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Том 239, № 2 (2019)
- Год: 2019
- Статей: 9
- URL: https://bakhtiniada.ru/1072-3374/issue/view/15005
Article
Stability of an Interval Family of Differential-Algebraic Equations with Variable Coefficients
Аннотация
We consider a linear nonstationary system of ordinary differential equations with interval coefficients which is not solvable with respect to the derivative of the unknown vector-valued function for any matrix coefficients in a given interval family. We obtain conditions sufficient for the preservation of the internal structure of the system. Under conditions guaranteeing the structure preservation, we obtain sufficient and necessary robust stability conditions. A variable rank of matrix coefficients and an arbitrary high unsolvability index are allowed.
214-231
Algorithms for Nonlinear Optimal Control Problems Based on the First and Second Order Necessary Conditions
Аннотация
We develop an algorithmic support for some classes of nonlinear optimal control problems. We propose several algorithms based on the first- and second order necessary conditions for solving optimal control problems with constraints on control. Efficiency of the algorithms is confirmed by solving various test and applied optimal control problems.
185-196
Analytical and Numerical Construction of Heat Wave Type Solutions to the Nonlinear Heat Equation with a Source
Аннотация
For a nonlinear parabolic heat equation we construct a heat wave type solution composed of the zero and nonnegative solutions joined continuously along the wave front. We prove the existence and uniqueness of an analytic solution to the problem with a given wave front in the cases of plane, circular, and spherical symmetry. The solution is constructed in the form of a characteristic series with recurrently defined coefficients. In the case of a power source, we show that the original problem can be reduced to the Cauchy problem for a second order ordinary differential equation and the solution is invariant. We present numerical results verified by using the constructed analytic solutions.
111-122
Analytical Synthesis of Aggregated Regulators for Unmanned Aerial Vehicles
Аннотация
We study analytical synthesis of control systems for nonlinear objects of high dimension. We propose to apply a synergetic approach owing to which it is not necessary to linearize the model and it is possible to synthesize regulators of high dimension by successively solving problems of small dimension. For an example of a nonlinear multidimensional control object we consider an unmanned aerial vehicle of helicopter type.
135-145
Stability of Difference Scheme for a Semilinear Differential Algebraic System of Index (k, 0)
Аннотация
We consider a semilinear differential-algebraic system of partial differential equations of index (k, 0). We numerically solve this system by applying the spline-collocation method based on splitting the matrix pencil. The method has high accuracy coinciding with the smallest order of the approximating spline.
172-184
Robust Controllability of Nonstationary Differential-Algebraic Equations with Unstructured Uncertainty
Аннотация
We consider a nonstationary system of differential-algebraic equations, i.e., the first order ordinary differential equations with variable coefficients and identically singular matrix at the derivative of the unknown vector-valued function. We construct the structural form for the perturbed system and obtain sufficient conditions for the robust (complete, differential, R-) controllability of such systems of index 1 and 2.
123-134
Bang-Bang Theorem for a Coupled ODE-PDE Control System
Аннотация
An optimal control problem for a system of one dimensional hyperbolic conservation laws is considered. The system is controlled through the boundary condition, which depends on the state of a nonlinear control system. It is proven that the problem has an optimal solution; moreover, it admits ε-optimal solutions generated by bang-bang controls. Bibliography: 8 titles. Illustrations: 1 figure.
146-158
Generalization of the Power Sum Arising in the Theory of Integrable Hierarchies
Аннотация
We consider a class of multiple sums involving odd powers of natural numbers. Such sums appear while considering the continuous limit of the integrable hierarchy of evolution equations associated with the Itoh–Narita–Bogoyavlenskii lattice. We discuss the problem of constructing polynomials that allow us to calculate the values of the corresponding sums.
159-171
Tauberian Theorem for Games with Unbounded Running Cost
Аннотация
We study two game families with total payoffs that are defined either as the Cesàro average (the long run average game family) or Abel average (the discounting game family) of the running costs. We study value functions for all sufficiently small discounts and for all sufficiently large planning horizons (asymptotic approach), investigate a robust strategy that provides a near-optimal total payoff in this case (uniform approach). Assuming the Dynamic Programming Principle, we prove the corresponding Tauberian theorems without requiring the boundedness of the running cost.
197-213