arXiv subjects

Huynh Khanh

Publications and source records attributed to Huynh Khanh.

7 recordsLinked to original sources

arXiv2025

On space-time FEM for time-optimal control problems governed by parabolic equations with mixed and endpoints constraints

In this paper, we consider a class of time-optimal control problems governed by linear parabolic equations with mixed control-state constraints and end-point constraints, and without Tikhonov regularization term in the objective function. By the finite element method, we discretize the optimal control problem to obtain a sequence of mathematical programming problems in finite-dimensional spaces. Under certain conditions, we show that the optimal solutions of the discrete problems converge to an optimal solution of the original problem. Besides, we show that if the second-order sufficient condition is satisfied, then some error estimates of approximate solutions are obtained.

math.OC↗

arXiv2024

Second-order Optimality Conditions for Time-Optimal Control Problems Governed by Semilinear Parabolic Equations

A class of time-optimal control problems governed by semilinear parabolic equations with mixed pointwise constraints and final point constraints is considered. By introducing the so-called locally optimal solution to time-optimal control problems, we establish first and second-order necessary optimality conditions of KKT-type and second-order sufficient conditions for locally optimal solutions to the problem.

math.OC↗

arXiv2024

Regularity of multipliers and second-order optimality conditions for semilinear parabolic optimal control problems with mixed pointwise constraints

A class of optimal control problems governed by semilinear parabolic equations with mixed pointwise constraints is considered. We give some criteria under which the first and second-order optimality conditions are of KKT-type. We then prove that the Lagrange multipliers belong to $L^p$-spaces. Moreover, we show that if the initial value is good enough and boundary $\partialΩ$ has a property of positive geometric density, then multipliers and optimal solutions are Hölder continuous.

math.OC↗

arXiv2024

Locally Lipschitz stability of solutions to a parametric parabolic optimal control problem with mixed pointwise constraints

A class of parametric optimal control problems governed by semilinear parabolic equations with mixed pointwise constraints is investigated. The perturbations appear in the objective functional, the state equation and in mixed pointwise constraints. By analyzing regularity and establishing stability condition of Lagrange multipliers we prove that, if the strictly second-order sufficient condition for the unperturbed problem is valid, then the solutions of the problems as well as the associated Lagrange multipliers are locally Lipschitz continuous functions of parameters.

math.OC↗

arXiv2023

Regularity of multipliers and second-order optimality conditions of KKT-type for semilinear parabolic control problems

A class of optimal control problems governed by semilinear parabolic equations with mixed constraints and a box constraint for control variable is considered. We show that if the separation condition is satisfied, then both optimality conditions of KKT-type and regularity of multipliers are fulfilled. Moreover, we show that if the initial value is good enough and boundary $\partialΩ$ has a property of positive geometric density, then multipliers and optimal solutions are Hölder continuous.

math.OC↗

arXiv2021

Stein kernels for $q$-moment measures and new bounds for the rate of convergence in the central limit theorem

Given an isotropic probability measure $μ$ on ${\mathbb R}^d$ with ${\rm d}μ\left( x \right) = {\left( {\varrho \left( x \right)} \right)^{ - α}}{\rm d}x$, where $α> d + 1$ and $\varrho :{{\mathbb R}^d} \to \left( {0, + \infty } \right)$ is a continuous function and uniformly convex (${\nabla ^2}\varrho \ge {\varepsilon_0}{\rm {Id}}$). By using Stein kernels for $\left( {α- d} \right)$-moment measures, we prove that the rates of convergence in the central limit theorem with sequence of i.i.d. random variables ${X_1},{X_2},...,{X_n}$ of the law $μ$, to be of form $c_{{\varepsilon}_0}\,\sqrt {\dfrac{d}{n}} $. The general case (i.e., $\varrho$ is only convex and continuous) remains open.

math.PR↗

arXiv2020

q-Moment Measures and Applications: A New Approach via Optimal Transport

In 2017, Bo'az Klartag obtained a new result in differential geometry on the existence of affine hemisphere of elliptic type. In his approach, a surface is associated with every a convex function $Φ$ : R^n $\rightarrow$ (0, +$\infty$) and the condition for the surface to be an affine hemisphere involves the 2-moment measure of $Φ$ (a particular case of q-moment measures, i.e measures of the form ($\nabla$$Φ$) \# ($Φ$^{--(n+q)}) for q > 0). In Klartag's paper, q-moment measures are studied through a variational method requiring to minimize a functional among convex functions, which is studied using the Borell-Brascamp-Lieb inequality. In this paper, we attack the same problem through an optimal transport approach, since the convex function $Φ$ is a Kantorovich potential (as already done for moment measures in a previous paper). The variational problem in this new approach becomes the minimization of a local functional and a transport cost among probability measures and the optimizer turns out to be of the form $ρ$ = $Φ$^{--(n+q)}.

math.AP↗