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Stabilization of uncertain control systems through piecewise constant feedback. (English) Zbl 0891.93068
An uncertain system with dynamics defined by \[ \dot{x}(t)=f(x(t),u(t),d(t)),\quad x(t) \in R^n,\;u(t) \in U \subset R^m,\;d(t) \in D \subset R^l, \] where \(t\), \(u\), \(d\), and \(x\) are the time, the control, the disturbance and the state, is considered. The sets \(U\) and \(D\) are supposed to be closed. Let \(V:R^n \to R^+\) be a given function and \(a\) be a positive parameter. The following approximate stabilization problem is studied: Fix \(T>0\) and \(\eta>0\). The aim is to find a control law \(u=u(t,x)\in U\) such that for any measurable \(d(.)\) the system of ordinary differential equations \[ \dot{x}(t)=f(x(t),u(t,x(t)),d(t)),\;t \in [0,T] \] has at least one solution and any solution \(x(.)\) satisfies \[ V(x(t))\leq V(x(0))e^{-at}+\eta,\;t \in [0,T]. \] Assuming that \(V\) is Lipschitz continuous and satisfies \[ \inf_{u \in U} \sup_{d\in D} \;\langle f(x,u,d),p\rangle \leq aqV(x) \] for each proximal normal \((p,q)\) to the epigraph \(\text{epi}V\) of \(V\) at the point \((x,V(x))\), the existence of a piecewise constant feedback satisfying the approximate stabilization property is proved under natural technical assumptions on \(f\). This is the main result of the paper. Its proof relies on the fact that the conditions imposed on \(V\) imply viability of the set \(\text{epi}V\) with respect to the following augmented system: \[ \dot{x}(t)=f(x(t),u(t),d(t)),\;u(t) \in U,\;d(t) \in D,\;\dot{w}(t)=-aw(t). \] Thus, for any \(\varepsilon > 0\) there exists a piecewise control such that for any measurable disturbance \(d(.)\) the corresponding trajectory \((x(.),w(.))\) starting from \((x_0, V(x_0))\) is defined on \([0,T]\) and the distance between \((x(t),w(t)), \;t \in [0,T]\) and \(\text{epi}V\) is not greater than \(\varepsilon\).

MSC:
93D21 Adaptive or robust stabilization
93D15 Stabilization of systems by feedback
93C41 Control/observation systems with incomplete information
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