Controlled dynamic systems and Carleman operator
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CONTROLLED DYNAMIC SYSTEMS AND CARLEMAN OPERATOR
UDC 62-50
V. F. Zadorozhny
Controlled dynamic systems on a compact set and thus with invariant measure are considered. This allows reducing differential dynamic systems to Fredholm-type integral equations. An algorithm of constructing a vector field of maximum descent rate along a trajectory is presented. The algorithm is reduced to a numerical moment-type procedure. Keywords: dynamic system, compact set, control, stability, Hilbert–Schmidt and Carleman operators, measure. INTRODUCTION The concept of a dynamic system has occurred as a generalization of the concept of a mechanical system whose movement can be described by the Newton, Lagrange or Hamilton–Jacobi differential equations. Being developed, it was filled with a new deep mathematical content and united applied problems of different meanings. Nowadays, the concept of a dynamic system is rather wide. It envelopes systems of any nature: physical, chemical, biological, economic, etc. Dynamic systems (mathematical models) may also be described in a great variety of ways: differential equations or functions of the algebra of logic, or graphs, symbolic dynamics, Markov chains, etc. Let us define a continuous dynamic system [1, 2]. Definition 1. A continuous-time dynamic system or a flow on a set M n is a family of transformations Yt , t Î R 1 , on M n such that Y0 is an identity transformation and Yt + s = Yt o Ys . Additional constraints are also imposed: continuity, smoothness, etc. The transformation Yt is defined using a vector field X ( x ) or a nonlinear vector equation (1) x& = X ( x ), x Î M n , where M n is a C r -manifold. The vector field X ( x ), x Î M n , is sometimes treated as a dynamic system. In what follows, dynamic systems are considered on closed bounded (compact sets) C r -manifolds W Ì R n alone. Vector fields
X ( x ) on such manifolds form compact dynamic systems. Denote them by å . The Cauchy Theorem [2]. Let (1) be a compact dynamic å-system. Let the compact set W be a smooth
C r -manifold. Then the integral curve (which is unique) x t ( x ) : t ® x t ( x ) ÎW passes through any point x on W = int W as through the initial point. This curve belongs to the class C r . For each t Î [ 0, ¥ ) , the equality Yt ( x ) Ò x t ( x ) defines the mapping Yt : W ® W, Y0 being an identity mapping and Yt1 o Yt2 = Yt1 +
t2
( x t1 ( x t2 ( x )) = x t1 +
t2 ( x )).
The equality Y( t , x ) = x t ( x ) defines the mapping Yt : R ´ M n ® M n . This mapping belongs to the class C r . Let us use a kernel k ( x, y ) , x, y ÎW, to introduce an operator A, where k ( x, × ) Î L2 (Y ) for almost all x( 0) = x ÎW. Let k ( x, × ) be the Carleman kernel and let g Î L2 W, then k ( x, × )g ( × ) Î L1 (Y ) for almost all x. For such kernels, the domain of definition g consists
V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 54–61, September–October 2008. Original article submitted Janu
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