Abstract
Approximation of the sections of the set of trajectories of the control system described by a nonlinear Volterra integral equation is studied. The admissible control functions are chosen from the closed ball of the space L p , p > 1, with radius µ and centered at the origin. The set of admissible control functions is replaced by the set of control functions, which includes a finite number of control functions and generates a finite number of trajectories. It is proved that the sections of the set of trajectories can be approximated by the sections of the set of trajectories, generated by a finite number of control functions.
Highlights
Control systems arise in various fields of theory and applications
Depending on the type of equation which describes the system’s behavior, control systems can be classified as linear or nonlinear control systems, control systems described by ordinary differential equations or control systems described by partial differential equations or control systems described by integral equations and etc
Precompactness of the set of trajectories of the control systems described by a nonlinear Volterra integral equation with integral constraint on the control functions is considered in [14]
Summary
Control systems arise in various fields of theory and applications. Depending on the type of equation which describes the system’s behavior, control systems can be classified as linear or nonlinear control systems, control systems described by ordinary differential equations or control systems described by partial differential equations or control systems described by integral equations and etc. Precompactness of the set of trajectories of the control systems described by a nonlinear Volterra integral equation with integral constraint on the control functions is considered in [14]. In [13] an approximation of the sections of the set of trajectories of the control systems described by a nonlinear Volterra integral equation with integral constraint on the controls is investigated, where the sections of the set of trajectories is approximated by the sections of the set of trajectories generated by the compact set of control functions. The Hausdorff distance between the set of trajectories generated by the mixed constrained and piecewise constant control functions the norm of which are the node points of the given uniform mesh, and the set of trajectories consisting of a finite number of trajectories is evaluated (Proposition 4).
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