@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{marigonda_nguyen_2023, title={STOCHASTIC EQUILIBRIUM SOLUTION FOR A DEBT MANAGEMENT PROBLEM WITH CURRENCY DEVALUATION}, volume={5}, ISSN={["2156-8499"]}, DOI={10.3934/mcrf.2023014}, abstractNote={Consider a model of debt management, where a sovereign state trades some bonds to service the debt with a pool of risk-neutral competitive foreign investors. At each time, the government decides which fraction of the gross domestic product (GDP) should be used to repay the debt, and how much to devaluate its currency. Both these operations have the effect to reduce the actual size of the debt, but have a social cost in terms of welfare sustainability. When the debt-to-GDP ratio reaches a given size $ x^* $, bankruptcy instantly occurs. Moreover, at any time the sovereign state can declare bankruptcy by paying a correspondent bankruptcy cost. To offset the possible loss of part of their investment, the foreign investors buy bonds at a discounted price which is not given a priori. This leads to a nonstandard optimal control problem. For a given bankruptcy threshold $ x^* $, we show that the optimization problem admits an equilibrium solution. The paper also studies properties of optimal feedback strategies, and the asymptotic behaviour of the expected total cost to the borrower as $ x^* $ is pushed to infinity.}, journal={MATHEMATICAL CONTROL AND RELATED FIELDS}, author={Marigonda, Antonio and Nguyen, Khai T.}, year={2023}, month={May} }
 @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={Abstract 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 H 2 regularity away from the shocks plus a corrector term having an asymptotic behavior like 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={We establish a sharp estimate for a minimal number of binary digits (bits) needed to represent all bounded total generalized variation functions taking values in a general totally bounded metric space $(E,\rho)$ up to an accuracy of $\varepsilon>0$ with respect to the ${\bf L}^1$-distance. Such an estimate is explicitly computed in terms of doubling and packing dimensions of $(E,\rho)$. The obtained result is applied to provide an upper bound on the metric entropy for a set of entropy admissible weak solutions to scalar conservation laws in one-dimensional space with weakly genuinely nonlinear fluxes.}, 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={We provide global and semi-global controllability results for hyperbolic conservation laws on a bounded domain, with a general (not necessarily convex)flux and a time-dependent source term acting as a control. The results are achieved for, possibly critical, both continuously differentiable states and BV states. The proofs are based on a combination of the return method and on the analysis of the Riccati equaiton for the space derivative of the solution.}, 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 L1(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 $\Omega(t)\in\mathbb{R}^d$, originally motivated by a model where a dog controls a flock of sheep. Necessary conditions and sufficient conditions are given, in order that the evolution be completely controllable. Similar techniques are then applied to the approximation of a sweeping process. Under suitable assumptions, we prove that there exists a control function such that the corresponding evolution of the set $\Omega(t)$ is arbitrarily close to the one determined by the sweeping process.}, 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 $$ u_{t} + f(u)_{x}~=~0,\qquad t\geq 0, x\in\mathbb{R}, $$ with uniformly strictly convex flux $f$, quantitative compactness estimates - in terms of Kolmogorov entropy in ${\bf L}^{1}_{loc}$ - were established in~\cite{DLG,AON1} for sets of entropy weak solutions evaluated at a fixed time $t>0$, whose initial data have a uniformly bounded support and vary in a bounded subset of ${\bf L}^\infty$. These estimates reflect the irreversibility features of entropy weak discontinuous solutions of these nonlinear equations. We provide here an extension of such estimates to the case of scalar conservation laws with a smooth flux function $f$ that either is strictly (but not necessarily uniformly) convex or has a single inflection point with a polynomial degeneracy.}, 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 L1-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}, abstractNote={We consider a noncooperative game in infinite time horizon, with linear dynamics and exponentially discounted quadratic costs. Assuming that the state space is one-dimensional, we prove that the Nash equilibrium solution in feedback form is stable under nonlinear perturbations. The analysis shows that, in a generic setting, the linear-quadratic game can have either one or infinitely many feedback equilibrium solutions. For each of these, a nearby solution of the perturbed nonlinear game can be constructed.}, 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).In addition 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. 
   
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. 
   
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. 
   
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 ε-entropy of the image through the mapping S t of bounded sets in L 1 ∩ L ∞ , which is of the same order 1/ε as the ones established by the authors for scalar conservation laws.We also provide an upper estimate of order 1/ε for the Kolmogorov ε-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 Fréchet (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 the classical normality assumption, we improve the well known result on the strict convexity of the reachable set from the origin by giving a polynomial 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 from the autonomous system obtained by taking the time to be 0 in the dynamics. Using a linearization approach, we prove a bang-bang principle, valid in dimensions 2 and 3 for a class of nonlinear systems, affine and symmetric with respect to the control. Moreover we show that, for two dimensional systems, the reachable set from the origin satisfies 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 minimum time 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={Abstract}, 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} }