Trained in Control Theory, my main field of expertise consists in connecting methods from Analysis and Stochastic Processes in view of solving concrete problems of applied mathematics, not necessarily arising from Control.
In the course of my scientific career I have supervised around 20 PhD students.
2 DESCRIPTION OF RESULTS
2.1 STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
I was among the early initiators of the theory of stochastic PDE in the sense of Ito. I worked in particular, alone or in cooperation, on stochastic monotone equations, stochastic Navier-Stokes equations, stochastic variational inequalities, and more recently on stochastic inertial manifolds.
This theory has found some of its most interesting applications in the study of nonlinear filtering theory, where the work of my former student PARDOUX is a reference.
Note that numerical resolution is now within reach, which makes this field of research quite active and quite useful for applications.
2.2 FILTERING AND CONTROL OF STOCHASTIC DISTRIBUTED SYSTEMS
These problems have been at the origin of my research. In particular my thesis (under the supervision of J.L. LIONS) was devoted to Kalman filtering for linear systems with white noise inputs on both time and space. My main contribution was to make use of the theory of cylindrical measures of
Gelfand Vilenkin and L. Schwartz. This permits a rigorous approach to a large variety of problems, provided the model is linear.
I was in particular able to apply these techniques to a bidimensional model playing a role in image processing, or to the problem of optimal location of sensors.
In the problem of control, I contributed strongly to the theory of necessary conditions, and to a rigorous approach of Dynamic Programming via nonlinear semi-groups.
The main field of application concerns the stochastic control of systems with partial information, which naturally has a huge domain of applications.
2.3 OPTIMAL STOPPING, IMPULSE CONTROL, VARIATIONAL AND QUASI VARIATIONAL INEQUALITIES
I discovered, which is now standardly used, that variational inequalities corresponded to the dynamic programming treatment of optimal stopping, and then solved, with J. L. LIONS, the impulse control problem, via a new theory, called the theory of quasi variational inequalities. This has initiated a long and fruitful cooperation on this topic with J.L. LIONS and many colleagues and students. Many applications of this theory concern old and new problems of Operations Research and Management, as well as Physics and Mechanics.
2.4 REGULARITY of NONLINERAR SYSTEMS of ELLIPTIC and PARABOLIC PDE and APPLICATIONS
Stochastic Nash differential games lead to sytems of elliptic and parabolic PDE. The regularity of solutions is extremely important for obtaining a Nash point. This has motivated a longstanding cooperation with J. FREHSE, one of the worldwide specialists of the regularity of elliptic systems. I have also solved with him the ergodic case, which even in the case of one equation was open in its full generality.
2.5 ROBUST CONTROL and RISK-SENSITIVE CONTROL
My interest for this domain stems from its connection with the field of Risk sensitive stochastic control. In fact, I first solved with J. VAN SCHUPPEN the problem of finding a sufficient statistics(of the same size as the the state, which was an open problem) for the LEG (Linear ExponentialGaussian) control. More recently, in cooperation with J. BARAS and R. ELLIOTT, for the partial information case, and H. NAGAI and J. FREHSE for the full information case, I have contributed several results to justify some earlier formal treatment. The connection between robust control and risk sensitive control can be best seen through small noise introduction and singular perturbations. This is a field of very active research worldwide, by the broadness of the applications and the use of many mathematical techniques. It is one of my current research areas.
2.6 EXACT CONTROLLABILITY
After the introduction of the HUM method by J. L. LIONS, and its use by many authors, I got interested in developing a general theory of exact controllability for infinite dimensional systems,where the dynamics is driven by a skew-symmetric operator. In this way, one can unify most of the existing results concerning wave equations, Maxwell equations, ...
Initiated by the probabilistic interpretation of homogenization, my interest in this domain has widened into an important cooperation with J.L. LIONS and G. PAPANICOLAOU, where we have developed many general approaches to this very fruitful theory.
More recently with L. BOCCARDO and F. MURAT I have considered the homogenization of Bellman equations, and with G. BLANKENSHIP the case of random homogenization.
2.8 REGULAR and SINGULAR PERTURBATIONS
Homogenization can be viewed as a particular situation of singular perturbations. But of course many other situations can be considered. It was natural for me to be interested in their application to Control theory, both deterministic and stochastic, and in the latter case with full or partial information. Many particularly useful results can be obtained in the case of partial information, where one can derive approximate but accurate finite dimensional feedback laws, where the optimal one is infinite dimensional.
2.9 SPECIFIC PROBLEMS in OPERATIONS MANAGEMENT and in FINANCE
Since Control Theory has a lot of applications in many areas of Quantitative Management, I have cooperated with several specialists of the field to obtain the solution of concrete problems. My current interest lies in Operations Management and in Finance. I have been working in the design of complex options with M. CROUHY and D. GALAI. More recently, with my student H. Julien, I have investigated models of options for incomplete markets, where the incompleteness arises from ”frictions” in the management of portfolios. Another approach to this problem has been developed with N. TOUZI and J. L. MENALDI, using penalty approximations and Viscosity methods. Since I joined UTD, the University of Texas at Dallas, I have been working on Inventory Control problems, with S. Sethi, M. Cakanyldirim, and PhD students. Our research focuses on stochastic models with partial information ( a field where little is available ), on service constraint models, on s, S policies. I have been also involved , jointly with S. SETHI, in models of Economic Growth developed by K.J. ARROW and al., taking into account population growth aspects.