Article
Computer Science, Interdisciplinary Applications
Jerome Loheac, Vineeth Satheeskumar Varma, Irinel Constantin Morarescu
Summary: This paper proposes control design strategies to minimize the time required by a mobile robot to accomplish a certain task while transmitting/receiving a message. It first designs a minimal-time control for simplified robot dynamics using the Pontryagin maximum principle, and then demonstrates how these results can be used to efficiently control more complicated non-holonomic dynamics.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2022)
Article
Computer Science, Software Engineering
Loic Bourdin, Gaurav Dhar
Summary: In this paper, we derive a Pontryagin maximum principle for general nonlinear optimal sampled-data control problems in the presence of running inequality state constraints. We prove that, under certain general hypotheses, the optimal trajectory activates the running inequality state constraints at most at the sampling times, overcoming theoretical and numerical challenges.
MATHEMATICAL PROGRAMMING
(2022)
Article
Automation & Control Systems
Emmanuel Trelat
Summary: This paper investigates the turnpike phenomenon in finite-dimensional optimal control problems, establishing a linear turnpike theorem that shows a linear discrepancy between the optimal trajectory and the turnpike set.
MATHEMATICS OF CONTROL SIGNALS AND SYSTEMS
(2023)
Article
Mathematics, Applied
Manuel de Leon, Manuel Lainz, Miguel C. Munoz-Lecanda
Summary: This paper combines two main topics in mechanics and optimal control theory, namely contact Hamiltonian systems and Pontryagin maximum principle. One of the important results is the development of a contact Pontryagin maximum principle that can handle optimal control problems with dissipation. The paper also discusses the Herglotz optimal control problem and provides an application to the study of a thermodynamic system.
JOURNAL OF NONLINEAR SCIENCE
(2023)
Article
Mathematics
Leonid Berlin, Andrey Galyaev, Pavel Lysenko
Summary: The time-optimal control problem for a system consisting of two non-synchronous oscillators is examined in this study. By proposing necessary extremum conditions in the form of nonlinear matrix equalities, the relationship between the reachability set and control classes of the oscillators is described. The obtained analytical results are validated through mathematical modeling.
Article
Mathematics
Leishi Wang, Mingtao Li, Xin Pei, Juan Zhang
Summary: China's livestock industry has been growing rapidly, but it faces the challenge of supply-demand imbalance. This study proposes four optimal breeding strategies based on modeling and analyzing the behavior of livestock farmers, and identifies the most suitable strategy through profit comparison. The results indicate that livestock farmers should adjust farm productivity according to price changes in order to maximize profits.
Article
Mathematics, Applied
Jun Moon
Summary: This paper examines the terminal state-constrained optimal control problem for Volterra integral equations with singular kernels. The state equation covers various types of state dynamics, including classical Volterra integral equations with nonsingular kernels, (Caputo) fractional differential equations, and ordinary differential state equations. The maximum principle for the corresponding state-constrained optimal control problem is proved using the Ekeland variational principle, spike variation technique, and properties of the distance function and generalized Gronwall's inequality.
Article
Automation & Control Systems
Cristopher Hermosilla, Michele Palladino
Summary: In this paper, we prove a fully nonsmooth Pontryagin maximum principle for optimal control problems driven by a sweeping process with drift term x is an element of f(t, x, u) -N-C(t)(x). The novel exact penalization technique is used to exploit the controllability properties of the dynamics and prove the maximum principle in the case when the moving set C(t) is both nonsmooth and nonconvex.
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2022)
Article
Mathematics, Applied
Mathias Oster, Leon Sallandt, Reinhold Schneider
Summary: This paper discusses the finite horizon control problem in ordinary differential equation systems, and presents two different methods for solving it: policy iteration and model predictive control. For high-dimensional systems, low-rank tensor approximation and high-dimensional quadrature methods are used for numerical solution, and the effectiveness of the methods is verified through examples.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Lei-Shi Wang, Ming -Tao Li, Xin Pei, Juan Zhang, Gui-Quan Sun, Zhen Jin
Summary: In recent years, there has been a resurgence of brucellosis outbreaks between humans and animals in China. It is crucial to find an optimal control strategy that maximizes farmers' profits while effectively managing the disease. To design a more effective and reasonable control strategy, it is important to understand the profit mechanism of farmers and the transmission mechanism of brucellosis. A dynamic model combining economic factors, livestock farmers' behaviors, and the transmission mechanism of brucellosis is proposed. The model is simulated using data from Ningxia Hui Autonomous Region, and the results indicate the need for adjusting the current control policy and increasing financial support for disease control.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Automation & Control Systems
E. R. Avakov, G. G. Magaril-Il'yaev
Summary: This paper proves the necessary second-order conditions for a local infimum in an optimal control problem, thereby strengthening the Pontryagin maximum principle and the known necessary second-order optimality conditions.
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2022)
Article
Mathematics, Applied
Mathias Oster, Leon Sallandt, Reinhold Schneider
Summary: The paper discusses the common occurrence of controlling systems of ordinary differential equations in science and engineering. It focuses on the local optimal control problems in finite horizon control systems, with two methods being applied to solve these problems - policy iteration and open-loop control methods inspired by model predictive control. The use of low-rank hierarchical tensor product approximation and high-dimensional quadrature is also explored for high-dimensional systems, with linear error propagation demonstrated with numerical evidence on diffusion and Allen-Cahn equations.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Xin Liu, Jason Frank
Summary: The article discusses a regularization of the forward-backward sweep iteration for solving the Pontryagin maximum principle in optimal control problems, proving its global convergence in the continuous time case. The authors further extend the proof to numerical discretization using symplectic Runge-Kutta pairs, demonstrating convergence through a simple numerical experiment.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics
Luis Almeida, Michel Duprez, Yannick Privat, Nicolas Vauchelet
Summary: This article studies the optimal release strategies for controlling disease transmission by using the sterile insect technique. Numerical simulations are presented to illustrate the results.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Automation & Control Systems
Victor Ayala, Philippe Jouan, Maria Luisa Torreblanca, Guilherme Zsigmond
Summary: This paper focuses on the time optimal control of linear systems on Lie groups. General necessary conditions for the existence of time minimizers are provided when controls are unbounded. The findings are then applied to systems on various Lie groups.
SYSTEMS & CONTROL LETTERS
(2021)
Article
Mathematics, Applied
Jerome Loheac, Emmanuel Trelat, Enrique Zuazua
Summary: This paper investigates the controllability problem for finite-dimensional linear autonomous control systems with nonnegative controls. It analyzes the condition of positive minimal controllability time and proves the existence and uniqueness of a minimal time control under certain conditions.
ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE
(2021)
Article
Automation & Control Systems
Riccardo Bonalli, Bruno Herisse, Emmanuel Trelat
IEEE TRANSACTIONS ON AUTOMATIC CONTROL
(2020)
Article
Automation & Control Systems
J. -C. Cortes, A. Navarro-Quiles, J. -V. Romero, M. -D. Rosello, Enrique Zuazua
JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS
(2020)
Article
Mathematics, Applied
Dongnam Ko, Enrique Zuazua
Summary: In this study, a solution to the guiding problem in an optimal control framework is proposed, utilizing the Random Batch Method (RBM) to reduce computational costs for a large number of interacting agents. By considering interactions within randomly divided batches of particles in subintervals, the RBM approximation converges to the exact dynamics in the L-2-expectation norm.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2021)
Article
Automation & Control Systems
Noboru Sakamoto, Enrique Zuazua
Summary: This paper uses dynamical system theory to investigate the turnpike property in nonlinear optimal control. It provides sufficient conditions for the occurrence of the turnpike behavior and discusses the relationship between the turnpike property and stability. The research offers new insights through geometric approaches and attempts to remove smallness restrictions for initial and target states.
Article
Mathematics, Applied
Gengsheng Wang, Yubiao Zhang, Enrique Zuazua
Summary: This article presents a decomposition for the flow generated by the heat equation with a real analytic memory kernel. The decomposition consists of three components: a parabolic component, a hyperbolic component with zero velocity of propagation, and a component with finite smoothing effect. This decomposition reveals the hybrid parabolic-hyperbolic nature of the flow and demonstrates the significant impact of the memory term on the parabolic behavior of the system in the absence of memory terms.
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
(2022)
Article
Operations Research & Management Science
Yesim Sarac, Enrique Zuazua
Summary: This article analyzes the sidewise controllability for the variable coefficients one-dimensional wave equation and reformulates the problem as a dual observability property for the corresponding adjoint system. The feasibility in the class of BV-coefficients is proved using sidewise energy propagation arguments over a sufficiently large time. Additionally, several open problems and perspectives for further research are presented.
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Shi Jin, Yuhua Zhu, Enrique Zuazua
Summary: In this study, we consider the Vlasov-Fokker-Planck equation with a random electric field and prove that the best N approximation in the random space has a convergence rate higher than the Monte-Carlo method. We also develop a residual based adaptive sparse polynomial interpolation method for more efficient solution of multi-scale linear kinetic equations.
NUMERISCHE MATHEMATIK
(2022)
Article
Automation & Control Systems
Jon Asier Barcena-Petisco, Enrique Zuazua
Summary: This paper examines the averaged dynamics for heat equations in the degenerate case where the diffusivity coefficient can be zero, demonstrating its analytical nature and discussing its null controllability depending on the behavior of the averaging density. The critical density threshold reveals similarities to the 1/2-fractional Laplacian, impacting the controllability in the null diffusivity regime. Null controllability fails or holds depending on the weights of the density in comparison to the critical regime.
SYSTEMS & CONTROL LETTERS
(2021)
Article
Automation & Control Systems
Thibault Liard, Enrique Zuazua
Summary: This article studies the identification problem for the 1-D Burgers equation and provides an alternative proof based on generalized backward characteristics to characterize the set of initial data leading to a given target. This offers hope for researching conservation laws systems in 1-D where the classical Lax-Hopf formula is no longer applicable.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL
(2022)
Article
Mathematics, Applied
D. W. M. Veldman, E. Zuazua
Summary: This paper proposes and analyzes a framework for randomized time-splitting in linear-quadratic optimal control, inspired by the successes of stochastic algorithms in the training of deep neural networks and the simulation of interacting particle systems. The study shows that the proposed method can achieve similar dynamics, minimal values of the cost functional, and optimal control as those in the original problem when the time grid is refined. Numerical experiments validate the derived convergence rates and indicate a reduction in computational cost for large-scale linear dynamical systems with the proposed method.
NUMERISCHE MATHEMATIK
(2022)
Article
Automation & Control Systems
Borjan Geshkovski, Enrique Zuazua
Summary: This article presents a reformulation of the problem of finding the actuator design that minimizes the controllability cost for finite-dimensional linear systems. By using the Brunovsky normal form, the restriction of working with diagonalizable system dynamics is removed and the problem reduces to the minimization of the norm of the inverse of a change of basis matrix. This reformulation allows for an easier deduction of solution existence and provides a clearer picture of the problem's intrinsic symmetries.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL
(2022)
Article
Mathematics, Applied
Giuseppe Maria Coclite, Nicola De Nitti, Alexander Keimer, Lukas Pflug, Enrique Zuazua
Summary: This paper studies the long-time behavior of the unique weak solution of a nonlocal regularization of the Burgers equation. The key of the proof lies in the suitable scaling argument and a nonlocal Oleinik-type estimate.
Article
Automation & Control Systems
Jan Heiland, Enrique Zuazua
Summary: The concept of turnpike connects optimal control problems with steady-state optimal controls, focusing on the linear quadratic regulator problem and the convergence of the associated differential Riccati equation. The analysis extends classical system theoretic results to investigate turnpike properties of standard state space systems and descriptor systems, establishing conditions for turnpike phenomena in certain cases. Additionally, the existence and convergence of solutions to a generalized differential Riccati equation are established for impulse controllable descriptor systems.
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2021)
Article
Automation & Control Systems
Borjan Geshkovski, Enrique Zuazua
Summary: This work investigates the local controllability of a one-dimensional free boundary problem for a fluid governed by the viscous Burgers equation. It shows that the fluid can be steered to constant velocity by controlling along the fixed boundary, in addition to prescribing the free boundary's position, under the condition that the initial velocities and interface positions are close enough.
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2021)
Article
Mathematics
Daniele Cassani, Zhisu Liu, Giulio Romani
Summary: This article investigates the strongly coupled nonlinear Schrodinger equation and Poisson equation in two dimensions. The existence of solutions is proved using a variational approximating procedure, and qualitative properties of the solutions are established through the moving planes technique.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Giovanni Alessandrini, Romina Gaburro, Eva Sincich
Summary: This paper considers the inverse problem of determining the conductivity of a possibly anisotropic body Ω, subset of R-n, by means of the local Neumann-to-Dirichlet map on a curved portion Σ of its boundary. Motivated by the uniqueness result for piecewise constant anisotropic conductivities, the paper provides a Hölder stability estimate on Σ when the conductivity is a priori known to be a constant matrix near Σ.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Nuno Costa Dias, Cristina Jorge, Joao Nuno Prata
Summary: This article studies the time dependent Euler-Bernoulli beam equation with discontinuous and singular coefficients, and obtains an explicit formulation of the differential problem using an extension of the Hormander product of distributions. The dynamics of the Euler-Bernoulli beam model with discontinuous flexural stiffness and structural cracks are further explored, and the relationship between the characteristic frequencies of the beam and the singularities in the flexural stiffness is investigated.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Baoquan Zhou, Hao Wang, Tianxu Wang, Daqing Jiang
Summary: This paper is Part I of a two-part series that presents a mathematical framework for approximating the invariant probability measures and density functions of stochastic generalized Kolmogorov systems with small diffusion. It introduces two new approximation methods and demonstrates their utility in various applications.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Yun Li, Danhua Jiang, Zhi-Cheng Wang
Summary: In this study, a nonlocal reaction-diffusion equation is used to model the growth of phytoplankton species in a vertical water column with changing-sign advection. The species relies solely on light for metabolism. The paper primarily focuses on the concentration phenomenon of phytoplankton under conditions of large advection amplitude and small diffusion rate. The findings show that the phytoplankton tends to concentrate at certain critical points or the surface of the water column under these conditions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Monica Conti, Stefania Gatti, Alain Miranville
Summary: The aim of this paper is to study a perturbation of the Cahn-Hilliard equation with nonlinear terms of logarithmic type. By proving the existence, regularity and uniqueness of solutions, as well as the (strong) separation properties of the solutions from the pure states, we finally demonstrate the convergence to the Cahn-Hilliard equation on finite time intervals.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Qi Qiao
Summary: This paper investigates a volume-filling chemotaxis model with a small cell diffusion coefficient and chemotactic sensitivity. By using the geometric singular perturbation theory, the existence of a positive traveling wave connecting two constant steady states is confirmed. The monotonicity of the wave is analyzed for different parameter ranges, and spectral instability is observed in some exponentially weighted spaces.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Xiaolong He
Summary: This article employs the CWB method to construct quasi-periodic solutions for nonlinear delayed perturbation equations, and combines the techniques of Green's function estimate and the reducibility method in KAM theory to solve the linear equation, thus extending the applicability of the CWB method. As an application, it studies the positive quasi-periodic solutions for a class of Lotka-Volterra equations with quasi-periodic coefficients and time delay.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Nicolas Camps, Louise Gassot, Slim Ibrahim
Summary: In this paper, we consider the probabilistic local well-posedness problem for the Schrodinger half-wave equation with a cubic nonlinearity in quasilinear regimes. Due to the lack of probabilistic smoothing in the Picard's iterations caused by high-low-low nonlinear interactions, we need to use a refined ansatz. The proof is an adaptation of Bringmann's method on the derivative nonlinear wave equation [6] to Schrodinger-type equations. In addition, ill-posedness results for this equation are discussed.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Elie Abdo, Mihaela Ignatova
Summary: In this study, we investigate the Nernst-Planck-Navier-Stokes system with periodic boundary conditions and prove the exponential nonlinear stability of constant steady states without constraints on the spatial dimension. We also demonstrate the exponential stability from arbitrary large data in the case of two spatial dimensions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Peter De Maesschalck, Joan Torregrosa
Summary: This paper provides the best lower bound for the number of critical periods of planar polynomial centers known up to now. The new lower bound is obtained in the Hamiltonian class and considering a single period annulus. The key idea is the perturbation of a vector field with many cusp equilibria, which is constructed using elements of catastrophe theory.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Leyi Jiang, Taishan Yi, Xiao-Qiang Zhao
Summary: This paper studies the propagation dynamics of a class of integro-difference equations with a shifting habitat. By transforming the equation using moving coordinates and establishing the spreading properties of solutions and the existence of nontrivial forced waves, the paper contributes to the understanding of the propagation properties of the original equation.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Mckenzie Black, Changhui Tan
Summary: This article investigates a family of nonlinear velocity alignments in the compressible Euler system and shows the asymptotic emergent phenomena of alignment and flocking. Different types of nonlinearity and nonlocal communication protocols are studied, resulting in a variety of different asymptotic behaviors.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Lorenzo Cavallina
Summary: In this paper, the concept of variational free boundary problem is introduced, and a unified functional-analytical framework is provided for constructing families of solutions. The notion of nondegeneracy of a critical point is extended to this setting.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Ying-Chieh Lin, Kuan-Hsiang Wang, Tsung-Fang Wu
Summary: In this study, we investigate a linearly coupled Schrodinger system and establish the existence of positive ground states under suitable assumptions and by using variational methods. We also relax some of the conditions and provide some results on the existence of positive ground states to a linearly coupled Schrodinger system in a bounded domain.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)