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This construction of strict Lyapunov functions is an ongoing engineering challenge. This book contains a broad repertoire of Lyapunov constructions for nonlinear systems, focusing on methods for transforming non-strict Lyapunov functions into strict ones.

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Converse Lyapunov function theory guarantees the existence of strict Lyapunov functions in many situations, but the functions it provides are often abstract and nonexplicit, and therefore may not lend themselves to engineering applications. Often, even when a system is known to be stable, one still needs explicit Lyapunov functions; however, once an appropriate strict Lyapunov function has been constructed, many robustness and stabilization problems can be solved through standard feedback designs or robustness arguments. Non-strict Lyapunov functions are often readily constructed.

This book contains a broad repertoire of Lyapunov constructions for nonlinear systems, focusing on methods for transforming non-strict Lyapunov functions into strict ones. Their explicitness and simplicity make them suitable for feedback design, and for quantifying the effects of uncertainty. Readers will benefit from the authors´ mathematical rigor and unifying, design-oriented approach, as well as the numerous worked examples.

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Background.- Background on Nonlinear Systems.- Review of Lyapunov Functions.- Time-Invariant Case.- Matrosov Conditions: Simple Case.- Jurdjevic-Quinn Conditions.- Systems Satisfying the Conditions of LaSalle.- Time-Varying Case.- Strictification: Basic Results.- Backstepping for Time-Varying Systems.- Matrosov Conditions: General Case.- Adaptively Controlled Systems.- Systems with Multiple Time Scales.- Rapidly Time-Varying Systems.- Slowly Time-Varying Systems.- Hybrid Time-Varying Systems.

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