"The same mathematical tool can be used in both neurophysiology and quantum physics"
Mario Sigalotti - © Inria
One year after the launch of the Geco team, it's time to take stock of its research work with the team's leader, Mario Sigalotti.
The team's work concerns the geometric control of systems. For us, a system is a family of laws governing evolutions over time. Controlling it consists in selecting a series of evolutionary laws in such a way that the final state is reached and certain criteria are met. To do this, we use differential geometric tools. These are well-suited to the study of evolutions under constraints: the evolution of a clock with two hands, with two angles as parameters, can therefore be envisaged like a curve on a doughnut. Our specificity in this area is that we focus our analyses on trajectories and families of trajectories. Optimising a trajectory, for example, can mean choosing one that reduces a distance, but also much more than that! Other criteria such as time or energy can also be taken into account. What is of interest is that the mathematical tool we are working on can be applied to three completely different fields of research.
Combining mathematics and neurophysiology may allow us to help people with limited mobility
One surprising application, for example, in the field of neurophysiology, consists in identifying the "costs" optimised by the brain during certain activities like, for example, moving the arms, legs or eyes... We unconsciously perform movements that are optimised according to our aim and therefore we study what comes naturally in order to reproduce it artificially. To do this, we analyse test subjects as they perform specific tasks, like pointing at an object with their arm, for example. Within this movement different muscles are activated and deactivated one after the other to achieve fluid locomotion. We try to find the criteria the brain uses to perform these movements. In the future, our work with neurophysiologists could help people with limited mobility by allowing them to effectively coordinate the activation of deficient muscles.
An image that has been adulterated and automatically reconstructed by a computer
Another form of optimisation concerns understanding vision and above all how the visual cortex works. For example, when we see a curve interrupted by a blank or a triangle with a missing angle our brain tends to fill in the image. It does this in an intelligent way and not by merely using a criteria minimising the distance between the closest points. In these cases we create algorithms inspired by how the brain works in order to recreate these images. Thanks to this biomimicry, a computer can also intelligently interpret interrupted or blurred images we submit to it.
The two other applications of our mathematical approach based on trajectory analysis are more abstract. We contribute significant help in the particular case of control systems based on quantum physics where control tools that require measurements cannot be applied because any measurement of the system influences its state. Thanks to the action of lasers pointed at molecular components we can modify its evolution by developing ad hoc tools.
We are also working on controls for switched systems , a class of systems that mix continuous and discrete dynamics. To understand the interaction between these two dynamics the simplest example is the automobile. The different speeds of the stick shift are in the discrete dynamic, i.e. there is a finite number of possibilities, while the directions the steering wheel can take are in the continuous dynamic, i.e. we cannot assign them a precise value; there is a continuity of values. For a system that combines these two dynamics it is essential to control the entire system by taking into account all of its parameters. We are particularly focused on its stability.