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Zbl 0911.93046
Ryan, E.P.
An integral invariance principle for differential inclusions with applications in adaptive control.
(English)
[J] SIAM J. Control Optimization 36, No.3, 960-980 (1998). ISSN 0363-0129; ISSN 1095-7138/e

The following extension of the invariance principle by {\it C. I. Byrnes} and {\it C. F. Martin} [IEEE Trans. Autom. Control 40, No. 6, 983-994 (1995; Zbl 0831.93058)] for differential inclusion is obtained. \par Theorem: Let $x:[0,\infty)\to \Bbb {R}^n$ be a maximal solution of Cauchy problem $$\dot{x}(t)\in X(x(t)),\qquad x(0)=x^0,$$ such that $x(t)\in U\subset G$ for all $t\geq 0$. Here $X(\cdot)$ is a compact-valued multifunction. Assume that $l:G\to R$ is a lower semi-continuous scalar function which is nonnegative on $U\subset G$. If $l(x(t))\in L^1([0,\infty))$, then the trajectory $x(t)$ approaches the largest weakly-invariant set in $\Sigma = \{z\in \text{cl} U\cap G: l(z)\leq 0 \}$, as $t\to\infty$. \par As implications of this general result, stability of discontinuous feedback control is demonstrated for an adaptive system, for an adaptive servomechanism and for a dynamically perturbed system characterized by a differential equation of first or second order.
[D.Silin (Berkeley)]
MSC 2000:
*93D15 Stabilization of systems by feedback
34A60 ODE with multivalued right-hand sides
93D05 Lyapunov and other classical stabilities of control systems
93D09 Robust stability of control systems
93D21 Adaptive and robust stabilization

Keywords: integral invariance principle; differential inclusion; adaptive control; weakly invariant set; stabilization; discontinuous feedback

Citations: Zbl 0831.93058

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