@article {2004,
title = {Semi-cooperative strategies for differential games},
journal = {Internat. J. Game Theory 32 (2004) 561-593},
number = {SISSA;103/2003/M},
year = {2004},
publisher = {Springer},
abstract = {The 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.},
doi = {10.1007/s001820400180},
url = {http://hdl.handle.net/1963/2893},
author = {Alberto Bressan and Wen Shen}
}
@article {2004,
title = {Small BV solutions of hyperbolic noncooperative differential games},
journal = {SIAM J. Control Optim. 43 (2004) 194-215},
number = {SISSA;21/2003/M},
year = {2004},
publisher = {SIAM},
abstract = {The 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.},
doi = {10.1137/S0363012903425581},
url = {http://hdl.handle.net/1963/2917},
author = {Alberto Bressan and Wen Shen}
}
@article {2000,
title = {BV estimates for multicomponent chromatography with relaxation},
journal = {Discrete Contin. Dynam. Systems 6 (2000) 21-38},
number = {SISSA;122/99/M},
year = {2000},
publisher = {SISSA Library},
abstract = {We 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.},
url = {http://hdl.handle.net/1963/1336},
author = {Alberto Bressan and Wen Shen}
}
@article {1998,
title = {Uniqueness for discontinuous ODE and conservation laws},
journal = {Nonlinear Analysis 34 (1998) 637-652},
number = {SISSA;26/97/M},
year = {1998},
publisher = {Elsevier},
abstract = {Consider 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 $\\\\L^\\\\infty$.},
doi = {10.1016/S0362-546X(97)00590-7},
url = {http://hdl.handle.net/1963/3699},
author = {Alberto Bressan and Wen Shen}
}