Nonlinear Dynamics 36: 107–122, 2004. C  2004 Kluwer Academic Publishers. Printed in the Netherlands. Symmetries and Conserved Quantities of Stochastic Dynamical Control Systems GAZANFER ¨ UNAL 1,∗ and JIAN-QIAO SUN 2 1 Faculty of Sciences, Istanbul Technical University, Maslak 80626, Istanbul, Turkey; 2 Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, U.S.A.; ∗ Author for correspondence (e-mail: gunal@itv.edu.tr) (Received: 20 September 2003; accepted: 10 November 2003) Abstract. A new definition is given for both exact and quasi symmetries of Itˆ o and Stratonovich dynamical control systems. Determining systems of symmetries for these systems have been obtained and their relation is discussed. It is shown that conserved quantities can be found from both exact and quasi symmetries of stochastic dynamical control systems, which includes Hamiltonian control systems as a special case. Systems which can be controlled via conserved quantities have been investigated. Results have been applied to the control of an N -species stochastic Lotka–Volterra system. Key words: conserved quantity, exact-quasi symmetry, integrating factor, stochastic dynamical control system 1. Introduction The Unifying features of the Lie theory have led to a vast number of successful advances, both theoretical and applied, in deterministic differential equations (see [1–3]). Recently, there has been growing interest in the extension of Lie theory to stochastic systems (see [4–8]). One of the outcomes of this approach is the derivation of conserved quantities which have crucial importance in both analytical [9] and numerical treatments [10] of stochastic dynamical systems. The purpose of this paper is to extend the theory given in [8] to nonlinear stochastic control systems in such a way that conserved quantities become the focus of attention. Application of Lie’s theory to deterministic nonlinear controlled systems was first reported in [11], where symmetries are used to decompose the systems to simpler subsystems. Our approach allows us study nonlinear stochastic controlled systems and to develop invariant control laws. We consider a controlled Itˆ o dynamical system of the form dx i = f i (x, u, t )dt + g i α (x, u, t )dB α , x(0) = x o , (i = 1,..., n; α = 1,..., r ), (1) where f i (x, u, t ) and g i α (x, u, t ) are drift vector and diffusion matrix with control u(x, t) ∈ R m re- spectively, and dB α is a vector Wiener process. (The summation convention applies to repeated indices hereafter.) This equation does indeed map a vector Wiener process into a controlled Itˆ o process. Al- though almost all sample functions of a Wiener process are continuous, in accordance with a theorem of Wiener, they are nowhere differentiable. Furthermore, almost all sample functions of B (t ) are of unbounded variation on any finite interval [0, T ]. This means that sup t n  i =1 | B (t i ) − B (t i −1 )|=∞, where the supremum is taken over all possible partitions τ :0 = t 0 <...< t n = T of [0, T ]. This