\quad (\Game_\tau v)(t) = \begin{cases} 
v(\tau) & \text{when} 0\le t < \tau\,, \\ 
v(t) & \text{when} t \ge \tau\,. 
\end{cases}$$ We will also use the notation $\Omega^\tau_a$ for all the paths in $\Omega^\tau$ staring at the point $a$, i.e., $v(\tau) = a$ if $v\in \Omega^\tau_a$. On occasion, it will be convenient to specify $\tau$ and $a$ in the notation for a path, e.g., $v(t; \tau, a)$.

The integral funnels $S(t_0, a)$ will be subsets of the set $\Omega^{t_0}_a$. Denote by $P_{cl}[\Omega]$ the space of all (bounded) closed subsets of $\Omega$ endowed with the Vietoris topology (= exponential topology), see [@Kuratowski], [@B], and [@A-B] for details on set-valued maps, their properties and, in particular, measurability. In this presentation, all set-valued maps from $X$ to $\Omega$ will have non-empty closed values and will be viewed as maps from the measurable space $(X, {\cal B}_X)$ to $P_{cl}[\Omega]$ with the Vietoris topology. If $\Gamma$ is such a map and $A\subset \Omega$, define $$\Gamma^{-}(A) = \{x\in X : \Gamma(x)\cap A \neq\emptyset\}\,.$$ For historical reasons, there are several confusingly similar notions of measurability of set-valued maps. A map $\Gamma: X\to P_{cl}[\Omega]$ is *weakly measurable* if $\Gamma^{-}(G)\in {\cal B}_X$ for every open set $G\subset\Omega$. $\Gamma$ is *measurable* if $\Gamma^{-}(F)\in {\cal B}_X$ for every closed set $F\subset\Omega$. Since $\Omega$ is metric, if $\Gamma$ is measurable, it is weakly measurable, [@A-B Lemma 18.2]. Since $\Omega$ is in addition separable, if $\Gamma$ is *compact-valued* and weakly measurable, it is measurable, [@A-B Theorem 18.10]. Thus, in our setting, for compact-valued $\Gamma: X\to P_{cl}[\Omega]$, there is no difference between weak measurability and measurability.

The fundamental result of Kuratowski and Ryll-Nardzewski, [@K-R Theorem 1, p. 398], implies that if $\Gamma: X\to P_{cl}[\Omega]$ is weakly measurable, then $\Gamma$ has a measurable selection, i.e., there exists a single-valued map $\gamma : X\to \Omega$ such that $\gamma(x)\in \Gamma(x)$ for all $x\in X$ and $\gamma$ is $({\cal B}_X, {\cal B}_\Omega)$-measurable: for every Borel set $A\subset \Omega$, $\gamma^{-1}(A)$ is a Borel set in $X$.

We introduce now abstract integral funnels $S(t_0, a)$ that have the properties prompted by the properties of integral funnels of solutions of equation .

$S(t_0, a)$, where $t_0\in [0, +\infty)$ and $a\in X$, is a family of abstract integral funnels on the space $X$ if, for every $t_0\ge 0$, $S(t_0, \cdot) : X\to P_{cl}[\Omega]$ is a set-valued map with the following properties.

1.  For every $a\in X$, $S(t_0, a)$ is a non-empty compact subset of $\Omega^{t_0}_a$. Every path $u$ in $S(t_0, a)$ is parametrized as $u(t; t_0, a)$, where $t\ge t_0$.

2.  Each map $S(t_0, \cdot)$ is measurable, i.e., for every closed set $C\subset \Omega^{t_0}$, $$\{x\in X: S(t_0, x)\cap C \neq \emptyset\}\in {\cal B}_X\,.$$

3.  If $u\in S(t_0, a)$, then $\sigma_\tau u \in S(t_0+\tau, u(t_0+\tau))$.

4.  If $u\in S(t_0, a)$ and $v\in S(t_0+\tau, u(t_0+\tau))$, then the spliced path $w = u \underset{t_0+\tau}{\bowtie} v$, defined as in , belongs to the funnel $S(t_0, a)$.

\[thm:main\] Every family of abstract integral funnels $S(t_0, a)$, $t_0\ge 0$, $a\in X$, has, for every $t_0$, a measurable selection $ a\to u(\cdot; t_0, a)\in S(t_0, a)$ with the semigroup property . As a corollary, there is a Borel measurable (semi)process $U(t_1, t_0) : X\to X$ whose orbits are $U(t_1, t_0)(a) = u(t_l; t_0, a)$, for all $t_1 \ge t_0\ge 0$ and for all $a\in X$.

We modify our proof for the autonomous case from [@C-K]. The plan is to successively reduct each funnel $S(t_0, a)$ while preserving the properties [**S1 - S4**]{} and so that the limiting funnel would contain just one path; property [**S3**]{} for the limiting funnel then spells . The reduction of the funnels is based on an idea from optimization theory that was used by N. V. Krylov in his proof of the Markov selection theorem, [@Krylov].

Let $\vp: X\to [0, 1]$ be a continuous function and let $\lambda$ be a positive real number. For $t_0 \ge 0$ and $a\in X$, define the functional $\zeta$ on $S(t_0, a)$ via the formula \[zeta\] (w) = \_0\^e\^[-t]{} (w(t\_0 + t))dt. This is a continuous functional and it attains its maximum on the compact set $S(t_0, a)$. Denote this maximum by $m_\zeta(t_0, a)$, \[m\] m\_(t\_0, a) = \_[wS(t\_0, a)]{} (w) Define \[V\] V\_= {vS(t\_0, a): (v) = m\_(t\_0, a)}. By the so-called measurable maximum theorem, [@A-B Theorem 18.19], $V_\zeta[S(t_0, a)]$ is a non-empty compact subset of $S(t_0, a)$, and the set-valued map $a \mapsto V_\zeta[S(t_0, a)]$ is measurable. Thus, the family of sets $V_\zeta[S(t_0, a)]$ has properties [**S1**]{} and [**S2**]{} of the abstract funnels. Let us check that it has the remaining two properties [**S3**]{} and [**S4**]{}. Suppose $u\in V_\zeta[S(t_0, a)]$ and consider the shifted path $\sigma_\tau u$. By property [**S3**]{} of the family $S$, $\sigma_\tau u\in S(t_0+\tau, u(t_0 + \tau))$. We have to show that $\sigma_\tau u$ maximizes $\zeta$ in the set $S(t_0+\tau, u(t_0 + \tau))$. Pick any path $v$ in $S(t_0+\tau, u(t_0 + \tau))$ and consider the spliced path $w = u \underset{t_0+\tau}{\bowtie} v$, which, by property [**S4**]{}, belongs to the funnel $S(t_0, a)$. Since $u$ maximizes $\zeta$ over $S(t_0, a)$, $\zeta(u) \ge \zeta(w)$, i.e., $$\int_0^\infty e^{-\lambda t} \vp(u(t_0 + t))\,dt \ge \int_0^\infty e^{-\lambda t} \vp(w(t_0 + t))\,dt\,.$$ But $$\begin{aligned}
& \int_0^\infty e^{-\lambda t} \vp(w(t_0 + t))\,dt = \int_0^\tau e^{-\lambda t} \vp(u(t_0 + t))\,dt + 
\int_\tau^\infty e^{-\lambda t} \vp(v(t_0 + t))\,dt = \\ 
& \int_0^\tau e^{-\lambda t} \vp(u(t_0 + t))\,dt + e^{-\lambda \tau}\,\int_0^\infty e^{-\lambda t} \vp(v(t_0 + \tau + t))\,dt\,,
\end{aligned}$$ while $$\int_0^\infty e^{-\lambda t} \vp(u(t_0 + t))\,dt = \int_0^\tau e^{-\lambda t} \vp(u(t_0 + t))\,dt + e^{-\lambda \tau}\,\int_0^\infty e^{-\lambda t} \vp(u(t_0 + \tau + t))\,dt\,.$$ Hence, $$\int_0^\infty e^{-\lambda t} \vp(u(t_0 + \tau + t))\,dt \ge \int_0^\infty e^{-\lambda t} \vp(v(t_0 + \tau + t))\,dt\,,$$ which means $\sigma_\tau u$ is a maximizer in $S(t_0+\tau, u(t_0 + \tau))$. To check property [**S4**]{} for $V_\zeta[S]$, pick $u\in V_\zeta[S(t_0, a)]$ and $v\in V_\zeta[S(t_0+\tau, u(t_0+\tau))]$ and consider the spliced path $w = u \underset{t_0+\tau}{\bowtie} v$. We have to show that $w$ maximizes $\zeta$ over $S(t_0, a)$ and hence belongs to $V_\zeta[S(t_0, a)]$. This follows from a simple calculation that takes into account what we have just shown, that $\zeta(\sigma_\tau u) = m_\zeta(t_0 + \tau, u(t_0 + \tau))$. If $v\in V_\zeta[S(t_0+\tau, u(t_0+\tau))]$, then $\zeta(v) = m_\zeta(t_0 + \tau, u(t_0 + \tau))$ as well. Thus, $$\begin{aligned}
& \int_0^\infty e^{-\lambda t} \vp(w(t_0 + t))\,dt = \int_0^\tau e^{-\lambda t} \vp(u(t_0 + t))\,dt + 
\int_\tau^\infty e^{-\lambda t} \vp(v(t_0 + t))\,dt = \\ 
& \int_0^\tau e^{-\lambda t} \vp(u(t_0 + t))\,dt + e^{-\lambda \tau} \zeta(v) = 
\int_0^\tau e^{-\lambda t} \vp(u(t_0 + t))\,dt + e^{-\lambda \tau} \zeta(\sigma_\tau u) = \zeta(u)\,,
\end{aligned}$$ i.e., $\zeta(w) = m_\zeta(t_0, a)$, i.e., $w\in V_\zeta[S(t_0, a)]$. To summarize, for any functional $\zeta$ of the form , $V_\zeta[S(t_0, a)]$ is a family of abstract integral funnels.

Now, choose a countable family $\Phi$ of continuous functions $\vp : X\to [0, 1]$ that separates the points of $X$, see [@B-K]. Choose some enumeration $(\lambda_n, \vp_n)$ of the countable set of pairs $(\lambda, \vp)$, where $\lambda$ runs through positive rational numbers and $\vp$ runs through $\Phi$. To each pair $(\lambda_n, \vp_n)$ corresponds the functional $\zeta_n$ via . Define recursively the shrinking families of abstract integral funnels $$S^0(t_0, a) = S(t_0, a), \quad S^{n}(t_0, a) = V_{\zeta_n}[S^{n-1}(t_0, a)], n = 1, 2, \dots\,.$$ For each $(t_0, a)$, $S^{n}(t_0, a)$ is a sequence of nested compacta in $\Omega^{t_0}$. The intersection, $$S^\infty(t_0, a) = \bigcap_{n = 0}^\infty S^{n}(t_0, a)\,,$$ is not empty and compact. In fact, $S^\infty(t_0, a)$ is an abstract family of integral funnels. Indeed, it is easy to see that properties [**S1**]{}, [**S3**]{}, and [**S4**]{}, are satisfied. As the intersection of compact-valued maps into a Polish space, $S^\infty(t_0, \cdot)$ is measurable by [@A-B Lemma 18.4].

It turns out that each funnel $S^\infty(t_0, a)$ is a singleton. Indeed, if $u, v\in S^\infty(t_0, a)$, then, for every $\vp\in \Phi$, $$\int_0^\infty e^{-\lambda t} \vp(u(t_0 + t))\,dt = \int_0^\infty e^{-\lambda t} \vp(v(t_0 + t))\,dt\,,\quad \forall \lambda\in {{\mathbb{Q}}}_+\,.$$ By the uniqueness of the Laplace transform, $\vp(u(t_0 + t)) = \vp(v(t_0 + t))$ for all $t\ge 0$. Because this is true for every $\vp\in\Phi$ and the family $\Phi$ separates the points of $X$, we obtain $u(t_0 + t) = v(t_0 + t)$ for all $t\ge 0$, i.e., $u = v$ as paths.

For $u\in S^\infty(t_0, a)$, we will use the notation $u(t; t_0, a)$, where $t\ge t_0$. If $t_1 > t_0$, then $(\sigma_{t_1 - t_0} u)(t) = u(t; t_1, u(t_1; t_0, a))$ for $t\ge t_1$, by property [**S3**]{}. Thus, if $t_2 > t_1$, $(\sigma_{t_1 - t_0} u)(t_2) = u(t_2; t_1, u(t_1; t_0, a))$. On the other hand, $(\sigma_{t_1 - t_0} u)(t_2) = u(t_2; t_0, a)$ by the definition of the shift operator $\sigma_\tau$. This establishes the semigroup property of the measurable selection $u(t; t_0, a)$.

Once the (measurable) selection $u$ is found, we define the process $U(t_1, t_0): X\to X$ for $t_1\ge t_0\ge 0$, by the formula $U(t_1, t_0)(a) = u(t_1; t_0, a)$. As we have shown, the map $a\mapsto \{u(\cdot; t_0, a)\}$ from $X$ to $2^\Omega$ is measurable and singleton-valued. By the Kuratowski-Ryll-Nardzewski selection theorem the map $a\mapsto u(\cdot; t_0, a)$ from $X$ to $\Omega^{t_0}$ is $({\cal B}_X,{\cal B}_\Omega)$-measurable. The map $U(t_1, t_0): X\to X$ is the composition of the map $a\mapsto u(\cdot; t_0, a)$ and the evaluation map $\pi_{t_1} : \Omega^{t_0}\owns w\to \pi_{t_1}(w) = w(t_1)\in X$, which is continuous. Consequently, $U(t_1, t_0)$ is Borel measurable. This completes the proof.

Local processes {#loc-proc}
===============

Let $X$ be a separable complete metric space as in the previous section. The integral funnels $S(t_0, a)$ will now be local. This means that to every initial state $a\in X$ and every initial moment $t_0$ corresponds a strictly positive number $T(t_0, a)$, the *terminal time*. The paths in the funnel $S(t_0, a)$ form a subset in $C([t_0, t_0 + T(t_0, a))\to X)$.

\[loc-fun\] A family $S(t_0, a)$, $t_0\in [0, +\infty)$, $a\in X$, will be called a family of abstract local integral funnels with terminal times $T(t_0, a)$ if they satisfy the following conditions.

1.  $T(t_0, a)$ is a lower semi-continuous function on $[0, +\infty)\times X$, i.e., if $(t_n, a_n)\to (t_0, a)$, then $T(t_0, a) \le \liminf T(t_n, a_n)$.

2.  Every set $S(t_0, a)$ is a non-empty compact in the space $C([t_0, t_0 + T(t_0, a))\to X)$ with the topology of uniform convergence on every closed subinterval $[\alpha, \beta]$ of $[t_0, t_0 + T(t_0, a))$. Every path $w(\cdot; t_0, a)\in S(t_0, a)$ is a continuous map from $[t_0, t_0 + T(t_0, a))$ into $X$, and $w(t_0; t_0, a) = a$.

3.  For every $t_0\ge 0$, the set-valued map $a\mapsto S(t_0, a)$ is measurable in the following sense. Each path $w$ in $S(t_0, a)$ can be re-parametrized as ${\tilde w}(s) = w(t_0 + s (T(t_0, a) - t_0))$ and then viewed as an element of the space ${\tilde \Omega} = C([0, 1)\to X)$. Denote by ${\tilde S}(t_0, a)$ the set $S(t_0, a)$ after such re-parametrization. We say that the map $a\mapsto S(t_0, a)$ is measurable if, for any closed subset $F$ of ${\tilde \Omega}$ (with the compact-open topology), $$\{a\in X: {\tilde S}(t_0, a) \cap F \neq \emptyset\}\in {\cal B}_X\,.$$

4.  If $u\in S(t_0, a)$ and $\tau < T(t_0, a)$, then $T(t_0 + \tau, u(t_0 + \tau; t_0, a)) \ge 
    T(t_0, a) - \tau$ and $\sigma_\tau u \in S(t_0+\tau, u(t_0+\tau; t_0, a))$.

5.  If $u\in S(t_0, a)$, $\tau < T(t_0, a)$, and $v\in S(t_0+\tau, u(t_0+\tau; t_0, a))$, then the spliced path $w = u \underset{t_0+\tau}{\bowtie} v$, defined by analogy with , belongs to the funnel $S(t_0, a)$.

\[thm:2\] Every family of local abstract integral funnels $S(t_0, a)$, $t_0\in [0, +\infty)$, $a\in X$, with terminal times $T(t_0, a)$, has a selection ${\frak u}(\cdot; t_0, a)$ with the following properties.

1.  For every $t_0\ge 0$, the map $ X \owns a \mapsto {\frak u}(\cdot; t_0, a)\in C([t_0, t_0 + T(t_0, a))\to X)$ is measurable.

2.  ${\frak u}(t_0; t_0, a) = a$,

3.  ${\frak u}(t_2; t_1, {\frak u}(t_1; t_0, a)) = {\frak u}(t_2; t_0, a)$ for all $t_1\in [t_0, t_0 + T(t_0, a))$ and $t_2\in [t_1, t_0 + T(t_0, a))$.

We mimic the proof of Theorem \[thm:main\] with a few modifications. Let $\vp: X\to [0, 1]$ be a continuous function and let $\lambda$ be a positive real number. For $t_0 \ge 0$ and $a\in X$, define the functional $\zeta$ on $S(t_0, a)$ via the formula \[zeta-2\] (w) = \_0\^[T(t\_0, a)]{} e\^[-t]{} (w(t\_0 + t))dt. This is a continuous functional and it attains its maximum on the compact set $S(t_0, a)$. Denote this maximum by $m_\zeta(t_0, a)$, \[m\] m\_(t\_0, a) = \_[wS(t\_0, a)]{} (w) and define \[V\] V\_= {vS(t\_0, a): (v) = m\_(t\_0, a)}. The way we treat measurability by re-parametrizing the paths (see property [**LS2**]{}), allows us to apply the measurable maximum theorem, [@A-B Theorem 18.19], and conclude that $V_\zeta[S(t_0, a)]$ is a non-empty compact subset of $S(t_0, a)$, and the set-valued map $a \mapsto V_\zeta[S(t_0, a)]$ is measurable. This shows that the family of sets $V_\zeta[S(t_0, a)]$ has properties [**LS1**]{} and [**LS2**]{}. Suppose $u(\cdot; t_0, a)\in V_\zeta[S(t_0, a)]$ and consider the shifted path $\sigma_\tau u$. By property [**LS3**]{}, $\sigma_\tau u\in S(t_0+\tau, u(t_0 + \tau; t_0, a))$. Let us show that $\sigma_\tau u$ maximizes $\zeta$ in the set $S(t_0+\tau, u(t_0 + \tau; t_0, a))$. Pick any path $v$ in $S(t_0+\tau, u(t_0 + \tau; t_0, a))$ and consider the spliced path $w = u \underset{t_0+\tau}{\bowtie} v$, which, by property [**LS4**]{}, belongs to the funnel $S(t_0, a)$. Since $u$ maximizes $\zeta$ over $S(t_0, a)$, $\zeta(u) \ge \zeta(w)$, i.e., $$\int_0^{T(t_0, a)} e^{-\lambda t} \vp(u(t_0 + t; t_0, a))\,dt \ge \int_0^{T(t_0, a)} e^{-\lambda t} \vp(w(t_0 + t; t_0, a))\,dt\,.$$ Now compute $$\begin{aligned}
& \int_0^{T(t_0, a)} e^{-\lambda t} \vp(w(t_0 + t; t_0, a))\,dt = \\ 
& \int_0^\tau e^{-\lambda t} \vp(u(t_0 + t; t_0, a))\,dt + 
\int_\tau^{T(t_0, a)} e^{-\lambda t} \vp(v(t_0 + t; t_0 + \tau, u(t_0 + \tau; t_0, a))\,dt = \\ 
& \int_0^\tau e^{-\lambda t} \vp(u(t_0 + t; t_0, a))\,dt + e^{-\lambda \tau}\,\int_0^{T(t_0, a) 
- \tau} e^{-\lambda t} \vp(v(t_0 + \tau + t; t_0 + \tau, u(t_0 + \tau; t_0, a) ))\,dt\,,
\end{aligned}$$ and $$\begin{aligned}
& \int_0^{T(t_0, a)} e^{-\lambda t} \vp(u(t_0 + t; t_0, a))\,dt = \\ 
& \int_0^\tau e^{-\lambda t} \vp(u(t_0 + t; t_0, a))\,dt + e^{-\lambda \tau}\,\int_0^{T(t_0, a) - \tau} e^{-\lambda t} \vp(u(t_0 + \tau + t; t_0, a))\,dt\,.
\end{aligned}$$ Hence, $$\begin{aligned}
& \zeta(\sigma_\tau u) = \int_0^{T(t_0, a) - \tau} e^{-\lambda t} \vp(u(t_0 + \tau + t; t_0, a))\,dt \ge 
 \\ 
& \int_0^{T(t_0, a) 
- \tau} e^{-\lambda t} \vp(v(t_0 + \tau + t; t_0 + \tau, u(t_0 + \tau; t_0, a) ))\,dt  = \zeta(v),
\end{aligned}$$ which means $\sigma_\tau u$ is a maximizer in $S(t_0+\tau, u(t_0 + \tau; t_0, a))$. With similar modifications, following the proof of Theorem \[thm:main\], one verifies property [**LS4**]{} for $V_\zeta[S]$. Thus, for any functional $\zeta$ of the form , $V_\zeta[S(t_0, a)]$ is a family of abstract local integral funnels.

Choose a countable family $\Phi$ of continuous functions $\vp : X\to [0, 1]$ that separates the points of $X$, and choose some enumeration $(\lambda_n, \vp_n)$ of the countable set of pairs $(\lambda, \vp)$, where $\lambda$ runs through positive rational numbers and $\vp$ runs through $\Phi$. To each pair $(\lambda_n, \vp_n)$ corresponds the functional $\zeta_n$ via . Define recursively the shrinking families of abstract integral funnels $S^0(t_0, a) = S(t_0, a)$, and $S^{n}(t_0, a) = V_{\zeta_n}[S^{n-1}(t_0, a)], n = 1, 2, \dots $. Again, the intersection, $$S^\infty(t_0, a) = \bigcap_{n = 0}^\infty S^{n}(t_0, a)\,,$$ is an abstract family of local integral funnels. To show that each funnel $S^\infty(t_0, a)$ is a singleton, assume $u, v\in S^\infty(t_0, a)$ Then, for every $\vp\in \Phi$, $$\int_0^{T(t_0, a)}  e^{-\lambda t} \vp(u(t_0 + t; t_0, a))\,dt = \int_0^{T(t_0, a)} e^{-\lambda t} \vp(v(t_0 + t; t_0, a))\,dt\,,\quad \forall \lambda\in {{\mathbb{Q}}}_+\,.$$ By the uniqueness of the Laplace transform, $\vp(u(t_0 + t; t_0, a)) = \vp(v(t_0 + t; t_0, a))$ for all $t\in [0, T(t_0, a))$. Because this is true for every $\vp\in\Phi$ and the family $\Phi$ separates the points of $X$, we obtain $u(t_0 + t; t_0, a) = v(t_0 + t; t_0, a)$ for all $t\in [0, T(t_0, a))$, i.e., $u = v$ as paths.

For the unique path in the funnel $S^\infty(t_0, a)$, let us use the notation ${\frak u}(t; t_0, a)$. If $t_0 < t_1 < T(t_0, a)$, then $(\sigma_{t_1 - t_0} {\frak u})(t) = {\frak u}(t; t_1, {\frak u}(t_1; t_0, a))$ for $t_1\le t < T(t_0, a)$, by property [**LS3**]{}. Thus, if $t_2 > t_1$, $(\sigma_{t_1 - t_0} {\frak u})(t_2) = {\frak u}(t_2; t_1, {\frak u}(t_1; t_0, a))$. On the other hand, $(\sigma_{t_1 - t_0} {\frak u})(t_2) = {\frak u}(t_2; t_0, a)$ by the definition of the shift operator $\sigma_\tau$. This establishes the semigroup property of the measurable selection ${\frak u}(t; t_0, a)$. The fact that the maps $X\owns a \mapsto {\frak u}(\cdot; t_0, a)\in C([t_0, T(t_0, a))\to X$ are measurable is established by an argument with the distance function similar to the one in the end of the proof of Theorem \[thm:main\]. This completes the proof.

There are differential equations for which not all initial conditions $x(t_0)=x_0$ are possible: there is a proper subset $C\subset (-\infty, +\infty)\times X$ of allowed initial conditions. This is the case, i.e., for the Clairaut equations. We discuss a particular, illustrative example in the next section. However, it is not hard to see that Theorems \[thm:main\] and \[thm:2\] can be adapted for such situations.

We can introduce the local process maps $U(t_1, t_0)$ as in the previous section: $U(t_1, t_0)(a) = {\frak u}(t_1; t_0, a)$. However, now $U(t_1, t_0)$ may not be defined on all of $X$ and, for every $a$, the range of admissible $t_1$ will be different. If we restrict $t_0$ to a compact set $[\alpha, \beta]\subset [0, +\infty)$ and $a$ to a compact set $K\subset X$, then, thanks to assumption [**TT**]{}, the infimum of $ T(t_0, a)$ over $[\alpha, \beta]\times K$ is strictly positive, and this gives a non-trivial admissible interval for $t_1$ when $(t_0, a)\in [\alpha, \beta]\times K$. At the moment we do not see any benefits in constructing an abstract theory of such local (semi)-processes.

Example {#sec:ex}
=======

The Clairaut equation is a scalar ODE of the form \[eq:cl-gen\] x = t[x]{} + ([x]{}) with some function $\psi$. The standard method of solution is as follows. Differentiate , $${\dot x} = {\dot x} + t\,{\ddot x}  + \psi^\prime({\dot x})\,{\ddot x}\,,$$ and simplify the result to obtain $${\ddot x}\,\left(t + \psi^\prime({\dot x})\right) = 0\,.$$ This offers two possibilities: \[eq:ddot\] [x]{} = 0 and/or \[eq:other\] t + \^([x]{}) = 0. Solve each of the equations with the initial condition $x(t_0) = x_0$. The first equation implies \[sol:ddot\] x(t) = x\_0 + c(t - t\_0) with a constant $c$. Substitute this into the original equation : $$x_0 + c\,(t - t_0) = t\,c + \psi(c)\,.$$ This yields the equation for the possible value(s) of $c$: \[sol:c\] x\_0 = c t\_0 + (c). Assuming $c(t_0, x_0)$ is a solution of (there may be many solutions), we can re-write the solution as follows: \[sol:ddot-2\] x(t) = (c(t\_0, x\_0)) + c(t\_0, x\_0)t. The second equation requires some kind of invertibility of the function $\psi^\prime$ (the simplest case is when $\psi^{\prime\prime} \neq 0$, i.e., $\psi$ is convex or concave). Assuming the inverse function $(\psi^\prime)^{-1}$ makes sense, we obtain \[eq:other-2\] [x]{} = (\^)\^[-1]{}(-t). After integration, \[sol:other\] x(t) = x\_0 + \_[t\_0]{}\^t (\^)\^[-1]{}(-s)ds. Substitute this expression into : \[eq:temp-11\] x\_0 + \_[t\_0]{}\^t (\^)\^[-1]{}(-s)ds = t (\^)\^[-1]{}(-t) + ((\^)\^[-1]{}(-t)) Notice that $$\frac{d\hfil}{ds} \left[s \,(\psi^\prime)^{-1}(-s) + \psi\left((\psi^\prime)^{-1}(-s)\right)\right] = 
(\psi^\prime)^{-1}(-s)$$ Thus, equation can be simplified to \[eq:restr-2\] x\_0 - = 0 It is useful to introduce the following version of the Legendre transform of the function $\psi$: \[Leg\] (t\_\*) = \_t. Then equation can be written in the form \[eq:restr-21\] x\_0 = (t\_0), and the solution takes the form $$x(t) = x_0 + {\tilde \psi}(t) - {\tilde \psi}(t_0)\,.$$ In view of , we obtain \[sol:other-2\] x(t) = (t). This is the so-called singular solution. We see that the singular solution corresponds to the value of $c = c(t_0, x_0)$ in that minimizes $[c\cdot t + \psi(c)]$.

Consider the special case $\psi(s) = s^2$, i.e., consider the equation \[eq:square\] x = t [x]{} + ([x]{})\^2. The corresponding equation has two solutions for $c$, \[c\_0\] c\_(t\_0, x\_0) = - , provided $x_0 + {t_0^2}/{4} > 0$, has one solution when $x_0 + {t_0^2}/{4} = 0$, and has no solutions when $x_0 + {t_0^2}/{4} < 0$. Since ${\tilde \psi}(t_*) = - t_*^2/4$, the singular solution is \[sol:sing\] x(t) = - . Thus, the region $C\subset (-\infty,+\infty)\times {{\mathbb{R}}}$ of the allowed initial data $(t_0, x_0)$ is the parabola $x = - t^2/4$ and the points above it: $$C = \{(t, x): x \ge - \frac{t^2}{4}\}\,.$$ The parabola is the envelope of the straight lines or, equivalently, : \[line\] x(t) = c\^2 + ct. Through every point $(t_0, x_0)$ in the interior of $C$ pass two straight lines (solutions) corresponding to $c_+(t_0, x_0)$ and $c_-(t_0, x_0)$. It is useful to notice that one of the lines touches the parabola in the past (at some moment $t = {\mathfrak t}_p(t_0, x_0) < t_0$) and the other touches the parabola in the future (at some moment $t = {\mathfrak t}_f(t_0, x_0) > t_0$). Let us denote these lines (solutions) ${\mathfrak x}_p(t; t_0, x_0)$ and ${\mathfrak x}_f(t; t_0, x_0)$, respectively. They are defined for all $t\in{{\mathbb{R}}}$. The integral funnel $S(t_0, x_0)$ contains the rays ${\mathfrak x}_p(t; t_0, x_0)$ and ${\mathfrak x}_f(t; t_0, x_0)$ for $t\ge t_0$. In addition, it contains infinitely many solutions that branch off ${\mathfrak x}_f(t; t_0, x_0)$: once the line touches the parabola at $t = {\mathfrak t}_f(t_0, x_0)$, the trajectory may continue along the parabola forever, or, at any time $r > {\mathfrak t}_f(t_0, x_0)$ it may take off along the tangent line.

To every point $(t_0, x_0)$ on the boundary of $C$ (on the parabola) correspond infinitely many solutions of that can be divided into three categories: 1) there is the singular solution ; 2) there is a ray tangent to the parabola: \[tan\] [x]{}\_[pf]{}(t; t\_0, - ) = - t + ,tt\_0; 3) there is a one-parameter family of solutions that follow the parabola for some time and then get off on the tangent line: \[branch\] x\_r(t) =

- & t\_0t r\
- t + & tr.

All these solutions form the funnel $S(t_0, x_0)$ when $(t_0, x_0)$ is on the parabola.

It is not hard to see that the trajectories that stay on the parabola for some time and then leave it cannot be part of the semi-process. Thus, there are three choices of a semi-process corresponding to equation . The first choice is completely determined by the initial conditions on the parabola: for $(t_*, x_*)\in\partial C$, (t; t\_\*, x\_\*) = - t + tt\_\*. When the point $(t_*, x_*)$ slides along the parabola clockwise, the corresponding rays ${\mathfrak x}_{pf}(t; t_*, x_*) = {\mathfrak u}(t; t_*, x_*)$ sweep the interior of $C$. For every point $(t_0, x_0)$ in the interior of $C$, there is a unique point $(t_*, x_*)$ (with $t_* = {\mathfrak t}_p(t_0, x_0)$) on the parabola such that ${\mathfrak u}(t_0; t_*, x_*) = x_0$. As a consequence, ${\mathfrak u}(t; t_0, x_0) = {\mathfrak u}(t; t_*, x_*)$ describes the ray-solution starting at $(t_0, x_0)$.

The second choice of the semi-process coincides with the first one on the interior of $C$, while on the parabola we pick the singular solution: ${\mathfrak u}(t; t_0, - t_0^2/4) = - t^2/4$.

The third choice is to select the trajectories (rays) that touch the parabola in the future and then continue along the parabola. In other words, (t; t\_0, x\_0) = - x\_0 = - , and, if $(t_0, x_0)$ is in the interior of $C$, then (t; t\_0, x\_0) =

[x]{}\_f(t; t\_0, x\_0) & t\_0t\_f(t\_0, x\_0),\
- & t &gt; [t]{}\_f(t\_0, x\_0).

All three semi-processes are Borel measurable. Depending on the choice of a countable family $\Phi$ of continuous bounded functions that separate the points of $X = {{\mathbb{R}}}$, the maximization procedure described in the proof of Theorem \[thm:main\] will pick out one of the three semi-processes.

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