@article {2007,
title = {Viscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients},
journal = {J. Hyperbolic Differ. Equ. 4 (2007) 771-795},
number = {SISSA;23/2003/M},
year = {2007},
publisher = {World Scientific},
abstract = {We consider Hamilton--Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main results are the existence and well--posedness of a viscosity solution to the Cauchy problem. We define a viscosity solution by treating the discontinuities in the coefficients analogously to \\\"internal boundaries\\\". By defining an appropriate penalization function, we prove that viscosity solutions are unique. The existence of viscosity solutions is established by showing that a sequence of front tracking approximations is compact in $L^\\\\infty$, and that the limits are viscosity solutions.},
doi = {10.1142/S0219891607001355},
url = {http://hdl.handle.net/1963/2907},
author = {Giuseppe Maria Coclite and Nils Henrik Risebro}
}
@article {2005,
title = {On the attainable set for Temple class systems with boundary controls},
journal = {SIAM J. Control Optim. 43 (2005) 2166-2190},
number = {SISSA;10/2002/M},
year = {2005},
publisher = {SISSA Library},
abstract = {Consider the initial-boundary value problem for a strictly hyperbolic, genuinely nonlinear, Temple class system of conservation laws \% $$ u_t+f(u)_x=0, \\\\qquad u(0,x)=\\\\ov u(x), \\\\qquad {{array}{ll} \&u(t,a)=\\\\widetilde u_a(t), \\\\noalign{\\\\smallskip} \&u(t,b)=\\\\widetilde u_b(t), {array}. \\\\eqno(1) $$ on the domain $\\\\Omega =\\\\{(t,x)\\\\in\\\\R^2 : t\\\\geq 0, a \\\\le x\\\\leq b\\\\}.$ We study the mixed problem (1) from the point of view of control theory, taking the initial data $\\\\bar u$ fixed, and regarding the boundary data $\\\\widetilde u_a, \\\\widetilde u_b$ as control functions that vary in prescribed sets $\\\\U_a, \\\\U_b$, of $\\\\li$ boundary controls. In particular, we consider the family of configurations $$ \\\\A(T) \\\\doteq \\\\big\\\\{u(T,\\\\cdot); ~ u {\\\\rm is a sol. to} (1), \\\\quad \\\\widetilde u_a\\\\in \\\\U_a, \\\\widetilde u_b \\\\in \\\\U_b \\\\big\\\\} $$ that can be attained by the system at a given time $T>0$, and we give a description of the attainable set $\\\\A(T)$ in terms of suitable Oleinik-type conditions. We also establish closure and compactness of the set $\\\\A(T)$ in the $lu$ topology.},
doi = {10.1137/S0363012902407776},
url = {http://hdl.handle.net/1963/1581},
author = {Fabio Ancona and Giuseppe Maria Coclite}
}
@article {2005,
title = {Conservation laws with time dependent discontinuous coefficients},
journal = {SIAM J. Math. Anal. 36 (2005) 1293-1309},
number = {SISSA;96/2002/M},
year = {2005},
publisher = {SISSA Library},
abstract = {We consider scalar conservation laws where the flux function depends discontinuously on both the spatial and temporal location. Our main results are the existence and well-posedness of an entropy solution to the Cauchy problem. The existence is established by showing that a sequence of front tracking approximations is compact in L1, and that the limits are entropy solutions. Then, using the definition of an entropy solution taken form [11], we show that the solution operator is L1 contractive. These results generalize the corresponding results from [16] and [11].},
doi = {10.1137/S0036141002420005},
url = {http://hdl.handle.net/1963/1666},
author = {Giuseppe Maria Coclite and Nils Henrik Risebro}
}
@article {2005,
title = {Stability of solutions of quasilinear parabolic equations},
journal = {J. Math. Anal. Appl. 308 (2005) 221-239},
number = {SISSA;51/2003/M},
year = {2005},
publisher = {Elsevier},
abstract = {We bound the difference between solutions $u$ and $v$ of $u_t = a\\\\Delta u+\\\\Div_x f+h$ and $v_t = b\\\\Delta v+\\\\Div_x g+k$ with initial data $\\\\phi$ and $ \\\\psi$, respectively, by $\\\\Vert u(t,\\\\cdot)-v(t,\\\\cdot)\\\\Vert_{L^p(E)}\\\\le A_E(t)\\\\Vert \\\\phi-\\\\psi\\\\Vert_{L^\\\\infty(\\\\R^n)}^{2\\\\rho_p}+ B(t)(\\\\Vert a-b\\\\Vert_{\\\\infty}+ \\\\Vert \\\\nabla_x\\\\cdot f-\\\\nabla_x\\\\cdot g\\\\Vert_{\\\\infty}+ \\\\Vert f_u-g_u\\\\Vert_{\\\\infty} + \\\\Vert h-k\\\\Vert_{\\\\infty})^{\\\\rho_p} \\\\abs{E}^{\\\\eta_p}$. Here all functions $a$, $f$, and $h$ are smooth and bounded, and may depend on $u$, $x\\\\in\\\\R^n$, and $t$. The functions $a$ and $h$ may in addition depend on $\\\\nabla u$. Identical assumptions hold for the functions that determine the solutions $v$. Furthermore, $E\\\\subset\\\\R^n$ is assumed to be a bounded set, and $\\\\rho_p$ and $\\\\eta_p$ are fractions that depend on $n$ and $p$. The diffusion coefficients $a$ and $b$ are assumed to be strictly positive and the initial data are smooth.},
doi = {10.1016/j.jmaa.2005.01.026},
url = {http://hdl.handle.net/1963/2892},
author = {Giuseppe Maria Coclite and Helge Holden}
}
@article {2005,
title = {Traffic flow on a road network},
journal = {SIAM J. Math. Anal. 36 (2005) 1862-1886},
number = {SISSA;13/2002/M},
year = {2005},
publisher = {SISSA Library},
abstract = {This paper is concerned with a fluidodynamic model for traffic flow. More precisely, we consider a single conservation law, deduced from conservation of the number of cars,\\ndefined on a road network that is a collection of roads with junctions. The evolution problem is underdetermined at junctions, hence we choose to have some fixed rules for the distribution of traffic plus an optimization criteria for the flux. We prove existence, uniqueness and stability of solutions to the Cauchy problem. Our method is based on wave front tracking approach, see [6], and works also for boundary data and time dependent coefficients of traffic distribution at junctions, so including traffic lights.},
doi = {10.1137/S0036141004402683},
url = {http://hdl.handle.net/1963/1584},
author = {Giuseppe Maria Coclite and Benedetto Piccoli and Mauro Garavello}
}
@article {2004,
title = {Solitary waves for Maxwell Schrodinger equations},
journal = {Electron. J. Differential Equations (2004) 94},
number = {SISSA;11/2002/M},
year = {2004},
publisher = {SISSA Library},
abstract = {In this paper we study solitary waves for the coupled system of Schrodinger-Maxwell equations in the three-dimensional space. We prove the existence of a sequence of radial solitary waves for these equations with a fixed L^2 norm. We study the asymptotic behavior and the smoothness of these solutions. We show also that the eigenvalues are negative and the first one is isolated.},
url = {http://hdl.handle.net/1963/1582},
author = {Giuseppe Maria Coclite and Vladimir Georgiev}
}
@mastersthesis {2003,
title = {Control Problems for Systems of Conservation Laws},
year = {2003},
school = {SISSA},
keywords = {Asymptotic Stabilization},
url = {http://hdl.handle.net/1963/5325},
author = {Giuseppe Maria Coclite}
}
@article {2003,
title = {An interior estimate for a nonlinear parabolic equation},
journal = {J.Math.Anal.Appl. 284 (2003) no.1, 49},
number = {SISSA;52/2002/M},
year = {2003},
publisher = {SISSA Library},
doi = {10.1016/S0022-247X(03)00157-4},
url = {http://hdl.handle.net/1963/1622},
author = {Giuseppe Maria Coclite}
}
@article {2003,
title = {Some results on the boundary control of systems of conservation laws},
journal = {SIAM J.Control Optim. 41 (2003),no.2, 607},
number = {SISSA;44/2002/M},
year = {2003},
publisher = {SISSA Library},
doi = {10.1137/S0363012901392529},
url = {http://hdl.handle.net/1963/1615},
author = {Alberto Bressan and Fabio Ancona and Giuseppe Maria Coclite}
}
@article {2002,
title = {On the Boundary Control of Systems of Conservation Laws},
journal = {SIAM J. Control Optim. 41 (2002) 607-622},
number = {SISSA;50/2001/M},
year = {2002},
publisher = {SIAM},
abstract = {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.},
doi = {10.1137/S0363012901392529},
url = {http://hdl.handle.net/1963/3070},
author = {Alberto Bressan and Giuseppe Maria Coclite}
}
@article {2002,
title = {A multiplicity result for the Schrodinger-Maxwell equations with negative potential},
journal = {Ann. Pol. Math. 79 (2002) 21-30},
number = {SISSA;20/2001/M},
year = {2002},
publisher = {IMPAN},
abstract = {We prove the existence of a sequence of radial solutions with negative energy of the Schr{\"o}dinger-Maxwell equations under the action of a negative potential.},
url = {http://hdl.handle.net/1963/3053},
author = {Giuseppe Maria Coclite}
}