Download Analysis, Controllability and Optimization of Time-Discrete by Prof. Dr. Werner Krabs, Dr. Stefan Wolfgang Pickl (auth.), PDF

oo 3. asymptotically st abl e, if (x n = Fn(XO))n EN'l! Xo E E , is stable and attractive. 1': The following st atements are equivalent: (1) All sequences (x n = Fn(XO)) nEN()! Xo E E , are stable. (2) One sequence (x n = Fn(XO))n ENo , Xo E E , is stable. (3) The sequence (x n = An °A n - 1 0 ·· · 0 A1(GE)) nEN = (:r n = G E)nENo is stable.

32) is satisfied for every N E No and t herefore t he sequence (A n 0 A n- 1 0 .. 0 A1(GE)nENo) is stable. Further there exists J = J (N) > 0 such that for every sequ ence (x n = Fn(XO)) nENo , Xo E E , with II xNl1< J it follows that lim IIxn - GE II :::; lim IIA n 0 A n- 1 0 n-+ oo n~oo Thus the sequence (An 0 A n- 1 0 . . 0 · ·· 0 AN+lll llxNl1 = O. A1(GE ))nENo is attra ctiv e. o Now we specialize to t he case E = IRk equipped with any norm. T hen we have f n(x ) = Anx + bn , x E IRk , where (An)nEN is a sequence of real k x k - matrices and (bn)nEN a sequ ence of vectors bn E IRk .