Stability analysis via coupled Hamilton-Jacobi equations (Reference: ESR9) **** this position has been filled ****

• Title: Stability analysis via coupled Hamilton-Jacobi equations
• Location:  University of Bayreuth and/or University of Würzburg, Germany
• Advisor: Lars Grüne and Fabian Wirth
• Secondment: Università degli Studi di Padova
• Contact: Lars Grüne
• Reference: ESR9

• Key Words: Hamilton-Jacobi equations, Zubov's equation, input-to-state stability, small gain theorems

• Application deadline: November 30, 2010, will be extended until the position is filled
• Description of the subject: Hamilton-Jacobi equations are connected to Lyapunov functions and thus to stability analysis of nonlinear systems. The Zubov equation is a Hamilton-Jacobi partial differential equation which provides a systematic way to characterize Lyapunov functions via optimal control problems. Optimal control and viscosity solution techniques have been used over the past decade to extend the method to both deterministically and stochastically perturbed and controlled systems. It is now revealed as a versatile tool for the stability analysis of nonlinear systems as well as for the computation of stabilizing controllers. However, for large scale systems these equations become difficult to solve both analytically and numerically. Here a decomposition into subsystems and a stability analysis based on a generalized small gain theorem may provide a remedy.   In this project, the corresponding decoupling of the related Zubov equation and the application to the stability analysis will be investigated. The relation to a generalization of Zubov’s method for differential games will also be studied.
   

• Requirements: The candidate for the PhD position in this project should preferably have experience in one or more of the following areas:

- Hamilton-Jacobi equations and viscosity solutions
- stability analysis of nonlinear systems, in particular Lyapunov function techniques
- input-to-state stability and small gain based stability analysis