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Global stabilization of nonlinear systems with a class of unmatched uncertainties. (English) Zbl 0759.93060
The paper studies global stability properties of dynamical systems described by \[ \dot x(t)=f(x,t)+\tilde\xi(x,t)+B(x,t)[\xi(x,t)+u],\;x(t_ 0)=x_ 0, \] where \(x(t)\in\mathbb{R}^ n\) denotes the state, \(u(t)\in\mathbb{R}^ n\) the control, \(\tilde\xi(x,t)\) the unmatched constraints and \(\xi(x,t)\) the matched constraints. Under the assumption of global asymptotic stability of \(\dot x(t)=f(x,t)\) and under certain unboundedness assumptions on the uncertainties of the constraints as well as of \(f\) and \(B\), the author constructs a state feedback controller \(u(x,t)\) which stabilizes the closed loop system in the large. The techniques make use of the fact that Lyapunov functions are in general not unique.

MSC:
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C10 Nonlinear systems in control theory
70K20 Stability for nonlinear problems in mechanics
93C15 Control/observation systems governed by ordinary differential equations
93D09 Robust stability
93D30 Lyapunov and storage functions
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