Le Centre pour la Communication Scientifique Directe - HAL - Université Paris Descartes, Feb 16, 2022
In this paper, we study the problem of initial data identification for the one-dimensional Burger... more In this paper, we study the problem of initial data identification for the one-dimensional Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the time-irreversibility of the Burgers equation, some target functions are unattainable from solutions of this equation, making the inverse problem under consideration ill-posed. To get around this issue, we introduce an non-smooth optimization problem which consists in minimizing the difference between the predictions of the Burgers equation and the observations of the system at a final time in L 2 (R) norm. The two main contributions of this work are as follows. • We fully characterize the set of minimizers of the aforementioned non-smooth optimization problem. • A wave-front tracking method is implemented to construct numerically all of them. One of minimizers is the backward entropy solution, constructed using a backward-forward method.
This paper deals with the stabilization of a linear hyperbolic system subject to a boundary distu... more This paper deals with the stabilization of a linear hyperbolic system subject to a boundary disturbance. Our feedback design relies on a super-twisting control algorithm, which leads to a feedback that is continuous with respect to the state, in contrast with the classical sliding mode design. Our first result is the existence of solutions of the closed-loop system. Moreover, the global asymptotic stability, that is our second result, is proved together with the guarantee that the disturbance is rejected.
Observation and control for some conservative systems
Dans cette thèse, nous nous intéressons à la contrôlabilité interne et à son coût pour une ou plu... more Dans cette thèse, nous nous intéressons à la contrôlabilité interne et à son coût pour une ou plusieurs équations aux dérivées partielles conservatives. Dans la première partie, nous introduisons et détaillons deux méthodes permettant d'estimer le coût du contrôle (et par dualité, de la constante d'observabilité) de l'équation des ondes avec potentiel L∞ en dimension un d'espace. La première utilise la propagation des ondes le long des caractéristiques en s'appuyant sur le rôle symétrique de la variable de temps et d'espace. La deuxième méthode repose sur la décomposition spectrale de l'équation des ondes et sur l'utilisation des inégalités d'ingham. L'estimation de la constante d'observabilité se ramène alors à l'étude d'un problème d'optimisation faisant intervenir les vecteurs propres du laplacien-dirichlet avec potentiel. Nous fournissons ensuite des propriétés qualitatives sur le minimiseurs ainsi qu'une estimation du ...
Autonomous vehicles (AV) offer new avenues for transportation applications and call for a new und... more Autonomous vehicles (AV) offer new avenues for transportation applications and call for a new understanding of traffic dynamics when both regular cars and AV coexist. Many works looked at this question with a microscopic approach where both the AVs and the regular cars are modelled as ODEs. Here, present a second order model describing the interaction between a macroscopic traffic flow described by PDEs and autonomous vehicles (AV) described by ODEs. This model is inspired by recent development on moving bottlenecks for scalar conservation laws and is intrinsically multiscale. We give an analysis of this model and we show the existence of weak solutions with bounded variations.
Le Centre pour la Communication Scientifique Directe - HAL - Université Paris Descartes, Feb 27, 2020
We introduce a coupled PDE-ODEs model to describe mixed traffic with humans and autonomous vehicl... more We introduce a coupled PDE-ODEs model to describe mixed traffic with humans and autonomous vehicles. The partial differential equation describes the bulk of human traffic while the ordinary differential equations characterize the trajectories of possibly many autonomous vehicles. The coupled PDE-ODE model is introduced, and existence of solutions for this model is shown, along with a proposed algorithm to construct approximate solutions. We propose a control strategy for the speeds of the autonomous vehicles to minimize total fuel consumption. Existence of solutions for the optimal control problem is proved, and show numerically that a greater reduction in total fuel consumption is possible with more AVs acting as moving bottlenecks.
In this paper, we study the problem of identification for the one-dimensional Burgers equation. T... more In this paper, we study the problem of identification for the one-dimensional Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the property of non-backward uniqueness of Burgers equation, there may exist multiple initial data leading to the same given target. In [12], [16], the authors fully characterize the set of initial data leading to a given target using the classical Lax-Hopf formula. In this note, an alternative proof based only on generalized backward characteristics is given. This leads to the hope of investigate systems of conservation laws in one dimension where the classical Lax-Hopf formula doesn't hold anymore. Moreover, numerical illustrations are presented using as a target, a function optimized for minimum pressure rise in the context of sonic-boom minimization problems. All of initial data leading to this given target are constructed using a wave-front tracking algorithm.
Dans cette these, nous nous interessons a la controlabilite interne et a son cout pour une ou plu... more Dans cette these, nous nous interessons a la controlabilite interne et a son cout pour une ou plusieurs equations aux derivees partielles conservatives. Dans la premiere partie, nous introduisons et detaillons deux methodes permettant d'estimer le cout du controle (et par dualite, de la constante d'observabilite) de l'equation des ondes avec potentiel L∞ en dimension un d'espace. La premiere utilise la propagation des ondes le long des caracteristiques en s'appuyant sur le role symetrique de la variable de temps et d'espace. La deuxieme methode repose sur la decomposition spectrale de l'equation des ondes et sur l'utilisation des inegalites d'ingham. L'estimation de la constante d'observabilite se ramene alors a l'etude d'un probleme d'optimisation faisant intervenir les vecteurs propres du laplacien-dirichlet avec potentiel. Nous fournissons ensuite des proprietes qualitatives sur le minimiseurs ainsi qu'une estimation du ...
Existence of solutions for scalar conservation laws with moving flux constraints
We consider a coupled PDE-ODE model representing a slow moving vehicle immersed in vehicular traf... more We consider a coupled PDE-ODE model representing a slow moving vehicle immersed in vehicular traffic. The PDE consists of a scalar conservation law modeling the evolution of vehicular traffic and the trajectory of a slow moving vehicle is given by an ODE depending on the downstream traffic density. The slow moving vehicle may be regarded as a moving bottleneck influencing the bulk traffic flow via a moving flux pointwise constraint. We prove existence of solutions with respect to initial data of bounded variation. Approximate solutions are constructed via the wave-front tracking method and their limit are solutions of the Cauchy problem PDE-ODE.
In this paper, we study the problem of inverse design for the one-dimensional Burgers equation. T... more In this paper, we study the problem of inverse design for the one-dimensional Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the time-irreversibility of the Burgers equation, some target functions are unattainable from solutions of this equation, making the inverse problem under consideration ill-posed. To get around this issue, we introduce an optimal control problem which consists in minimizing the difference between the predictions of the Burgers equation and the observations of the system at a final time in L 2 (R) norm. The two main contributions of this work are the following: • We fully characterize the set of minimizers of the aforementioned optimal control problem. • A wave-front tracking method is implemented to construct numerically all of them. One of minimizers is the backward entropy solution, constructed using a backward-forward method.
We consider a coupled PDE-ODE model representing a slow moving vehicle immersed in vehicular traf... more We consider a coupled PDE-ODE model representing a slow moving vehicle immersed in vehicular traffic. The PDE consists of a scalar conservation law modeling the evolution of vehicular traffic and the trajectory of a slow moving vehicle is given by an ODE depending on the downstream traffic density. The slow moving vehicle may be regarded as a moving bottleneck influencing the bulk traffic flow via a moving flux pointwise constraint. We prove existence of solutions with respect to initial data of bounded variation. Approximate solutions are constructed via the wave-front tracking method and their limit are solutions of the Cauchy problem PDE-ODE.
This paper introduces a new sliding mode methodology for a system of two hyperbolic equations. Th... more This paper introduces a new sliding mode methodology for a system of two hyperbolic equations. The main difference with the existing literature is the definition of the sliding variable, given here by the gradient of a Lyapunov functional. We state and prove an existence theorem and a global asymptotic stability result. The efficiency of our feedback-law is illustrated by some numerical simulations relying on implicit schemes.
This article considers the possibility of using a small number of autonomous vehicles (AV) for tr... more This article considers the possibility of using a small number of autonomous vehicles (AV) for traffic control of the predominantly human-piloted traffic. Specifically, we consider the control of the AV to act as a moving bottleneck, which will be used to optimize traffic flow properties such as fuel consumption of the combined human-piloted and autonomous traffic flow. We use a coupled partial differential equation (PDE)-ordinary differential equation (ODE) framework to model the bulk traffic flow using a PDE, and the trajectory of an autonomous vehicle in the flow using an ODE, depending on the downstream traffic density. The autonomous vehicle acts on the traffic flow as a moving bottleneck via a moving flux constraint. Using this modeling framework, we consider an optimal control problem which consists in finding the optimal AV trajectory to minimize fuel consumption of the entire traffic flow. We prove existence of optimal AV trajectories and we present two different optimal driving strategies depending on the initial traffic conditions.
Autonomous vehicles (AVs) allow new ways of regulating the traffic flow on road networks. Most of... more Autonomous vehicles (AVs) allow new ways of regulating the traffic flow on road networks. Most of available results in this direction are based on microscopic approaches, where ODEs describe the evolution of regular cars and AVs. In this paper, we propose a multiscale approach, based on recently developed models for moving bottlenecks. Our main result is the proof of existence of solutions for time-varying bottleneck speed, which corresponds to open-loop controls with bounded variation.
In this article, we investigate a non-localization property of the eigenfunctions of Sturm-Liouvi... more In this article, we investigate a non-localization property of the eigenfunctions of Sturm-Liouville operators Aa = −∂xx + a(•) Id with Dirichlet boundary conditions, where a(•) runs over the bounded nonnegative potential functions on the interval (0, L) with L > 0. More precisely, we address the extremal spectral problem of minimizing the L 2-norm of a function e(•) on a measurable subset ω of (0, L), where e(•) runs over all eigenfunctions of Aa, at the same time with respect to all subsets ω having a prescribed measure and all L ∞ potential functions a(•) having a prescribed essentially upper bound. We provide some existence and qualitative properties of the minimizers, as well as precise lower and upper estimates on the optimal value. Several consequences in control and stabilization theory are then highlighted.
Computational Intelligence and Optimization Methods for Control Engineering, 2019
This article considers the problem of traffic control in which an autonomous vehicle is used to r... more This article considers the problem of traffic control in which an autonomous vehicle is used to regulate human piloted traffic to dissipate stop and go traffic waves. We first investigate the controllability of well-known microscopic traffic flow models, namely i) the Bando model (also known as the optimal velocity model), ii) the follow-the-leader model, and iii) a combined optimal velocity follow the leader model. Based on the controllability results, we propose three control strategies for an autonomous vehicle to stabilize the other, human-piloted traffic. We subsequently simulate the control effects on the microscopic models of human drivers in numerical experiments to quantify the potential benefits of the controllers. Based on the simulations, finally we conduct a field experiment with 22 human drivers and a fully autonomous-capable vehicle, to assess the feasibility of autonomous vehicle based traffic control on real human piloted traffic. We show that both in simulation and in the field test that an autonomous vehicle is able to dampen waves generated by 22 cars, and that as a consequence, the total fuel consumption of all vehicles is reduced by up to 20%.
We consider a partial differential equation-ordinary differential equation system to describe the... more We consider a partial differential equation-ordinary differential equation system to describe the dynamics of traffic flow with autonomous vehicles. In the model the bulk flow is represented by a scalar conservation law, while each autonomous vehicle is described by a car following model. The autonomous vehicles act as tracer vehicles in the flow and collect measurements along their trajectory to estimate the bulk flow. The main result is to prove theoretically and show numerically how to reconstruct the correct traffic density using only the measurements from the autonomous vehicles.
We consider a strongly coupled ODE-PDE system representing moving bottlenecks immersed in vehicul... more We consider a strongly coupled ODE-PDE system representing moving bottlenecks immersed in vehicular traffic. The PDE consists of a scalar conservation law modeling the traffic flow evolution and the ODE models the trajectory of a slow moving vehicle. The moving bottleneck influences the bulk traffic flow via a point flux constraint, which is given by an inequality on the flux at the slow vehicle position. We prove uniqueness and continuous dependence of solutions with respect to initial data of bounded variation. The proof is based on a new backward in time method established to capture the values of the norm of generalized tangent vectors at every time.
Mathematics of Control, Signals, and Systems, 2017
In this article, we give a necessary and sufficient condition of Kalman type for the indirect con... more In this article, we give a necessary and sufficient condition of Kalman type for the indirect controllability of systems of groups of linear operators, under some "regularity and locality" conditions on the control operator that will be made precise later and fit very well the case of distributed controls. Moreover, in the case of first order in time systems, when the Kalman rank condition is not satisfied, we characterize exactly the initial conditions that can be controlled. Some applications to the control of systems of Schrödinger or wave equations are provided. The main tool used here is the fictitious control method coupled with the proof of an algebraic solvability property for some related underdetermined system and some regularity results.
For a given bounded connected domain in IR n , the issue of computing the observability constant ... more For a given bounded connected domain in IR n , the issue of computing the observability constant associated to a wave operator, an observation time T and a generic observation subdomain constitutes in general a hard task, even for one-dimensional problems. In this work, we introduce and describe two methods to provide precise (and even sharp in some cases) estimates of observability constants for general one dimensional wave equations: the first one uses a spectral decomposition of the solution of the wave equation whereas the second one is based on a propagation argument along the characteristics. Both methods are extensively described and we then comment on the advantages and drawbacks of each one. The discussion is illustrated by several examples and numerical simulations. As a byproduct, we deduce from the main results estimates of the cost of control (resp. the decay rate of the energy) for several controlled (resp. damped) wave equations.
Le Centre pour la Communication Scientifique Directe - HAL - Université Paris Descartes, Feb 16, 2022
In this paper, we study the problem of initial data identification for the one-dimensional Burger... more In this paper, we study the problem of initial data identification for the one-dimensional Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the time-irreversibility of the Burgers equation, some target functions are unattainable from solutions of this equation, making the inverse problem under consideration ill-posed. To get around this issue, we introduce an non-smooth optimization problem which consists in minimizing the difference between the predictions of the Burgers equation and the observations of the system at a final time in L 2 (R) norm. The two main contributions of this work are as follows. • We fully characterize the set of minimizers of the aforementioned non-smooth optimization problem. • A wave-front tracking method is implemented to construct numerically all of them. One of minimizers is the backward entropy solution, constructed using a backward-forward method.
This paper deals with the stabilization of a linear hyperbolic system subject to a boundary distu... more This paper deals with the stabilization of a linear hyperbolic system subject to a boundary disturbance. Our feedback design relies on a super-twisting control algorithm, which leads to a feedback that is continuous with respect to the state, in contrast with the classical sliding mode design. Our first result is the existence of solutions of the closed-loop system. Moreover, the global asymptotic stability, that is our second result, is proved together with the guarantee that the disturbance is rejected.
Observation and control for some conservative systems
Dans cette thèse, nous nous intéressons à la contrôlabilité interne et à son coût pour une ou plu... more Dans cette thèse, nous nous intéressons à la contrôlabilité interne et à son coût pour une ou plusieurs équations aux dérivées partielles conservatives. Dans la première partie, nous introduisons et détaillons deux méthodes permettant d'estimer le coût du contrôle (et par dualité, de la constante d'observabilité) de l'équation des ondes avec potentiel L∞ en dimension un d'espace. La première utilise la propagation des ondes le long des caractéristiques en s'appuyant sur le rôle symétrique de la variable de temps et d'espace. La deuxième méthode repose sur la décomposition spectrale de l'équation des ondes et sur l'utilisation des inégalités d'ingham. L'estimation de la constante d'observabilité se ramène alors à l'étude d'un problème d'optimisation faisant intervenir les vecteurs propres du laplacien-dirichlet avec potentiel. Nous fournissons ensuite des propriétés qualitatives sur le minimiseurs ainsi qu'une estimation du ...
Autonomous vehicles (AV) offer new avenues for transportation applications and call for a new und... more Autonomous vehicles (AV) offer new avenues for transportation applications and call for a new understanding of traffic dynamics when both regular cars and AV coexist. Many works looked at this question with a microscopic approach where both the AVs and the regular cars are modelled as ODEs. Here, present a second order model describing the interaction between a macroscopic traffic flow described by PDEs and autonomous vehicles (AV) described by ODEs. This model is inspired by recent development on moving bottlenecks for scalar conservation laws and is intrinsically multiscale. We give an analysis of this model and we show the existence of weak solutions with bounded variations.
Le Centre pour la Communication Scientifique Directe - HAL - Université Paris Descartes, Feb 27, 2020
We introduce a coupled PDE-ODEs model to describe mixed traffic with humans and autonomous vehicl... more We introduce a coupled PDE-ODEs model to describe mixed traffic with humans and autonomous vehicles. The partial differential equation describes the bulk of human traffic while the ordinary differential equations characterize the trajectories of possibly many autonomous vehicles. The coupled PDE-ODE model is introduced, and existence of solutions for this model is shown, along with a proposed algorithm to construct approximate solutions. We propose a control strategy for the speeds of the autonomous vehicles to minimize total fuel consumption. Existence of solutions for the optimal control problem is proved, and show numerically that a greater reduction in total fuel consumption is possible with more AVs acting as moving bottlenecks.
In this paper, we study the problem of identification for the one-dimensional Burgers equation. T... more In this paper, we study the problem of identification for the one-dimensional Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the property of non-backward uniqueness of Burgers equation, there may exist multiple initial data leading to the same given target. In [12], [16], the authors fully characterize the set of initial data leading to a given target using the classical Lax-Hopf formula. In this note, an alternative proof based only on generalized backward characteristics is given. This leads to the hope of investigate systems of conservation laws in one dimension where the classical Lax-Hopf formula doesn't hold anymore. Moreover, numerical illustrations are presented using as a target, a function optimized for minimum pressure rise in the context of sonic-boom minimization problems. All of initial data leading to this given target are constructed using a wave-front tracking algorithm.
Dans cette these, nous nous interessons a la controlabilite interne et a son cout pour une ou plu... more Dans cette these, nous nous interessons a la controlabilite interne et a son cout pour une ou plusieurs equations aux derivees partielles conservatives. Dans la premiere partie, nous introduisons et detaillons deux methodes permettant d'estimer le cout du controle (et par dualite, de la constante d'observabilite) de l'equation des ondes avec potentiel L∞ en dimension un d'espace. La premiere utilise la propagation des ondes le long des caracteristiques en s'appuyant sur le role symetrique de la variable de temps et d'espace. La deuxieme methode repose sur la decomposition spectrale de l'equation des ondes et sur l'utilisation des inegalites d'ingham. L'estimation de la constante d'observabilite se ramene alors a l'etude d'un probleme d'optimisation faisant intervenir les vecteurs propres du laplacien-dirichlet avec potentiel. Nous fournissons ensuite des proprietes qualitatives sur le minimiseurs ainsi qu'une estimation du ...
Existence of solutions for scalar conservation laws with moving flux constraints
We consider a coupled PDE-ODE model representing a slow moving vehicle immersed in vehicular traf... more We consider a coupled PDE-ODE model representing a slow moving vehicle immersed in vehicular traffic. The PDE consists of a scalar conservation law modeling the evolution of vehicular traffic and the trajectory of a slow moving vehicle is given by an ODE depending on the downstream traffic density. The slow moving vehicle may be regarded as a moving bottleneck influencing the bulk traffic flow via a moving flux pointwise constraint. We prove existence of solutions with respect to initial data of bounded variation. Approximate solutions are constructed via the wave-front tracking method and their limit are solutions of the Cauchy problem PDE-ODE.
In this paper, we study the problem of inverse design for the one-dimensional Burgers equation. T... more In this paper, we study the problem of inverse design for the one-dimensional Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the time-irreversibility of the Burgers equation, some target functions are unattainable from solutions of this equation, making the inverse problem under consideration ill-posed. To get around this issue, we introduce an optimal control problem which consists in minimizing the difference between the predictions of the Burgers equation and the observations of the system at a final time in L 2 (R) norm. The two main contributions of this work are the following: • We fully characterize the set of minimizers of the aforementioned optimal control problem. • A wave-front tracking method is implemented to construct numerically all of them. One of minimizers is the backward entropy solution, constructed using a backward-forward method.
We consider a coupled PDE-ODE model representing a slow moving vehicle immersed in vehicular traf... more We consider a coupled PDE-ODE model representing a slow moving vehicle immersed in vehicular traffic. The PDE consists of a scalar conservation law modeling the evolution of vehicular traffic and the trajectory of a slow moving vehicle is given by an ODE depending on the downstream traffic density. The slow moving vehicle may be regarded as a moving bottleneck influencing the bulk traffic flow via a moving flux pointwise constraint. We prove existence of solutions with respect to initial data of bounded variation. Approximate solutions are constructed via the wave-front tracking method and their limit are solutions of the Cauchy problem PDE-ODE.
This paper introduces a new sliding mode methodology for a system of two hyperbolic equations. Th... more This paper introduces a new sliding mode methodology for a system of two hyperbolic equations. The main difference with the existing literature is the definition of the sliding variable, given here by the gradient of a Lyapunov functional. We state and prove an existence theorem and a global asymptotic stability result. The efficiency of our feedback-law is illustrated by some numerical simulations relying on implicit schemes.
This article considers the possibility of using a small number of autonomous vehicles (AV) for tr... more This article considers the possibility of using a small number of autonomous vehicles (AV) for traffic control of the predominantly human-piloted traffic. Specifically, we consider the control of the AV to act as a moving bottleneck, which will be used to optimize traffic flow properties such as fuel consumption of the combined human-piloted and autonomous traffic flow. We use a coupled partial differential equation (PDE)-ordinary differential equation (ODE) framework to model the bulk traffic flow using a PDE, and the trajectory of an autonomous vehicle in the flow using an ODE, depending on the downstream traffic density. The autonomous vehicle acts on the traffic flow as a moving bottleneck via a moving flux constraint. Using this modeling framework, we consider an optimal control problem which consists in finding the optimal AV trajectory to minimize fuel consumption of the entire traffic flow. We prove existence of optimal AV trajectories and we present two different optimal driving strategies depending on the initial traffic conditions.
Autonomous vehicles (AVs) allow new ways of regulating the traffic flow on road networks. Most of... more Autonomous vehicles (AVs) allow new ways of regulating the traffic flow on road networks. Most of available results in this direction are based on microscopic approaches, where ODEs describe the evolution of regular cars and AVs. In this paper, we propose a multiscale approach, based on recently developed models for moving bottlenecks. Our main result is the proof of existence of solutions for time-varying bottleneck speed, which corresponds to open-loop controls with bounded variation.
In this article, we investigate a non-localization property of the eigenfunctions of Sturm-Liouvi... more In this article, we investigate a non-localization property of the eigenfunctions of Sturm-Liouville operators Aa = −∂xx + a(•) Id with Dirichlet boundary conditions, where a(•) runs over the bounded nonnegative potential functions on the interval (0, L) with L > 0. More precisely, we address the extremal spectral problem of minimizing the L 2-norm of a function e(•) on a measurable subset ω of (0, L), where e(•) runs over all eigenfunctions of Aa, at the same time with respect to all subsets ω having a prescribed measure and all L ∞ potential functions a(•) having a prescribed essentially upper bound. We provide some existence and qualitative properties of the minimizers, as well as precise lower and upper estimates on the optimal value. Several consequences in control and stabilization theory are then highlighted.
Computational Intelligence and Optimization Methods for Control Engineering, 2019
This article considers the problem of traffic control in which an autonomous vehicle is used to r... more This article considers the problem of traffic control in which an autonomous vehicle is used to regulate human piloted traffic to dissipate stop and go traffic waves. We first investigate the controllability of well-known microscopic traffic flow models, namely i) the Bando model (also known as the optimal velocity model), ii) the follow-the-leader model, and iii) a combined optimal velocity follow the leader model. Based on the controllability results, we propose three control strategies for an autonomous vehicle to stabilize the other, human-piloted traffic. We subsequently simulate the control effects on the microscopic models of human drivers in numerical experiments to quantify the potential benefits of the controllers. Based on the simulations, finally we conduct a field experiment with 22 human drivers and a fully autonomous-capable vehicle, to assess the feasibility of autonomous vehicle based traffic control on real human piloted traffic. We show that both in simulation and in the field test that an autonomous vehicle is able to dampen waves generated by 22 cars, and that as a consequence, the total fuel consumption of all vehicles is reduced by up to 20%.
We consider a partial differential equation-ordinary differential equation system to describe the... more We consider a partial differential equation-ordinary differential equation system to describe the dynamics of traffic flow with autonomous vehicles. In the model the bulk flow is represented by a scalar conservation law, while each autonomous vehicle is described by a car following model. The autonomous vehicles act as tracer vehicles in the flow and collect measurements along their trajectory to estimate the bulk flow. The main result is to prove theoretically and show numerically how to reconstruct the correct traffic density using only the measurements from the autonomous vehicles.
We consider a strongly coupled ODE-PDE system representing moving bottlenecks immersed in vehicul... more We consider a strongly coupled ODE-PDE system representing moving bottlenecks immersed in vehicular traffic. The PDE consists of a scalar conservation law modeling the traffic flow evolution and the ODE models the trajectory of a slow moving vehicle. The moving bottleneck influences the bulk traffic flow via a point flux constraint, which is given by an inequality on the flux at the slow vehicle position. We prove uniqueness and continuous dependence of solutions with respect to initial data of bounded variation. The proof is based on a new backward in time method established to capture the values of the norm of generalized tangent vectors at every time.
Mathematics of Control, Signals, and Systems, 2017
In this article, we give a necessary and sufficient condition of Kalman type for the indirect con... more In this article, we give a necessary and sufficient condition of Kalman type for the indirect controllability of systems of groups of linear operators, under some "regularity and locality" conditions on the control operator that will be made precise later and fit very well the case of distributed controls. Moreover, in the case of first order in time systems, when the Kalman rank condition is not satisfied, we characterize exactly the initial conditions that can be controlled. Some applications to the control of systems of Schrödinger or wave equations are provided. The main tool used here is the fictitious control method coupled with the proof of an algebraic solvability property for some related underdetermined system and some regularity results.
For a given bounded connected domain in IR n , the issue of computing the observability constant ... more For a given bounded connected domain in IR n , the issue of computing the observability constant associated to a wave operator, an observation time T and a generic observation subdomain constitutes in general a hard task, even for one-dimensional problems. In this work, we introduce and describe two methods to provide precise (and even sharp in some cases) estimates of observability constants for general one dimensional wave equations: the first one uses a spectral decomposition of the solution of the wave equation whereas the second one is based on a propagation argument along the characteristics. Both methods are extensively described and we then comment on the advantages and drawbacks of each one. The discussion is illustrated by several examples and numerical simulations. As a byproduct, we deduce from the main results estimates of the cost of control (resp. the decay rate of the energy) for several controlled (resp. damped) wave equations.
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Papers by Thibault Liard