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Tan H. Cao

Publications and source records attributed to Tan H. Cao.

11 recordsLinked to original sources
arXiv2026

Second-Order Sensitivity of Efficient Solution Maps in Parametric Vector Optimization with Set Constraints

We study second-order fixed-ray Dini sensitivity of the efficient solution map \(S\) for the parametric vector problem \(\min_C f(p,x)\) subject to \(x\in H(p)\). Under a value-to-decision error bound (VDB) -- an inverse stability estimate -- the second-order semi-derivative of the marginal map \(Φ\) transfers to an exact second-order Dini formula for \(S\). The result separates value sensitivity from the inverse recovery of efficient decisions. This mechanism allows a nonsingleton efficient-value fiber and requires neither injectivity nor strict monotonicity of \(\nabla_x f\). For structured feasible maps \(H(p)=\{x\inΩ: g(p,x)\in D\}\), we derive primitive-data formulas for \(D^2H\), \(D^2Φ\), and \(D^2S\) under Robinson metric regularity, second-order regularity of \(Ω\) and \(D\), objective-aware recession conditions, and directional second-order semi-derivability of the data. Polyhedral inequality/equality systems give explicit specializations. A parametric multi-objective portfolio family verifies (VDB) with an explicit constant, obtained directly from the model's feasible directions, while a two-branch example shows how the formula operates on a nonsingleton decision fiber.

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arXiv2026

Convergence and Stability of a Catching-Up Algorithm for Differential Inclusions with Maximal Monotone Operators

We study a catching-up algorithm for a class of differential inclusions driven by maximal monotone operators with continuous perturbations. Using a decomposition of the monotone operator into the closed convex hull of its single-valued part and the normal cone to a closed convex set, we establish existence of solutions and derive global energy bounds under a mild tangent dissipativity assumption. Under an additional local Lipschitz assumption on the perturbation, we also obtain uniqueness and stability with respect to the initial data. We then analyze a time-discretized catching-up scheme with variable step sizes and approximate projections. On every finite horizon, we prove convergence of the discrete trajectories to solutions of the continuous problem. A discrete velocity decomposition together with a discrete energy inequality yields uniform boundedness of the iterates, quantitative stability estimates, and explicit error bounds. We also establish asymptotic feasibility of the predictor step in an $L^2$ sense, as well as a Cesàro-type averaged feasibility property, showing that the constraint violations generated by the free step vanish as the discretization is refined. Finally, we illustrate the theory on explicit examples, including a fully explicit one--dimensional test case and a multidimensional constrained dry-friction system.

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arXiv2025

Optimal Control of Nonconvex Sweeping Processes with Variable Time via Finite-Difference Approximations

The paper is devoted to the study of a new class of optimal control problems for nonsmooth dynamical systems governed by nonconvex discontinuous differential inclusions of the sweeping type with involving variable time into optimization. We develop a novel version of the method of discrete approximations of its own qualitative and numerical importance with establishing its well-posedness and strong convergence to optimal solutions of the controlled sweeping process. Using advanced tools of variational analysis and generalized differentiation leads us to deriving new necessary conditions for optimal solutions to discrete approximation problems, which serve as suboptimality conditions for the original continuous-time controlled sweeping process. The obtained results are applied to a class of motion models of practical interest, where the established necessary conditions are used to investigate the agents' interactions and to develop an algorithm for calculating optimal solutions.

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arXiv2025

Optimal Control of Several Motion Models

This paper is devoted to the study of the dynamic optimization of several controlled crowd motion models in the general planar settings, which is an application of a class of optimal control problems involving a general nonconvex sweeping process with perturbations. A set of necessary optimality conditions for such optimal control problems involving the crowd motion models with multiple agents and obstacles is obtained and analyzed. Several effective algorithms based on such necessary optimality conditions are proposed and various nontrivial illustrative examples together with their simulations are also presented. The implementation of all the considered motion models can be found via the link: https://github.com/tancao1128/Optimal_Control_of_Several_Motion_Models with the instruction and demonstration video uploaded at https://www.youtube.com/watch?v=B8DQ0wvCtIQ.

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arXiv2021

Crowd motion paradigm modeled by a bilevel sweeping control problem

This article concerns an optimal crowd motion control problem in which the crowd features a structure given by its organization into N groups (participants) each one spatially confined in a set. The overall optimal control problem consists in driving the ensemble of sets as close as possible to a given point (the 'exit') while the population in each set minimizes its control effort subject to its sweeping dynamics with a controlled state dependent velocity drift. In order to capture the conflict between the goal of the overall population and those of the various groups, the problem is cast as a bilevel optimization framework. A key challenge of this problem consists in bringing together two quite different paradigms: bilevel programming and sweeping dynamics with a controlled drift. Necessary conditions of optimality in the form of a Maximum Principle of Pontryagin in the Gamkrelidze framework are derived. These conditions are then used to solve a simple illustrative example with two participants, emphasizing the interaction between them.

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arXiv2021

Optimization of Fully Controlled Sweeping Processes

The paper is devoted to deriving necessary optimality conditions in a general optimal control problem for dynamical systems governed by controlled sweeping processes with hard-constrained control actions entering both polyhedral moving sets and additive perturbations. By using the first-order and mainly second-order tools of variational analysis and generalized differentiation, we develop a well-posed method of discrete approximations, obtain optimality conditions for solutions to discrete-time control systems, and then establish by passing to the limit verifiable necessary optimality conditions for local minimizers of the original controlled sweeping process that are expressed entirely in terms of its given data. The efficiency of the obtained necessary optimality conditions for the sweeping dynamics is illustrated by solving three nontrivial examples of their own interest.

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arXiv2020

Optimization and Discrete Approximation of Sweeping Processes with Controlled Moving Sets and Perturbations

This paper addresses a new class of optimal control problems for perturbed sweeping processes with measurable controls in additive perturbations of the dynamics and smooth controls in polyhedral moving sets. We develop a constructive discrete approximation procedure that allows us to strongly approximate any feasible trajectory of the controlled sweeping process by feasible discrete trajectories and also establish a $W^{1,2}$-strong convergence of optimal trajectories for discretized control problems to a given local minimizer of the original continuous-time sweeping control problem of the Bolza type. Employing advanced tools of first-order and second-order variational analysis and generalized differentiation, we derive necessary optimality conditions for discrete optimal solutions under fairly general assumptions formulated entirely in terms of the given data. The obtained results give us efficient suboptimality ("almost optimality") conditions for the original sweeping control problem that are illustrated by a nontrivial numerical example.

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arXiv2017

Optimal control of a nonconvex perturbed sweeping process

The paper concerns optimal control of discontinuous differential inclusions of the normal cone type governed by a generalized version of the Moreau sweeping process with control functions acting in both nonconvex moving sets and additive perturbations. This is a new class of optimal control problems in comparison with previously considered counterparts where the controlled sweeping sets are described by convex polyhedra. Besides a theoretical interest, a major motivation for our study of such challenging optimal control problems with intrinsic state constraints comes from the application to the crowd motion model in a practically adequate planar setting with nonconvex but prox-regular sweeping sets. Based on a constructive discrete approximation approach and advanced tools of first-order and second-order variational analysis and generalized differentiation, we establish the strong convergence of discrete optimal solutions and derive a complete set of necessary optimality conditions for discrete-time and continuous-time sweeping control systems that are expressed entirely via the problem data.

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arXiv2015

Optimal control of a perturbed sweeping process via discrete discrete approximations

The paper addresses an optimal control problem for a perturbed sweeping process of the rate-independent hysteresis type described by a controlled "play and stop" operator with separately controlled perturbations. This problem can be reduced to dynamic optimization of a state-constrained unbounded differential inclusion with highly irregular data that cannot be treated by means of known results in optimal control theory for differential inclusions. We develop the method of discrete approximations, which allows us to adequately replace the original optimal control problem by a sequence of well-posed finite-dimensional optimization problems whose optimal solutions strongly converge to that of the controlled perturbed sweeping process. To solve the discretized control systems, we derive effective necessary optimality conditions by using second-order generalized differential tools of variational analysis that explicitly calculated in terms of the given problem data.

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arXiv2015

Optimality conditions for a controlled sweeping process with applications to the crowd motion model

The paper concerns the study and applications of a new class of optimal control problems governed by a perturbed sweeping process of the hysteresis type with control functions acting in both play-and-stop operator and additive perturbations. Such control problems can be reduced to optimization of discontinuous and unbounded differential inclusions with pointwise state constraints, which are immensely challenging in control theory and prevent employing conventional variation techniques to derive necessary optimality conditions. We develop the method of discrete approximations married with appropriate generalized differential tools of modern variational analysis to overcome principal difficulties in passing to the limit from optimality conditions for finite-difference systems. This approach leads us to nondegenerate necessary conditions for local minimizers of the controlled sweeping process expressed entirely via the problem data. Besides illustrative examples, we apply the obtained results to an optimal control problem associated with of the crowd motion model of traffic flow in a corridor, which is formulated in this paper. The derived optimality conditions allow us to develop an effective procedure to solve this problem in a general setting and completely calculate optimal solutions in particular situations.

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