Here is my updated
list of publications.
Books
-
Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem
(bookcover.html).
-
Introduction to the Mathematical Theory of Control
(leaflet.pdf)
(a list of misprints.pdf)
.
-
Lecture Notes on Functional Analysis, with Applications to Linear Partial Differential Equations
(bookcover.html)
(a list of misprints.pdf).
For instructors, a complete set of solutions to the 180 homework problems is available upon request.
Research papers
- On hyperbolic conservation laws
- On fluid flow
- On nonlinear wave equations
- On controlled growth and optimal shapes in biology
- On conservation law models of traffic flow
- On non-cooperative games
- On mathematical models in finance
- On control theory
- On differential inclusions and discontinuous ODEs
- On dynamic blocking models of fire confinement
On hyperbolic conservation laws
-
A posteriori error estimates for numerical solutions to
hyperbolic conservation laws.
Arch. Rational Mech. Anal. 241 (2021), 357–402 (with M.T.Chiri and W.Shen).
-
Vanishing viscosity solutions for conservation laws with regulated
flux,
J. Differential Equations 299 (2019), 312--351 (with G.Guerra and W.Shen).
-
On finite time BV blow-up for the p-system,
Comm. Partial Diff. Equat. 43 (2018), 1242--1280 (with G.Chen and Q.Zhang).
-
No BV bounds for approximate solutions to p-system with
general pressure law,
J. Hyperbolic Diff. Equat. 12 (2015), 1--18 (with G.Chen, Q.Zhang, and S.Zhu).
-
A semigroup approach to an integro-differential equation modeling slow erosion
J. Differential Equations 257 (2014), 2360--2403 (with W.Shen).
- Non-existence and
non-uniqueness for multidimensional sticky particle systems.
Kinetic Rel. Models, 7 (2014), 205--218 (with T.V.Nguyen).
- On the convergence of viscous approximations
after shock interactions.
Discr. Cont. Dynam. Syst. 23 (2009), 29--48 (with C.Donadello).
- Optimality conditions for
solutions to hyperbolic balance laws,
in ``Control Methods
in PDE - Dynamical Systems'',
F.Ancona, I.Lasieka, W.Littman, and R.Triggiani eds., AMS Contemporary Mathematics 426 (2007), 129--152 (with W.Shen).
-
BV instability of the Godunov scheme,
Comm. Pure Appl. Math. 59 (2006), 1604-1638 (with P.Baiti and H.K.Jenssen).
-
Instability of travelling wave profiles for
the Lax-Friedrichs scheme
Discr. Cont. Dynam. Syst., 13 (2005), 877-899 (with P.Baiti and H.K.Jenssen).
-
Vanishing viscosity solutions of nonlinear hyperbolic systems,
Annals of Mathematics, 161 (2005), 223-342 (with S.Bianchini).
-
On the rate of convergence of vanishing viscosity
approximations,
Comm. Pure Appl. Math. 57 (2004), 1075-1109 (with T.Yang).
-
A sharp decay estimate for nonlinear positive waves,
SIAM J. Math. Analysis, 36 (2004), 659-677 (with T.Yang).
-
An ill posed Cauchy problem for a
hyperbolic system in two space dimensions,
Rend. Sem. Mat. Univ. Padova, 110 (2003), 103--117.
- A lemma and a conjecture on the cost of rearrangements,
Rend. Sem. Mat. Univ. Padova, 110 (2003), 97--102.
-
On the boundary control of systems of conservation laws,
SIAM J. Control Optim. 41 (2002), 607--622 (with G.~M.~Coclite).
- Error bounds for a deterministic version of the Glimm scheme,
Arch. Rational Mech. Anal. 142 (1998), 155-176 (with A.Marson).
- A maximum principle for optimally controlled
systems of conservation laws,
Rend. Sem. Mat. Univ. Padova 94 (1995), 79-94 (with A.Marson).
- A variational calculus
for discontinuous
solutions of conservative systems,
Comm. Part. Diff. Equat. 20 (1995), 1491-1552 (with A.Marson).
- A locally contractive metric for systems of conservation laws,
Ann. Scuola Normale Sup. Pisa, Serie IV, Vol. XXII (1995), 109-135.
On fluid flow
-
Numerical study of non-uniqueness for 2D compressible isentropic Euler equations,
J. Comput. Physics 445 (2021) 110588 (with Y.Jiang and H.Liu).
-
On self-similar solutions to the incompressible Euler
equations
J. Differential Equations 269 (2020), 5142--5203 (with R.Murray).
-
A posteriori error estimates for self-similar solutions to the Euler
equations
Discr. Cont. Dyn. Syst., to appear (with W.Shen).
-
The deflection angle of surface ocean currents from the wind direction,
J. Geophysical Research - Oceans (2019) (with A.Constantin).
On nonlinear wave equations
-
Shock interactions for the Burgers-Hilbert equation.
Preprint 2022 (with S.Galtung, K.Grunert, and K.Nguyen).
-
Piecewise smooth solutions to the Burgers-Hilbert equation.
Comm. Math. Sci. 15 (2017), 165--184 (with T.Zhang).
-
Generic regularity of conservative solutions
to a nonlinear wave equation,
Ann. Inst. H.Poincar\'e Anal. Nonlin. 34 (2017), 335--354 (with G.Chen).
-
Lipschitz metrics for a class of nonlinear
wave equations,
Arch. Rational Mech. Anal. 226 (2017), 1303--1343 (with G.Chen).
-
Representation of dissipative solutions to a nonlinear
variational wave equation,
Comm. Math. Sci. 14 (2016), 31--53 (with T.Huang).
-
Structurally stable singularities
for a nonlinear wave equation.
Bull. Inst. Math. Acad. Sinica, 10 (2015), 449--478 (with T.Huang and F.Yu).
-
Unique conservative solutions to a variational wave equation,
Archive Rational Mech. Anal., 21 (2015), 1069--1101 (with G.Chen and Q.Zhang).
- Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics
Discr. Cont. Dynam. Syst. 35 (2015), 25--42 (with G.Chen and Q.Zhang).
-
Global existence of weak solutions for the Burgers-Hilbert equation,
SIAM J. Math. Anal. 46 (2014), 2884--2904 (with K.Nguyen).
-
Lipschitz metric for the Hunter-Saxton equation,
J. Mathematiques Pures Appliques, 94 (2010), 68--92 (with H.Holden and X.Raynaud).
-
Asymptotic variational wave equations
Arch. Rat. Mech. Anal., 183 (2007), 163-185 (with P.Zhang and Y.Zheng).
-
An optimal transportation metric for solutions of the
Camassa-Holm equation,
Methods and Applications of Analysis, 12 (2005), 191--220 (with M.Fonte).
- Global dissipative solutions
to the Camassa-Holm equation,
Analysis and Applications 5 (2007), 1--27 (with A.Constantin).
- Global conservative solutions to the Camassa-Holm equation,
Arch. Rat. Mech. Anal. 183 (2007), 215-239 (with A.Constantin).
-
Conservative solutions to a nonlinear variational wave equation,
Comm. Math. Phys. 266 (2006), 471-497 (with Y.Zheng).
On conservation law models of traffic flow
-
Entropy admissibility of the limit solution for a nonlocal model of traffic flow.
Comm. Math. Sci. 19 (2021), 1447–1450 (with W.Shen).
-
On traffic flow with nonlocal flux: a relaxation representation
Archive Rational Mech. Anal. 237 (2020), 1213--1236 (with W.Shen).
-
Globally optimal departure rates for several groups of drivers,
Mathematics in Engineering 1 (2019), 583--613 (with Y.Huang).
-
The Riemann Solver for traffic flow at an intersection
with buffer of vanishing size.
Netw. Heter. Media 12 (2017), 173--189 (with A.Nordli).
-
Optima and equilibria for traffic
flow on networks with backward propagating queues.
Netw. Heter. Media 10 (2015), 717--748 (with K. Nguyen).
-
Continuous Riemann solvers for traffic flow at a junction.
Discr. Cont. Dyn. Syst. 35 (2015), 4149--4171 (with F.Yu).
-
Conservation law models for traffic flow on a network of roads.
Netw. Heter. Media 10} (2015), 255--293 (with K.Nguyen).
-
Existence of optima
and equilibria for traffic flow on networks
Networks & Heter. Media 8 (2013), 627--648 (with Ke Han).
-
Nash
equilibria for a model of traffic flow
with several groups of drivers,
ESAIM; Control, Optim. Calc. Var., 18 (2012), 969--986 (with Ke Han).
-
Variational analysis of Nash equilibria for a model of traffic flow
Quarterly Appl. Math. 70 (2012), 495-515 (with C.J.Liu, W.Shen, and F.Yu).
-
Optima and equilibria for a model of traffic flow
SIAM J. Math. Anal., 43 (2011), 2384--2417 (with Ke Han).
On controlled growth and shape optimization in biology
-
Optimal shapes for tree roots,
preprint 2021 (with S.Galtung and Q.Sun).
-
Weighted irrigation plans,
Comm. Math. Sci., to appear (with Q.Sun).
-
A 2-dimensional shape optimization problem for tree branches,
Networks and Heterogeneous Media 16 (2021), 1--29 (with S.Galtung).
-
Soil searching by an artificial root,
submitted (with F.Ancona and M.T.Chiri).
-
Competition models for plant stems,
J. Differential Equations 269 (2020), 1571--1611 (with S.Galtung, A.Reichstad, and J.Ridder).
-
Variational problems for tree roots and branches,
Calc. Var. PDE 57, 7 (2020), (with M.Palladino and Q.Sun).
-
On the optimal shape of tree roots and branches,
Math. Models & Methods in the Applied Sciences 28 (2018), 2763-2801. (with Q.Sun).
-
Well-posedness of a model for the growth of
tree stems and vines,
Discr. Cont. Dyn. Syst. 38 (2018), 2047--2064 (with M.Palladino).
-
Feedback stabilization of stem growth,
J. Dyn. Diff. Equat. 31 (2019), 1079--1106 (with F.Ancona, O.Glass, and W.Shen).
-
A model of controlled growth,
Arch. Rational Mech. Anal. 227 (2018), 1223-1266 (with M. Lewicka).
-
Growth models for tree stems and vines,
J. Differential Equations 263 (2017), 2280--2316 (with M.Palladino and W.Shen).
On non-cooperative games
-
Self-consistent feedback Stackelberg equibria for infinite horizon
stochastic games,
Dynam. Games Appl., 10 (2020), 328--360 (with Y.Jiang).
-
On the generic structure and stability of Stackelberg equilibria,
J. Optim. Theory Appl. 183 (2019), 840--880 (with Y.Jiang).
-
On the competitive harvesting of marine resources,
SIAM J. Control Optim. 57 (2019), 3961--3984 (with V.Staicu).
-
Stability of feedback solutions
for infinite horizon noncooperative differential games,
Dyn. Games Appl. 8 (2018), 42--78 (with K.Nguyen).
-
Stackelberg solutions of feedback type for differential games with random initial data.
Dynamic Games Appl. 3 (2013), 341--358 (with D.Wei).
-
Measure valued solutions to a harvesting game
with several players.
Advances in Dynamic Games, 11 (2011), 399--423 (with W.Shen).
- Semi-cooperative strategies for differential games,
Intern. J. Game Theory, 32 (2004), 561-593 (with W.Shen).
- Small BV solutions of
hyperbolic non-cooperative differential games.
SIAM J. Control Optim. 43 (2004), 104-215 (with W.Shen).
On mathematical models in finance
-
A dynamic model of the limit order book,
Discr. Cont. Dyn. Syst. - Series B, 25 (2020), 1015–1041 (with M.Mazzola and H.Wei).
-
The vanishing viscosity limit for a system of H-J
equations related to a debt management problem,
Discr. Cont. Dyn. Syst. - Series S 11 (2018), 793--824 (with Y.Jiang).
-
Optimal open-loop strategies in a debt management problem,
Analysis & Appl. 16 (2018), 133--157 (with Y.Jiang).
-
A stochastic model of optimal debt management
and bankruptcy,
SIAM J. Financial Math. 8 (2017), 841--873 (with A.Marigonda, K.Nguyen, and M.Palladino).
-
An equilibrium model of debt and bankruptcy,
ESAIM: Control, Optim. Calc. Variat. 22 (2016), 953--982 (with K.Nguyen).
-
Dynamic stability of the Nash equilibrium for a bidding game,
Analysis and Applications 14 (2016), 591--614 (with H.Wei).
-
A bidding game with heterogeneous players,
J. Optim. Theory Appl. 163 (2014), 1018--1048 (with D.Wei).
-
Discrete bidding strategies for a random incoming order
SIAM J. Financial Math. 5 (2014), 50--70 (with G.Facchi).
-
A bidding game in a continuum limit order
book
SIAM J. Control Optim. 51 (2013), 3459--3485 (with G.Facchi).
On control theory
-
Optimal control of moving sets,
Preprint, 2021. (with M.T.Chiri and N.Salehi).
-
On the optimal control of propagation fronts,
Math. Models & Methods in the Applied Sciences, to appear (with M.T.Chiri and N.Salehi).
-
Approximation of sweeping processes and controllability
for a set valued evolution,
SIAM J. Control Optim. 57 (2019), 2487--2514 (with M.Mazzola and K.Nguyen).
-
The Bang-Bang theorem via Baire category. A Dual Approach.
Nonlin. Diff. Equat. Appl. 23 (2016), (with M.Mazzola and K.Nguyen).
- Graph completions for impulsive
feedback controls,
J. Math. Anal. Appl. 412 (2014), 976--988 (with M.Mazzola).
-
On the control
of non-holonomic systems by active constraints
Discr. Cont. Dyn. Syst. 33 (2013), 3329--3353 (with Ke Han and F.Rampazzo).
-
A multi-dimensional optimal harvesting problem with measure valued solutions.
SIAM J. Control Optim. 51 (2013), 1186--1202 (with G.Coclite and W.Shen).
-
Control problems for a class of set valued evolutions.
Set Valued, Var. Anal. 20 (2012), 581-601 (with D.Zhang).
- On the controllability of
Lagrangian systems
by active constraints.
J. Differential Equations, 247 (2009), 543--563 (with Z.Wang).
- Nearly optimal patchy
feedbacks,
Discr. Cont. Dynam. Syst. 21 (2008), 687--701 (with F.S.Priuli).
-
Optimal control problems on stratified domains,
Networks and Heterogeneous Media 2 (2007), 313-331 (with Y.Hong).
- On the intersection of a
Clarke cone with a Boltyanskii cone,
SIAM J. Control Optim. 45 (2007), 2054--2064.
-
Flow stability of patchy vector fields,
SIAM J. Control Optim., 41 (2003), 1455-1476. (with F.Ancona).
-
Patchy vector fields and asymptotic stabilization,
ESAIM; Control, Optimization and Calculus of Variations 4 (1999), 445-471 (with F.Ancona).
- A Baire category
approach to the bang-bang property,
J. Differential Equations 116 (1995), 318-337 (with B. Piccoli).
-
On differential systems with vector-valued impulsive controls,
Boll. Un. Matematica Italiana 2-B, (1988), 641-656 (with F.Rampazzo).
On differential inclusions and discontinuous ODEs
-
Diffusion approximations of Markovian solutions to discontinuous
ODEs,
submitted (with M. Mazzola and K. T. Nguyen).
-
Markovian solutions to discontinuous ODEs,
J. Dynam. Diff. Equat., to appear (with M.Mazzola and K.T.Nguyen).
-
Random extremal solutions of differential inclusions.
Nonlin. Diff. Equat. Appl. 23 (2016), 1--16 (with V.Staicu).
-
Extremal solutions to differential inclusions via Baire category:
a dual approach.
J. Differential Equations 255 (2013), 2392--2399.
-
Multidimensional graph completions and
Cellina approximable multifunctions,
Rocky Mountain J. Math. 41 (2011), 411--444 (with R.DeForest).
- Trajectories of
differential inclusions with state constraints,
J. Differential Equations 250 (2011), 2267--2281 (with G.Facchi).
- Classical solutions to differential
inclusions with totally
disconnected right hand side.
J. Differential Equations, 246 (2009), 629--640 (with Z.Wang).
-
On discontinuous differential equations,
in ``Differential Inclusions and Optimal Control'', J.Andres, L.Gorniewicz and P.Nistri Eds.,
J.Schauder Center, Lecture Notes in Nonlinear Analysis 2 (1998), 73-87 (with W.Shen).
- Extremal selections of multifunctions generating a continuous flow,
Annal. Polon. Mat. LX.2 (1994) (with G.Crasta).
-
On total differential inclusions,
Rend. Sem. Mat. Univ. Padova 92 (1994), 9-16 (with F.Flores).
-
A class of absolute retracts in spaces of integrable functions.
Proc. Amer. Math. Soc. 112 (1991), 413--418 (with A.Cellina and A.Fryszkowski)
-
The most likely path of a differential inclusion,
J. Differential Equations 88 (1990), 155-174.
-
Existence and continuous dependence
for discontinuous O.D.E.'s,
Boll. Un. Mat. Ital. 4-B (1990), 295-311 (with G.Colombo).
-
Generalized Baire category and differential inclusions in Banach spaces,
J. Differential Equations, 76 (1988), 135-158 (with G.Colombo).
-
Unique solutions for a class of discontinuous differential equations,
Proc. Amer. Math. Soc. 104 (1988), 772-778.
-
Extensions and selections of maps with decomposable values,
Studia Mathematica, XC (1988), 69-86 (with G.Colombo).
On dynamic blocking models of fire confinement
-
On the regularity of optimal dynamic
blocking strategies. Calc. Variat. & PDEs, to appear
(with M.T.Chiri).
-
Dynamic blocking problems
for a model of fire propagation In: Advances in Applied Mathematics, Modeling,
and Computational Science, pp.11--40.
R.Melnik and I.Kotsireas editors. Fields Institute Communications, Springer, New York, 2013.
- On the optimal strategy for an isotropic blocking problem.
Calculus of Variations & PDE, 45 (2012), 125--145 (with T.Wang).
-
Global necessary conditions for a dynamic
blocking problem,
ESAIM; Control, Optimization and Calculus of Variations, 18 (2012), 124--156 (with T.Wang).
- Equivalent formulation and numerical analysis
of a fire confinement problem,
ESAIM; Control, Optimization and Calculus of Variations, 16 (2010), 974--1001 (with T.Wang).
- Existence of optimal strategies
for a fire confinement problem.
Comm. Pure Appl. Math. 62 (2009), 789--830 (with C.De Lellis).
- The minimum speed for a blocking problem on the half plane.
J. Math. Anal. Appl. 356 (2009), 133-144 (with T.Wang).
- Blocking strategies for
a fire control problem,
Analysis and Applications, 6 (2008), 229--246 (with M.Burago, A.Friend, and J.Jou).
- Differential inclusions and the control of forest fires,
J. Differential Equations (special volume in honor of A.Cellina and J.Yorke), 243 (2007), 179-207.