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On the Hamilton–Jacobi theory for nonsmooth variational problems A. D. Ioffe Technion – Israel Institute of Technology, Haifa, Israel
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    § 1. IntroductionThe problem we consider is stated as the classical Bolza problem: minimize
$$
\begin{equation}
J(x(\,{\cdot}\,))= \int_0^T L(t,x(t),\dot x(t))\,dt + g(x(T))
\end{equation}
\tag{1}
$$
but under substantially more general assumptions on $g$ and $L$. It is an easy matter to see that, under these assumptions, the problem includes many optimal control problems (see, for example, [7]). Apart from the simplicity of formulation, this is the main factor that makes the problem quite attractive. The term ‘generalized Bolza problem’ was introduced by Clarke [8], although the problem of the minimization of $J$ with the integrand $L$ convex in $(x,y)$, but otherwise under assumptions substantially weaker than in the classical calculus of variations was previously considered by Rockafellar [12]. As follows from the title, our work aims at extending the Hamilton–Jacobi theory to such problems under fairly general assumptions on $L$ and $g$. Specifically, we assume that
In the course of further discussions we can add more assumptions on $L$. We must observe that the generalization of the Hamilton–Jacobi theory to a certain class of optimal control problems was initiated by Vinter (see [15] and [1]). A key assumption in those works was that the set of feasible controls was bounded. This excludes possible applications of results to problems in the calculus of variations. It seems that assumptions similar to ours were for the first time made by Rockafellar [13] and Clarke [8] (although a similar convex problem was considered by Rockafellar a few years earlier [12]). The first of these papers was basically devoted to the existence of solutions, while the second — to necessary conditions for a minimum. The main result in [13] was a theorem on the lower semicontinuity of $J$ in the weak topology of $W^{1,1}$. This is a key property in our paper too. It must also be said that sufficient conditions for lower semicontinuity of the functional were first obtained by Olech [10] in 1976. Necessary and sufficient conditions for this property were obtained by this author in [3], shortly after the appearance of Rockaffelar’s work. I dedicate this paper to V.M. Tikhomirov, my teacher and then my closest friend for almost 60 years. § 2. Preliminary results2.1. Integral functionalsTo begin with, we consider the simplest integral functional with variable lower limit and the integrand $\varphi$ assuming values in $(-\infty,\infty]$. We say that $\varphi$ satisfies the growth condition if
$$
\begin{equation*}
\int_0^T\varphi^*(t,p)\,dt<\infty \quad \forall\, p\in \mathbb R^n.
\end{equation*}
\notag
$$
Here $\varphi^*(t,\,\cdot\,)$ is the function conjugate to $\varphi(t,\,\cdot\,)$:
$$
\begin{equation*}
\varphi^*(t,p) = \sup_{y}(\langle p,y\rangle - \varphi(t,y)).
\end{equation*}
\notag
$$
A proof of the following result can be found in [7], § 9.1, Theorem 1. Proposition 2.1. Consider a function $\varphi(t,x)$ on $[0,T]\times\mathbb{R}^n$ that is measurable with respect to the $\sigma$-algebra generated by the products $\Delta\times Q$, where $\Delta$ is a measurable subset of $[0,T]$ and $Q$ is a Borel subset of $\mathbb{R}^n$. If $\varphi$ satisfies the growth condition, then for each $C\in \mathbb{R}$ the set is either empty or relatively compact in the weak topology of $L^1$. In connection with this proposition, it is useful to observe that the function conjugate to
$$
\begin{equation*}
S(r)= \inf\biggl\{ \int_0^T\varphi(t,x(t))\,dt\colon \int_0^Tx(t)\,dt = r \biggr\},
\end{equation*}
\notag
$$
coincides with $\displaystyle\int_0^T\varphi^*(t,p)\,dt$ (see [7], § 8.3, Theorem 2), no matter whether or not $\varphi$ satisfies the growth condition. However, as follows from Proposition 2.1, if $\varphi$ satisfies the growth condition, then the function $x(\,{\cdot}\,)\to \displaystyle\int_0^T\varphi(t,x(t))\,dt$ is bounded below on $L^1$ and assumes a finite value at least at one point. In what follows we assume that the function $\varphi(t,0)$ is summable. A slightly more precise version of Proposition 2.1 is stated below. Proposition 2.2. Again, let $\varphi (t,x)$ satisfy the assumptions of Proposition 2.1. Also let $\beta(\,{\cdot}\,)$ be a nonnegative summable function on $[0,T]$. Then for any $C\in \mathbb{R}$ and $r\geqslant 0$ the set is either empty or relatively compact in the weak topology of $W^{1,1}$. Proof. Let $\displaystyle\beta = \int_0^T\beta(t)\,dt$. In what follows let $x(\,{\cdot}\,)\in W^{1,1}$ and $r=|x(0)|$. Then
$$
\begin{equation*}
\begin{aligned} \, \int_0^T(\varphi(t,\dot x(t))-\beta (t)|x(t)|)\,dt &\geqslant \int_0^T\biggl(\varphi(t,\dot x(t)) - \beta(t)\biggl(r+\int_0^t|\dot x(s)|\,ds\biggr)\biggr)dt\\ &=\int_0^T\biggl(\varphi(t,\dot x(t)) - |\dot x(t)|\int_0^t\beta(s)\,ds\biggr)dt-\beta r \\ &\geqslant\int_0^T\bigl(\varphi(t,\dot x(t)) - \beta |\dot x(t)|\bigr)\,dt-\beta r. \end{aligned}
\end{equation*}
\notag
$$
It is easy to see that the function $\psi(t,y)= \varphi(t,y)-k|y|$ satisfies the growth condition for any $k$. Indeed,
$$
\begin{equation*}
\begin{aligned} \, \psi^*(t,p)&=\sup_y(\langle p,y\rangle -\varphi(t,y) + k|y|) \\ &=\sup_y\sup_{| q|=1}(\langle p,y\rangle -\varphi(t,y) + k\langle q,y\rangle) \\ &=\sup_{|q|=1}\sup_y(\langle p+ kq,y\rangle - \varphi(t,y)) =\sup_{|q|=1}\varphi^*(t,p+ kq) \end{aligned}
\end{equation*}
\notag
$$
and the ball of radius $k$ around $p$ lies in the convex hull of the $2n$ points $p+\pm k\sqrt{n}\,e_i$, where the $e_i$ are elements of some orthonormal basis of $\mathbb{R}^n$. Now the proof follows easily from Proposition 2.1. The proposition is proved. In what follows we assume that the following condition is satisfied: Consider the problems of the minimization of the functional
$$
\begin{equation*}
J_t(x(\,{\cdot}\,))=\int_t^TL(t,x(t),\dot x(t))\,dt + g(x(T))
\end{equation*}
\notag
$$
on the set $X(t,x)$ of absolutely continuous functions on $[t,T]$ such that $x(t)=x$. Proposition 2.3. In addition to conditions ($\mathrm A_1$)–($\mathrm A_3$), assume that $L(t,x,\,\cdot\,)$ is a convex function for all $t$ and $x$. Then for any $t$ the functional $J_t$ is lower semicontinuous in the weak topology of the space $W^{1,1}[t,T]$ and attains its minimum on every nonempty set $X(t,x)$. Proof. By ($\mathrm A_1$) the first statement holds true if the functional
$$
\begin{equation*}
I_t(x(\,{\cdot}\,))= \int_t^T L(s,x(s),\dot x(s))\,ds
\end{equation*}
\notag
$$
is lower semicontinuous. But this is a well-known fact (see, for instance, [8] or [3], Theorem 5). In turn, the second statement follows from the first and Proposition 2.2.
The proposition is proved. We must emphasize that the convexity of $L(t,x,\,\cdot\,)$ is a necessary condition for the last statement to hold. The verification of this fact is elementary. On the other hand, under very weak assumptions the lower closure of $I_t$ (that is, the functional whose epigraph is the closure of the epigraph of $I_t$) coincides with the integral functional associated with the convexification of $L(t,x,\,\cdot\,)$. 2.2. SubdifferentialsThere are several types of subdifferentials studied in nonsmooth analysis. We need the simplest one known as the Dini–Hadamard subdifferential or directional subdifferential [5], [11]. So let $X$ be a Banach space and $f$ a function on $X$ which is finite at $x\in X$. The subdifferential we need (of $f$ at $x$) is the set $\partial^{-}f(x)$ of linear functionals $x^*\in X^*$ such that
$$
\begin{equation*}
\liminf_{t\to +0,\, u\to h} t^{-1}(f(x+tu)- f(x))\geqslant \langle x^*,h\rangle
\end{equation*}
\notag
$$
for all $h\in X$. It immediately follows from the definition that the Dini–Hadamard subdifferential (if it is nonempty) is a convex set closed in the weak$^*$ topology of the dual space. We need the following properties of the Dini–Hadamard subdifferential:
2.3. Pseudo-Lipschitz mappingsConsider a set-valued mapping $Q(t)$ from $[0,T]$ to $\mathbb{R}^n$, and let $Q$ be the graph of $Q(t)$. Definition 2.4. $L$ is pseudo-Lipschitz on $Q$ if there are $\beta>0$ and a strictly positive summable function $k(t)$ such that for any $t\in [0,T]$ and $x\in Q(t)$ and any $y\in \mathbb{R}^n$ for which $L(t,x,y)<\infty$ there exist $x'\in Q(t)$ and $y'$ such that and The latter is, in particular, true if for all $t\in [0,T]$ and $y$ the function $L(t,\,{\cdot}\,,y)$ is $(k(t)+\beta|y|)$-Lipschitz on $Q(t)$. An easily verifiable consequence of this property is that the set-valued mapping $(t,x)\to F(t,x):= \operatorname{epi} L(t,x,\,\cdot\,)$ has the global pseudo-Lipschitz property introduced in [4]: if $x,x'\in Q(t)$, then the inclusion holds for all $N>0$. (An earlier version of this property was introduced in [9].) Here $\operatorname{epi} f=\{(\alpha,u)\colon \alpha\geqslant f(u)\}$ is the epigraph of the function $f$.The origins of the following result (sufficiently well known for the standard problem of the classical calculus of variations) go back to the 1930 work of Bogolyubov [2]. Proposition 2.5. Assume that $L$ satisfies ($\mathrm A_2$) and ($\mathrm A_3$) and is, in addition, pseudo-Lipschitz in a neighbourhood of the graph of a mapping $x(\,{\cdot}\,)\in W^{1,1}$. Let $\widehat L(t,x,\,\cdot\,)$ be the convexification of $L(t,x,\,\cdot\,)$, and let Then for each $t$ there is a sequence of $x_m(\,{\cdot}\,)\in X(t,x)$ (that is, such that $x_m(t)=x(t)$) weakly converging to $x(\,{\cdot}\,)$ and such that Proof. Our proof follows basically the proof of Theorem 4.1 in [4]. The theorem itself cannot be used here because of an additional assumption (($\mathrm A_4$) in [4]) which is not needed here.
Clearly, it is sufficient to prove the proposition only for $t=0$. Given any $\varepsilon>0$, we can find measurable $y_i(\,{\cdot}\,)$ and $\alpha_i(\,{\cdot}\,)$, $i=1,\dots,n+1$, such that for each $t$
$$
\begin{equation*}
\alpha_i(t)\geqslant 0, \qquad \sum_{i=1}^n \alpha_i(t) = 1, \qquad \sum_{i=1}^n\alpha_i(t)y_i(t) = \dot x(t)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\sum_{i=1}^n\alpha_i(t) L(t,x(t),y_i(t))\leqslant \widehat L(t,x(t),\dot x(t)) +\varepsilon.
\end{equation*}
\notag
$$
Now we can find a set $\Delta\subset [0,T]$ with measure not smaller than $T-\varepsilon$ and such that the values of the functions $|y_i(t)|$ and $L(t,x(t),\dot x(t))$ do not exceed certain $\rho$ almost everywhere on $\Delta$. Once $\Delta$ is chosen, we can modify the $y_i(\,{\cdot}\,)$ outside $\Delta$ by setting them equal to $\dot x(t)$. Finally, we can find sequences of functions $\alpha_{im}(\,{\cdot}\,)$ assuming values $0$ and $1$, satisfying $\sum_i\alpha_{im}(t)=1$ almost everywhere and converging weakly in $L^{\infty}$ to functions coinciding with $\alpha_i(\cdot)$ on $\Delta$ and equal to $(n+1)^{-1}$ outside $\Delta$. It is clear that the functions $z_m(\,{\cdot}\,)$ defined by
$$
\begin{equation*}
z_m(0)= x(0)\quad\text{and} \quad \dot z_m(t) = \begin{cases} \displaystyle \sum\alpha_{im}(t)y_i(t)& \text{if } t\in\Delta, \\ \dot x(t) &\text{otherwise} \end{cases}
\end{equation*}
\notag
$$
converge to $x(\,{\cdot}\,)$ in the weak topology of $W^{1,1}$ and therefore uniformly.
As $L$ is pseudo-Lipschitz near $x(\,{\cdot}\,)$, the distance of $(y_i(t),L(t,x(t),y_i(t)))$ to $\operatorname{epi} L(t,z_m(t),\,\cdot\,)$ (as a function of $t$) tends to zero in the $L^1$-metric. By Theorem 3.1 in [4] (for $z_m(\,{\cdot}\,)$ playing the role of $x(\,{\cdot}\,)$) there exist $a_m(\,{\cdot}\,)\in L^1$ and $x_m(\,{\cdot}\,)\in W^{1,1}$ such that $a_m(t)\geqslant L(t,x_m(t),\dot x_m(t))$ almost everywhere and $a_m(\,{\cdot}\,)- L(\,\cdot\,,x(\,{\cdot}\,),\dot z_m(\,{\cdot}\,))\to 0$ and $x_m(\,{\cdot}\,)-z_m(\,{\cdot}\,)\to 0$ in $L^1$ and $W^{1,1}$, respectively. This completes the proof of the proposition. § 3. Value function and HamiltonianWe return to the minimization of the functional
$$
\begin{equation*}
J_t(x(\,{\cdot}\,))= \int_t^T L(t,x(t),\dot x(t))\,dt + g(x(T))
\end{equation*}
\notag
$$
on the set $X(t,x)$ of absolutely continuous functions on $[t,T]$ such that $x(t)=x$. Let $V(t,x)$ be the value function, that is, the minimum value of the functional in the problem. By Proposition 2.5 the value function in the problem associated with the convexified integrand $\widehat L$ coincides with $V$, at least if $L$ is pseudo-Lipscitz on $\mathbb{R}^n$. Hence without any loss of generality we may assume in what follows that $L(t,x,\,\cdot\,)$ is a convex function for all $t$ and $x$. Theorem 3.1. In addition to ($\mathrm A_1$)–($\mathrm A_3$), assume that $L(t,x,\,\cdot\,)$ is a convex function. Then the value function $V(t,x)$ is lower semicontinuous. Proof. Note first that by ($\mathrm A_3$) the inequality
$$
\begin{equation}
L(t,x,y)+\beta(t)|x|\geqslant \varphi(t,y)\geqslant - \varphi^*(t,0)
\end{equation}
\tag{3}
$$
holds for all $t$, $x$ and $y$, and the function $\varphi^*(t,0)$ is summable as $\varphi$ satisfies the growth condition. By Proposition 2.3, for any $t\in [0,T]$ the functional $J_t$ is lower semicontinuous in the weak topology of $W^{1,1}$. Furthermore, since the function $t\to \varphi^*(t,p)$ is summable for any $p$, the function $\varphi(s,y(s))$ must be summable for some $y(\,{\cdot}\,)\in L^1$. This follows from the simple fact mentioned above that the function $\displaystyle p\to \int \varphi^*(t,p)\,dt$ is conjugate to We may assume without any loss of generality that the function $\varphi(s,0)$ is summable.
Let $(t_i,x_i)\to (t,x)$. If $V(t_i,x_i)\to \infty$, then, of course, $\liminf V(t_i,x_i)\geqslant V(t,x)$. So we may assume that $V(t_i,x_i)\leqslant C<\infty$. By Proposition 2.3, $J_{t_i}$ attains its minimum on $X(t_i,x_i)$ at some $x_i(\,{\cdot}\,)$, that is $J_{t_i}(x_i(\,{\cdot}\,))= V(t_i,x_i)$. The sequence $\{g_i(x_i(T))\}$ is bounded, which is immediate from Proposition 2.3. Setting $x_i(s)= x_i(t_i)$ for $s\in [0,t_i]$, we extend every $x_i(\,{\cdot}\,)$ to the whole of $[0,T]$. Then the integrals
$$
\begin{equation*}
\int_0^T (\varphi(t,\dot x_i(t)) - \beta(t)|x_i(t)|)\,dt
\end{equation*}
\notag
$$
are uniformly bounded, so that the sequence $(x_i(\,{\cdot}\,))$ is weakly compact in $W^{1,1}$. Therefore, we can assume that this sequence converges weakly in $W^{1,1}$ to some $x(\,{\cdot}\,)$. We can also be sure that the functions $x_i(\,{\cdot}\,)$ are uniformly bounded, that is, there is $r>0$ such that $|x_i(s)|\leqslant r$ for all $s$.
Consider first the case when $t_i\leqslant t$ for all $i$. In this case
$$
\begin{equation*}
\liminf_{i\to\infty} J_{t_i}(x_i(\,{\cdot}\,)) \geqslant \liminf_{i\to\infty}\int_{t_i}^tL(s,x_i(s),\dot x_i(s))\,ds + \liminf_{i\to\infty}J_t(x_i(\,{\cdot}\,)).
\end{equation*}
\notag
$$
The first term in the right side of this inequality is nonnegative. This follows from (3) and the fact that the $x_i(\,{\cdot}\,)$ are uniformly bounded. Now the inequality $\liminf J_{t_i}(x_i(\,{\cdot}\,))\geqslant J_t(x(\,{\cdot}\,))$ follows from Proposition 2.3.
Now let $t_i>t$ for all $i$. By Proposition 2.3, in this case
$$
\begin{equation*}
\liminf_{i\to\infty} J_{\tau} (x_i(\,{\cdot}\,))\geqslant J_{\tau}(x(\,{\cdot}\,))
\end{equation*}
\notag
$$
for any $\tau>t$. So to prove the inequality $\lim_{\tau\searrow t} J_{\tau}(x(\,{\cdot}\,))\geqslant J_{t}(x(\,{\cdot}\,))$ we only need to refer to Fatou’s lemma, having in mind (3) along with the fact that $x(\,{\cdot}\,)$ is a bounded mapping. This completes the proof of the theorem. Next consider the function called the Hamiltonian of $L$. In our case $H(t,x,\,\cdot\,)$ is everywhere finite for all $t$ and $x$. This follows from the obvious inequality $H(t,x,p)\leqslant\varphi^*(t,p)+\beta|x|$, which, in turn, immediately follows from (3).Theorem 3.2. Assume that ($\mathrm A_1$) and ($\mathrm A_2$) hold. Also assume that $L(t,x,\,\cdot\,)$ is convex for all $t$ and $x$, and let $(\alpha,p)\in\partial^{-} V(t,x)$. (a) If $L(\,{\cdot}\,{,}\,{\cdot}\,,u)$ is continuous at $(t,x)$ from the left in the sense that $L(s,w,u)\to L(t,x,u)$ as $(s,w)\to (t,x)$, $s<t$ (where the case $L(t,x,u)=\infty$ is not ruled out), then (b) If $H(\,{\cdot}\,{,}\,{\cdot}\,,p)$ is continuous at $(t,x)$ from the right, then Proof. (a) For any $t>0$ and $\lambda >0$ the inequality
$$
\begin{equation*}
V(t-\lambda,x-\lambda u)\leqslant V(t,x)+ \int_{t-\lambda}^tL(s,x(s),\dot x(s))\,ds,
\end{equation*}
\notag
$$
where $x(t-\lambda)= x-\lambda u$ and $x(t)=x$, is valid for all $u$. In particular,
$$
\begin{equation*}
V(t-\lambda,x-\lambda u)\leqslant V(t,x)+ \int_{t-\lambda}^t L(s,su+x-tu,u)\,ds.
\end{equation*}
\notag
$$
It follows that for any $\lambda >0$ where $r(\lambda)\to 0$ as $\lambda\to 0$. Since $L(\,{\cdot}\,{,}\,{\cdot}\,,u)$ is a continuous function, we get that $-\alpha\leqslant L(t,x,u)- \langle -p,u\rangle$. Hence
(b) Now let $0\leqslant t<T$. We choose some $\overline x(\,{\cdot}\,)$ defined on $[t,T]$ such that $\overline{x}(t)=x$ and $V(t,x)= J_t(\overline{x}(\,{\cdot}\,))$. By Proposition 2.3 such a function $\overline{x}(\,{\cdot}\,)$ does exist. Set $\overline{u}(s)= \overline{x}(t+s) - \overline{x}(t)$. Then for any sufficiently small $\lambda>0$
$$
\begin{equation*}
V(t,x)= V(t+\lambda,x+\overline u(\lambda)) +\int_0^{\lambda}L(t+s,x+\overline u(s),\dot{\overline u}(s))\,ds.
\end{equation*}
\notag
$$
Thus if $(\alpha,p)\in \partial^{-}V(t,x)$, then
$$
\begin{equation*}
0\geqslant \lambda(\alpha+\langle p,\overline u(\lambda)\rangle) +\int_0^{\lambda}L(t+s,x+\overline u(s),\dot{\overline u}(s))\,ds + o(\lambda).
\end{equation*}
\notag
$$
It follows that for any sufficiently small $\lambda>0$
$$
\begin{equation*}
\begin{aligned} \, 0 &\leqslant -\alpha+\lambda^{-1}\int_t^{t+\lambda}\sup_u[\langle-p,u\rangle-L(s,\overline x(s),u)]\,ds+r(\lambda) \\ &= -\alpha+\lambda^{-1}\int_t^{t+\lambda}H(s,\overline x(s),-p)\,ds + r(\lambda) \end{aligned}
\end{equation*}
\notag
$$
(where $r(\lambda)\to 0$ as $\lambda\to 0$), which immediately implies the required inequality.
The theorem is proved. § 4. Characterization of the value functionIn this section we plan to prove a result partially converse to Theorem 3.2 in the case when $L$ does not depend on $t$. Assume first that a set-valued mapping $\mathcal F\colon \mathbb R^n\rightrightarrows \mathbb R^m$ is fixed. Set
$$
\begin{equation*}
\mathcal H(x,q)= \sup\{\langle q,u\rangle\colon u\in \mathcal F(x)\}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathcal U(x,q)= \{u\in\mathcal F(x)\colon \langle q,u\rangle = \mathcal H(x,q) \}.
\end{equation*}
\notag
$$
Lemma 4.1 ([6], Proposition 4.8). Let $\mathcal{F}$ be a convex-valued mapping, and let $\overline{y}\in\mathcal{F}(\overline{x})$. Also assume that there are $\varepsilon>0$ and a nondecreasing strictly positive function $\rho(t)$ on $\mathbb{R}_+$ such that the relation holds for any $x,x'\in B(\overline{x},\varepsilon)$ and $N>0$. If the set $\mathcal{U}(\overline{x},\overline{q})$ is nonempty and bounded, that is to say, there exists $r>0$ such that $\mathcal{U}(\overline{x},\overline{q})\subset B(\overline{y},r)$, then for any $\eta>0$ there exists $\delta>0$ such that $|y-\overline{y}|\leqslant r+\eta$ whenever $y\in \mathcal{U}(x,q)$, provided that $|x-\overline{x}|\leqslant\delta$ and $|q-\overline{q}|<\delta$. In other words, $\mathcal{H}$ is a Lipschitz mapping in a neighbourhood of $(\overline{x},\overline{q})$. The following assumption allows us to apply this lemma to our problem:
This is obviously true if and the norms of $x$, $x'$ and $y$ do not exceed $m$. Note that, unlike in ($\mathrm A_4$), if this condition is satisfied, then the domains of definition of the mappings $F(x,\,\cdot\,)$ are independent of $x$.If $L$ is a convex function of the last argument, then applying the lemma to
$$
\begin{equation*}
\mathcal F\colon x\to \operatorname{epi} L(x,\,\cdot\,) = \{(\alpha,y)\colon \alpha\geqslant L(x,y)\},
\end{equation*}
\notag
$$
we obtain the following result. Corollary 4.2. Assume that ($\mathrm A_1$)–($\mathrm A_4$) hold. Then $H(\cdot,\,\cdot\,)$ satisfies the Lipschitz condition in a neighbourhood of any point $(x,p)$ at which it is finite. Proof. By definition $\mathcal{F}$ is a set-valued mapping from $\mathbb{R}^n$ into $\mathbb{R}^{n+1}$ whose values are convex closed sets. Set $u= (\alpha,y)$ and $q=(\xi,p)$. Then $\xi\leqslant 0$ if $\mathcal{H}(x,q)<\infty$, and for $\xi=-1$ we obtain and
$$
\begin{equation*}
\mathcal U(x,q)= \{(\alpha,y)\colon \alpha=L(x,y),\,\langle p,y\rangle -\alpha=H(x,p)\}.
\end{equation*}
\notag
$$
However, $|y|^{-1}L(x,y)\to \infty$ as $|y|\to\infty$ by ($\mathrm A_3$). Hence $\mathcal{U}(x,q)$ is a bounded set and, as follows from (4), $\mathcal{F}(x')\subset \mathcal{F} (x) + k|x-x'|B$ for some $k>0$.
The corollary is proved. The main result in this section is contained in the following statement. Theorem 4.3. Let conditions ($\mathrm A_1$)–($\mathrm A_4$) be satisfied, and let $\varphi(t,x)$ be a lower semicontinuous function on $[0,T]\times\mathbb{R}^n$ with values in $(-\infty,\infty]$ and such that for some $\rho >0$ and $k(t)\geqslant 0$ summable on $[0,T]$. Assume also that $H(x,-p)\leqslant 0$ if $(\alpha,p)\in\partial^{-}\varphi(t,x)$. Then for any $t\in [0,T)$ the inequality holds for all absolutely continuous mappings $x(\,{\cdot}\,)\colon [t,T]\to \mathbb{R}^n$ with bounded derivative. Proof. It is obviously sufficient to prove the theorem only for $t=0$. Note also that $\varphi(t,\,\cdot\,)$ is differentiable almost everywhere on $B(\overline{x}(t),\rho)$.
Let $\mathcal{N}_m$ be the system of functions $\eta(t,x)\geqslant 0$ on $[0,T]\times\mathbb{R}^n$ that are measurable in $t$, continuously differentiable with respect to $x$ and such that for each $t$
$$
\begin{equation*}
\int_{\mathbb R^n}\eta(t,x)\,dx= 1 \quad\text{and}\quad \eta(t,x)=0 \quad\text{if } |x|\geqslant m^{-1}.
\end{equation*}
\notag
$$
Fix some $\eta_m(\,{\cdot}\,)\in\mathcal{N}_m$, and let
$$
\begin{equation}
\psi_m(t,x) = \int_{\mathbb R^n} \varphi(t,x-w)\eta_m(t,w)\,dw.
\end{equation}
\tag{7}
$$
Then (see [14], the proof of Theorem 9.67)
$$
\begin{equation*}
\psi_m'(t,x) = \int_{\mathbb R^n} \varphi(t,x-w)\eta_m(t,w)\,dw,
\end{equation*}
\notag
$$
the functions $\psi_m(t,x)$ are continuously differentiable and converge uniformly to $\varphi(t,x)$ on compact sets Here $\psi_m'(t,x)$ is the derivative of the function $\psi_m(t,\,\cdot\,)$ at $x$. Note that by our assumptions $H(x,\varphi'(t,x))\leqslant 0$ whenever $(\lambda,\varphi'(t,x))\in\partial^{-}\varphi(t,x)$.
According to our assumptions, the inequality $|\varphi'(t,x)|\leqslant k(t)$ holds for any $t\in [0,T]$ and $x\in B(\overline{x}(t),\rho)$ such that $\varphi(t,\,\cdot\,)$ is differentiable at $x$. As $H$ is a locally Lipschitz function (Corollary 4.2), we can be sure that for any $t$ there exists $k_1(t)>0$ such that $H$ satisfies the Lipschitz condition with constant $k_1(t)$ on the set of pairs $(x,p)$ such that $x\in B(\overline x(t),\rho)$ and $|p|\leqslant rk(t)$. Note next that every vector $\psi'((t,x))$ belongs to the closed convex hull of the set $\{\varphi'(t,x-w)\}$, $|w|\leqslant 1/m$. In other words, for any $t$ and any $x\in B(\overline x,\rho)$ there exist $w_i$ satisfying $|w_i|\leqslant 1/m$ and $\alpha_i\geqslant 0$, $i=1,\dots,n+1$, such that $\sum\alpha_i=1$, $\varphi(t,\,\cdot\,)$ is differentiable at $x-w_i$ and
$$
\begin{equation*}
\biggl|\psi_m'(t,x) - \sum\alpha_i\varphi'(t,x-w_i)\biggr|\leqslant m^{-1}.
\end{equation*}
\notag
$$
Since $H(t,x,\,\cdot\,)$ is a convex function, it follows that
$$
\begin{equation*}
\begin{aligned} \, H(x,-\psi_m'(t,x)) &\leqslant H\Bigl(x,,\sum\alpha_i\varphi'(t,x-w_i)\Bigr) +k_1(t)m^{-1} \\ &\leqslant \sum\alpha_iH(x,-\varphi'(x-w_i)) + k_1(t)m^{-1} \\ &\leqslant \sum\alpha_i H(x-w_i,-\varphi'(x-w_i)) + 2k_1(t)m^{-1}. \end{aligned}
\end{equation*}
\notag
$$
The last inequality implies that for all $y$
$$
\begin{equation*}
-\langle \psi_m'(t,x),y\rangle - L(t,x,y)\leqslant 2k_1(t) m^{-1},
\end{equation*}
\notag
$$
so that
$$
\begin{equation*}
\psi_m(t,x(t)) - \psi_m(T,x(T))\leqslant \int_t^T L(x(s),\dot x(s))\,ds +2kTm^{-1}.
\end{equation*}
\notag
$$
But $\psi_m(t,x)\to \varphi(t,x)$ as $m\to \infty$ by definition, and the required result follows.
Theorem 4.3 is proved. 
Citation in format AMSBIB
\Bibitem{Iof25} |
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