Hasnaa Zidani

• Professor at Institut National des Sciences Appliquées Rouen Normandie

The paper gives a characterization of the uniform robust domain of attraction for a finite non-linear controlled system subject to perturbations and state constraints. We extend the Zubov approach to characterize this domain by means of the value function of a suitable infinite horizon state-constrained control problem which at the same time is a L...

An optimal finite-time horizon feedback control problem for (semi-linear) wave equations is presented. The feedback law can be derived from the dynamic programming principle and requires to solve the evolutionary Hamilton−Jacobi Bellman (HJB) equation. Classical discretization methods based on finite elements lead to approximated problems governed...

This paper deals with a state-constrained control problem. It is well known that, unless some compatibility condition between constraints and dynamics holds, the Value Function has not enough regularity, or can fail to be the unique constrained viscosity solution of a Hamilton–Jacobi–Bellman (HJB) equation. Here, we consider the case of a set of co...

This work is concerned with stochastic optimal control for a running maximum cost. A direct approach based on dynamic programming techniques is studied leading to the characterization of the value function as the unique viscosity solution of a second order Hamilton-Jacobi-Bellman (HJB) equation with an oblique derivative boundary condition. A gener...

We establish a comparison principle for a Hamilton–Jacobi–Bellman equation, more appropriately a system, related to an infinite horizon problem in presence of an interface. Namely a low dimensional subset of the state variable space where discontinuities in controlled dynamics and costs take place. Since corresponding Hamiltonians, at least for the...

In this paper we consider an optimal control problem in large time horizon and solve it numerically. More precisely, we are interested in an aerial vehicle guidance problem: launched from a ground platform, the vehicle aims at reaching a ground/sea target under specified terminal conditions while minimizing a cost modelling some performance and con...

This article studies the problem of estimating the state variable of non-smooth subdifferential dynamics constrained in a bounded convex domain given some real-time observation. On the one hand, we show that the value function of the estimation problem is a viscosity solution of a Hamilton Jacobi Bellman equation whose sub and super solutions have...

This work focuses on a control problem in the Wasserstein space of probability measures over $\mathbb{R}^d$. Our aim is to link this control problem to a suitable Hamilton-Jacobi-Bellman (HJB) equation. We explore a notion of viscosity solution using test functions that are locally Lipschitz and locally semiconvex or semiconcave functions. This reg...

In this paper we investigate optimal control problems perturbed by random events. We assume that the control has to be decided prior to observing the outcome of the perturbed state equations. We investigate the use of probability functions in the objective function or constraints to define optimal or feasible controls. We provide an extension of di...

In this article, we develop a novel notion of viscosity solutions for first order Hamilton-Jacobi equations in proper \(\mathrm {CAT(0)}\) spaces. The notion of viscosity is defined by taking test functions that are locally Lipschitz and can be represented as a difference of two semiconvex functions. Under mild assumptions on the Hamiltonian, we re...

This paper is concerned with the relationship between the maximum principle and dynamic programming for a large class of optimal control problems with maximum running cost. Inspired by a technique introduced by Vinter in the 1980s, we are able to obtain jointly a global and a partial sensitivity relation that link the coextremal with the value func...

This manuscript aims to study finite horizon, first order Hamilton Jacobi Bellman (HJB) equations on stratified domains. This problem is related to optimal control problems with discontinuous dynamics. We use nonsmooth analysis techniques to derive a strong comparison principle as in the classical theory and deduce that the value function is the un...

We survey the main numerical techniques for finite-dimensional nonlinear optimal control. The chapter is written as a guide to practitioners who wish to get rapidly acquainted with the main numerical methods used to efficiently solve an optimal control problem. We consider two classical examples, simple but significant enough to be enriched and gen...

We investigate the large time behavior of the value function associated to an optimal control problem in the finite dimensional case. We first establish the large time asymptotic expansion in the linear quadratic (LQ) case using the Hamiltonian structure of the Pontryagin Maximum Principle (PMP) extremal equations and some basic results of LQ theor...

This paper concerns an optimal control problem on the space of probability measures over a compact Riemannian manifold. The motivation behind it is to model certain situations where the central planner of a deterministic controlled system has only a probabilistic knowledge of the initial condition. The lack of information here is very specific. In...

We survey on numerics for finite-dimensional nonlinear optimal control. The chapter is written as a guide to practitioners who wish to get rapidly acquainted with the main numerical methods used to efficiently solve an optimal control problem. We consider throughout two classical examples, quite simple but representative enough to be complexified a...

In this work, we study optimistic planning methods to solve some state-constrained optimal control problems in finite horizon. While classical methods for calculating the value function are generally based on a discretization in the state space, optimistic planning algorithms have the advantage of using adaptive discretization in the control space....

We study the Hamilton-Jacobi (HJ) approach for a two-person zero-sum differential game with state constraints and where controls of the two players are coupled within the dynamics, the state constraints, and the cost functions. It is known for such problems that the value function may be discontinuous and its characterization by means of an HJ equa...

In this paper, we consider a class of optimal control problems governed by a differential system. We analyse the sensitivity relations satisfied by the co-state arc of the Pontryagin maximum principle and the value function that associates the optimal value of the control problem to the initial time and state. Such a relationship has been already i...

Hybrid control systems are dynamical systems that can be controlled by a combination of both continuous and discrete actions. In this paper we study the approximation of optimal control problems associated to this kind of systems, and in particular of the quasi-variational inequality which characterizes the value function. Our main result features...

Intermittent sources of energy represent a challenge for electrical networks, particularly regarding demand satisfaction at peak times. Energy management tools such as load shaving or storage systems can be used to mitigate intermittency. In this work, the value of different mechanisms to move energy through time is examined through a multi-objecti...

This paper deals with a trajectory optimization problem for a three-stage launcher with the aim to minimize the consumption of propellant needed to steer the launcher from the Earth to the GEO (geostationary orbit). Here we use a global optimization procedure based on Hamilton-Jacobi-Bellman approach and consider a complete model including the tran...

This article is devoted to the study of a controlled population of cells. The modeling of the problem leads to a mathematical formulation of stability and reachability properties of some controlled systems under uncertainties. We use the Hamilton‐Jacobi approach to address these problems and to design a numerical method that we analyze on several n...

This paper studies optimal control problems with state constraints by imposing structural assumptions on the constraint domain coupled with a tangential restriction with the dynamics. These assumptions replace pointing or controllability assumptions that are common in the literature, and provide a framework under which feasible boundary trajectorie...

The goal of this paper is to show how nonparametric statistics can be used to solve some chance constrained optimization and optimal control problems. We use the kernel density estimation method to approximate the probability density function of a random variable with unknown distribution from a relatively small sample. We then show how this techni...

The aim of this article is to study the Hamilton Jacobi Bellman (HJB) approach for state-constrained control problems with maximum cost. In particular, we are interested in the characterization of the value functions of such problems and the analysis of the associated optimal trajectories, without assuming any controllability assumption. The rigoro...

The resolution of the launcher ascent trajectory problem by the so-called Hamilton–Jacobi–Bellman (HJB) approach, relying on the Dynamic Programming Principle, has been investigated. The method gives a global optimum and does not need any initialization procedure. Despite these advantages, this approach is seldom used because of the dicculties of c...

This work aims at studying some optimal control problems with convex state constraint sets. It is known that for state constrained problems, and when the state constraint set coincides with the closure of its interior, the value function satisfies a Hamilton–Jacobi equation in the constrained viscosity sense. This notion of solution has been introd...

We study a model based on the so called SIR model to control the spreading of a disease in a varying population via vaccination and treatment. Since we assume that medical treatment is not immediate we add a new compartment, $M$, to the SIR model. We work with the normalized version of the proposed model. For such model we consider the problem of s...

https://hal.archives-ouvertes.fr/hal-01585766

This paper studies some optimal control problems on networks. The Value Function associated with the control problem is characterized as solution to a system of Hamilton-Jacobi equations with appropriate junction conditions. The novel feature of the result lies in that the controllability conditions are not needed and the characterization remains v...

This paper deals with junction conditions for Hamilton-Jacobi-Bellman (HJB) equations for finite horizon control problems on multi-domains. We consider two different cases where the final cost is continuous or lower semi-continuous. In the continuous case we extend the results of "Hamilton-Jacobi-Bellman equations on multi-domains" by the second an...

This paper deals with junction conditions for Hamilton-Jacobi-Bellman (HJB) equations for finite horizon control problems on multi-domains. We consider two different cases where the final cost is continuous or lower semi-continuous. In the continuous case we extend the results of "Hamilton-Jacobi-Bellman equations on multi-domains" by the second an...

This paper deals with a payload optimization problem for three-stage space launcher. The mission of the launch vehicle is to put the payload on a sun-synchronous (SSO) orbit. The considered flight sequence includes two boosts. The first one steers the launcher to a transfer orbit. Then, after a ballistic flight, a second boost is used to perform th...

This work deals with an optimization problem for three-stage space launcher. The mission of the launcher is to put a given payload on the GEO orbit with the minimal propellant consumption. The considered flight sequence performs two boosts. The first one steers the launcher to a given GTO orbit. Then, after a ballistic flight, a second boost is use...

This paper deals with a class of stochastic optimal control problems (SOCPs) in the presence of state constraints. It is well known that for such problems the value function is, in general, discontinuous, and its characterization by a Hamilton-Jacobi equation requires additional assumptions involving an interplay between the boundary of the set of...

This work deals with numerical approximations of unbounded and discontinuous value functions associated to some stochastic control problems. We derive error estimates for monotone schemes based on a Semi-Lagrangian method (or more generally in the form of a Markov chain approximation). A motivation of this study consists in approximating chance-con...

This special volume gathers a number of new contributions addressing various topics related to the field of optimal control theory and sensitivity analysis. The field has a rich and varied mathematical theory, with a long tradition and a vibrant body of applications. It has attracted a growing interest across the last decades, with the introduction...

This special volume gathers a number of new contributions addressing various topics related to the field of optimal control theory and sensitivity analysis. The field has a rich and varied mathematical theory, with a long tradition and a vibrant body of applications. It has attracted a growing interest across the last decades, with the introduction...

This paper deals with a problem of trajectory optimization of the flight phases of a three-stage launcher. The aim of this optimization problem is to minimize the consumption of ergols that is need to steer the launcher from the Earth to the GEO. Here we use a global optimization procedure based on Hamilton-Jacobi-Bellman approach and consider a co...

The invention relates to a method for managing energy consumption for an automobile having an electric battery and a heat engine, said method making it possible to select the use phases of said engine along a given route so as to minimize the fuel consumption of said vehicle. The main characteristic of the method according to the invention is that...

Hybrid systems are a general framework which can model a large class of control systems arising whenever a collection of continuous and discrete dynamics are put together in a single model. In this paper, we study the convergence of monotone numerical approximations of value functions associated to control problems governed by hybrid systems. We di...

We present an abstract convergence result for the fixed point approximation of station-ary Hamilton–Jacobi equations. The basic assumptions on the discrete operator are invariance with respect to the addition of constants, ε-monotonicity and consistency. The result can be applied to various high-order approximation schemes which are illustrated in...

The present paper studies an optimal control problem governed by measure driven differential systems and in presence of state constraints. The first result shows that using the graph completion of the measure, the optimal solutions can be obtained by solving a reparametrized control problem of absolutely continuous trajectories but with time-depend...

The invention relates to a method for managing energy consumption for an automobile having an electric battery and a heat engine, said method making it possible to select the use phases of said engine along a given route so as to minimize the fuel consumption of said vehicle. The main characteristic of the method according to the invention is that...

A system of Hamilton-Jacobi (HJ) equations on a partition of ℝ d is considered, and a uniqueness and existence result for viscosity solutions is analyzed. While the notion of viscosity solution is by now well-known, the question of uniqueness of solution, when the Hamiltonian is discontinuous, remains an important issue. A uniqueness result has bee...

L'invention se rapporte à un procédé de gestion d'énergie pour un véhicule électrique, comprenant un calculateur, un moteur électrique, une batterie électrique et un moteur thermique d'appoint utilisant un carburant et conçu pour aider la batterie à faire fonctionner ledit moteur électrique. La principale caractéristique d'un procédé selon l'invent...

The method involves selecting a trajectory ranging between a starting point and a destination place. An initial energy state of the vehicle is evaluated. A strategy of energy consumption along the selected trajectory is applied by a controller based on optimized use of an auxiliary internal combustion engine and battery so as to support overall len...

This study aims at characterizing a reachable set of a hybrid dynamical system with a lag constraint in the switch control. The setting does not consider any controllability assumptions and uses a level-set approach. The approach consists in the introduction of on adequate hybrid optimal control problem with lag constraints on the switch control wh...

The {\em Sense and Avoid} capacity of Unmanned Aerial Vehicles (UAV) is one of the key elements to open the access to airspace for UAVs. In order to replace a pilot's {\em See and Avoid} capacity such a system has to be certified "as safe as a human pilot on-board". The problem is to prove that an unmanned aircraft equipped with a S\&A system can c...

The Sense and Avoid capacity of Unmanned Aerial Vehicles (UAV) is one of the key elements to open the access to airspace for UAVs. In order to replace a pilot's See and Avoid capacity such a system has to be certified "as safe as a human pilot on-board". The problem is to prove that an unmanned aircraft equipped with a S&A system can comply with th...

The paper deals with deterministic optimal control problems with state constraints and non-linear dynamics. It is known for such problems that the value function is in general discontinuous and its characterization by means of a Hamilton-Jacobi equation requires some controllability assumptions involving the dynamics and the set of state constraint...

This work presents a stochastic dynamic programming (SDP) algorithm that aims at minimizing an economic criteria based on the total energy consumption of a range extender electric vehicle (REEV). This algorithm integrates information from the REEV's navigation system in order to obtain some information about future expected vehicle speed. The model...

We consider approximation schemes for monotone systems of fully nonlinear second order partial differential equations. We first prove a general convergence result for monotone, consistent and regular schemes. This result is a generalization to the well known framework of Barles-Souganidis, in the case of scalar nonlinear equation. Our second main r...

This paper aims to investigate a control problem governed by differential equations with Radon measure as data and with final state constraints. By using a known reparametrization method (by Dal Maso and Rampazzo [18]), we obtain that the value function can be characterized by means of an auxiliary control problem of absolutely continuous trajector...

We consider minimal time problems governed by nonlinear systems under general time dependent state constraints and in the two-player games setting. In general, it is known that the characterization of the minimal time function, as well as the study of its regularity properties, is a difficult task in particular when no controllability assumption is...

This paper deals with some optimal control problems governed by ordinary differential equations with Radon measures as data and with pointwise state constraints. We study the properties of the value function and obtain its characterization by means of an auxiliary control problem of absolutely continuous trajectories. For this, we use some known te...

This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a "junction", that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular...

This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a ``junction'', that is to say the union of a finite number of half-lines with a unique common point. The main results are a comparison principle, existence and stability of solutions. The two challenging difficulties are the singular geometry o...

Because of their importance in many applications, questions of path planning and reachability analysis for nonlinear dynamical systems have been studied extensively in the control theory. Here we focus on the cases when the controlled systems are constrained to evolve in a certain known set (e.g avoidance of obstacles). We study general framework b...

This study aims to investigate the Hamilton-Jacobi-Bellman approach for solving an optimization problem for space launchers. We consider a simplied (realistic) ight mis sion of the European launcher Ariane V to the Geostationary Transfer Orbit, and aim t minimize the fuel consumption. We consider the complete ight including stage separa tions and d...

In the present paper, we consider nonlinear optimal control problems with constraints on the state of the system. We are interested in the characterization of the value function without any controllability assumption. In the unconstrained case, it is possible to derive a characterization of the value function by means of a Hamilton-Jacobi-Bellman (...

The goal of this paper is to study some numerical approximations of particular Hamilton-Jacobi-Bellman equations in dimension 1 and with possibly discontinuous initial data. We investigate two anti-diffusive numerical schemes, the first one is based on the Ultra-Bee scheme and the second one is based on the Fast Marching Method. We prove the conver...

In this paper, we are interested in some front propagation problems coming from control problems in d-dimensional spaces, with d≥2. As opposed to the usual level set method, we localize the front as a discontinuity of a characteristic function. The evolution of the front is computed by solving an Hamilton-Jacobi-Bellman equation with discontinuous...

We prove the convergence of a non-monotonous scheme for a one-dimensional first order Hamilton–Jacobi–Bellman equation of the form v t + maxα (f(x, α)v x ) = 0, v(0, x) = v 0(x). The scheme is related to the HJB-UltraBee scheme suggested in Bokanowski and Zidani (J Sci Comput 30(1):1–33, 2007). We show for general discontinuous initial data a first...

We consider a target problem for a nonlinear system under state constraints. We give a new continuous level-set approach for characterizing the optimal times and the backward-reachability sets. This approach leads to a characterization via a Hamilton-Jacobi equation, without assuming any controllability assumption. We also treat the case of time-de...

In this paper we develop a general efficient sparse storage technique suitable to coding front evolutions in d>= 2 space dimensions. This technique is mainly applied here to deal with deterministic target problems with constraints, and solve the associated minimal time problems. To this end we consider an Hamilton-Jacobi-Bellman equation and use an...

On étudie un schéma non monotone pour l'équation Hamilton Jacobi Bellman du premier ordre, en dimension 1. Le schéma considèré est lié au schéma anti-diffusif, appellé UltraBee, proposé par Bokanowski-Zidani (publié en 2007 dans J. Sci. Compt.). Ici, on prouve la convergence, en norme $L^1$, à l'ordre 1, pour une condition initiale discontinue. Le...

In this paper, we investigate a minimum time problem for controlled non-autonomous differential systems, with a dynamics depending on the final time. The minimal time function associated to this problem does not satisfy the dynamic programming principle. However, we will prove that it is related to a standard front propagation problem via the reach...

We study a superreplication problem of European options with gamma constraints, in mathematical finance. The initially unbounded control problem is set back to a problem involving a viscosity PDE solution with a set of bounded controls. Then a numerical approach is introduced, inconditionnally stable with respect to the mesh steps. A generalized fi...

This paper deals with convergence results of Howard's algorithm for the resolution of $\min_{a\in \cA} (B^a x - b^a)=0$ where $B^a$ is a matrix, $b^a$ is a vector (possibly of infinite dimension), and $\cA$ is a compact set. We show a global super-linear convergence result, under a monotonicity assumption on the matrices $B^a$. In the particular ca...

We propose two new antidiffusive schemes for advection (or linear transport), one of them being a mixture of Roe's Super-Bee scheme and of the "Ultra-Bee" scheme. We show how to apply these schemes to treat time-dependent first or- der Hamilton-Jacobi-Bellman equations with discontinuous initial data, pos- sibly infinitely-valued. Numerical tests a...

We obtain error bounds for monotone approximation schemes of a stochastic impulse control problem. This is an extension of the theory for error estimates for the Hamilton-Jacobi-Bellman equation. For obtaining these bounds we build a sequence of stochastic impulse control problems, and a sequence of monotone approximation schemes. Extending methods...

International audience We deal with a numerical method for HJB equations coming from optimal control problems with state constraints. More precisely, we present here an antidissipative scheme applied on an adaptative grid. The adaptative grid is generated using linear quadtree structure. This technique of adaptation facilitates stocking data and de...

This paper is concerned with the numerical approximation of viability kernels. The method described here provides an alternative approach to the usual viability algorithm. We first consider a characterization of the viability kernel as the value function of a related optimal control problem, and then use a specially relevant numerical scheme for it...

We obtain error bounds for monotone approximation schemes of a particular Isaacs equation. This is an extension of the theory for estimating errors for the Hamilton-Jacobi-Bellman equation. To obtain the upper error bound, we consider the "Krylov regularization" of the Isaacs equation to build an approximate sub-solution of the scheme. To get the l...

We deal with a numerical method for HJB equations coming from optimal control problems with state constraints. More precisely, we present here an antidissipative scheme applied on an adaptative grid.The adaptative grid is generated using linear quadtree structure. This technique of adaptation facilitates stocking data and dealing with large numeric...

This Note presents an approximation scheme for second-order Hamilton–Jacobi–Bellman equations arising in stochastic optimal control. The scheme is based on a Markov chain approximation method. It is easy to implement in any dimension. The consistency of the scheme is proved, which guarantees its convergence. To cite this article: R. Munos, H. Zidan...

We consider the problem of optimal design of hybrid car engines which combine thermal and electric power. The optimal configuration of the different motors composing the hybrid system involves the choice of certain design parameters. For a given configuration, the goal is to minimize the fuel consumption along a trajectory. This is an optimal contr...

This paper analyses the implementation of the generalized finite differences method for the HJB equation of stochastic control, introduced by two of the authors in [Bonnans and Zidani, SIAM J. Numer. Anal. 41, 1008–1021 (2003; Zbl 1130.49307)]. The computation of coefficients needs to solve at each point of the grid (and for each control) a linear...

This paper discusses the numerical resolution of the Hamilton-Jacobi-Bellman equation associated with optimal control problem when the state equation is of algebraic differential type. We discuss two numerical sche­mes. The first reduces to the standard framework, while the second does not suppose any knowledge of the Jacobian of the data. We obtai...

We analyze a class of numerical schemes for solving the HJB equation for stochastic control problems, which enters the framework of Markov chain approximations and generalizes the usual finite difference method. The latter is known to be monotonic, and hence valid, only if the scaled covariance matrix is dominant diagonal. We generalize this result...

In this work we consider a numerical approximation of an optimal control problem governed by variational inequalities. We use a total discretization scheme: implicite Euler discretization with respect to the time variable and finite element method for the space variable, and give convergence results.

This paper deals with optimal control problems of semilinear parabolic equations with pointwise state constraints and coupled integral state-control constraints. We obtain necessary optimality conditions in the form of a Pontryagin's minimum principle for local solutions in the sense of Lp, p ≤ +∞.

We consider time optimal control problems governed by semilinear parabolic equations with pointwise state constraints and unbounded controls. We derive a Pontryagin's principle for boundary controls. We prove a regularity result for the gradient of the state variable and by this way we are able to define a Hamiltonian functional which intervenes in...

In this work we consider a numerical approximation of an optimal control problem governed by variational inequalities. We use a total discretization scheme: implicit Euler discretization with respect to the time variable and finite element method for the space variable, and give convergence results.