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In this work, we give sufficient conditions for the almost global asymptoticstability of a cascade in which the subsystems are only almost globallyasymptotically stable. The result is extended to upper triangular systems ofarbitrary size. In particular, if the unforced subsystems are almost globallyasymptotically stable and their only chain recurrent points are hyperbolicequilibria, then the boundedness of forward trajectories is sufficient for thealmost global asymptotic stability of the full upper triangular system. We showthat unboundedness of such cascades is prohibited by growth rate conditions onthe interconnection term and a Lyapunov function for the unforced outersubsystem, and the required structure for the chain recurrent set is enjoyed byclasses of systems common in geometric control e.g. dissipative mechanicalsystems. Our results stand in contrast to prior works that require either timescale separation, prohibitively strong disturbance robustness properties, orglobal asymptotic stability in the subsystems.

A Compositional Approach to Certifying the Almost Global Asymptotic Stability of Cascade Systems