Stabilization of two-dimensional persistently excited linear control systems with arbitrary rate of convergence

Abstract : We study the control system $\dot x = A x + \alpha(t) b u$ where the pair $(A, b)$ is controllable, $x \in \mathbb R^2$, $u \in \mathbb R$ is a scalar control and the unknown signal $\alpha: \mathbb R_+ \to [0, 1]$ is $(T,\mu)$-persistently exciting (PE), i.e., there exists $T \geq \mu > 0$ such that, for all $t \in \mathbb R_+$, $\int_t^{t + T} \alpha(s) ds \geq \mu$. We are interested in the stabilization problem of this system by a linear state feedback $u = - K x$. In this paper, we positively answer a question asked in \cite{YacineMario} and prove the following: Assume that the class of $(T,\mu)$-PE signals is restricted to those which are $M$\nobreakdash-Lipschitzian, where $M>0$ is a positive constant. Then, given any $C>0$, there exists a linear state feedback $u = - K x$ where $K$ only depends on $(A,b)$ and $T,\mu,M$ so that, for every $M$-Lipschitzian $(T,\mu)$-PE signal, the rate of exponential decay of the time-varying system $\dot x = (A -\alpha(t) bK)x $ is greater than $C$.
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Yacine Chitour, Guilherme Mazanti, Mario Sigalotti. Stabilization of two-dimensional persistently excited linear control systems with arbitrary rate of convergence. SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2013, 51 (2), pp.801-823. ⟨10.1137/110848153 ⟩. ⟨inria-00610345v2⟩

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