We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish non branching geodesic space. We show that we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce an assumption on the transport problem π which implies that the conditional probabilities of the first marginal on each geodesic are continuous. It is known that this regularity is sufficient for the construction of an optimal transport map.

%B Nonlinear Conservation Laws and Applications %I Springer US %C Boston, MA %P 217–233 %@ 978-1-4419-9554-4 %G eng %0 Journal Article %J Discrete Contin. Dyn. Syst. 23 (2009) 29-48 %D 2009 %T On the convergence of viscous approximations after shock interactions %A Alberto Bressan %A Carlotta Donadello %X We consider a piecewise smooth solution to a scalar conservation law, with possibly interacting shocks. We show that, after the interactions have taken place, vanishing viscosity approximations can still be represented by a regular expansion on smooth regions and by a singular perturbation expansion near the shocks, in terms of powers of the viscosity coefficient. %B Discrete Contin. Dyn. Syst. 23 (2009) 29-48 %I American Institute of Mathematical Sciences %G en_US %U http://hdl.handle.net/1963/3412 %1 923 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2009-01-09T12:23:11Z\\nNo. of bitstreams: 1\\n019.pdf: 273198 bytes, checksum: 4513df79a27da75ea57c4374c83b0cf6 (MD5) %R 10.3934/dcds.2009.23.29 %0 Journal Article %J Arch. Ration. Mech. Anal. 183 (2007) 163-185 %D 2007 %T Asymptotic variational wave equations %A Alberto Bressan %A Zhang Ping %A Zheng Yuxi %X We investigate the equation $(u_t + (f(u))_x)_x = f\\\'\\\'(u) (u_x)^2/2$ where $f(u)$ is a given smooth function. Typically $f(u)= u^2/2$ or $u^3/3$. This equation models unidirectional and weakly nonlinear waves for the variational wave equation $u_{tt} - c(u) (c(u)u_x)_x =0$ which models some liquid crystals with a natural sinusoidal $c$. The equation itself is also the Euler-Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.\\nWe establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function $f$ has Lipschitz continuous second-order derivative. In the case where $f$ is convex, the Cauchy problem is well-posed also within the class of dissipative solutions. However, when $f$ is not convex, we show that the dissipative solutions do not depend continuously on the initial data. %B Arch. Ration. Mech. Anal. 183 (2007) 163-185 %G en_US %U http://hdl.handle.net/1963/2182 %1 2062 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-05T07:08:27Z\\nNo. of bitstreams: 1\\n0502124v1.pdf: 205978 bytes, checksum: a1efb4667d7315509947b4d7a4999bd1 (MD5) %R 10.1007/s00205-006-0014-8 %0 Report %D 2007 %T BV instability for the Lax-Friedrichs scheme %A Paolo Baiti %A Alberto Bressan %A Helge Kristian Jenssen %X It is proved that discrete shock profiles (DSPs) for the Lax-Friedrichs scheme for a system of conservation laws do not necessarily depend continuously in BV on their speed. We construct examples of $2 \\\\times 2$-systems for which there are sequences of DSPs with speeds converging to a rational number. Due to a resonance phenomenon, the difference between the limiting DSP and any DSP in the sequence will contain an order-one amount of variation. %G en_US %U http://hdl.handle.net/1963/2335 %1 1681 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-11-05T09:55:21Z\\nNo. of bitstreams: 1\\n0502043v1.pdf: 257155 bytes, checksum: 5055aa260394ac1339fa775b656f3b15 (MD5) %0 Journal Article %J Ann. Inst. H. Poincare Anal. Non Lineaire 24 (2007) 279-310 %D 2007 %T Nearly time optimal stabilizing patchy feedbacks %A Fabio Ancona %A Alberto Bressan %X We consider the time optimal stabilization problem for a nonlinear control system $\\\\dot x=f(x,u)$. Let $\\\\tau(y)$ be the minimum time needed to steer the system from the state $y\\\\in\\\\R^n$ to the origin, and call $\\\\A(T)$ the set of initial states that can be steered to the origin in time $\\\\tau(y)\\\\leq T$. Given any $\\\\ve>0$, in this paper we construct a patchy feedback $u=U(x)$ such that every solution of $\\\\dot x=f(x, U(x))$, $x(0)=y\\\\in \\\\A(T)$ reaches an $\\\\ve$-neighborhood of the origin within time $\\\\tau(y)+\\\\ve$. %B Ann. Inst. H. Poincare Anal. Non Lineaire 24 (2007) 279-310 %G en_US %U http://hdl.handle.net/1963/2185 %1 2059 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-05T08:22:02Z\\nNo. of bitstreams: 1\\n0512531v1.pdf: 428805 bytes, checksum: 8eae0ca68a7339938991d987a677d6f9 (MD5) %R 10.1016/j.anihpc.2006.03.010 %0 Journal Article %J Comm. Math. Phys. 266 (2006) 471-497 %D 2006 %T Conservative Solutions to a Nonlinear Variational Wave Equation %A Alberto Bressan %A Zheng Yuxi %X We establish the existence of a conservative weak solution to the Cauchy problem for the nonlinear variational wave equation $u_{tt} - c(u)(c(u)u_x)_x=0$, for initial data of finite energy. Here $c(\\\\cdot)$ is any smooth function with uniformly positive bounded values. %B Comm. Math. Phys. 266 (2006) 471-497 %G en_US %U http://hdl.handle.net/1963/2184 %1 2060 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-05T08:06:00Z\\nNo. of bitstreams: 1\\n0502058v1.pdf: 265359 bytes, checksum: 77983b4834210b17aa4c758dbf189d4a (MD5) %R 10.1007/s00220-006-0047-8 %0 Report %D 2006 %T Infinite Horizon Noncooperative Differential Games %A Alberto Bressan %A Fabio Simone Priuli %X For a non-cooperative differential game, the value functions of the various players satisfy a system of Hamilton-Jacobi equations. In the present paper, we consider a class of infinite-horizon games with nonlinear costs exponentially discounted in time. By the analysis of the value\\nfunctions, we establish the existence of Nash equilibrium solutions in feedback form and provide results and counterexamples on their uniqueness and stability. %B J. Differential Equations 227 (2006) 230-257 %G en_US %U http://hdl.handle.net/1963/1720 %1 2431 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-01-24T09:08:57Z\\nNo. of bitstreams: 1\\nmath.AP0505212.pdf: 267217 bytes, checksum: 4c2f5eb11250fa9b31d9b46983c201e0 (MD5) %R 10.1016/j.jde.2006.01.005 %0 Journal Article %J Comm. Pure Appl. Math. 59 (2006) 1604-1638 %D 2006 %T An instability of the Godunov scheme %A Alberto Bressan %A Helge Kristian Jenssen %A Paolo Baiti %X We construct a solution to a $2\\\\times 2$ strictly hyperbolic system of conservation laws, showing that the Godunov scheme \\\\cite{Godunov59} can produce an arbitrarily large amount of oscillations. This happens when the speed of a shock is close to rational, inducing a resonance with the grid. Differently from the Glimm scheme or the vanishing viscosity method, for systems of conservation laws our counterexample indicates that no a priori BV bounds or $L^1$ stability estimates can in general be valid for finite difference schemes. %B Comm. Pure Appl. Math. 59 (2006) 1604-1638 %G en_US %U http://hdl.handle.net/1963/2183 %1 2061 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-05T07:52:50Z\\nNo. of bitstreams: 1\\n0502125v1.pdf: 337578 bytes, checksum: 41193faac363c00e9f80d6c05c0b098c (MD5) %R 10.1002/cpa.20141 %0 Journal Article %J Discrete Contin. Dyn. Syst. 13 (2005) 1-12 %D 2005 %T On the Blow-up for a Discrete Boltzmann Equation in the Plane %A Alberto Bressan %A Massimo Fonte %X We study the possibility of finite-time blow-up for a two dimensional Broadwell model. In a set of rescaled variables, we prove that no self-similar blow-up solution exists, and derive some a priori bounds on the blow-up rate. In the final section, a possible blow-up scenario is discussed. %B Discrete Contin. Dyn. Syst. 13 (2005) 1-12 %G en_US %U http://hdl.handle.net/1963/2244 %1 2000 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-17T09:31:05Z\\nNo. of bitstreams: 1\\n0403047v2.pdf: 116770 bytes, checksum: a7432c5b660f4e4672c952003dd0210f (MD5) %0 Journal Article %J SIAM J. Math. Anal. 37 (2005) 996-1026 %D 2005 %T Global solutions of the Hunter-Saxton equation %A Alberto Bressan %A Adrian Constantin %X We construct a continuous semigroup of weak, dissipative solutions to a nonlinear partial differential equations modeling nematic liquid crystals. A new distance functional, determined by a problem of optimal transportation, yields sharp estimates on the continuity of solutions with respect to the initial data. %B SIAM J. Math. Anal. 37 (2005) 996-1026 %G en_US %U http://hdl.handle.net/1963/2256 %1 1991 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-18T10:16:52Z\\nNo. of bitstreams: 1\\n0502059v1.pdf: 276204 bytes, checksum: ba0bfd37b8aa3a5ef2c0ef7358081cf9 (MD5) %R 10.1137/050623036 %0 Report %D 2005 %T An Optimal Transportation Metric for Solutions of the Camassa-Holm Equation %A Alberto Bressan %A Massimo Fonte %X In this paper we construct a global, continuous flow of solutions to the Camassa-Holm equation on the entire space H1. Our solutions are conservative, in the sense that the total energy int[(u2 + u2x) dx] remains a.e. constant in time. Our new approach is based on a distance functional J(u, v), defined in terms of an optimal transportation problem, which satisfies d dtJ(u(t), v(t)) ≤ κ · J(u(t), v(t)) for every couple of solutions. Using this new distance functional, we can construct arbitrary solutions as the uniform limit of multi-peakon solutions, and prove a general uniqueness result. %B Methods Appl. Anal. 12 (2005) 191-219 %G en_US %U http://hdl.handle.net/1963/1719 %1 2432 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-01-24T09:03:54Z\\nNo. of bitstreams: 1\\nmath.AP0504450.pdf: 261370 bytes, checksum: 75945d031343a82836b46ab9705ed6de (MD5) %0 Journal Article %J Ann. of Math. 161 (2005) 223-342 %D 2005 %T Vanishing viscosity solutions of nonlinear hyperbolic systems %A Stefano Bianchini %A Alberto Bressan %X We consider the Cauchy problem for a strictly hyperbolic, $n\\\\times n$ system in one space dimension: $u_t+A(u)u_x=0$, assuming that the initial data has small total variation.\\nWe show that the solutions of the viscous approximations $u_t+A(u)u_x=\\\\ve u_{xx}$ are defined globally in time and satisfy uniform BV estimates, independent of $\\\\ve$. Moreover, they depend continuously on the initial data in the $\\\\L^1$ distance, with a Lipschitz constant independent of $t,\\\\ve$. Letting $\\\\ve\\\\to 0$, these viscous solutions converge to a unique limit, depending Lipschitz continuously on the initial data. In the conservative case where $A=Df$ is the Jacobian of some flux function $f:\\\\R^n\\\\mapsto\\\\R^n$, the vanishing viscosity limits are precisely the unique entropy weak solutions to the system of conservation laws $u_t+f(u)_x=0$. %B Ann. of Math. 161 (2005) 223-342 %I Annals of Mathematics %G en_US %U http://hdl.handle.net/1963/3074 %1 1259 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-10T10:46:16Z\\nNo. of bitstreams: 1\\n0111321v1.pdf: 746310 bytes, checksum: f3b7c9e76e33050e9f367ba2c57e2161 (MD5) %0 Journal Article %J Comm. Pure Appl. Math. 57 (2004) 1075-1109 %D 2004 %T On the convergence rate of vanishing viscosity approximations %A Alberto Bressan %A Tong Yang %X Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound $\\\\big\\\\|u(t,\\\\cdot)-u^\\\\ve(t,\\\\cdot)\\\\big\\\\|_{\\\\L^1}= \\\\O(1)(1+t)\\\\cdot \\\\sqrt\\\\ve|\\\\ln\\\\ve|$ on the distance between an exact BV solution $u$ and a viscous approximation $u^\\\\ve$, letting the viscosity coefficient $\\\\ve\\\\to 0$. In the proof, starting from $u$ we construct an approximation of the viscous solution $u^\\\\ve$ by taking a mollification $u*\\\\phi_{\\\\strut \\\\sqrt\\\\ve}$ and inserting viscous shock profiles at the locations of finitely many large shocks, for each fixed $\\\\ve$. Error estimates are then obtained by introducing new Lyapunov functionals which control shock interactions, interactions between waves of different families and by using sharp decay estimates for positive nonlinear waves. %B Comm. Pure Appl. Math. 57 (2004) 1075-1109 %I Wiley %G en_US %U http://hdl.handle.net/1963/2915 %1 1785 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-09-11T13:30:25Z\\nNo. of bitstreams: 1\\nmath.AP0307141.pdf: 265243 bytes, checksum: 795adebd067228364ac1240add5f7b02 (MD5) %R 10.1002/cpa.20030 %0 Journal Article %J Internat. J. Game Theory 32 (2004) 561-593 %D 2004 %T Semi-cooperative strategies for differential games %A Alberto Bressan %A Wen Shen %X 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. %B Internat. J. Game Theory 32 (2004) 561-593 %I Springer %G en_US %U http://hdl.handle.net/1963/2893 %1 1807 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-09-11T08:11:56Z\\nNo. of bitstreams: 1\\n069.pdf: 389569 bytes, checksum: 3cea0d9a5adfd0be673a7d69218e238a (MD5) %R 10.1007/s001820400180 %0 Journal Article %J SIAM J. Math. Anal. 36 (2004) 659-677 %D 2004 %T A sharp decay estimate for positive nonlinear waves %A Alberto Bressan %A Tong Yang %X 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. %B SIAM J. Math. Anal. 36 (2004) 659-677 %I SIAM %G en_US %U http://hdl.handle.net/1963/2916 %1 1784 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-09-11T13:37:38Z\\nNo. of bitstreams: 1\\nmath.AP0307140.pdf: 167236 bytes, checksum: 9e7a6fecd3de67843ca5c73f13bd841d (MD5) %R 10.1137/S0036141003427774 %0 Journal Article %J SIAM J. Control Optim. 43 (2004) 194-215 %D 2004 %T Small BV solutions of hyperbolic noncooperative differential games %A Alberto Bressan %A Wen Shen %X 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. %B SIAM J. Control Optim. 43 (2004) 194-215 %I SIAM %G en_US %U http://hdl.handle.net/1963/2917 %1 1783 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-09-11T13:46:04Z\\nNo. of bitstreams: 1\\n021.pdf: 296267 bytes, checksum: 16f3c148f2a00551188fee22ac90f666 (MD5) %R 10.1137/S0363012903425581 %0 Journal Article %J Rend. Mat. Acc. Lincei, s. 9, v. 15 (2004) 3-4, pp. 225 - 233 %D 2004 %T Some remarks on multidimensional systems of conservation laws %A Alberto Bressan %X 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. %B Rend. Mat. Acc. Lincei, s. 9, v. 15 (2004) 3-4, pp. 225 - 233 %I Accademia Nazionale dei Lincei %G en_US %U http://hdl.handle.net/1963/3642 %1 662 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2009-05-21T10:01:16Z\\nNo. of bitstreams: 1\\nM2004-34-21.pdf: 187400 bytes, checksum: 045bd5d68458b90865a3a4c132a90163 (MD5) %0 Journal Article %J ESAIM COCV 10 (2004) 168-200 %D 2004 %T Stability rates for patchy vector fields %A Fabio Ancona %A Alberto Bressan %X 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. %B ESAIM COCV 10 (2004) 168-200 %I EDP Sciences %G en_US %U http://hdl.handle.net/1963/2959 %1 1741 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-09-22T08:49:39Z\\nNo. of bitstreams: 1\\n0111109v1.pdf: 336899 bytes, checksum: 851fc204f3e9f254b9122e0bd1f8894f (MD5) %R 10.1051/cocv:2004003 %0 Journal Article %D 2003 %T An ill posed Cauchy problem for a hyperbolic system in two space dimensions %A Alberto Bressan %X The theory of weak solutions for nonlinear conservation laws is now well developed in the case of scalar equations [3] and for one-dimensional hyperbolic systems [1, 2]. For systems in several space dimensions, however, even the global existence of solutions to the Cauchy problem remains a challenging open question. In this note we construct a conterexample showing that, even for a simple class of hyperbolic systems, in two space dimensions the Cauchy problem can be ill posed. %I Università di Padova %G en_US %U http://hdl.handle.net/1963/2913 %1 1787 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-09-11T13:18:40Z\\nNo. of bitstreams: 1\\nmath.AP0302227.pdf: 126071 bytes, checksum: 3536c415c6e375bb55591a913ced63c2 (MD5) %0 Journal Article %J Rend. Sem. Mat. Univ. Padova 110 (2003) 97-102 %D 2003 %T A lemma and a conjecture on the cost of rearrangements %A Alberto Bressan %X Consider a stack of books, containing both white and black books. Suppose that we want to sort them out, putting the white books on the right, and the black books on the left (fig.~1). This will be done by a finite sequence of elementary transpositions. In other words, if we have a stack of all black books of length $a$ followed by a stack of all white books of length $b$, we are allowed to reverse their order at the cost of $a+b$. We are interested in a lower bound on the total cost of the rearrangement. %B Rend. Sem. Mat. Univ. Padova 110 (2003) 97-102 %I Università di Padova %G en_US %U http://hdl.handle.net/1963/2914 %1 1786 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-09-11T13:23:07Z\\nNo. of bitstreams: 1\\nmath.AP0302228.pdf: 114252 bytes, checksum: 85213b6876996792213f6032c6fe2035 (MD5) %0 Journal Article %J SIAM J.Control Optim. 41 (2003),no.2, 607 %D 2003 %T Some results on the boundary control of systems of conservation laws %A Alberto Bressan %A Fabio Ancona %A Giuseppe Maria Coclite %B SIAM J.Control Optim. 41 (2003),no.2, 607 %I SISSA Library %G en %U http://hdl.handle.net/1963/1615 %1 2503 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T13:05:20Z (GMT). No. of bitstreams: 1\\nmath.AP0205246.pdf: 155904 bytes, checksum: 294663b00a9e6b6ae91096fd628a942a (MD5)\\n Previous issue date: 2002 %R 10.1137/S0363012901392529 %0 Journal Article %J SIAM J. Control Optim. 41 (2002) 607-622 %D 2002 %T On the Boundary Control of Systems of Conservation Laws %A Alberto Bressan %A Giuseppe Maria Coclite %X The paper is concerned with the boundary controllability of entropy weak solutions to hyperbolic systems of conservation laws. We prove a general result on the asymptotic stabilization of a system near a constant state. On the other hand, we give an example showing that exact controllability in finite time cannot be achieved, in general. %B SIAM J. Control Optim. 41 (2002) 607-622 %I SIAM %G en_US %U http://hdl.handle.net/1963/3070 %1 1263 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-10T10:05:11Z\\nNo. of bitstreams: 1\\n0108224v1.pdf: 198822 bytes, checksum: b08b507c4b17c8b48aaa3aafdc4d7360 (MD5) %R 10.1137/S0363012901392529 %0 Journal Article %J Commun. Pure Appl. Anal. 1 (2002) 161-190 %D 2002 %T A center manifold technique for tracing viscous waves %A Stefano Bianchini %A Alberto Bressan %X In this paper we introduce a new technique for tracing viscous travelling profiles. To illustrate the method, we consider a special 2 x 2 hyperbolic system of conservation laws with viscosity, and show that any solution can be locally decomposed as the sum of 2 viscous travelling profiles. This yields the global existence, stability and uniform BV bounds for every solution with suitably small BV data. %B Commun. Pure Appl. Anal. 1 (2002) 161-190 %I American Institute of Mathematical Sciences %G en_US %U http://hdl.handle.net/1963/3075 %1 1258 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-10T10:55:34Z\\nNo. of bitstreams: 1\\npreprint2001_53.pdf: 444727 bytes, checksum: 26cee63284464e4850e1e6426216fe70 (MD5) %0 Journal Article %J SIAM J. Control Optim. 41 (2002) 1455-1476 %D 2002 %T Flow Stability of Patchy Vector Fields and Robust Feedback Stabilization %A Fabio Ancona %A Alberto Bressan %X The paper is concerned with patchy vector fields, a class of discontinuous, piecewise smooth vector fields that were introduced in AB to study feedback stabilization problems. We prove the stability of the corresponding solution set w.r.t. a wide class of impulsive perturbations. These results yield the robusteness of patchy feedback controls in the presence of measurement errors and external disturbances. %B SIAM J. Control Optim. 41 (2002) 1455-1476 %I SIAM %G en_US %U http://hdl.handle.net/1963/3073 %1 1260 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-10T10:37:20Z\\nNo. of bitstreams: 1\\n0111107v1.pdf: 227962 bytes, checksum: 76bf14ce144f1803bb403d8781e00a0d (MD5) %R 10.1137/S0363012901391676 %0 Journal Article %J Nonlinear Anal. 51 (2002) 649-662 %D 2002 %T On a Lyapunov functional relating shortening curves and viscous conservation laws %A Stefano Bianchini %A Alberto Bressan %X We study a nonlinear functional which controls the area swept by a curve moving in the plane in the direction of curvature. In turn, this yields a priori estimates on solutions to a class of parabolic equations and of scalar viscous conservation laws. A further application provides an estimate on the \\\"change of shape\\\" of a BV solution to a scalar conservation law. %B Nonlinear Anal. 51 (2002) 649-662 %I Elsevier %G en %U http://hdl.handle.net/1963/1337 %1 3118 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:56:28Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1999 %R 10.1016/S0362-546X(01)00848-3 %0 Journal Article %J Discrete Cont. Dyn. Syst. 7 (2001) 449-476 %D 2001 %T A case study in vanishing viscosity %A Stefano Bianchini %A Alberto Bressan %B Discrete Cont. Dyn. Syst. 7 (2001) 449-476 %I American Institute of Mathematical Sciences %G en_US %U http://hdl.handle.net/1963/3091 %1 1242 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-14T08:16:28Z\\nNo. of bitstreams: 1\\ncasestudy.pdf: 289769 bytes, checksum: d1ec5d9048e85bd7ebbd1e3de3259336 (MD5) %0 Journal Article %J Discrete Contin. Dynam. Systems 6 (2000) 21-38 %D 2000 %T BV estimates for multicomponent chromatography with relaxation %A Alberto Bressan %A Wen Shen %X 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. %B Discrete Contin. Dynam. Systems 6 (2000) 21-38 %I SISSA Library %G en %U http://hdl.handle.net/1963/1336 %1 3119 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:56:27Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1999 %0 Journal Article %J Indiana Univ. Math. J. 49 (2000) 1673-1714 %D 2000 %T BV solutions for a class of viscous hyperbolic systems %A Stefano Bianchini %A Alberto Bressan %B Indiana Univ. Math. J. 49 (2000) 1673-1714 %I Indiana University Mathematics Journal %G en_US %U http://hdl.handle.net/1963/3194 %1 1107 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-27T16:30:56Z\\nNo. of bitstreams: 1\\nBVsolutions.pdf: 250249 bytes, checksum: e7f587fcd03629f68d93b0b14058f3e6 (MD5) %R 10.1512/iumj.2000.49.1776 %0 Journal Article %J Chinese Ann. Math. B, 2000, 21, 269 %D 2000 %T On the convergence of Godunov scheme for nonlinear hyperbolic systems %A Alberto Bressan %A Helge Kristian Jenssen %B Chinese Ann. Math. B, 2000, 21, 269 %I SISSA Library %G en %U http://hdl.handle.net/1963/1473 %1 2690 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T13:02:54Z (GMT). No. of bitstreams: 0\\n Previous issue date: 2000 %0 Journal Article %J Differential Integral Equations 13 (2000) 1503-1528 %D 2000 %T Stability of L^infty Solutions of Temple Class Systems %A Alberto Bressan %A Paola Goatin %XLet $u_t+f(u)_x=0$ 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.

%B Differential Integral Equations 13 (2000) 1503-1528 %I Khayyam Publishing %G en_US %U http://hdl.handle.net/1963/3256 %1 1445 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-11-06T13:16:51Z\\nNo. of bitstreams: 1\\nTemple.pdf: 221742 bytes, checksum: a13773198af84c04068cf3021f12d3c8 (MD5) %0 Journal Article %J Discrete Contin. Dynam. Systems 6 (2000) 673-682 %D 2000 %T A Uniqueness Condition for Hyperbolic Systems of Conservation Laws %A Alberto Bressan %A Marta Lewicka %X Consider the Cauchy problem for a hyperbolic $n\\\\times n$ system of conservation laws in one space dimension: $$u_t+f(u)_x=0, u(0,x)=\\\\bar u(x).\\\\eqno(CP)$$ Relying on the existence of a continuous semigroup of solutions, we prove that the entropy admissible solution of (CP) is unique within the class of functions $u=u(t,x)$ which have bounded variation along a suitable family of space-like curves. %B Discrete Contin. Dynam. Systems 6 (2000) 673-682 %I American Institute of Mathematical Sciences %G en_US %U http://hdl.handle.net/1963/3195 %1 1106 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-27T16:44:05Z\\nNo. of bitstreams: 1\\n033.pdf: 151263 bytes, checksum: 7e5335ead21fcf20991edff341d1f424 (MD5) %0 Book %B Mem. Amer. Math. Soc. 146 (2000), no. 694, 134 p. %D 2000 %T Well-posedness of the Cauchy problem for n x n systems of conservation laws %A Alberto Bressan %A Graziano Crasta %A Benedetto Piccoli %B Mem. Amer. Math. Soc. 146 (2000), no. 694, 134 p. %I American Mathematical Society %G en_US %U http://hdl.handle.net/1963/3495 %1 769 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2009-02-06T11:34:33Z\\nNo. of bitstreams: 1\\nbcpams.pdf: 323551 bytes, checksum: 9bc3f06472e13a47266da010edb58ba1 (MD5) %0 Journal Article %J Rev. Mat. Complut. 12 (1999) 135-200 %D 1999 %T Hyperbolic Systems of Conservation Laws %A Alberto Bressan %X This is a survey paper, written in the occasion of an invited talk given by the author at the Universidad Complutense in Madrid, October 1998. Its purpose is to provide an account of some recent advances in the mathematical theory of hyperbolic systems of conservation laws in one space dimension. After a brief review of basic concepts, we describe in detail the method of wave-front tracking approximation and present some of the latest results on uniqueness and stability of entropy weak solutions. %B Rev. Mat. Complut. 12 (1999) 135-200 %G en %U http://hdl.handle.net/1963/1855 %1 77 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-09-25T15:15:53Z\\nNo. of bitstreams: 1\\nhyperbolic.pdf: 538861 bytes, checksum: 9f2c0c2ce6d8dc618aefeec91ce6293c (MD5) %0 Journal Article %J Arch. Ration. Mech. Anal. 149 (1999), no. 1, 1--22 %D 1999 %T L-1 stability estimates for n x n conservation laws %A Alberto Bressan %A Tai-Ping Liu %A Tong Yang %X Let $u_t+f(u)_x=0$ be a strictly hyperbolic $n\\\\times n$ system of conservation laws, each characteristic field being linearly degenerate or genuinely nonlinear. In this paper we explicitly define a functional $\\\\Phi=\\\\Phi(u,v)$, equivalent to the $L^1$ distance, which is `almost decreasing\\\', i.e., $\\\\Phi(u(t),v(t))-\\\\Phi(u(s),v(s))\\\\leq\\\\break O (\\\\epsilon)·(t-s)$ for all $t>s\\\\geq 0$, for every pair of $\\\\epsilon$-approximate solutions $u,v$ with small total variation, generated by a wave-front-tracking algorithm. The small parameter $\\\\epsilon$ here controls the errors in the wave speeds, the maximum size of rarefaction fronts and the total strength of all non-physical waves in $u$ and in $v$. From the above estimate, it follows that front-tracking approximations converge to a unique limit solution, depending Lipschitz continuously on the initial data, in the $L^1$ norm. This provides a new proof of the existence of the standard Riemann semigroup generated by an $n\\\\times n$ system of conservation laws.\\\'\\\' %B Arch. Ration. Mech. Anal. 149 (1999), no. 1, 1--22 %I Springer %G en_US %U http://hdl.handle.net/1963/3373 %1 957 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-11-28T18:35:48Z\\nNo. of bitstreams: 1\\nBressan_Liu.pdf: 1602231 bytes, checksum: fe5990668e708e14f93a4f78715b929c (MD5) %R 10.1007/s002050050165 %0 Journal Article %J J. Differential Equations 156 (1999), no. 1, 26--49 %D 1999 %T Oleinik type estimates and uniqueness for n x n conservation laws %A Alberto Bressan %A Paola Goatin %X Let $u_t+f(u)_x=0$ be a strictly hyperbolic $n\\\\times n$ system of conservation laws in one space dimension. Relying on the existence of a semigroup of solutions, we first establish the uniqueness of entropy admissible weak solutions to the Cauchy problem, under a mild assumption on the local oscillation of $u$ in a forward neighborhood of each point in the $t\\\\text{-}x$ plane. In turn, this yields the uniqueness of weak solutions which satisfy a decay estimate on positive waves of genuinely nonlinear families, thus extending a classical result proved by Oleĭnik in the scalar case. %B J. Differential Equations 156 (1999), no. 1, 26--49 %I Elsevier %G en_US %U http://hdl.handle.net/1963/3375 %1 955 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-12-01T09:26:02Z\\nNo. of bitstreams: 1\\nOleinik_type.pdf: 1567166 bytes, checksum: ba588e7f2b587d26f5b613a53557bb2f (MD5) %R 10.1006/jdeq.1998.3606 %0 Journal Article %J Indiana Univ. Math. J. 48 (1999), no. 1, 43--84 %D 1999 %T Structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws %A Alberto Bressan %A Philippe G. LeFloch %X 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. %B Indiana Univ. Math. J. 48 (1999), no. 1, 43--84 %I Indiana University %G en_US %U http://hdl.handle.net/1963/3374 %1 956 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-12-01T09:02:14Z\\nNo. of bitstreams: 1\\nStructural_stability.pdf: 3175042 bytes, checksum: c04760cbf353315764f5aa8e879b0a62 (MD5) %R 10.1512/iumj.1999.48.1524 %0 Journal Article %D 1999 %T Vanishing viscosity solutions of hyperbolic systems on manifolds %A Stefano Bianchini %A Alberto Bressan %I SISSA Library %G en %U http://hdl.handle.net/1963/1238 %1 2705 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:55:09Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1999 %0 Journal Article %J Arch. Rational Mech. Anal. 142 (1998), no. 2, 155-176 %D 1998 %T Error bounds for a deterministic version of the Glimm scheme %A Andrea Marson %A Alberto Bressan %X Consider the hyperbolic system of conservation laws $u_t F(u)_x=0. Let $u$ be the unique viscosity solution with initial condition $u(0,x)=\\\\bar u(x)$ and let $u^\\\\varepsilon$ be an approximate solution constructed by the Glimm scheme, corresponding to the mesh sizes $\\\\Delta x,\\\\Delta t=O(\\\\Delta x). With a suitable choise of the sampling sequence, we prove the estimate $$ \\\\left\\\\Vert u^\\\\varepsilon(t,\\\\cdot)-u(t,\\\\cdot) \\\\right\\\\Vert_1=o(1)\\\\cdot\\\\sqrt{\\\\Delta x}\\\\vert\\\\ln\\\\Delta x\\\\vert. $$ %B Arch. Rational Mech. Anal. 142 (1998), no. 2, 155-176 %I Springer %G en %U http://hdl.handle.net/1963/1045 %1 2811 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:42:05Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1995 %R 10.1007/s002050050088 %0 Journal Article %J SIAM J. Control Optim. 36 (1998) 12-32 %D 1998 %T A generic classification of time-optimal planar stabilizing feedbacks %A Alberto Bressan %A Benedetto Piccoli %X Consider the problem of stabilization at the origin in minimum time for a planar control system affine with respect to the control. For a family of generic vector fields, a topological equivalence relation on the corresponding time-optimal feedback synthesis was introduced in a previous paper [Dynamics of Continuous, Discrete and Impulsive Systems, 3 (1997), pp. 335--371]. The set of equivalence classes can be put in a one-to-one correspondence with a discrete family of graphs. This provides a classification of the global structure of generic time-optimal stabilizing feedbacks in the plane, analogous to the classification of smooth dynamical systems developed by Peixoto. %B SIAM J. Control Optim. 36 (1998) 12-32 %I SISSA Library %G en %U http://hdl.handle.net/1963/998 %1 2858 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:41:28Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1995 %R 10.1137/S0363012995291117 %0 Journal Article %J Nonlinear Analysis 34 (1998) 637-652 %D 1998 %T Uniqueness for discontinuous ODE and conservation laws %A Alberto Bressan %A Wen Shen %X 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 $\\\\L^\\\\infty$. %B Nonlinear Analysis 34 (1998) 637-652 %I Elsevier %G en_US %U http://hdl.handle.net/1963/3699 %1 606 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2009-07-28T09:39:33Z\\nNo. of bitstreams: 1\\nbressan.pdf: 182373 bytes, checksum: 0eb82866d65dea9685ab7ff75c16c0d2 (MD5) %R 10.1016/S0362-546X(97)00590-7 %0 Journal Article %J Differential Integral Equations 10 (1997), no. 3, 401-418 %D 1997 %T The semigroup generated by a temple class system with large data %A Paolo Baiti %A Alberto Bressan %X 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. %B Differential Integral Equations 10 (1997), no. 3, 401-418 %I SISSA Library %G en %U http://hdl.handle.net/1963/1023 %1 2833 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:41:49Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1995 %0 Journal Article %J Discrete Contin. Dynam. Systems 3 (1997), no. 1, 35--58. %D 1997 %T Shift-differentiability of the flow generated by a conservation law %A Alberto Bressan %A Graziano Guerra %X 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. %B Discrete Contin. Dynam. Systems 3 (1997), no. 1, 35--58. %I SISSA Library %G en %U http://hdl.handle.net/1963/1033 %1 2823 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:41:57Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1995 %0 Journal Article %J Dynam. Contin. Discrete Impuls. Systems 3 (1997), no. 3, 335--371 %D 1997 %T Structural stability for time-optimal planar sytheses %A Alberto Bressan %A Benedetto Piccoli %B Dynam. Contin. Discrete Impuls. Systems 3 (1997), no. 3, 335--371 %I SISSA Library %G en %U http://hdl.handle.net/1963/997 %1 2859 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:41:28Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1995 %0 Journal Article %J Mat. Contemp. 10 (1996) 21-74 %D 1996 %T The semigroup approach to systems of conservation laws %A Alberto Bressan %B Mat. Contemp. 10 (1996) 21-74 %I Sociedade Brasileira de Matematica %G en %U http://hdl.handle.net/1963/1037 %1 2819 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:42:00Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1995 %0 Journal Article %J Indiana Univ. Math. J. 44 (1995), no. 3, 677-725 %D 1995 %T Unique solutions of 2x2 conservation laws with large data %A Alberto Bressan %A Rinaldo M. Colombo %X For a 2x2 hyperbolic system of conservation laws, we first consider a Riemann problem with arbitrarily large data. A stability assumption is introduced, which yields the existence of a Lipschitz semigroup of solutions, defined on a domain containing all suitably small BV perturbations of the Riemann data. We then establish a uniqueness result for large BV solutions, valid within the same class of functions where a local existence theorem can be proved. %B Indiana Univ. Math. J. 44 (1995), no. 3, 677-725 %I Indiana University Mathematics Journal %G en %U http://hdl.handle.net/1963/975 %1 3479 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:41:11Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1995 %R 10.1512/iumj.1995.44.2004 %0 Journal Article %J Proc.Amer.Math.Soc. 112 (1991), no.2, 413 %D 1991 %T A class of absolute retracts of dwarf spheroidal galaxies %A Alberto Bressan %A Arrigo Cellina %A Andrzej Fryszkowski %B Proc.Amer.Math.Soc. 112 (1991), no.2, 413 %I SISSA Library %G en %U http://hdl.handle.net/1963/837 %1 2954 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:38:33Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1989 %0 Journal Article %J Boll. Un. Mat. Ital. B (7) 4 (1990), no. 2, 295--311 %D 1990 %T Existence and continuous dependence for discontinuous O.D.E.s %A Alberto Bressan %A Giovanni Colombo %B Boll. Un. Mat. Ital. B (7) 4 (1990), no. 2, 295--311 %I SISSA Library %G en %U http://hdl.handle.net/1963/716 %1 3210 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:36:40Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1988 %0 Journal Article %J Proc. Amer. Math. Soc. 106 (1989), no. 3, 771-775 %D 1989 %T Upper semicontinuous differential inclusions without convexity %A Alberto Bressan %A Arrigo Cellina %A Giovanni Colombo %B Proc. Amer. Math. Soc. 106 (1989), no. 3, 771-775 %I SISSA Library %G en %U http://hdl.handle.net/1963/670 %1 3256 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:36:07Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1988 %0 Journal Article %J Boll. Un. Mat. Ital. B (7) 2 (1988), no. 3, 641-656 %D 1988 %T On differential systems with vector-valued impulsive controls. %A Alberto Bressan %A Franco Rampazzo %B Boll. Un. Mat. Ital. B (7) 2 (1988), no. 3, 641-656 %I SISSA Library %G en %U http://hdl.handle.net/1963/535 %1 3369 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:34:05Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1987 %0 Journal Article %J J. Differential Equations 76 (1988), no. 1, 135-158. %D 1988 %T Generalized Baire category and differential inclusions in Banach spaces. %A Alberto Bressan %A Giovanni Colombo %B J. Differential Equations 76 (1988), no. 1, 135-158. %I SISSA Library %G en %U http://hdl.handle.net/1963/538 %1 3366 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:34:07Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1987