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Khai T. Nguyen

Publications and source records attributed to Khai T. Nguyen.

At least 19 recordsLinked to original sources
arXiv2026

On scalar nonlinear balance laws with singular nonlocal sources

We investigate one-dimensional scalar balance laws with singular convolution-type source terms. Under appropriate convexity and kernel assumptions, we establish the global existence of entropy weak solutions in ${\bf L}^2(\mathbb{R})$, together with two partial uniqueness results, in the ${\bf L}^2$-periodic setting and non-periodic setting with ${\bf L}^1(\mathbb{R})$ kernel. In the ${\bf L}^1$-kernel case, the characteristic speed satisfies an Oleinik-type estimate, and entropy weak solutions possess locally bounded fractional variation for all positive times. Furthermore, we derive a simple criterion characterizing local smoothness and wave breaking of solutions, which, in particular, includes both the Burger-Poisson and the Burgers-Hilbert equation as special cases.

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arXiv2026

Wave breaking for perturbed Burgers equations

We establish a simple and explicit criterion for wave breaking for a general class of perturbed Burgers equations that cover several Burgers-type models, including the Fractional KdV equation, the Whitham equation, and the Fornberg-Whitham equation. The proof is both rigorous and straightforward.

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arXiv2025

Two-point boundary value problems for quasi-monotone dynamical systems

This paper studies the existence of minimal solutions to two-point boundary value problems for quasi-monotone dynamical systems. Specifically, the pointwise infimum of all supersolutions is shown to coincide with the minimal solution. This result is then applied to establish a non-uniqueness result for strong stable solutions to a class of mean field games with a continuum of players.

math.CA↗
arXiv2025

On the rate of convergence in superquadratic Hamilton--Jacobi equations with state constraints

In this paper, we investigate the convergence rate in the vanishing viscosity limit for solutions to superquadratic Hamilton--Jacobi equations with state constraints. For every $p>2$, we establish the rate of convergence for nonnegative Lipschitz data vanishing on the boundary to be of order $ \mathcal{O}(\varepsilon^{1/2}) $ and obtain an improved upper rate of order $ \mathcal{O}\big(\varepsilon^{\frac{p}{2(p-1)}}\big)$ for semiconcave data.

math.AP↗
arXiv2025

Generic uniqueness and conjugate points for optimal control problems

The paper is concerned with an optimal control problem on $\mathbb{R}^n$, where the dynamics is linear w.r.t.~the control functions. For a terminal cost $ψ$ in a $mathcal{G}_δ$ set of $\mathcal{C}^4(\mathbb{R}^n)$ (i.e., in a countable intersection of open dense subsets), two main results are proved.Namely: the set $Γ_ψ\subset\mathbb{R}^n$ of conjugate points is closed, with locally bounded $(n-2)$-dimensional Hausdorff measure. Moreover, the set of initial points $y\in \mathbb{R}^n\setminusΓ_ψ$, which admit two or more globally optimal trajectories, is contained in the union of a locally finite family of embedded manifolds. In particular, the value function is continuously differentiable on an open, dense subset of $\mathbb{R}^n$.

math.OC↗
arXiv2024

Generic Solutions to Controlled Balance Laws

The paper is concerned with a scalar balance law, where the source term depends on a control function $α(t)$. Given a control $α\in \mathbf{L}^\infty\bigl([0,T]\bigr)$, it is proved that, for generic initial data $\bar u \in \mathcal{C}^3(\mathbb{R})$, the solution has finitely many shocks, interacting at most two at a time. Moreover, at the terminal time $T$ no shock interaction occurs, and no new shock is formed. In addition, a family of optimal control problems is considered, including a running cost and a terminal cost. An example is constructed where the optimal solution contains two shocks merging exactly at the terminal time $T$. Such behavior persists under any suitably small perturbation of the flux, source, and cost functions, and of the initial data. This shows that generic solutions of optimization problems have different qualitative properties, compared with generic solutions to Cauchy problems.

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arXiv2024

On the structure of the value function of optimal exit time problems

In this paper, we study an optimal exit time problem with general running and terminal costs and a target $\mathcal{S}\subset\mathbb{R}^d$ having an inner ball property for a nonlinear control system that satisfies mild controllability assumptions. In particular, Petrov's condition at the boundary of $\mathcal{S}$ is not required and the value function $V$ may fail to be locally Lipschitz. In such a weakened set-up, we first establish a representation formula for proximal (horizontal) supergradients of $V$ by using transported proximal normal vectors. This allows us to obtain an external sphere condition for the hypograph of $V$ which yields several regularity properties. In particular, $V$ is almost everywhere twice differentiable and the Hausdorff dimension of its singularities is not greater than $d-1/2$. Furthermore, besides optimality conditions for trajectories of the optimal control problem, we extend the analysis to propagation of singularities and differentiability properties of the value function. An upper bound for the Hausdorff measure of the singular set is also studied, which implies that $V$ is a function of special bounded variation.

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arXiv2024

A sharp quantitative estimate of critical sets

The paper establishes a sharp quantitative estimate for the $(d-1)$-Hausdorff measure of the critical set of $\mathcal{C}^1$ vector-valued functions on $\mathbb{R}^d$. Additionally, we prove that for a generic $\mathcal{C}^2$ function where ``generic" is understood in the topological sense of Baire category, the critical set has a locally finite $(d-1)$-Hausdorff measure.

math.FA↗
arXiv2024

Hausdorff measure of zeros of polynomials

The paper provides an elementary proof establishing a sharp universal bound on the $(d-1)$-Hausdorff measure of the zeros of any nontrivial multivariable polynomial $p:\mathbb{R}^d\to\mathbb{R}$ within a $d$-dimensional cube of size $r$. This bound depends solely on the parameter $r$, the dimension $d$, and the degrees of $p$.

math.CA↗
arXiv2024

Generic Properties of Conjugate Points in Optimal Control Problems

The first part of the paper studies a class of optimal control problems in Bolza form, where the dynamics is linear w.r.t.~the control function. A necessary condition is derived, for the optimality of a trajectory which starts at a conjugate point. The second part is concerned with a classical problem in the Calculus of Variations, with free terminal point. For a generic terminal cost $ψ\in \C^4(\mathbb{R}^n)$, applying the previous necessary condition we show that the set of conjugate points is contained in the image of an $(n-2)$-dimensional manifold, and has locally bounded $(n-2)$-dimensional Hausdorff measure.

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arXiv2023

A lower bound on the quantitative version of the transversality theorem

The present paper studies a quantitative version of the transversality theorem. More precisely, given a continuous function $f\in \mathcal{C}([0,1]^d,\mathbb{R}^m)$ and a manifold $W\subset \mathbb{R}^m$ of dimension $p$, a sharpness result on the upper quantitative estimate of the $(d+p-m)$-dimensional Hausdorff measure of the set $\mathcal{Z}_{W}^{f}=\left\{x\in [0,1]^d: f(x)\in W\right\}$, which was achieved in [8], will be proved in terms of power functions.

math.FA↗
arXiv2022

Generic Properties of First Order Mean Field Games

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.

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arXiv2022

Shock interactions for the Burgers-Hilbert Equation

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 |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.

math.AP↗
arXiv2022

A quantitative version of the transversality theorem

The present paper studies a quantitative version of the transversality theorem. More precisely, given a continuous function $g\in \mathcal{C}([0,1]^d,\mathbb{R}^m)$ and a global smooth manifold $W\subset \mathbb{R}^m$ of dimension $p$, we establish a quantitative estimate on the $(d+p-m)$-dimensional Hausdorff measure of the set $\mathcal{Z}_{W}^{g}=\left\{x\in [0,1]^d: g(x)\in W\right\}$. The obtained result is applied to quantify the total number of shock curves in weak entropy solutions to scalar conservation laws with uniformly convex fluxes in one space dimension.

math.FA↗
arXiv2022

Metric entropy for Hamilton-Jacobi equation with uniformly directionally convex Hamiltonian

The present paper first aims to study the BV-type regularity for viscosity solutions of the Hamilton-Jacobi equation \[ u_t(t,x)+H\big(D_{x} u(t,x)\big)~=~0\qquad\forall (t,x)\in ]0,\infty[\times\mathbb{R}^d \] with a coercive and uniformly directionally convex Hamiltonian $H\in\mathcal{C}^{1}(\mathbb{R}^d)$. More precisely, we establish a BV bound on the slope of backward characteristics $DH(u(t,\cdot))$ starting at a positive time $t>0$. Relying on the BV bound, we quantify the metric entropy in ${\bf W}^{1,1}_{\mathrm{loc}}(\mathbb{R}^d)$ for the map $S_t$ that associates to every given initial data $u_0\in{\bf Lip}\big(\mathbb{R}^d\big)$, the corresponding solution $S_tu_0$. Finally, a counter example is constructed to show that both $D_xu(t,\cdot)$ and $DH(D_xu(t,\cdot))$ fail to be in $BV_{\mathrm{loc}}$ for a general strictly convex and coercive $H\in\mathcal{C}^2(\mathbb{R}^d)$.

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arXiv2021

Diffusion Approximations of Markovian Solutions to Discontinuous ODEs

In a companion paper, the authors have characterized all deterministic semigroups, and all Markov semigroups, whose trajectories are Carathe'odory solutions to a given ODE x'=f(x), with f possibly discontinuous. The present paper establishes two approximation results. Namely, every deterministic semigroup can be obtained as the pointwise limit of the flows generated by a sequence of ODEs $x'=f_n(x) with smooth right hand sides. Moreover, every Markov semigroup can be obtained as limit of a sequence of diffusion processes with smooth drifts and with diffusion coefficients approaching zero.

math.CA↗
arXiv2021

Solutions to a system of first order H-J equations related to a debt management problem

The paper studies a system of first order Hamilton-Jacobi equations with discontinuous coefficients, arising from a model of deterministic optimal debt management in infinite time horizon, with exponential discount and currency devaluation. The existence of an equilibrium solution is obtained by a suitable concatenation of backward solutions to the system of Hamilton-Jacobi equations. A detailed analysis of the behavior of the solution as the debt-ratio-income $x^*\to +\infty$ is also provided.

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