On the Minimax Problem of Beam Dynamics Optimization

The problem of simultaneous optimization of the ensemble of trajectories and some selected trajectory arises in the research of the charged particle beam dynamics [1–8]. The present work suggests the use of a smooth functional for the evaluation of the selected trajectories and a minimax functional for the evaluation of the dynamics of the beam of trajectories. A combination of those functionals is considered. INTRODUCTION In the present work a new approach to the beam dynamics optimization, based on the use of smooth and nonsmooth functionals for the evaluation of the dynamics of the charged particles, is developed. The problem of simultaneous optimization of the program motion and the ensemble of trajectories is formulated. The dynamics of the program motion is evaluated using a smooth integral functional and the dynamics of the ensemble of disturbed motions is evaluated using a non-smooth functional. In this paper the analytical form of the variation for the combination of a smooth and non-smooth functionals is presented, allowing to develop various methods of optimization. Those methods can be implemented, for instance, to the optimization of particle dynamics in a RFQ structure. It should be noted that the problems of analysis and optimization of the particle dynamics in RFQ accelerators in an equivalent running wave were explored in numerous works [9–14], but those did not utilize non-smooth functionals. MATHEMATICAL MODEL Let us consider the following system of differential equations dx dt = f(t, x, u), x(0) = x0. (1) Here t ∈ [0, T ] — independent variable, T > 0 is a fixed moment of time; x — n–dimensional phase-vector; u = u(t) — r–dimentional piecewise continuous control vector-function from a class D; f(t, x, u) — n– dimentional reasonably smooth vector-function . Let us call the solution of system (1) a program motion. At the same time we consider the so-called disturbed motions, which are the solutions of the following system of equations [1] dy dt = F (t, x, y, u), y(0) = y0 ∈M0. (2) ∗m.mizintseva@spbu.ru Here y — n–dimensional phase-vector; F (t, x, y, u) — ndimensional reasonably smooth vector-function; M0 — a compact set. The trajectories of system (2) are vector-functions y = y(t, x(t, x0, u), y0, u), continuously dependent on the program motion x(t, x0, u) and initial conditions y0 ∈M0. Let us introduce the set of terminal positions of the system (2) Y = {y(T, x0, y0, u) | u ∈ D,x(0) = x0, y0 ∈M0}. On the solutions of system (1) let us introduce a functional