STOCHASTIC CONTROL OF BEAM DYNAMICS N. Cufaro-Petroni, Dip. di Fisica, Universit` a and INFN, Bari,Italy S. De Martino, S. De Siena, F. Illuminati, Dip. di Fisica, Universit´ a, INFM, and INFN, Salerno, Italy R. Fedele, Dip. di Scienze Fisiche, Universit´ a ”Federico II” and INFN, Napoli, Italy S.I. Tzenov, Fermilab, Batavia, USA Abstract The methods of stochastic control theory are proposed in the context of charged-particle beam dynamics. The stochastic dynamics that is introduced here is invariant for time reversal and can be easily recast in the form of a Schr¨ odinger-like equation where Planck’s constant is re- placed by the beam emittance. It changes a bilinear control problem for Schr ¨ odinger equation in a linear control prob- lem, then resulting more adequate to our aim. This point of view seems to be in agreement with accelerators physics modus operandi. 1 INTRODUCTION The macroscopic state of a particle bunch in an accelerat- ing machine is essentially the result of the collective in- teraction of the particle among them as well as with the surroundings. However, this macroscopic dynamics in- volves both several coherent and incoherent microscopic processes whose nature is deterministic or stochastic. The sum of all these processes determines the above macro- scopic state whose nature is essentially classical. For exam- ple, coherent oscillations of the beam density that are man- ifested through some mechanism of local correlation and loss of statistical independence may be described by taking into account all the interactions as a whole. Within the con- text of the conventional descriptions of the beam dynamics, it must be recognized that the study of statistical effects on the dynamics of electron (positron) colliding beams with Fokker–Planck equation for the beam density has received a great deal of attention in literature, stimulating very much the description of the noise sources and dissipation in par- ticle accelerators by standard classical probabilistic tech- niques [1],[2]. Nevertheless, approaches alternative to the conventional ones should be mentioned for their natural ap- plications to the descriptions of the interaction betwen the beam as a whole and the surroundings. In particular, three approaches are based on a quantum-like formalism which takes into account the diffusion among the beam particles. One of these is known as Thermal Wave Model (TWM) [3] which assumes that the beam dynamics as whole is gov- erned by a Schr¨ odinger-like equation whose diffraction- like term describes the thermal spreading among the elec- tronic rays (diffusion). Another approach is based on a stochastic quantization ala Nelson of the beam dynamics in a thermal bath with the environment [4]. Finally, a more recent approach, is based on the simulation of semiclas- sical corrections to classical dynamics by suitable classical stochastic fluctuations with a suitably defined random kine- matics by replacing the classical deterministic trajectories [5]. Additionally, it is worth mentioning that recent exper- iments on confined classical systems with special phase– space boundary conditions seem to be well described by a quantum–like formalism (Schr¨ odinger-like equation) [6]. In this paper, we use the stochastic formalisms to intro- duce, as a novel concept, stochastic control theory in beam dynamics. This is done by giving the description of the sta- bility regime for the beam, when thermal dissipative effects are balanced on average by the RF energy pumping, and the overall dynamics is conservative and time–reversal invari- ant in mean. To this end, we observe that, according to the stochastic formalism, the diffusion process describes the effective motion at the mesoscopic level (interplay of ther- mal equilibrium, classical mechanical stability, and fun- damental quantum noise) and therefore the diffusion co- efficient is set to be the semiclassical unit of emittance provided by qualitative dimensional analysis. In the next section we model the random kinematics with a particular class of diffusion processes, the Nelson diffusions, that are nondissipative and time–reversal invariant [7]. This allows us to introduce briefly the hydrodynamic equations for the collective stochastic dynamics, and, in turn, to develop con- trol tecniquesfor the beams. In particular, the dynamical equations are derived via variational principle of classical dynamics, with the only crucial difference that the kinemat- ical rules and the dynamical quantities, such as the Action and the Lagrangian, are now random. The stochastic vari- ational principle formally reproduces the equations of the Madelung fluid (hydrodynamic) representation of quantum mechanics with Planck’s constant replaced by emittance. In this sense, the present scheme allows us for a quantum– like formulation equivalent to the probabilistic one. 2 STOCHASTIC DYNAMICS The above quantum-like approaches of beam dynamics are formulated, starting from different physical point of view, but they have the common feature that one can model spa- tially coherent fluctuations by a random kinematics per- formed by some collective degree of freedom q(t) repre- sentative of the beam. This way, the random kinematics provides an effective description of the space–time varia- tions of the particle beam density ρ(x, t) as it coincides with the probability density of the diffusion process per- formed by q(t). Then, in suitable units, the basic stochastic 1259