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An estimation from within of the reachable set of nonlinear R. Brockett integrator with small nonlinearity

In this paper, the nonlinear R. Brockett integrator with small nonlinear addition to the right-hand side of the corresponding differential equations is considered. More precisely, investigating the possibility to estimate from within the corresponding reachable set, we have obtained an efficient form of the ellipsoidal estimation from within. We used our previous results on the similar theme. | Yugoslav Journal of Operations Research 23 (2013) Number 3, 355-365 DOI: 10.2298/YJOR121005026M AN ESTIMATION FROM WITHIN OF THE REACHABLE SET OF NONLINEAR R. BROCKETT INTEGRATOR WITH SMALL NONLINEARITY Aboubacar MOUSSA Department of Mathematics and Computer Science Faculty of Sciences, Abdou Moumouni University BP: 10662 Niamey-NIGER moussa@mail.ru Mikhail Sergeevich NIKOLSKII Steklov Mathematical Institute, Russian Academy of Sciences 119991, Moscow-RUSSIA, Gubkina str., 8. mni@mi.ras.ru Received: October 2012 / Accepted: Novembar 2012 Abstract: In this paper, the nonlinear R. Brockett integrator with small nonlinear addition to the right-hand side of the corresponding differential equations is considered. More precisely, investigating the possibility to estimate from within the corresponding reachable set, we have obtained an efficient form of the ellipsoidal estimation from within. We used our previous results on the similar theme. Keywords: Control problem, reachable set, R. Brockett integrator, estimation from within. MSC: 49J10. 1. INTRODUCTION The important object of interest in Optimal Control Theory is the reachable set. If linear case is in study of reachable sets, convex analysis is successfully applied, and in nonlinear case, nonlinear analysis is successfully applied. In this paper, we obtain a non-trivial estimation from within for the reachable set of the nonlinear R. Brockett integrator with small nonlinearity. Such estimations are the object of interest for the Theory of Optimal Control and its Applications. 356 A. Moussa, M.S. Nikolskii / An Estimation From Within of The Reachable Set 2. PROBLEM FORMULATION We consider the nonlinear R. Brockett integrator with small nonlinearity of the form x&1 = u1 x&2 = u2 (1) x&3 = u2 x1 − u1 x2 + ε g ( x, u ) with zero initial condition x(0) = 0. (2) Where x ∈ R 3 , ε > 0, g ( x, u ) is a continuous function on R 3 × U . We impose on the control vector u the constrain { } u ∈ U = u ∈ R 2 : ui ≤