| Russian Mathematical Surveys |
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This article is cited in 1 scientific paper (total in 1 paper) Brief communications On the minimax solution of path-dependent Hamilton–Jacobi equations for neutral-type systems M. I. Gomoyunovab, N. Yu. Lukoyanovab a Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences b Ural Federal University
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 Let $h > 0$, $T > 0$, $n \in \mathbb{N}$, and let $\operatorname{Lip}=\operatorname{Lip}([-h,T],\mathbb{R}^n)$ be the space of Lipschitz continuous functions $x \colon [-h,T] \to \mathbb{R}^n$ with the norm $\|x(\cdot)\|_\infty=\max_{\tau \in [-h,T]} \|x(\tau)\|$, where $\|\cdot\|$ is the Euclidean norm in $\mathbb{R}^n$. A functional $\varphi \colon [0,T] \times \operatorname{Lip} \to \mathbb{R}$ is called: (i) non-anticipative if, for all $(t, x(\cdot)) \in [0,T) \times \operatorname{Lip}$ and $y(\cdot) \in \operatorname{Lip}(t,x(\cdot))=\{\bar{y}(\cdot) \in\operatorname{Lip} \colon \bar{y}(\tau)=x(\tau), \, \tau \in [-h,t]\}$, the equality $\varphi(t,x(\cdot))=\varphi(t,y(\cdot))$ holds; (ii) coinvariantly ($ci$-) differentiable at a point $(t,x(\cdot)) \in [0,T) \times \operatorname{Lip}$ if there exist $\partial_t \varphi(t,x(\cdot)) \in \mathbb{R}$ and $\nabla \varphi(t,x(\cdot)) \in \mathbb{R}^n$ such that the relation below is satisfied for every function $y(\cdot) \in \operatorname{Lip}(t,x(\cdot))$:
$$
\begin{equation*}
\varphi(\tau, y(\cdot))-\varphi(t,x(\cdot))=\partial_t \varphi(t,x(\cdot))(\tau-t)+ \langle\nabla\varphi(t,x(\cdot)),y(\tau)-x(t)\rangle+o(\tau-t),
\end{equation*}
\notag
$$
where $\tau \in (t,T]$ and the function $o(\cdot)$ can depend on $y(\cdot)$, $o(\delta) / \delta \to 0$ as $\delta \to 0^+$; the quantities $\partial_t \varphi(t,x(\cdot))$ and $\nabla \varphi(t,x(\cdot))$ are called the $ci$-derivatives of $\varphi$ at the point $(t,x(\cdot))$.
Let the mappings $g \colon [0,T] \times \operatorname{Lip} \to \mathbb{R}^n$, $H \colon [0,T] \times \operatorname{Lip} \times \mathbb{R}^n \to \mathbb{R}$, and $\sigma \colon \operatorname{Lip} \to \mathbb{R}$ be given and satisfy the following assumptions (A): (i) $g$, $H$, and $\sigma$ are continuous; (ii) there exists $h_0 \in (0,h]$ such that for any $\mu > 0$ we can find $\lambda_g > 0$ for which
$$
\begin{equation*}
\|g(t_1, x_1(\cdot))-g(t_2, x_2(\cdot))\|\leqslant \lambda_g \bigl(|t_1-t_2|+\|x_1(\,\cdot\,\wedge (t_1-h_0))- x_2(\,\cdot\,\wedge (t_2-h_0))\|_\infty \bigr)
\end{equation*}
\notag
$$
for all $(t_1,x_1(\cdot)),(t_2,x_2(\cdot)) \in [0,T] \times X_\mu$; (iii) there exists $c_H > 0$ such that
$$
\begin{equation*}
|H(t,x(\cdot),s_1)-H(t,x(\cdot),s_2)|\leqslant c_H\bigl(1+\|x(\,\cdot\,\wedge t)\|_\infty\bigr)\|s_1-s_2\|
\end{equation*}
\notag
$$
for all $t \in [0,T]$, $x(\cdot) \in \operatorname{Lip}$, and $s_1,s_2 \in \mathbb{R}^n$; (iv) for any $\mu > 0$ there exists $\lambda_H > 0$ such that for all $t \in [0,T]$, $x_1(\cdot),x_2(\cdot) \in X_\mu$, and $s \in \mathbb{R}^n$,
$$
\begin{equation*}
|H(t,x_1(\cdot),s)-H(t,x_2(\cdot), s)|\leqslant \lambda_H (1+\|s\|)\|x_1(\,\cdot\,\wedge t)- x_2(\,\cdot\,\wedge t)\|_\infty.
\end{equation*}
\notag
$$
Here $a \wedge b=\min\{a,b\}$ and $X_\mu$ is the set of functions $x(\cdot) \in \operatorname{Lip}$ such that $\|x(\cdot)\|_\infty \leqslant \mu$ and $\|x(\tau_1)-x(\tau_2)\| \leqslant \mu|\tau_1-\tau_2|$ for all $\tau_1,\tau_2 \in [-h,T]$.
Consider the Cauchy problem (CP) for the path-dependent Hamilton–Jacobi equation
$$
\begin{equation*}
\partial_t \varphi(t,x(\cdot))+ \langle \partial_t g(t,x(\cdot)),\nabla \varphi(t,x(\cdot)) \rangle+ H \bigl(t,x(\cdot),\nabla \varphi(t,x(\cdot)) \bigr)=0,\quad (t,x(\cdot)) \in X_\ast,
\end{equation*}
\notag
$$
under the right-end boundary condition $\varphi(T,x(\cdot))=\sigma(x(\cdot))$, $x(\cdot) \in \operatorname{Lip}$. Here the non-anticipative functional $\varphi \colon [0,T] \times \operatorname{Lip} \to \mathbb{R}$ is to be found, $X_\ast$ denotes the set of points $(t,x(\cdot)) \in [0,T) \times \operatorname{Lip}$ at which all the coordinates $g_i \colon [0,T] \times \operatorname{Lip} \to \mathbb{R}$, $i = {1,\dots,n}$, of the mapping $g=(g_1,\dots,g_n)$ are $ci$-differentiable, $\partial_t g(t,x(\cdot))=(\partial_t g_1 (t,x(\cdot)),\dots,\partial_t g_n(t,x(\cdot)))$; $\langle\,\cdot\,{,}\,\cdot\,\rangle$ is the inner product in $\mathbb{R}^n$.
Cauchy problems of the type under consideration arise in the study of optimal control problems and differential games for dynamical systems whose motion is described by functional differential equations of neutral type in Hale’s form. As noted in the recent survey paper [1], under fairly natural and general assumptions (A) no results on the existence and uniqueness of a generalized (minimax) solution of the Cauchy problem (CP) have been obtained previously. The present brief communication fills this gap. A minimax solution of the Cauchy problem (CP) is defined as a non-anticipative and continuous functional $\varphi \colon [0,T] \times \operatorname{Lip} \to \mathbb{R}$ that satisfies the boundary condition and has the following property: for any $(t,x(\cdot)) \in [0, T) \times \operatorname{Lip}$, $\vartheta \in (t,T]$, $s \in \mathbb{R}^n$, and $\varepsilon > 0$ there exist pairs $(y_1(\cdot),z_1(\cdot)), (y_2(\cdot),z_2(\cdot)) \in \operatorname{Sol}(t,x(\cdot),s)$ such that
$$
\begin{equation*}
\varphi(\vartheta,y_1(\cdot))-z_1(\vartheta) \leqslant \varphi(t,x(\cdot))+\varepsilon\quad\text{and}\quad \varphi(\vartheta,y_2(\cdot))-z_2(\vartheta) \geqslant \varphi(t,x(\cdot))-\varepsilon.
\end{equation*}
\notag
$$
Here $\operatorname{Sol}(t,x(\cdot),s)$ denotes the set of pairs $(y(\cdot),z(\cdot)) \in \operatorname{Lip}(t,x(\cdot)) \times \operatorname{Lip}([-h,T],\mathbb{R})$ such that $z(\tau)=0$, $\tau \in [-h,t]$, and for almost all $\tau \in [t,T]$
$$
\begin{equation*}
\frac{\mathrm{d}}{\mathrm{d} \tau}\bigl(y(\tau)- g(\tau,y(\cdot)),z(\tau)\bigr)\in E(\tau,y(\cdot),s)
\end{equation*}
\notag
$$
where $E(\tau,y(\cdot),s)=\{(f,\chi) \in \mathbb{R}^n \times \mathbb{R} \colon \|f\| \leqslant c_H(1+\|y(\,\cdot\,\wedge \,\tau)\|_\infty)$, and $\chi=\langle s,f \rangle- H(\tau,y(\cdot),s)\}$.
Theorem 1. Under assumptions (A) a minimax solution of the Cauchy problem (CP) exists and is unique. Previously, theorems on the existence and uniqueness of a minimax solution of the Cauchy problem (CP) were proved under more restrictive conditions: in [2] the positive homogeneity of $H$ with respect to the third variable was assumed; in [3] and [4] the case where $g(t,x(\cdot))=g(t,x(t- h))$ and $H(t,x(\cdot),s)=H(t,x(t- h),s)$ was considered. The general case of assumptions (A) is covered by reasoning in accordance with the scheme from these works, but with the choice of a Lyapunov–Krasovskii functional described below. Let $(t,x(\cdot)) \in [0,T) \times \operatorname{Lip}$. Choose $\mu > 0$ so that the inclusion $y(\cdot) \in X_\mu$ is valid for all $s \in \mathbb{R}^n$ and $(y(\cdot),z(\cdot)) \in \operatorname{Sol}(t,x(\cdot),s)$, define numbers $\lambda_g$ and $\lambda_H$ in accordance with assumptions (ii) and (iv), and take $m \in \mathbb{N}$ satisfying $m h_0 \geqslant T-t$. Set $\lambda_0=2\lambda_H(1+\lambda_g)^{m-1}/(3-\sqrt{5}\,)$ and fix $\varepsilon_0 \in (0,e^{-\lambda_0(T-t)})$. The Lyapunov–Krasovskii functional suitable for Theorem 1 has the following form:
$$
\begin{equation*}
\nu_\varepsilon(\tau,y_1(\cdot),y_2(\cdot))= \frac{e^{-\lambda_0(\tau-t)}-\varepsilon}{\varepsilon} \sqrt{\varepsilon^4+ \gamma\bigl(\tau,w(\,\cdot\,;y_1(\cdot),y_2(\cdot))\bigr)}\,.
\end{equation*}
\notag
$$
Here $\tau \in [t,T]$, $y_1(\cdot),y_2(\cdot) \in \operatorname{Lip}(t,x(\cdot))$, $\varepsilon \in (0,\varepsilon_0]$ is a parameter; $w(\xi; y_1(\cdot),y_2(\cdot))=0$ if $\xi \in [-h,t)$, and $w(\xi;y_1(\cdot),y_2(\cdot))=y_1(\xi)-g(\xi,y_1(\cdot))-y_2(\xi)+g(\xi,y_2(\cdot))$ if $\xi \in [t,T]$; the functional $\gamma \colon [0,T] \times \operatorname{Lip} \to \mathbb{R}$ is defined as follows (see, for instance, [5]):
$$
\begin{equation*}
\gamma(\tau,w(\cdot))=\frac{\bigl(\|w(\,\cdot\,\wedge \tau)\|_\infty^2-\|w(\tau)\|^2\bigr)^2}{\|w(\,\cdot\,\wedge \tau)\|_\infty^2}+\|w(\tau)\|^2
\end{equation*}
\notag
$$
if $\|w(\,\cdot\,\wedge \tau)\|_\infty > 0$, and $\gamma(\tau,w(\cdot))=0$ if $\|w(\,\cdot\,\wedge \tau)\|_\infty=0$, where $(\tau,w(\cdot)) \in [0,T] \times \operatorname{Lip}$.
Citation in format AMSBIB
\Bibitem{GomLuk24} |
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