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We study the chemostat model for one species competing for one nutrient using a Lyapunov-type analysis. We design the dilution rate function so that all solutions of the chemostat converge to a prescribed periodic solution. In terms of chemostat biology, this means that no matter what positive initial levels for the species concentration and nutrient are selected, the long term species concentration and substrate levels closely approximate a prescribed oscillatory behavior. This is significant because it reproduces the realistic ecological situation where the species and substrate concentrations oscillate. We show that the stability is maintained when the model is augmented by additional species that are being driven to extinction. We also give an input-to-state stability result for the chemostat tracking equations for cases where there are small perturbations acting on the dilution rate and initial concentration. This means that the long term species concentration and substrate behavior enjoys a highly desirable robustness property, since it continues to approximate the prescribed oscillation up to a small error when there are small unexpected changes in the dilution rate function., Comment: 20 pages, 3 figures, accepted for publication in Mathematical Biosciences and Engineering on December 5, 2006

We provide general methods for explicitly constructing strict Lyapunov functions for fully nonlinear slowly time-varying systems. Our results apply to cases where the given dynamics and corresponding frozen dynamics are not necessarily exponentially stable. This complements our previous Lyapunov function constructions for rapidly time-varying dynamics. We also explicitly construct input-to-state stable Lyapunov functions for slowly time-varying control systems. We illustrate our findings by constructing explicit Lyapunov functions for a pendulum model, an example from identification theory, and a perturbed friction model., Comment: Accepted for publication in Mathematics of Control, Signals, and Systems (MCSS) on November 20, 2006

We provide explicit closed form expressions for strict Lyapunov functions for time-varying discrete time systems. Our Lyapunov functions are expressed in terms of known nonstrict Lyapunov functions for the dynamics and finite sums of persistency of excitation parameters. This provides a discrete time analog of our previous continuous time Lyapunov function constructions. We also construct explicit strict Lyapunov functions for systems satisfying nonstrict discrete time analogs of the conditions from Matrosov's Theorem. We use our methods to build strict Lyapunov functions for time-varying hybrid systems that contain mixtures of continuous and discrete time evolutions., Comment: 14 pages. Accepted for publication in Nonlinear Analysis: Hybrid Systems and Applications on September 6, 2006

We explicitly construct global strict Lyapunov functions for rapidly time-varying nonlinear control systems. The Lyapunov functions we construct are expressed in terms of oftentimes more readily available Lyapunov functions for the limiting dynamics which we assume are uniformly globally asymptotically stable. This leads to new sufficient conditions for uniform global exponential, uniform global asymptotic, and input-to-state stability of fast time-varying dynamics. We also construct strict Lyapunov functions for our systems using a strictification approach. We illustrate our results using a friction control example., Comment: 10 pages, 0 figues, revised and accepted for publication as a regular paper in Automatica in May 2006. To appear in October 2006 issue

We study the low-bias stabilization of active magnetic bearings (AMBs) subject to voltage saturation based on a recently proposed model for the AMB switching mode of operation. Using a forwarding-like approach, we construct a stabilizing controller of arbitrarily small amplitude and a control-Lyapunov function for the AMB dynamics. We illustrate our construction using a numerical example., Comment: 9 pages, 2 figures. IEEE Transactions on Control Systems Technology, accepted for publication in January 2006

For a broad class of nonlinear systems, we construct smooth control-Lyapunov functions whose derivatives along the trajectories of the systems can be made negative definite by smooth control laws that are arbitrarily small in norm. We assume our systems satisfy appropriate generalizations of the Jurdjevic-Quinn conditions. We also design state feedbacks of arbitrarily small norm that render our systems integral-input-to-state stable to actuator errors., Comment: 15 pages, 0 figures, accepted for publication in IEEE Transactions on Automatic Control in October 2005

We study the stability properties of a class of time-varying nonlinear systems. We assume that non-strict input-to-state stable (ISS) Lyapunov functions for our systems are given and posit a mild persistency of excitation condition on our given Lyapunov functions which guarantee the existence of strict ISS Lyapunov functions for our systems. Next, we provide simple direct constructions of explicit strict ISS Lyapunov functions for our systems by applying an integral smoothing method. We illustrate our constructions using a tracking problem for a rotating rigid body., Comment: 6 pages, submitted for publication in June 2004