Optimal control in nonlinear system with no ideal constraints

Abstract: Main idea of the phase plane method applied to the optimal control in nonlinear dynamical systems with triggers of a coupled singularities, and with one degree of freedom, is reconsidered. Theory and models of the heavy mass particle optimal control motion along rough curvilinear line are presented. Introduction of the paper presents a short review of the author’s previously published results containing series of special cases of optimal control in nonlinear dynamical systems with trigger of coupled singularities important for engineering applications. Task of the defined dynamical system optimal control is: by using controlling force acting on the system, transfer initial kinetic state of the nonlinear dynamics of the system into the final terminate kinetic state of the nonlinear dynamics in the minimal time. Use, previously published results in paper written by author, the mathematical description of the heavy mass particle motion along rough curvilinear line with Coulomb’s type friction in generalized form, the optimal control of such system dynamics analytically is considered. Three cases of a rough line are considered in the phase plane and analytical expressions for the phase trajectories of the mass particle dynamics along rough line loaded by controlling force are obtained. For considered cases of mass particle motion along rough lines, we can identify a member in the differential double equation, proportional to the square of the generalized velocity. This corresponds to the known case of system in the turbulent damping field. Also, for general case, as well as for a special case we separate a corresponding fictive conservative system with two alternate one side equilibrium positions defined by an angle of friction corresponding to the coefficient of Coulomb friction. [ABSTRACT FROM AUTHOR]