@article{bressan_nguyen_2023, title={Generic Properties of First-Order Mean Field Games}, volume={1}, ISSN={["2153-0793"]}, DOI={10.1007/s13235-022-00487-3}, abstractNote={We consider a class of deterministic mean field games, where the state associated with each player evolves according to an ODE which is linear w.r.t. the control. Existence, uniqueness, and stability of solutions are studied from the point of view of generic theory. Within a suitable topological space of dynamics and cost functionals, we prove that, for “nearly all” mean field games (in the Baire category sense) the best reply map is single-valued for a.e. player. As a consequence, the mean field game admits a strong (not randomized) solution. Examples are given of open sets of games admitting a single solution, and other open sets admitting multiple solutions. Further examples show the existence of an open set of MFG having a unique solution which is asymptotically stable w.r.t. the best reply map, and another open set of MFG having a unique solution which is unstable. We conclude with an example of a MFG with terminal constraints which does not have any solution, not even in the mild sense with randomized strategies.}, journal={DYNAMIC GAMES AND APPLICATIONS}, author={Bressan, Alberto and Nguyen, Khai T. T.}, year={2023}, month={Jan} }
 @article{bressan_galtung_grunert_nguyen_2022, title={Shock interactions for the Burgers-Hilbert equation}, volume={6}, ISSN={["1532-4133"]}, DOI={10.1080/03605302.2022.2084628}, abstractNote={This paper provides an asymptotic description of a solution to the Burgers-Hilbert equation in a neighborhood of a point where two shocks interact. The solution is obtained as the sum of a function with H2 regularity away from the shocks plus a corrector term having an asymptotic behavior like |x| ln |x| close to each shock. A key step in the analysis is the construction of piecewise smooth solutions with a single shock for a general class of initial data.}, journal={COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS}, author={Bressan, Alberto and Galtung, Sondre T. and Grunert, Katrin and Nguyen, Khai T.}, year={2022}, month={Jun} }
 @article{capuani_dutta_nguyen_2021, title={METRIC ENTROPY FOR FUNCTIONS OF BOUNDED TOTAL GENERALIZED VARIATION}, volume={53}, ISSN={["1095-7154"]}, url={http://dx.doi.org/10.1137/20m1310953}, DOI={10.1137/20m1310953}, abstractNote={Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 6 January 2020Accepted: 07 December 2020Published online: 22 February 2021Keywordsmetric entropy, doubling dimension, total generalized variationAMS Subject Headings41A30, 35L02, 35L65Publication DataISSN (print): 0036-1410ISSN (online): 1095-7154Publisher: Society for Industrial and Applied MathematicsCODEN: sjmaah}, number={1}, journal={SIAM JOURNAL ON MATHEMATICAL ANALYSIS}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Capuani, Rossana and Dutta, Prerona and Nguyen, Khai T.}, year={2021}, pages={1168–1190} }
 @article{metric entropy for functions of bounded total generalized variation_2021, journal={SIAM J. Math. Anal.}, year={2021}, month={May} }
 @article{ancona_nguyen_2021, title={ON THE GLOBAL CONTROLLABILITY OF SCALAR CONSERVATION LAWS WITH BOUNDARY AND SOURCE CONTROLS}, volume={59}, ISSN={["1095-7138"]}, DOI={10.1137/20M1369221}, abstractNote={Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 24 September 2020Accepted: 13 July 2021Published online: 15 November 2021Keywordsconservation laws, source control, global exact controllability, return methodAMS Subject Headings35L65, 35Q93, 93B05, 93C20Publication DataISSN (print): 0363-0129ISSN (online): 1095-7138Publisher: Society for Industrial and Applied MathematicsCODEN: sjcodc}, number={6}, journal={SIAM JOURNAL ON CONTROL AND OPTIMIZATION}, author={Ancona, Fabio and Nguyen, Khai T.}, year={2021}, pages={4314–4338} }
 @article{gilmore_nguyen_2021, title={SBV regularity for Burgers-Poisson equation}, volume={500}, ISSN={["1096-0813"]}, DOI={10.1016/j.jmaa.2021.125095}, abstractNote={The SBV regularity of weak entropy solutions to the Burgers-Poisson equation for initial data in L 1 ( R ) is considered. We show that the derivative of a solution consists of only the absolutely continuous part and the jump part.}, number={1}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Gilmore, Steven and Nguyen, Khai T.}, year={2021}, month={Aug} }
 @article{markovian solutions to discontinuous odes_2020, year={2020}, month={Sep} }
 @article{metric entropy for hamilton-jacobi equation with uniformly directionally convex hamiltonian_2020, year={2020}, month={Dec} }
 @article{on the global controllability of scalar conservation laws with boundary and source controls_2020, year={2020}, month={Sep} }
 @article{sbv regularity for burgers-poisson equation_2020, year={2020}, month={Dec} }
 @article{a model of debt with bankruptcy risk and currency devaluation_2019, year={2019}, month={Oct} }
 @article{bressan_mazzola_nguyen_2019, title={APPROXIMATION OF SWEEPING PROCESSES AND CONTROLLABILITY FOR A SET-VALUED EVOLUTION}, volume={57}, ISSN={["1095-7138"]}, url={http://dx.doi.org/10.1137/18m1189610}, DOI={10.1137/18m1189610}, abstractNote={We consider a controlled evolution problem for a set $\Omega(t)\subset\mathbb{R}^d$, originally motivated by a model where a dog controls a flock of sheep. Necessary conditions and sufficient condi...}, number={4}, journal={SIAM JOURNAL ON CONTROL AND OPTIMIZATION}, author={Bressan, Alberto and Mazzola, Marco and Nguyen, Khai T.}, year={2019}, pages={2487–2514} }
 @book{mazzola_nguyen_2019, title={Lyapunov’s Theorem via Baire Category}, volume={32}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85068738460&partnerID=MN8TOARS}, DOI={10.1007/978-3-030-17949-6_10}, abstractNote={Lyapunov’s theorem is a classical result in convex analysis, concerning the convexity of the range of nonatomic measures. Given a family of integrable vector functions on a compact set, this theorem allows to prove the equivalence between the range of integral values obtained considering all possible set decompositions and all possible convex combinations of the elements of the family. Lyapunov type results have several applications in optimal control theory: they are used to prove bang-bang properties and existence results without convexity assumptions. Here, we use the dual approach to the Baire category method in order to provide a “quantitative” version of such kind of results applied to a countable family of integrable functions.}, journal={Springer INdAM Series}, author={Mazzola, M. and Nguyen, K.T.}, year={2019}, pages={181–194} }
 @article{ancona_glass_nguyen_2019, title={ON KOLMOGOROV ENTROPY COMPACTNESS ESTIMATES FOR SCALAR CONSERVATION LAWS WITHOUT UNIFORM CONVEXITY}, volume={51}, ISSN={["1095-7154"]}, url={http://dx.doi.org/10.1137/18m1198090}, DOI={10.1137/18m1198090}, abstractNote={In the case of scalar conservation laws $u_{t} + f(u)_{x}~=~0, t\geq 0, x\in\mathbb{R}$, with uniformly strictly convex flux $f$, quantitative compactness estimates---in terms of Kolmogorov entropy...}, number={4}, journal={SIAM JOURNAL ON MATHEMATICAL ANALYSIS}, author={Ancona, Fabio and Glass, Olivier and Nguyen, Khai T.}, year={2019}, pages={3020–3051} }
 @article{a debt management problem with currency devaluation_2018, year={2018}, month={May} }
 @article{dutta_nguyen_2018, title={Covering numbers for bounded variation functions}, volume={468}, ISSN={0022-247X}, url={http://dx.doi.org/10.1016/j.jmaa.2018.08.062}, DOI={10.1016/j.jmaa.2018.08.062}, abstractNote={In this paper, we provide upper and lower estimates for the minimal number of functions needed to represent a bounded variation function with an accuracy of epsilon with respect to L 1 -distance.}, number={2}, journal={Journal of Mathematical Analysis and Applications}, publisher={Elsevier BV}, author={Dutta, Prerona and Nguyen, Khai T.}, year={2018}, month={Dec}, pages={1131–1143} }
 @article{bressan_nguyen_2018, title={Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games}, volume={8}, ISSN={["2153-0793"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85042207992&partnerID=MN8TOARS}, DOI={10.1007/s13235-016-0206-2}, number={1}, journal={DYNAMIC GAMES AND APPLICATIONS}, author={Bressan, Alberto and Nguyen, Khai T.}, year={2018}, month={Mar}, pages={42–78} }
 @article{bressan_marigonda_nguyen_palladino_2017, title={A Stochastic Model of Optimal Debt Management and Bankruptcy}, volume={8}, ISSN={["1945-497X"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85041586490&partnerID=MN8TOARS}, DOI={10.1137/16m1095019}, abstractNote={A problem of optimal debt management is modeled as a noncooperative interaction between a borrower and a pool of lenders, in an infinite time horizon with exponential discount. The yearly income of the borrower is governed by a stochastic process. When the debt-to-income ratio $x(t)$ reaches a given size $x^*$, bankruptcy instantly occurs. The interest rate charged by the risk-neutral lenders is precisely determined in order to compensate for this possible loss of their investment. For a given bankruptcy threshold $x^*$, existence and properties of optimal feedback strategies for the borrower are studied, in a stochastic framework as well as in a limit deterministic setting. The paper also analyzes how the expected total cost to the borrower changes, depending on different values of $x^*$.}, number={1}, journal={SIAM JOURNAL ON FINANCIAL MATHEMATICS}, author={Bressan, Alberto and Marigonda, Antonio and Nguyen, Khai T. and Palladino, Michele}, year={2017}, pages={841–873} }
 @article{cavagnari_marigonda_nguyen_priuli_2017, title={Generalized Control Systems in the Space of Probability Measures}, volume={26}, ISSN={1877-0533 1877-0541}, url={http://dx.doi.org/10.1007/s11228-017-0414-y}, DOI={10.1007/s11228-017-0414-y}, number={3}, journal={Set-Valued and Variational Analysis}, publisher={Springer Nature}, author={Cavagnari, Giulia and Marigonda, Antonio and Nguyen, Khai T. and Priuli, Fabio S.}, year={2017}, month={May}, pages={663–691} }
 @article{bressan_nguyen_2016, title={An equilibrium model of debt and bankruptcy}, volume={22}, ISSN={1292-8119 1262-3377}, url={http://dx.doi.org/10.1051/cocv/2016030}, DOI={10.1051/cocv/2016030}, abstractNote={We study optimal strategies for a borrower, who services a debt in an infinite time horizon, taking into account the risk of possible bankruptcy. In a first model, the interest rate as well as the instantaneous bankruptcy risk are given, increasing functions of the total amount of debt. In a second model only the bankruptcy risk is given, while the interest rate is determined from a Nash equilibrium, in a game between the borrower and a pool of risk-neutral lenders. This yields a non-standard optimal control problem for the borrower, where the dynamics involves all future values of the control. For this model, optimal repayment strategies are constructed, in open-loop form. In addition, for optimal strategies in feedback form, our analysis shows that the value function should satisfy a new kind of nonlinear, degenerate elliptic equation.}, number={4}, journal={ESAIM: Control, Optimisation and Calculus of Variations}, publisher={EDP Sciences}, author={Bressan, Alberto and Nguyen, Khai T.}, year={2016}, month={Sep}, pages={953–982} }
 @article{nguyen_grunert_2016, title={Global existence of weak solutions for Burgers-Poisson equation}, volume={23}, number={4}, journal={Journal of Differential Equations}, author={Nguyen, Tien Khai and Grunert, K.}, year={2016}, pages={23–46} }
 @article{grunert_nguyen_2016, title={On the Burgers–Poisson equation}, volume={261}, ISSN={0022-0396}, url={http://dx.doi.org/10.1016/J.JDE.2016.05.028}, DOI={10.1016/j.jde.2016.05.028}, abstractNote={In this paper, we prove the existence and uniqueness of weak entropy solutions to the Burgers-Poisson equation for initial data in L^1(R). Additional an Oleinik type estimate is established and some criteria on local smoothness and wave breaking for weak entropy solutions are provided.}, number={6}, journal={Journal of Differential Equations}, publisher={Elsevier BV}, author={Grunert, K. and Nguyen, Khai T.}, year={2016}, month={Sep}, pages={3220–3246} }
 @article{bressan_mazzola_nguyen_2016, title={The bang–bang theorem via Baire category: a dual approach}, volume={23}, ISSN={1021-9722 1420-9004}, url={http://dx.doi.org/10.1007/s00030-016-0400-3}, DOI={10.1007/s00030-016-0400-3}, abstractNote={The paper develops a new approach to the classical bang–bang theorem in linear control theory, based on Baire category. Among all controls which steer the system from the origin to a given point $${\bar x}$$
 , we consider those which minimize an auxiliary linear functional $${\phi}$$
 . For all $${\phi}$$
 in a residual set, we show that the minimizing control is unique and takes values within a set of extreme points.}, number={4}, journal={Nonlinear Differential Equations and Applications NoDEA}, publisher={Springer Nature}, author={Bressan, Alberto and Mazzola, Marco and Nguyen, Khai T.}, year={2016}, month={Jul} }
 @article{nguyen_ancona_cannarsa_2016, title={The compactness estimates for Hamilton Jacobi Equations depending on space}, volume={11}, journal={Bulletin of the Institute of Mathematics, Academia Sinica}, author={Nguyen, Tien Khai and Ancona, F. and Cannarsa, P.}, year={2016}, pages={63–113} }
 @article{compactness estimates for hamilton-jacobi equations depending on space_2015, year={2015}, month={Apr} }
 @article{bressan_nguyen_2015, title={Conservation law models for traffic flow on a network of roads}, volume={10}, ISSN={1556-1801}, url={http://dx.doi.org/10.3934/nhm.2015.10.255}, DOI={10.3934/nhm.2015.10.255}, abstractNote={The paper develops a model of traffic flow near an intersection,
where drivers seeking to enter a congested road wait in a
buffer of limited capacity. Initial data comprise the vehicle density on each road,
together with the percentage of drivers approaching the intersection
who wish to turn into each of the outgoing roads.
 &nbsp 
If the queue sizes
within the buffer are known, then the initial-boundary value
problems become decoupled and can be independently solved
along each incoming road.
Three variational problems are introduced, related to different kind of
boundary conditions. From the
value functions, one recovers the traffic density
along each incoming or outgoing road by a Lax type formula.
 &nbsp 
Conversely,
if these value functions are known, then the queue sizes can be determined
by balancing the boundary fluxes of all incoming and outgoing roads.
In this way one obtains a contractive transformation, whose fixed point
 yields the unique solution of the Cauchy problem
for traffic flow in an neighborhood of the intersection.
 &nbsp 
The present model accounts for
 backward propagation of queues along roads leading to a crowded intersection,
it achieves well-posedness for general $L^\infty $ data, and continuity w.r.t. weak convergence of the initial densities.}, number={2}, journal={Networks and Heterogeneous Media}, publisher={American Institute of Mathematical Sciences (AIMS)}, author={Bressan, Alberto and Nguyen, Khai T.}, year={2015}, month={Apr}, pages={255–293} }
 @article{nguyen_ancona_glass_2015, title={On compactness estimates for general nonlinear system hyperbolic systems}, volume={32}, number={6}, journal={Annales de l'Institut Henri Poincaré, Analyse non Linéaire}, author={Nguyen, Tien Khai and Ancona, F. and Glass, O.}, year={2015}, pages={1229–1257} }
 @article{ancona_glass_nguyen_2015, title={On compactness estimates for hyperbolic systems of conservation laws}, volume={32}, ISSN={0294-1449}, url={http://dx.doi.org/10.1016/J.ANIHPC.2014.09.002}, DOI={10.1016/j.anihpc.2014.09.002}, abstractNote={We study the compactness in $L^{1}_{loc}$ of the semigroup mapping $(S_t)_{t > 0}$ defining entropy weak solutions of general hyperbolic systems of conservation laws in one space dimension. We establish a lower estimate for the Kolmogorov $\varepsilon$-entropy of the image through the mapping $S_t$ of bounded sets in $L^{1}\cap L^\infty$, which is of the same order $1/\varepsilon$ as the ones established by the authors for scalar conservation laws. We also provide an upper estimate of order $1/\varepsilon$ for the Kolmogorov $\varepsilon$-entropy of such sets in the case of Temple systems with genuinely nonlinear characteristic families, that extends the same type of estimate derived by De Lellis and Golse for scalar conservation laws with convex flux. As suggested by Lax, these quantitative compactness estimates could provide a measure of the order of "resolution" of the numerical methods implemented for these equations.}, number={6}, journal={Annales de l'Institut Henri Poincare (C) Non Linear Analysis}, publisher={Elsevier BV}, author={Ancona, Fabio and Glass, Olivier and Nguyen, Khai T.}, year={2015}, month={Nov}, pages={1229–1257} }
 @article{bressan_nguyen_2015, title={Optima and equilibria for traffic flow on networks with backward propagating queues}, volume={10}, ISSN={1556-1801}, url={http://dx.doi.org/10.3934/nhm.2015.10.717}, DOI={10.3934/nhm.2015.10.717}, abstractNote={This paper studies an optimal decision problem for several
groups of drivers on a network of roads. Drivers have different
origins and destinations,
and different costs, related to their departure and arrival time.
On each road the flow is governed by a conservation law, while intersections
are modeled using buffers of limited capacity, so that queues can spill backward
along roads leading to a crowded intersection.
Two main results are proved: (i)
the existence of a globally optimal solution, minimizing the sum of the costs
to all drivers, and (ii) the existence of
a Nash equilibrium solution, where no driver can lower his own cost by changing
his departure time or the route taken to reach destination.}, number={4}, journal={Networks and Heterogeneous Media}, publisher={American Institute of Mathematical Sciences (AIMS)}, author={Bressan, Alberto and Nguyen, Khai T.}, year={2015}, month={Oct}, pages={717–748} }
 @article{cannarsa_marigonda_nguyen_2015, title={Optimality conditions and regularity results for time optimal control problems with differential inclusions}, volume={427}, ISSN={0022-247X}, url={http://dx.doi.org/10.1016/j.jmaa.2015.02.027}, DOI={10.1016/j.jmaa.2015.02.027}, abstractNote={We study the time optimal control problem with a general target S for a class of differential inclusions that satisfy mild smoothness and controllability assumptions. In particular, we do not require Petrov's condition at the boundary of S. Consequently, the minimum time function T(⋅) fails to be locally Lipschitz—never mind semiconcave—near S. Instead of such a regularity, we use an exterior sphere condition for the hypograph of T(⋅) to develop the analysis. In this way, we obtain dual arc inclusions which we apply to show the constancy of the Hamiltonian along optimal trajectories and other optimality conditions in Hamiltonian form. We also prove an upper bound for the Hausdorff measure of the set of all non-Lipschitz points of T(⋅) which implies that the minimum time function is of special bounded variation (SBV).}, number={1}, journal={Journal of Mathematical Analysis and Applications}, publisher={Elsevier BV}, author={Cannarsa, Piermarco and Marigonda, Antonio and Nguyen, Khai T.}, year={2015}, month={Jul}, pages={202–228} }
 @article{ancona_cannarsa_nguyen_2015, title={Quantitative Compactness Estimates for Hamilton–Jacobi Equations}, volume={219}, ISSN={0003-9527 1432-0673}, url={http://dx.doi.org/10.1007/s00205-015-0907-5}, DOI={10.1007/s00205-015-0907-5}, abstractNote={We study quantitative compactness estimates in $\mathbf{W}^{1,1}_{loc}$ for the map $S_t$, $t>0$ that associates to every given initial data $u_0\in Lip(\mathbb{R}^N)$ the corresponding solution $S_t u_0$ of a Hamilton-Jacobi equation $$ u_t+H\big(\nabla_{/!x} u\big)=0\,, \qquad t\geq 0,\quad x\in \mathbb{R}^N, $$ with a uniformly convex Hamiltonian $H=H(p)$. We provide upper and lower estimates of order $1/\varepsilon^N$ on the the Kolmogorov $\varepsilon$-entropy in $\mathbf{W}^{1,1}$ of the image through the map $S_t$ of sets of bounded, compactly supported initial data. Estimates of this type are inspired by a question posed by P.D. Lax within the context of conservation laws, andcould provide a measure of the order of "resolution" of a numerical method implemented for this equation.}, number={2}, journal={Archive for Rational Mechanics and Analysis}, publisher={Springer Science and Business Media LLC}, author={Ancona, Fabio and Cannarsa, Piermarco and Nguyen, Khai T.}, year={2015}, month={Jul}, pages={793–828} }
 @inbook{marigonda_nguyen_vittone_2014, place={Berlin Heidelberg}, series={Lecture Notes in Computer Science}, title={BV Regularity and Differentiability Properties of a Class of Upper Semicontinuous Functions}, volume={8353 LNCS}, ISBN={9783662438794 9783662438800}, ISSN={0302-9743 1611-3349}, url={http://dx.doi.org/10.1007/978-3-662-43880-0_12}, DOI={10.1007/978-3-662-43880-0_12}, abstractNote={We study a class of upper semicontinuous functions \(f:\mathbb R^d\rightarrow \mathbb R\) whose hypograph \(\mathrm {hypo}\,f\) (see Definition 1) satisfies a geometric regularity property, namely: there exist \(c>0\), \(\theta \in ]0,1]\) such that for each \(P\) on the boundary of \(\mathrm {hypo}\,f\) there exists a unitary Frechet (outer) normal \(v\in N^F_{\mathrm {hypo}\,f}(P)\cap \mathbb S^d\) to \(\mathrm {hypo}\,f\) with \(\langle v,P-Q\rangle \le c \Vert P-Q\Vert ^{1+\theta }\qquad \mathrm{{for~every}}~Q\in \mathrm {hypo}\,f.\)}, booktitle={Large-Scale Scientific Computing}, publisher={Springer}, author={Marigonda, Antonio and Nguyen, Khai T. and Vittone, Davide}, editor={Lirkov, I. and Margenov, S. and Waśniewski, J.Editors}, year={2014}, pages={116–124}, collection={Lecture Notes in Computer Science} }
 @article{bressan_nguyen_2014, title={Global Existence of Weak Solutions for the Burgers--Hilbert Equation}, volume={46}, ISSN={0036-1410 1095-7154}, url={http://dx.doi.org/10.1137/140957536}, DOI={10.1137/140957536}, abstractNote={This paper establishes the global existence of weak solutions to the Burgers--Hilbert equation, for general initial data in ${\bf L}^2(\mathbb{R})$. For positive times, the solution lies in ${\bf L}^2\cap{\bf L}^\infty$. A partial uniqueness result is proved for spatially periodic solutions, as long as the total variation remains locally bounded.}, number={4}, journal={SIAM Journal on Mathematical Analysis}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Bressan, Alberto and Nguyen, Khai T.}, year={2014}, month={Jan}, pages={2884–2904} }
 @inproceedings{on quantitative compactness estimates for hyperbolic conservation laws_2014, booktitle={Hyperbolic problems: theory, numerics, applications, 249–257, AIMS Ser. Appl. Math., 8, Am. Inst. Math. Sci. (AIMS), Springfield, MO,}, year={2014} }
 @article{colombo_nguyen_nguyen_2013, title={Non-Lipschitz points and the SBV regularity of the minimum time function}, volume={51}, ISSN={0944-2669 1432-0835}, url={http://dx.doi.org/10.1007/s00526-013-0682-9}, DOI={10.1007/s00526-013-0682-9}, number={1-2}, journal={Calculus of Variations and Partial Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Colombo, Giovanni and Nguyen, Khai T. and Nguyen, Luong V.}, year={2013}, month={Nov}, pages={439–463} }
 @article{colombo_nguyen_2013, title={On the minimum time function around the origin}, volume={3}, ISSN={2156-8472}, url={http://dx.doi.org/10.3934/mcrf.2013.3.51}, DOI={10.3934/mcrf.2013.3.51}, abstractNote={We deal with finite dimensional linear and nonlinear control systems. If the system is linear and autonomous and satisfies theclassical normality assumption, we improve the well known result on the strict convexity of the reachable set from the origin by giving apolynomial estimate. The result is based on a careful analysis of the switching function.We extend this result to nonautonomous linear systems, provided the time dependent system is not too far fromthe autonomous system obtained by taking the time to be $0$ in the dynamics.  &nbspUsing a linearization approach, we prove a bang-bang principle, valid in dimensions $2$ and $3$ for a class of nonlinear systems, affine and symmetricwith respect to the control. Moreover we show that, for two dimensional systems, the reachable set from the originsatisfies the same polynomial strict convexity property as for the linearized dynamics, provided the nonlinearity is small enough.Finally, under the same assumptions we show that the epigraph of the minimumtime function has positive reach, hence proving the first result of this type in a nonlinear setting.In all the above results, we require that the linearization at the origin be normal.We provide examples showing the sharpness of our assumptions.}, number={1}, journal={Mathematical Control and Related Fields}, publisher={American Institute of Mathematical Sciences (AIMS)}, author={Colombo, Giovanni and Nguyen, Khai T.}, year={2013}, month={Feb}, pages={51–82} }
 @article{marigonda_nguyen_vittone_2013, title={Some regularity results for a class of upper semicontinuous functions}, volume={62}, ISSN={0022-2518}, url={http://dx.doi.org/10.1512/iumj.2013.62.4896}, DOI={10.1512/iumj.2013.62.4896}, abstractNote={We study regularity properties enjoyed by a class of real-valued upper semicontinuous functions f : R d ! R whose hypograph satis es a geometric property implying, for each point P on the boundary of hypo f, the existence of a sort of (uniform) subquadratic tangent hypersurface whose intersection with hypo f in a neighbourhood of P reduces to P. This geometric property generalizes both the concepts of semiconcave functions and functions whose hypograph has positive reach in the sense of Federer; the associated class of functions arises in the study of regularity properties for the minimum time function of certain classes of nonlinear control systems and di erential inclusions. We will prove that these functions share several regularity properties with semiconcave functions. In particular, they are locally BV and di erentiable a.e. Our approach consists in providing upper bounds for the dimension of the set of nondi erentiability points. Moreover, a ner classi cation of the singularities can be performed according to the dimension of the normal cone to the hypograph, thus generalizing a similar result proved by Federer for sets with positive reach. Techniques of nonsmooth analysis and geometric measure theory are used.}, number={1}, journal={Indiana University Mathematics Journal}, publisher={Indiana University Mathematics Journal}, author={Marigonda, Antonio and Nguyen, Khai Tien and Vittone, Davide}, year={2013}, pages={45–89} }
 @article{ancona_glass_nguyen_2012, title={Lower compactness estimates for scalar balance laws}, volume={65}, ISSN={0010-3640}, url={http://dx.doi.org/10.1002/cpa.21406}, DOI={10.1002/cpa.21406}, abstractNote={In this paper, we study the compactness in L of the semigroup (St)t≥0 of entropy weak solutions to strictly convex scalar conservation laws in one space dimension. The compactness of St for each t > 0 was established by P. D. Lax. Upper estimates for the Kolmogorov e-entropy of the image of bounded sets in L1 n L∞ through St were given by C. De Lellis and F. Golse. Here we provide lower estimates on this e-entropy of the same order as the one established by De Lellis and Golse, thus showing that such an e-entropy is of size ≈ 1/ε. Moreover, we extend these estimates of compactness to the case of convex balance laws. © 2012 Wiley Periodicals, Inc.}, number={9}, journal={Communications on Pure and Applied Mathematics}, publisher={Wiley}, author={Ancona, Fabio and Glass, Olivier and Nguyen, Khai T.}, year={2012}, month={May}, pages={1303–1329} }
 @article{nguyen_vittone_2012, title={Rectifiability of special singularities of non-lipschitz functions}, volume={19}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84856829336&partnerID=MN8TOARS}, number={1}, journal={Journal of Convex Analysis}, author={Nguyen, Tien Khai and Vittone, D.}, year={2012}, pages={159–170} }
 @article{albano_cannarsa_nguyen_sinestrari_2012, title={Singular gradient flow of the distance function and homotopy equivalence}, volume={356}, ISSN={0025-5831 1432-1807}, url={http://dx.doi.org/10.1007/s00208-012-0835-8}, DOI={10.1007/s00208-012-0835-8}, abstractNote={Let $$M$$ be a Riemannian manifold and let $$\varOmega $$ be a bounded open subset of $$M$$ . It is well known that significant information about the geometry of $$\varOmega $$ is encoded into the properties of the distance, $$d_{\partial \varOmega }$$ , from the boundary of $$\varOmega $$ . Here, we show that the generalized gradient flow associated with the distance preserves singularities, that is, if $$x_0$$ is a singular point of $$d_{\partial \varOmega }$$ then the generalized characteristic starting at $$x_0$$ stays singular for all times. As an application, we deduce that the singular set of $$d_{\partial \varOmega }$$ has the same homotopy type as $$\varOmega $$ .}, number={1}, journal={Mathematische Annalen}, publisher={Springer Science and Business Media LLC}, author={Albano, P. and Cannarsa, P. and Nguyen, Khai T. and Sinestrari, C.}, year={2012}, month={Jul}, pages={23–43} }
 @article{cannarsa_nguyen_2011, title={Exterior Sphere Condition and Time Optimal Control for Differential Inclusions}, volume={49}, ISSN={0363-0129 1095-7138}, url={http://dx.doi.org/10.1137/110825078}, DOI={10.1137/110825078}, abstractNote={The minimum time function $T(\cdot)$ of smooth control systems is known to be locally semiconcave provided Petrov's controllability condition is satisfied. Moreover, such a regularity holds up to the boundary of the target under an inner ball assumption. We generalize this analysis to differential inclusions, replacing the above hypotheses with the continuity of $T(\cdot)$ near the target, and an inner ball property for the multifunction associated with the dynamics. In such a weakened setup, we prove that the hypograph of $T(\cdot)$ satisfies, locally, an exterior sphere condition. As is well known, this geometric property ensures most of the regularity results that hold for semiconcave functions, without assuming $T(\cdot)$ to be Lipschitz.}, number={6}, journal={SIAM Journal on Control and Optimization}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Cannarsa, Piermarco and Nguyen, Khai T.}, year={2011}, month={Jan}, pages={2558–2576} }
 @article{nguyen_2010, title={Hypographs satisfying an external sphere condition and the regularity of the minimum time function}, volume={372}, ISSN={0022-247X}, url={http://dx.doi.org/10.1016/j.jmaa.2010.07.010}, DOI={10.1016/j.jmaa.2010.07.010}, abstractNote={We prove that if the hypograph of a continuous function f admits at every boundary point a supporting ball then it has “essentially” positive reach, i.e. the hypograph of the restriction of f outside a closed set of zero measure has (locally) positive reach. Hence such a function enjoys some properties of a concave function, in particular a.e. twice differentiability. We apply this result to a minimum time problem in the case of a nonlinear smooth dynamics and a target satisfying internal sphere condition.}, number={2}, journal={Journal of Mathematical Analysis and Applications}, publisher={Elsevier BV}, author={Nguyen, Khai T.}, year={2010}, month={Dec}, pages={611–628} }
 @article{colombo_nguyen_2010, title={On the Structure of the Minimum Time Function}, volume={48}, ISSN={0363-0129 1095-7138}, url={http://dx.doi.org/10.1137/090774549}, DOI={10.1137/090774549}, abstractNote={A minimum time problem with a nonlinear smooth dynamics and a target satisfying an internal sphere condition is considered. Under the assumption that the minimum time $T$ be continuous and the normal cone to the hypograph of $T$, $N_{hypo(T)}$, be pointed, we show that $hypo(T)$ is $\varphi$-convex, i.e., satisfies a strong external sphere condition. Consequently, $T$ is a.e. twice differentiable and satisfies some further regularity properties. Our results are based on a representation of Clarke generalized gradient of $T$. An example is provided, showing that if $N_{hypo(T)}$ is not pointed, then the result may fail.}, number={7}, journal={SIAM Journal on Control and Optimization}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Colombo, Giovanni and Nguyen, Khai T.}, year={2010}, month={Jan}, pages={4776–4814} }
 @article{colombo_nguyen_2009, title={Quantitative isoperimetric inequalities for a class of nonconvex sets}, volume={37}, ISSN={0944-2669 1432-0835}, url={http://dx.doi.org/10.1007/s00526-009-0256-z}, DOI={10.1007/s00526-009-0256-z}, number={1-2}, journal={Calculus of Variations and Partial Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Colombo, Giovanni and Nguyen, Khai T.}, year={2009}, month={Jun}, pages={141–166} }
 @article{critical points of non-c2 functionals_2007, journal={Topological Methods in Nonlinear Analysis}, year={2007} }
 @article{nguyen_duong_tran_2007, title={Critical points of non-C2 functional}, volume={29}, number={1}, journal={Topological Methods of Nonlinear Analysis}, author={Nguyen, Tien Khai and Duong, D.M. and Tran, H.V.}, year={2007}, pages={35–68} }
 @article{nguyen_duong_tran_2007, title={Morse-Palais Lemma for nonsmooth functionals on normed spaces}, volume={135}, number={3}, journal={Proceedings of the American Mathematical Society}, author={Nguyen, Tien Khai and Duong, D.M. and Tran, H.V.}, year={2007}, pages={921–927} }
 @article{duc_hung_khai_2007, title={Morse-palais lemma for nonsmooth functionals on normed spaces}, volume={135}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-77950647891&partnerID=MN8TOARS}, DOI={10.1090/S0002-9939-06-08662-X}, abstractNote={Using elementary differential calculus we get a version of the Morse-Palais lemma. Since we do not use powerful tools in functional analysis such as the implicit theorem or flows and deformations in Banach spaces, our result does not require the C 1 -smoothness of functions nor the completeness of spaces. Therefore it is stronger than the classical one but its proof is very simple.}, number={3}, journal={Proceedings of the American Mathematical Society}, author={Duc, D.M. and Hung, T.V. and Khai, N.T.}, year={2007}, pages={921–927} }