TY - JOUR T1 - Semi-cooperative strategies for differential games JF - Internat. J. Game Theory 32 (2004) 561-593 Y1 - 2004 A1 - Alberto Bressan A1 - Wen Shen AB - 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. PB - Springer UR - http://hdl.handle.net/1963/2893 U1 - 1807 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A sharp decay estimate for positive nonlinear waves JF - SIAM J. Math. Anal. 36 (2004) 659-677 Y1 - 2004 A1 - Alberto Bressan A1 - Tong Yang AB - We consider a strictly hyperbolic, genuinely nonlinear system of conservation laws in one space dimension. A sharp decay estimate is proved for the positive waves in an entropy weak solution. The result is stated in terms of a partial ordering among positive measures, using symmetric rearrangements and a comparison with a solution of Burgers\\\' equation with impulsive sources. PB - SIAM UR - http://hdl.handle.net/1963/2916 U1 - 1784 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Small BV solutions of hyperbolic noncooperative differential games JF - SIAM J. Control Optim. 43 (2004) 194-215 Y1 - 2004 A1 - Alberto Bressan A1 - Wen Shen AB - 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. PB - SIAM UR - http://hdl.handle.net/1963/2917 U1 - 1783 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Some remarks on multidimensional systems of conservation laws JF - Rend. Mat. Acc. Lincei, s. 9, v. 15 (2004) 3-4, pp. 225 - 233 Y1 - 2004 A1 - Alberto Bressan AB - This note is concerned with the Cauchy problem for hyperbolic systems of conservation\\nlaws in several space dimensions. We first discuss an example of ill-posedness, for a special system\\nhaving a radial symmetry property. Some conjectures are formulated, on the compactness of the set of\\nflow maps generated by vector fields with bounded variation. PB - Accademia Nazionale dei Lincei UR - http://hdl.handle.net/1963/3642 U1 - 662 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Stability rates for patchy vector fields JF - ESAIM COCV 10 (2004) 168-200 Y1 - 2004 A1 - Fabio Ancona A1 - Alberto Bressan AB - This paper is concerned with the stability of the set of trajectories of a patchy vector field, in the presence of impulsive perturbations. Patchy vector fields are discontinuous, piecewise smooth vector fields that were introduced in Ancona and Bressan (1999) to study feedback stabilization problems. For patchy vector fields in the plane, with polygonal patches in generic position, we show that the distance between a perturbed trajectory and an unperturbed one is of the same order of magnitude as the impulsive forcing term. PB - EDP Sciences UR - http://hdl.handle.net/1963/2959 U1 - 1741 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Some results on the boundary control of systems of conservation laws JF - SIAM J.Control Optim. 41 (2003),no.2, 607 Y1 - 2003 A1 - Alberto Bressan A1 - Fabio Ancona A1 - Giuseppe Maria Coclite PB - SISSA Library UR - http://hdl.handle.net/1963/1615 U1 - 2503 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Stability of L^infty Solutions of Temple Class Systems JF - Differential Integral Equations 13 (2000) 1503-1528 Y1 - 2000 A1 - Alberto Bressan A1 - Paola Goatin AB -

Let $u_t+f(u)_x=0$ be a strictly hyperbolic, genuinely nonlinear system of conservation laws of Temple class. In this paper, a continuous semigroup of solutions is constructed on a domain of $L^\infty$ functions, with possibly unbounded variation. Trajectories depend Lipschitz continuously on the initial data, in the $L^1$ distance. Moreover, we show that a weak solution of the Cauchy problem coincides with the corresponding semigroup trajectory if and only if it satisfies an entropy condition of Oleinik type, concerning the decay of positive waves.

PB - Khayyam Publishing UR - http://hdl.handle.net/1963/3256 U1 - 1445 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws JF - Indiana Univ. Math. J. 48 (1999), no. 1, 43--84 Y1 - 1999 A1 - Alberto Bressan A1 - Philippe G. LeFloch AB - The paper is concerned with the qualitative structure of entropy solutions to a strictly hyperbolic, genuinely nonlinear system of conservation laws. We first give an accurate description of the local and global wave-front structure of a BV solution, generated by a front tracking algorithm. PB - Indiana University UR - http://hdl.handle.net/1963/3374 U1 - 956 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - The semigroup generated by a temple class system with large data JF - Differential Integral Equations 10 (1997), no. 3, 401-418 Y1 - 1997 A1 - Paolo Baiti A1 - Alberto Bressan AB - We consider the Cauchy problem $$u_t + [F(u)]_x=0, u(0,x)=\\\\bar u(x) (*)$$ for a nonlinear $n\\\\times n$ system of conservation laws with coinciding shock and rarefaction curves. Assuming the existence of a coordinates system made of Riemann invariants, we prove the existence of a weak solution of (*) that depends in a lipschitz continuous way on the initial data, in the class of functions with arbitrarily large but bounded total variation. PB - SISSA Library UR - http://hdl.handle.net/1963/1023 U1 - 2833 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Shift-differentiability of the flow generated by a conservation law JF - Discrete Contin. Dynam. Systems 3 (1997), no. 1, 35--58. Y1 - 1997 A1 - Alberto Bressan A1 - Graziano Guerra AB - The paper introduces a notion of \\\"shift-differentials\\\" for maps with values in the space BV. These differentials describe first order variations of a given functin $u$, obtained by horizontal shifts of the points of its graph. The flow generated by a scalar conservation law is proved to be generically shift-differentiable, according to the new definition. PB - SISSA Library UR - http://hdl.handle.net/1963/1033 U1 - 2823 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Structural stability for time-optimal planar sytheses JF - Dynam. Contin. Discrete Impuls. Systems 3 (1997), no. 3, 335--371 Y1 - 1997 A1 - Alberto Bressan A1 - Benedetto Piccoli PB - SISSA Library UR - http://hdl.handle.net/1963/997 U1 - 2859 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - The semigroup approach to systems of conservation laws JF - Mat. Contemp. 10 (1996) 21-74 Y1 - 1996 A1 - Alberto Bressan PB - Sociedade Brasileira de Matematica UR - http://hdl.handle.net/1963/1037 U1 - 2819 U2 - Mathematics U3 - Functional Analysis and Applications ER -