Nathalie Khalil
Doctor of Philosophy
(Ph.D.)
Research Interests
Control Theory
In control theory, a controller manipulates the inputs (known as controls) to a given system in order to obtain the desired effect on the output of the system. For instance, the driver controls the motion of the car by acting on the gas pedal and on the steering wheel, in order to achieve the desired behavior of steering the car into a parking spot.
Optimal Control
When we introduce a performance criterion (to minimize or to maximize) - known as objective function, coupled with a control system, we step into the optimal control theory field: the main objective of this theory is to provide tools for a choice among all possible strategies which accomplish the ‘best’ behavior of the system in question. This 'best behavior' is known as 'optimal solution'.
NonSmooth Analysis
Generalize the concepts of differentiability in convex analysis to a larger class of problems. A familiarity with nonsmooth analysis is, therefore, essential for an in-depth understanding of present day research in optimal control.
Bilevel Optimization
Bilevel optimization problems are of interest to application domains, such as, transportation, economics, decision sci-
ence, business, engineering, etc., requiring the nesting of optimization processes being the outer one designated by upper-level optimization problem, and the inner one by the lower-level optimization problem.
Calculus of Variations
Variational calculus had possibly its beginnings in 1696 with J. Bernoulli with his famous Brachistochrone Problem, from the Ancient Greek words brachistos and chronos, meaning ‘shortest’ and ‘time’. This problem consists in finding a curve connecting two points A and B such that a mass point moves from A to B as fast as possible (i.e. minimal time) in a downward directed constant gravitational field, disregarding any friction. The solution to this problem is a cycloid curve. In the 18th century, the calculus of variations became an independent discipline of mathematics and much of the formulation of this field of mathematics was developed by Euler, Lagrange and Laplace.
Sweeping Process
Sweeping processes (“processus de rafle”) were introduced by Jean-Jacques Moreau in the 1970s. They were originally motivated by applications to elastoplasticity, but also in hysteresis, electric circuits, traffic equilibria, populations motion in confined spaces, and other areas of applied sciences and operations research.
Indirect Computational Methods
Indirect methods are computational techniques based on the maximum principle to solve optimal control problems by finding the optimal solutions. These methods become more complicated when a state constraint is added to the problem.