New Mathematical Optimization Models for Rfq Structures

Conventional approach to the designing of controlled systems is to start with calculation of program motion and to continue afterwards by examining perturbed motions using equations in deviations. It does not always, however, result in desirable outcomes. Thus, while analysing perturbed motions, which depend significantly on the program motion, it can happen, that dynamical characteristics of obtained perturbed motions are not satisfactory. This paper suggests new mathematical models, which allow joint optimization of program motion and an ensemble of perturbed motions. These mathematical models include description of controlled dynamical process, choice of control functions or parameters of optimization as well as construction of quality functionals, which allow efficient evaluation of various characteristics of examined control motions. This optimization problem is considered as the problem of mathematical control theory. The suggested approach allows to develop various methods of directed search and to conduct parallel optimization of program and perturbed motions. Suggested approach is applied to the optimization of RFQ channel. Simple model for description of beam longitudinal motion in the equivalent running wave is suggested. For the estimation of beam dynamics corresponding functionals are suggested. 1 MATHEMATICAL MODELS OF OPTIMIZATION First we consider following dynamic system: dx dt f t x u = ( , , ) , (1) dy dt F t x y u = ( , , , ) , (2) with initial conditions: x x y M ( ) ( ) 0 0 0 0 0 = = ∈ , y ; (3) Here t is time, x is n-vector, u u t = ( ) is rdimensional control vector function, y m-vector, f t x u ( , , ) and F t x y u ( , , , ) are n-dimensional and mdimensional ___________________ Work supported in part by Russian Foundation for Fundamental Researches, project 99-01-00678 # Email: [email protected] vector functions correspondingly. We assume that f , F and div F F y y i i i m = = ∑ ∂ ∂ 1 are continuous with their partial derivatives. The set M0 is a given compact set in the phase space of nonzero measure in R m . We assume that admissible controls u t ( ) constitute certain class D of functions, that are piecewise continuous on interval [ , ] 0 T and have values in a compact set U R r ⊂ . Equation (1) describes dynamics of synchronous particle. Here in this paper we will consider this motion as the program one. Equation (2) is the derivations equation describing perturbed motions. Along with equations (1), (2) we consider changes of density of particles ρ( , ) t y , along the system (2) trajectory: d dt div F t x t y t u t y ρ ρ = − ⋅ ( , ( ), ( ), ( )) , (4) ρ ρ ( , ( )) ( ), 0 0 0 0 0 0 y y M = ∈ y , (5) where ρ0 0 ( ) y is density of particles distribution at the set M0 . We introduce following functionals: I u c t x t u t dt c g x T)) T 1 1 1 0 2 1 ( ) ( , ( ), ( )) ( ( = + ∫ φ , (6)