Adaptive exponential stabilization for a class of nonlinear retarded processes (Q751595)

Denote by \(C(J,R^ n)\), \(LL^ p(J,R^ n)\), and \(L^ p_{\alpha}(J,R^ n)\), the spaces of continuous, locally p-integrable, and p-integrable with weight exp(\(\alpha\) t), resp. \(R^ n\)-valued functions on an interval \(J\subset R\). Define an operator \(\pi_ t:\) \(LL^ p([a,\infty),R^ n)\to LL^ p([a,\infty),R^ n)\) by \((\pi_ tf)(\tau)=f(\tau)\), \(a\leq \tau \leq t\), \((\pi_ tf)(\tau)=0\) otherwise, and assume that given operators \(P_ i\), \(i=1,2\), of \(LL^ 1([- r,\infty),R^ n\) into \(LL^ 1([-r,\infty),R^ n)\), \(LL^ 1([- r,\infty),R)\), resp., satisfy \(\| \pi_ t(P_ if-P_ ig)\|_{1,\alpha}\leq \gamma_ i\| \pi_ t(f- g)\|_{1,\alpha}\) for some \(\gamma_ i>0\), \(P_ i(0)=0\). For a function \(A: [0,r]\to R^{n\times n}\quad of\) bounded variation, let \[ \begin{gathered} \dot x=P_ 1x+b(P_ 2x+u)+\int^{r}_{0}x(t-\tau)dA(\tau),\;x_{[-r,0]}=x_ 0\in C([-r,0],R^ n);\tag{2.1}\\ y=C^ Tx,\tag{2.2}\end{gathered} \] where \(b,c\in R^ n\), \(c^ Tb\neq 0\), and the matrix \[ \left[\begin{matrix} sI-\int^{r}_{0}\exp (-s\tau)dA(\tau) &, & - & b\\ c^ T &, && 0\end{matrix}\right] \] is regular for all Re \(s\geq - \alpha\), \(\alpha >0\). Then for sufficiently small \(\gamma_ i\) and each \(u\in LL^ 1([0,\infty),R^ n)\), the system (2.1) has a unique locally absolutely continuous solution on [-r,\(\infty)\). Let, for some \(0<\delta \leq \Delta\), functions \(f_ i:\) \([0,\infty)\times R\to R\), satisfy \(\delta x^ 2\leq | xf_ i(t,x)| \leq \Delta x^ 2\), \(t\geq 0\), \(x\in R\), \(i=1,2\), \(N\in LL^{\infty}(R,R)\) be a scaling invariant switching function, and \(k: [0,\infty)\to [\epsilon,\infty),\epsilon >0\). Consider actuator and sensor equations \((3.1)\quad u(t)=k(t)f_ 1(t,N(\xi))f_ 2(t,y),\) \((3.2)\quad {\dot \xi}(t)=k(t)\cdot | f_ 2(t,y)| \cdot \exp (\alpha t),\quad \xi (0)=\xi_ 0.\) Then the adaptive exponential stabilization nonlinear retarded system has unique solution \(x\in \cap^{\infty}_{p=1}L^ p_{\alpha}([0,\infty),R^ n)\) and \(\lim_{t\to \infty}\xi (t)\) exists and is finite.