01056nas a2200121 4500008004300000245005500043210005400098260001300152520069800165100002100863700001400884856003600898 2004 en_Ud 00aSemi-cooperative strategies for differential games0 aSemicooperative strategies for differential games bSpringer3 aThe paper is concerned with a non-cooperative differential game for two players. We first consider Nash equilibrium solutions in feedback form. In this case, we show that the Cauchy problem for the value functions is generically ill-posed. Looking at vanishing viscosity approximations, one can construct special solutions in the form of chattering controls, but these also appear to be unstable. In the second part of the paper we propose an alternative \\\"semi-cooperative\\\" pair of strategies for the two players, seeking a Pareto optimum instead of a Nash equilibrium. In this case, we prove that the corresponding Hamiltonian system for the value functions is always weakly hyperbolic.1 aBressan, Alberto1 aShen, Wen uhttp://hdl.handle.net/1963/289300818nas a2200121 4500008004300000245007100043210006900114260000900183520043300192100002100625700001400646856003600660 2004 en_Ud 00aSmall BV solutions of hyperbolic noncooperative differential games0 aSmall BV solutions of hyperbolic noncooperative differential gam bSIAM3 aThe paper is concerned with an n-persons differential game in one space dimension. We state conditions for which the system of Hamilton-Jacobi equations for the value functions is strictly hyperbolic. In the positive case, we show that the weak solution of a corresponding system of conservation laws determines an n-tuple of feedback strategies. These yield a Nash equilibrium solution to the non-cooperative differential game.1 aBressan, Alberto1 aShen, Wen uhttp://hdl.handle.net/1963/291701967nas a2200121 4500008004100000245006700041210006700108260001800175520158100193100002101774700001401795856003601809 2000 en d00aBV estimates for multicomponent chromatography with relaxation0 aBV estimates for multicomponent chromatography with relaxation bSISSA Library3 aWe consider the Cauchy problem for a system of $2n$ balance laws which arises from the modelling of multi-component chromatography: $$\\\\left\\\\{ \\\\eqalign{u_t+u_x&=-{1\\\\over\\\\ve}\\\\big( F(u)-v\\\\big),\\\\cr v_t&={1\\\\over\\\\ve}\\\\big( F(u)-v\\\\big),\\\\cr}\\\\right. \\\\eqno(1)$$ This model describes a liquid flowing with unit speed over a solid bed. Several chemical substances are partly dissolved in the liquid, partly deposited on the solid bed. Their concentrations are represented respectively by the vectors $u=(u_1,\\\\ldots,u_n)$ and $v=(v_1,\\\\ldots,v_n)$. We show that, if the initial data have small total variation, then the solution of (1) remains with small variation for all times $t\\\\geq 0$. Moreover, using the $\\\\L^1$ distance, this solution depends Lipschitz continuously on the initial data, with a Lipschitz constant uniform w.r.t.~$\\\\ve$. Finally we prove that as $\\\\ve\\\\to 0$, the solutions of (1) converge to a limit described by the system $$\\\\big(u+F(u)\\\\big)_t+u_x=0,\\\\qquad\\\\qquad v=F(u).\\\\eqno(2)$$ The proof of the uniform BV estimates relies on the application of probabilistic techniques. It is shown that the components of the gradients $v_x,u_x$ can be interpreted as densities of random particles travelling with speed 0 or 1. The amount of coupling between different components is estimated in terms of the expected number of crossing of these random particles. This provides a first example where BV estimates are proved for general solutions to a class of $2n\\\\times 2n$ systems with relaxation.1 aBressan, Alberto1 aShen, Wen uhttp://hdl.handle.net/1963/133601052nas a2200121 4500008004300000245005900043210005900102260001300161520068500174100002100859700001400880856003600894 1998 en_Ud 00aUniqueness for discontinuous ODE and conservation laws0 aUniqueness for discontinuous ODE and conservation laws bElsevier3 aConsider a scalar O.D.E. of the form $\\\\dot x=f(t,x),$ where $f$ is possibly discontinuous w.r.t. both variables $t,x$. Under suitable assumptions, we prove that the corresponding Cauchy problem admits a unique solution, which depends H\\\\\\\"older continuously on the initial data.\\nOur result applies in particular to the case where $f$ can be written in the form $f(t,x)\\\\doteq g\\\\big( u(t,x)\\\\big)$, for some function $g$ and some solution $u$ of a scalar conservation law, say $u_t+F(u)_x=0$. In turn, this yields the uniqueness and continuous dependence of solutions to a class of $2\\\\times 2$ strictly hyperbolic systems, with initial data in $\\\\L^\\\\infty$.1 aBressan, Alberto1 aShen, Wen uhttp://hdl.handle.net/1963/3699