Stability research of parametric systems with partially fixed boundary conditions

<ns0:p>General formulations of practical stability problems for nonlinear systems of differential equations dependent on parameters are formulated in the presence of additional boundary conditions that arise during the design of linear resonance accelerators, in particular, when coordinating different sections of the accelerator. Common types of boundary conditions are given in the form of additional conditions for the state vector at the initial and final moments of time, and their linear combination.Within the framework of the formulated problems, a linear inhomogeneous system with disturbances in the presence of boundary conditions on part of the coordinates of the state vector was investigated. Using the methods of practical stability of parametric systems, criteria for checking the corresponding stability qualities under partial restrictions on the vector of initial conditions and constantly acting disturbances were obtained.Linear and nonlinear dynamic constraints on the phase coordinates of the system under study are considered; additional conditions are given in specific structures such as ellipsoids.To record the achievable estimates, the general solution of the parametric system is presented in Cauchy form and the corresponding extremal problem is solved. For the case of nonlinear constraints on the state vector in dynamics, a preliminary approximation of a closed convex set by tangent hyperplanes is performed.As a special case, the stability region of a linear parametric system is estimated in the presence of partially fixed initial conditions and dynamic phase constraints of linear and nonlinear type. It is noted that such problems arise, for example, if it is known that the particles have a small spread in radial velocities.The problems of estimating the stability region for linear parametric systems with additional perturbations in the presence of boundary conditions on part of the state vector and partially fixed initial conditions characterizing the relationship of the particle bunch motion in a linear resonant accelerator at certain moments of time are investigated. In this case, part of the initial conditions vector and constantly acting perturbations are subject to specific constraints. The evaluation criteria for the stability of a parametric system are obtained for the case of linear and nonlinear dynamic constraints on the phase coordinate vector.</ns0:p>