Path-Dependent Hamilton–Jacobi Equations: The Minimax Solutions Revised

Motivated by optimal control problems and differential games for functional differential equations of retarded type, the paper deals with a Cauchy problem for a path-dependent Hamilton–Jacobi equation with a right-end boundary condition. Minimax solutions of this problem are studied. The existence and uniqueness result is obtained under assumptions that are weaker than those considered earlier. In contrast to previous works, on the one hand, we do not require any properties concerning positive homogeneity of the Hamiltonian in the impulse variable, and on the other hand, we suppose that the Hamiltonian satisfies a Lipshitz continuity condition with respect to the path variable in the uniform (supremum) norm. The progress is related to the fact that a suitable Lyapunov–Krasovskii functional is built that allows to prove a comparison principle. This functional is in some sense equivalent to the square of the uniform norm of the path variable and, at the same time, it possesses appropriate smoothness properties. In addition, the paper provides non-local and infinitesimal criteria of minimax solutions, their stability with respect to perturbations of the Hamiltonian and the boundary functional, as well as consistency of the approach with the non-path-dependent case. Connection of the problem statement under consideration with some other possible statements (regarding the choice of path spaces and derivatives used) known in the theory of path-dependent Hamilton–Jacobi equations is discussed. Some remarks concerning viscosity solutions of the studied Cauchy problem are given. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature

On the solution of a control problem for the motion of an object in the dense layers of the atmosphere

The paper deals with the construction of algorithms for solving an optimal control problem for a nonlinear dynamic system in the presence of phase constraints. The system under consideration describes the motion of a controlled object as a rigid body in the dense layers of the atmosphere under the gravitational and aerodynamic forces. The desired control must minimize a terminal performance index under a number of constraints on the control and the phase state of the dynamic system. The performance index characterizes the accuracy of bringing the center of mass of the object to a given set with a required direction of its velocity. The control is carried out by changing the spatial orientation of movable control elements of the object structure. A time-iterative procedure is proposed for the construction of admissible controls. The procedure is based on the sequential use of the aerodynamic force acting on the controlling elements, which provides the desired direction of the velocity vector of the center of mass under all the constraints. To determine the required moment, it is proposed to use a relation that connects it with the moment of the aerodynamic force acting on the remaining surface of the object with the desired direction of the velocity vector. For this moment, the values of the control parameters that implement it are calculated. The efficiency of the proposed algorithm for constructing admissible controls is illustrated by a model example of an applied optimal control problem. In this problem, the dynamic system describes the motion of a stage of a launch vehicle (recoverable block) in the atmospheric section of its trajectory, where the block moves to a specified landing area. The results of numerical simulation are presented. © 2021 Krasovskii Institute of Mathematics and Mechanics. All rights reserved

Methodology and design synthesis of source of the magnetic fields with complicated energy-frequency and polarization structure

The method of synthesis of the rotating magnetic fields, created by combination of three mutually perpendicular solenoidal contours power supplied by an alternating current, is discussed. Possibility of construction of the magnetoforming system with changing direction of the magnetic field rotation, allowing in practical realization to take into account the chiral dissymmetry of molecules of matters, subject to treatment by the magnetic field, is shown