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
Examples and conjectures on the regularity of solutions to balance laws.
Quarterly Appl. Math., to appear (with F.Ancona, S.Bianchini, R.M.Colombo, and K.T.Nguyen).
-
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. 41 (2021), 113--130 (with W.Shen).
-
The deflection angle of surface ocean currents from the wind direction,
J. Geophysical Research - Oceans, 124 (2019), 7412-7420 (with A.Constantin).
On nonlinear wave equations
-
Shock interactions for the Burgers-Hilbert equation.
Comm. Partial Diff. Equat., to appear (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,
SIAM J. Math. Anal. 54 (2022), 4757--4784. (with S.Galtung and Q.Sun).
-
Weighted irrigation plans,
Comm. Math. Sci., 20 (2022), 611-651 (with Q.Sun).
-
Soil searching by an artificial root,
SIAM J. Math. Anal., 82 (2022), 1423--1445. (with F.Ancona and M.T.Chiri).
-
A 2-dimensional shape optimization problem for tree branches,
Networks and Heterogeneous Media 16 (2021), 1--29 (with S.Galtung).
-
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
-
Generic properties of first order mean field games,
Dynam. Games Appl., to appear (with Khai T. Nguyen).
-
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
-
Infinite horizon models of a Limit Order Book,
submitted (with H.Wei).
-
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
-
Controlled traveling profiles for models of invasive biological species,
submitted (with M.Zhang).
-
Optimal control of moving sets,
J. Differential Equations, to appear (with M.T.Chiri and N.Salehi).
-
On the optimal control of propagation fronts,
Math. Models & Methods in the Applied Sciences, 32 (2022), 1109– 1140 (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,
J. Dynam. Diff. Equat., to appear (with M.Mazzola and K.T.Nguyen).
-
Markovian solutions to discontinuous ODEs,
J. Dynam. Diff. Equat. 35 (2023), 135–162 (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 (2022) 61:36
(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.