PHYSCON 2011, Le ´ on, Spain, September, 5–September, 8 2011 LIMIT SHAPES OF REACHABLE SETS OF SINGULARLY PERTURBED LINEAR CONTROL SYSTEMS Elena Goncharova Institute for System Dynamics and Control Theory, SB RAS Russian Federation goncha@icc.ru Alexander Ovseevich Institute for Problems in Mechanics Russian Academy of Sciences Russian Federation ovseev@ipmnet.ru Abstract We study shapes of reachable sets of singularly per- turbed linear control systems. The fast component of a phase vector is assumed to be governed by a hyper- bolic linear system. We show that the shapes of reach- able sets have a limit as the parameter of singular per- turbation tends to zero. We also find a sharp estimate for the rate of convergence. A precise asymptotics for the support function of the normalized reachable sets is presented. Key words Singular linear dynamic system, reachable set. 1 Singular Linear Dynamic Systems Consider the following dynamic system ˙ x = Ax + By + F u, ε ˙ y = Cx + Dy + Gu, u ∈ U, (1) where ε> 0 is a small parameter. Traditionally, the components x ∈ X = R m and y ∈ Y = R k of a phase vector z =(x, y) are said to be slow and fast, respectively. The feasible motions start from zero at zero time: z(0) = 0. The dynamic systems of this form arise in abundance from physics and engineering: For instance, assume that in the RC-network drawn below the resistance r = ε is much smaller than R. Then the network is described by dynamic equations of the form (1), where x = C 1 v 1 +C 2 v 2 C 1 +C 2 is the slow variable, and y = v 2 is the fast one. All the system data, i.e., the matrices A, ..., G, and the set U ⊂ U = R r are functions of time and the pa- rameter ε. In order to avoid unnecessary complications the sets U are assumed to be central symmetric con- vex bodies: U = −U , and U has nonempty interior. We hold the regularity and controllability hypotheses as stated in Sections 3, 4, and make the hyperbolicity assumption: For any t the matrix D| ε=0 has no purely imaginary eigenvalues. 2 Problem Statement Given an interval of time [0,T ], we study the reach- able set D ε (T ) of system (1) as ε → 0. Recall that the reachable set of a control dynamic system is the set of the ends at time T of all admissible trajectories. The reachable sets are central symmetric convex bodies in the phase space V = X × Y. The issue on the limit behavior of reachable sets as ε → 0 was addressed in [Dontchev and Slavov, 1988] under the assumption that D| ε=0 is a stable matrix for any t. The main result is that the sets D ε (T ) have a limit with respect to the Hausdorff metric as ε → 0, and the rate of convergence is O(ε α ), where 0 <α< 1 is arbitrary. In our recent paper [Goncharova and Ovseevich, 2009], under the same assumptions we have shown that the rate of convergence is ε log 1/ε. Moreover, we have isolated the main term of the form c ε log 1/ε in the as- ymptotics for the support function of D ε (T ) so that the remainder is o(ε log 1/ε). Now, we are to extend these results to the unstable, hyperbolic case. The direct gen- eralization of the above mentioned results is false: the sets D ε (T ) have no, in general, a natural limit. How- ever, the notion of a shape of a convex body [Ovsee- vich, 1991] is a sure remedy, and allows us to state and prove a similar asymptotics for shapes of the reachable sets. Our methods are based on the exact decomposition of slow, stable fast, and unstable fast variables. We also rely heavily on the averaging principle (ergodic theo-