The Back and Forth Nudging algorithm for data assimilation problems : theoretical results on transport equations

In many areas of science and engineering, it is important to estimate the initial (or the current) state of a complex system from output functions measured over some finite time interval. In oceanography and meteorology, this problem is called data assimilation and it is mainly concerned with determining the initial state of the system from existing measures. On the other hand, one of the important concepts of control theory, introduced by Kalman and Luenberger, is the notion of observer. An observer is a new dynamical system in which one injects the measured functions, so that the current state of this new system approaches the current state of the original one for large times. One of the funding ideas of back and forth nudging, introduced in Auroux and Blum (2005, 2008), is to consider both forward and backward in time observers to recover the initial state of the system. However, this procedure requires long time measures, which are generally not available in applications to oceanography and meteorology. The second leading idea of back and forth nudging methodology is to iterate forward and backward observers using only measures on a short time interval. This paper brings one of the first mathematically rigorous justifications of this method in the context of partial differential equations. The main considered applications are a linear viscous transport equation and the viscous and non-viscous Burgers' equation. A very interesting discussion on the competing roles of viscous and convective terms is provided in the concluding section.

The Back and Forth Nudging algorithm for data assimilation problems : theoretical results on transport equations
Didier Auroux and Maëlle Nodet
ESAIM: COCV 18 (2012) 318-342

Global optimality conditions for a dynamic blocking problem

Imagine a fire developing in a windy zone. The burned area will grow developing in all directions and being influenced by the wind. A model for this situation is naturally given by a differential inclusion, in dimension two, and the burned area is the reachable set. Firefighters will start constructing walls (or barriers) to limit the fire expansion. However, one has to take into account the time needed for that. The optimization problem considered in this paper corresponds to the minimization of burned area and of costs for construction of wall, modeled by a one-dimensional rectifiable set. Necessary conditions are derived, which are obviously quite complicated to state, but explicit enough to solve some simple yet interesting problems. Moreover, they open the way to address the problem numerically.

Global optimality conditions for a dynamic blocking problem
Alberto Bressanand Tao Wang
ESAIM: COCV 18 (2012) 124-156

On the null-controllability of diffusion equations

Controllability of Partial Differential Equations is a very active research field in the recent decades, especially after the classical paper by J.-L. Lions (SIAM Rev., 1988). In particular, H.O. Fattorini and D.L. Russell (1972) addressed the null controllability of linear parabolic equations in one space dimension, while the same controllability result in several space dimensions has been independently established by G. Lebeau and L. Robbiano (1995), and by A.V. Fursikov and O.Yu. Imanuvilov (1996). This article introduces a new abstract version of the Lebeau and Robbiano approach. In the special case of the heat equation in high dimensional rectangular domains, the authors provide an impressive alternative way, based on an inequality of P. Turan (1946), to check the key Lebeau-Robbiano spectral condition so that the corresponding null controllbility can be proved without using the usual Carleman-type estimate.

On the null-controllability of diffusion equations Gérald Tenenbaum and Marius Tucsnak
ESAIM: COCV 17 (2011) 1088-1100

Deterministic state-constrained optimal control problems without controllability assumptions

In the setting of optimal control problems of ordinary differential equations with state constraints, the paper aims at characterizing the value function using the Hamilton-Jacobi-Bellamn (HJB) equation. The answer is known when one assumes either the inward or the outward pointing qualification condition. In the present paper no controllability assumption is done. The value function can be discontinuous and the problem may be unfeasible, resulting in an infinite value function.

A first reduction consists in mapping the real line to a bounded interval, after which to an unfeasible problem is associated the value say 1. It appears then that the value function is the unique solution of the HJB equation with some elaborated boundary conditions. In order to get a more practical characterization (especially when having in view numerical methods), the authors consider the HJB equation on a slightly enlarged domain, with 'classical' boundary conditions. They show that any l.s.c. bounded viscosity solution of the latter converges punctually to the original value function.

Deterministic state-constrained optimal control problems without controllability assumptions
Olivier Bokanowski, Nicolas Forcadel and Hasnaa Zidani
ESAIM: COCV 17 (2011) 995-1015

Local semiconvexity of Kantorovich potentials on non-compact manifolds

In the theory of optimal transportation, one wants to pair sources with sinks, so as to minimize the average cost c(x,y) of transportation. Historically, there were two approaches to doing so: Monge's (1781) was to assign each source to a single sink, whereas Kantorovich's (1942) allowed the freedom of dividing the output of each source between many sinks. The latter problem is a linear program -- infinite-dimensional when sources or sinks are spread continuously over the landscape. It is solved through a dual linear program which assigns implicit values to the goods being transported, depending on the merits of their location. These values are called Kantorovich potentials; when they exist, their spatial derivatives often contain enough information to allow the problem of Monge to be solved also.

The contribution of Figalli and Gigli is to show such potentials have two derivatives almost everywhere -- the same regularity as concave functions. In fact, they differ from a concave function by a smooth function. For compactly supported measures and/or special cost functions, such results go back to Brenier (1987) and Gangbo and McCann (1996); they are key to solving variants of Monge's problem, and the Monge-Ampère equation. Figalli and Gigli show this extends to measures with non-compact support, assuming only mild growth conditions on the cost c in C2. This is a deep result with multifold applications, as examples include a wide variety of cost functions arising from Tonelli-Lagrangians, such as Riemannian distance squared costs, and natural mechanical actions.

Irène Fonseca
Corresponding Editor

Local semiconvexity of Kantorovich potentials on non-compact manifolds
Alessio Figalli and Nicola Gigli
ESAIM: COCV 17 (2011) 648-653
DOI: http://dx.doi.org/10.1051/cocv/2010011

Discrete mechanics and optimal control: An analysis

Finding a solution to optimal control problems for mechanical systems is of paramount importance in a number of applications, ranging from vehicle dynamics to bio-engineering. More precisely, in most cases real systems can be attacked only by feasible computational approaches. When dealing with a continuous model, there are many different possible patterns to reach a discrete algorithm. This paper focuses on the idea that discretization should be introduced at the very first stage, producing a new approach called Discrete Mechanics and Optimal Control. The latter has various advantages, in particular the discrete solution inherits structural properties, like symmetries and integrals of the motion, and may be convenient from computational point of view. The authors present the whole approach from theoretical properties all the way to simulations. They also add a very nice and honest conclusion section, where they point out the difficulty in finding global optima and some recipe to deal with that.

Benedetto Piccoli, Corresponding Editor

Discrete mechanics and optimal control: An analysis
Sina Ober-Blöbaum, Oliver Junge and Jerrold E. Marsden
ESAIM: COCV 17 (2011) 322-352

Weighted energy-dissipation functionals for gradient flows

This is an interesting gradient flow analysis combined with time-discretization that is applied to examples of microstructure evolution, previously addressed by Conti and Ortiz.

John Ball

Weighted energy-dissipation functionals for gradient flows
Alexander Mielke and Ulisse Stefanelli
ESAIM: COCV 17 (2011) 52-85
DOI: http://dx.doi.org/10.1051/cocv/2009043

Nonlinear feedback stabilization of a two-dimensional Burgers equation

This article investigates the important problem of nonlinear feedback synthesis for the stabilization of a two-dimensional Burgers equation. It is relevant in particular because it contains some key ingredients that arise in the control of two-dimensional Navier-Stokes equations

The article is a significant step in the direction of developing control laws for nonlinear partial differential equations that guarantee more than infinitesimally local stability results. The authors achieve stabilization for any stationary profile of the Burgers equation using a nonlinear feedback law. The numerical simulations show that this nonlinear feedback permits to stabilize the system in some situations where a classical linear one does not.

Emmanuel Trélat

Nonlinear feedback stabilization of a two-dimensional Burgers equation
Laetitia Thevenet, Jean-Marie Buchot and Jean-Pierre Raymond
ESAIM: COCV 16 (2010) 929-955
DOI: http://dx.doi.org/10.1051/cocv/2009028

An estimation of the controllability time for single-input systems on compact Lie Groups

Nowadays, quantum control appears to be one of the most challenging applications of control theory.

Indeed in many processes in modern technology it is necessary to induce a transition between energy levels of a quantum mechanical system, that usually is a molecule (in photochemistry or in laser spectroscopy) or a spin system (e.g. in nuclear magnetic resonance). In the experiments, excitation and ionization are induced by means of a sequence of external pulses (e.g. lasers or magnetic fields). Usually the transfer should be as fast as possible in order to minimize the effects of relaxation and decoherence that are always present. In most of the cases, the description of such processes translates into a multilinear control system on the unitary group SU(n) or on the group of unitary transformations on a complex infinite dimensional Hilbert space. One of the most promising application is in quantum information processing for the realization of quantum gates (namely the quantum counterpart of the elementary operations AND, OR, NOT).

This paper contains a crucial estimate for the minimum time necessary to induce a transition in a quantum mechanical system. More precisely, for a generic single-input control system on a compact semi-simple Lie group (and in particular on SU(n)), an upper and a lower bound on the minimum time necessary to steer (with unbounded controls) the identity to any other point of the group are provided. The most important byproduct of this paper is that the ideas presented generalize to infinite dimension, allowing to get controllability results for the bilinear Schroedinger equation (as a PDE) by means of Galerkin approximations, as obtained in a recent paper by one of the authors et al.

Enrique Zuazua, Editor-in-Chief,
Benedetto Piccoli, Corresponding Editor

An estimation of the controllability time for single-input systems on compact Lie Groups
Andrei Agrachev, Thomas Chambrion
ESAIM: COCV 12 (2006) 409-441
DOI: http://dx.doi.org/10.1051/cocv:2006007