Path Integrals and Stability

  • J. C. Willems


A path integral associated with a dynamical system is an integral of a memoryless function of the system variables which, when integrated along trajectories of the system, depends only on the value of the trajectory and its derivatives at the endpoints of the integration interval. In this chapter we study path independence for linear systems and integrals of quadratic differential forms. These notions and the results are subsequently applied to stability questions. This leads to Lyapunov stability theory for autonomous systems described by high-order differential equations, and to more general stability concepts for systems in interaction with their environment. The latter stability issues are intimately related to the theory of dissipative systems.


Path integrals path independence differential systems stability Lyapunov theory dissipative systems quadratic differential forms behaviors 


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© Springer Science+Business Media New York 1999

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