부산대학교 도서관

This paper deals with a class of nonlocal conservation laws modeling traffic flow, $$ \partial_t\rho(t,x)+\partial_x(\rho(t,x)v(t,x))=0,\quad t>0,\ x\in\Bbb{R}, $$ where the drivers are supposed to adapt the velocity $v(t,x)$ on the basis of both the perceived downstream density (``look-ahead'' effect) and the perceived upstream density (``nudging'' or ``look-behind'' effect): namely, $$ v(t,x)\coloneq f\left(\int_x^{x+\eta}\omega(s-x)\rho(t,s){\rm ds}\right)g\left(\int^x_{x-\zeta}\widetilde\omega(x-s) \rho(t,s){\rm ds}\right),\quad t\ge 0,\ x\in\Bbb{R}. $$ In the case of a ring-road, under suitable assumptions on the constants $\eta,\zeta>0$, the functions $f,g,\omega,\widetilde{\omega}\:\Bbb{R}_+\to\Bbb{R}_+$, and the initial data, the well-posedness of classical solutions to the Cauchy problem is established; moreover, a properly designed nudging effect is shown to yield local exponential stabilization of the system in the $L^2$-norm. Several numerical examples are provided to illustrate the results.