TY - JOUR
T1 - Viscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients
JF - J. Hyperbolic Differ. Equ. 4 (2007) 771-795
Y1 - 2007
A1 - Giuseppe Maria Coclite
A1 - Nils Henrik Risebro
AB - 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.
PB - World Scientific
UR - http://hdl.handle.net/1963/2907
U1 - 1793
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - On the attainable set for Temple class systems with boundary controls
JF - SIAM J. Control Optim. 43 (2005) 2166-2190
Y1 - 2005
A1 - Fabio Ancona
A1 - Giuseppe Maria Coclite
AB - 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.
PB - SISSA Library
UR - http://hdl.handle.net/1963/1581
U1 - 2537
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Conservation laws with time dependent discontinuous coefficients
JF - SIAM J. Math. Anal. 36 (2005) 1293-1309
Y1 - 2005
A1 - Giuseppe Maria Coclite
A1 - Nils Henrik Risebro
AB - 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].
PB - SISSA Library
UR - http://hdl.handle.net/1963/1666
U1 - 2452
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Stability of solutions of quasilinear parabolic equations
JF - J. Math. Anal. Appl. 308 (2005) 221-239
Y1 - 2005
A1 - Giuseppe Maria Coclite
A1 - Helge Holden
AB - 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.
PB - Elsevier
UR - http://hdl.handle.net/1963/2892
U1 - 1808
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Traffic flow on a road network
JF - SIAM J. Math. Anal. 36 (2005) 1862-1886
Y1 - 2005
A1 - Giuseppe Maria Coclite
A1 - Benedetto Piccoli
A1 - Mauro Garavello
AB - 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.
PB - SISSA Library
UR - http://hdl.handle.net/1963/1584
U1 - 2534
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Solitary waves for Maxwell Schrodinger equations
JF - Electron. J. Differential Equations (2004) 94
Y1 - 2004
A1 - Giuseppe Maria Coclite
A1 - Vladimir Georgiev
AB - 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.
PB - SISSA Library
UR - http://hdl.handle.net/1963/1582
U1 - 2536
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - THES
T1 - Control Problems for Systems of Conservation Laws
Y1 - 2003
A1 - Giuseppe Maria Coclite
KW - Asymptotic Stabilization
PB - SISSA
UR - http://hdl.handle.net/1963/5325
U1 - 5154
U2 - Mathematics
U3 - Functional Analysis and Applications
U4 - -1
ER -
TY - JOUR
T1 - An interior estimate for a nonlinear parabolic equation
JF - J.Math.Anal.Appl. 284 (2003) no.1, 49
Y1 - 2003
A1 - Giuseppe Maria Coclite
PB - SISSA Library
UR - http://hdl.handle.net/1963/1622
U1 - 2496
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 - On the Boundary Control of Systems of Conservation Laws
JF - SIAM J. Control Optim. 41 (2002) 607-622
Y1 - 2002
A1 - Alberto Bressan
A1 - Giuseppe Maria Coclite
AB - 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.
PB - SIAM
UR - http://hdl.handle.net/1963/3070
U1 - 1263
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - A multiplicity result for the Schrodinger-Maxwell equations with negative potential
JF - Ann. Pol. Math. 79 (2002) 21-30
Y1 - 2002
A1 - Giuseppe Maria Coclite
AB - We prove the existence of a sequence of radial solutions with negative energy of the SchrÃ¶dinger-Maxwell equations under the action of a negative potential.
PB - IMPAN
UR - http://hdl.handle.net/1963/3053
U1 - 1280
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -