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

Problems of Control and Information Theory, Vol. 17 (4), PP 731—238 (1988) ON THE LIPSCHITZEAN DEPENDENCE OF TRAJECTORIES OF MULTI-INPUT TIME DEPENDENT BILINEAR SYSTEMS ON CONTROLS S. Celikovsky {Prüft ue) (Received February 9, 1987) A multi-input time dependent bilinear system is considered. An estimate for the Lipschitzean dependence of trajectories of the bilinear systems on controls in certain norms is developed. This estimate is then used to study some interesting properties of attainable sets of bilinear systems. Possible numerical applications of the Lipschitzean dependence are briefly described. 1. Introduction Let us consider the following control system x = M(t)+ £ Bk(t)uk(t))x + C(t)u(t) k= 1 х('о) = *о> f6[t„,ti]cR, (1) which will be further denoted as a multi-input time dependent bilinear system. Here /4(f), Bk(t), k = t,m, are (n x n)-dimensional matrix-valued functions, C(f) is (n x m)-dimensional matrix-valued function. The control (or input) u = (u‘, ..., um)T e e Rn is assumed to be a measurable function on every finite time interval [t0, f,] and such that, for each k = 1, .. .,m almost everywhere (further a.e.) on [f0, r,], holds ak ^uk(t)^bk, where ak, bk, k=\, .. ., m are given real numbers. Such a control is further denoted as an admissible one. Finally, xe R" is the vector of the state variables, and .x0 e R" is the given initial state of the system. Bilinear systems have been studied widely and have been shown to be an important extension to linear systems with various possible applications (see e.q. [1], [2], [3]). The aim of this contribution is to derive an estimate which establishes Lipschitzean dependence of trajectories of bilinear systems on controls in certain norms, namely max ||x,(r)—x2(f)||*„^ f€[ío,<l] m Lk max <e[lo.«il I u\(s)ds — J uk2(s)ds . to to(2) Akadémiai Kiadó, Budapest Pergamon Press, Oxford

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