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CPDENL Report Summary

Project ID: 266907
Funded under: FP7-IDEAS-ERC
Country: France

Final Report Summary - CPDENL (Control of partial differential equations and nonlinearity)

New methods and tools were introduced and developed in this project to study the controllability and the stabilization of control systems. This includes

-Strategies to construct trajectories going to a given equilibrium, going back to it and having a linearized control which is controllable. This is useful to get controllability results (return method).

-A power series expansion method to stabilize nonlinear systems by using the nonlinearities.

-A new method to handle difficulties created by boundary layers when one wants to obtain global controllability in small time. This allowed us to solve an old conjecture, namely the global controllability in small time of the Navier-Stokes equations in the case where the fluid slips on the boundary according to Navier’s law.

-The use of the algebraic solvability to get the controllability of control systems modeled by means of partial differential equations, even if the systems do not have any smoothing properties (e.g. hyperbolic equations) by using a Nash-Moser inverse theorem due to M. Gromov.

-New approaches (Lyapunov and time-delay) for the study of the stability 1-D hyperbolic equations.

-The use of new linear transformations to get rapid stabilization.

-The use of a backstepping approach to produce feedback laws leading to stabilization in finite time for partial differential equations.

-The use of a backstepping approach to stabilize 1-D hyperbolic 1-D hyperbolic systems.

-A method to get obstruction to controllability in small time which can be applied to nonlinear partial differential equations.

We also discovered unexpected phenomena created by the nonlinearities, in particular the importance of the functional space for the stability of 1-D hyperbolic.

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