On the Solvability of a Nonlinear Tracking Problem Under Boundary Control for the Elastic Oscillations Described by Fredholm Integro-Differential Equations

Abstract

In the present paper we investigate nonlinear tracking problem under boundary control for the oscillation processes described by Fredholm integrodifferential equations. When we investigate this problem we use notion of a weak generalized solution of the boundary value problem. Based on the maximum principle for distributed systems we obtain optimality conditions from which follow the nonlinear integral equation of optimal control and the differential inequality. We have developed an algorithm to construct the optimization problem solution. This solving method of a nonlinear tracking problem is constructive and can be used in applications.

Research of Multiperiodic Solutions of Perturbed Linear Autonomous Systems With Differentiation Operator on the Vector Field

“…The foundations of the method used in this note were laid in [1,2], which were further developed in [3][4][5][6][7][8][9][10] and applied to the study of solutions different problems in the partial differential equations [11,12]. These methods with simple modifications extend to the study solutions of problems of the differential and integro-differential equations of different types [1][2][3][4][5][6][7][8][9][10][11][12], in particular, problems on multi-frequency solutions of equations from control theory [13]. The methods of research for multiperiodic solutions are successfully combined by methods for studying solutions of boundary value problems for equations of mathematical physics.…”

Research of Multiperiodic Solutions of Perturbed Linear Autonomous Systems With Differentiation Operator on the Vector Field

“…The foundations of the method used in this note were laid in [1,2], which were further developed in [3][4][5][6][7][8][9][10][11][12][13][14] and applied to the study of solutions different problems in the partial differential equations [15,16]. These methods with simple modifications extend to the study solutions of problems of the differential and integro-differential equations of different types [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], in particular, problems on multi-frequency solutions of equations from control theory [17]. Many oscillatory phenomena are described by systems with a differentiation operator with respect to toroidal vector fields, and new methods based on the ideas of the Fourier [18], Poincaré-Lyapunov and Hamilton-Jacobi methods [19,20] appear to establish their periodic oscillatory solutions.…”