Problems of Control and Information Theory 17. (Budapest, 1988)

1988 / 4. szám - Čelikovský, S.: On the Lipschitzean dependence of trajectories of multi-input time dependent bilinear systems on controls

232 CELIKOVSKY: lipschitzean dependence of multi-input time dependent bilinear systems Here x,(r) and x2(t), te [t0, (,] are trajectories corresponding to admissible controls “i(s) = (m1(s), ..u7(.s))' and u2(s) = (n^(s), ..u2(s))r, se[t0, t,], respectively; Lk, k = \, ..m, are certain constants, depending only on the parameters of system (1). Estimate (2) is then used to study some important properties of the so-called attainable set of bilinear systems and some ideas of further possible applications of this estimate are given. 2. Estimate for Lipschitzean dependence We derive estimate (2) by generalization of the corresponding result for single input time dependent bilinear systems. This case is considered in [4, Theorem 3]: Theorem 1. Let us consider the time-dependent bilinear system with single input: x = (A (f) + B(t)u(t))x + C(t)u(t) + f(t), *(*о) = *с» f6[f0,fi], x0 e Rn, u(t) e [a, h] a.e. on [t0,f,], (4) where A (r), B(t) are (n x n)-dimensional matrix-valued functions, c(i), /(f) are functions with values in R". We impose the following assumptions: (i) ß(f), c(t) are absolutely continuous and almost everywhere on [i0,r,] ||ß(t)|UßM, ||ß'(f)||s^ß0M, IHOIIa^c", lk'(t)IU^cßM. (ii) A(t), f(t) are essentially bounded measurable functions and a.e. on [t0,fi] \\A(t)\\süAM, \\f(t)\\R„üfMHere II . И, stands for the spectral matrix norm and || . ||х„ for the Euclidean vector norm in R". (iii) For every t', t" e [f0, f,] matrices ß(f') and B(t") commute, i.e., B(t')B(t") = = B(t")B(t'Y Let us x,(t) and x2(f) be trajectories of system (4) for admissible controls u,(t) and u2(f), respectively. Then max Hx1(t)-x2(t)llK„gK max le[lo,<il (e[lo,li) J u{(s)ds— j u2(s)ds(5) where K = K(||x0||#t„, up, AM, BM, cM, BDM, i0, i,) = = KlK2\\x0\\Rn + 2KiKl(K3 + KtKsup(tl-t0))KtKb(tl-t0) + + K2K4(\+KAK5(\+BMup(ti ~10)))Кь«1-10) +