Applicability of the mesh method and the method of straight lines for solving a class of optimal control problems

In the paper, the method of straight lines approximately solves one class of optimal control problems for systems, the behavior of which is described by a nonlinear equation of parabolic type and a set of ordinary differential equations. Control is carried out using distributed and lumped parameters. Distributed control is included in the partial differential equation, and lumped controls are contained both in the boundary conditions and in the right-hand side of the ordinary differential equation. The convergence of the solutions of the approximating boundary value problem to the solution of the original one is proved when the step of the grid of straight lines tends to zero, and on the basis of this fact, the convergence of the approximate solution of the approximating optimal problem with respect to the functional is established. A constructive scheme for constructing an optimal control by a minimizing sequence of controls is proposed. The control of the process in the approximate solution of a class of optimization problems is carried out on the basis of the Pontryagin maximum principle using the method of straight lines. For the numerical solution of the problem, a gradient projection scheme with a special choice of step is used, this gives a converging sequence in the control space. The numerical solution of one variational problem of the mentioned type related to a one-dimensional heat conduction equation with boundary conditions of the second kind is presented. An inequality-type constraint is imposed on the control function entering the right-hand side of the ordinary differential equation. The numerical results obtained on the basis of the compiled computer program are presented in the form of tables and figures. The described numerical method gives a sufficiently accurate solution in a short time and does not show a tendency to «dispersion». With an increase in the number of iterations, the value of the functional monotonically tends to zero

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Solving the problem of optimal control of fog diffusion

Hydrometeorological research and forecasting ◽

10.37162/2618-9631-2021-2-52-65 ◽

2021 ◽

Vol 2 ◽

pp. 52-65

Author(s):

V.V. / Klyomin ◽

◽

S.S. Suvorov ◽

Keyword(s):

Optimal Control ◽

Diffusion Equation ◽

Optimization Methods ◽

Diffusion Control ◽

Control Problems ◽

Existence Of A Solution ◽

Method Of Straight Lines ◽

Straight Lines ◽

Modern Optimization ◽

Control Diffusion

Solving the problem of optimal control of fog diffusion / Klyomin V.V., Suvorov S.S. // Hydrometeorological Research and Forecasting, 2021, no. 2 (380), pp. 52-65. The paper discusses a possibility of applying one of the fundamental modern optimization methods, namely, the Pontryagin’s method for solving process control problems, whose behavior is described by the diffusion equation. The parabolic diffusion equation is discretized by the method of straight lines and comes to a closed system of ordinary differential equations, which allow finding an optimal control impact in terms of operating speed. The existence of a solution to the problem of optimal control of fog diffusion is proved for the mentioned sampling. The methodology for finding control action switching points is substantiated, the calculations for the revealed two and three switching moments are performed. Keywords: Pontryagin’s method, fog diffusion control, diffusion equation

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On closed-loop equilibrium strategies for mean-field stochastic linear quadratic problems

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2019057 ◽

2020 ◽

Vol 26 ◽

pp. 41

Author(s):

Tianxiao Wang

Keyword(s):

Optimal Control ◽

Closed Loop ◽

Optimal Control Problems ◽

Mean Field ◽

Open Loop ◽

Time Inconsistency ◽

Control Problems ◽

Linear Quadratic ◽

Linear Quadratic Optimal Control ◽

Equilibrium Strategies

This article is concerned with linear quadratic optimal control problems of mean-field stochastic differential equations (MF-SDE) with deterministic coefficients. To treat the time inconsistency of the optimal control problems, linear closed-loop equilibrium strategies are introduced and characterized by variational approach. Our developed methodology drops the delicate convergence procedures in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. When the MF-SDE reduces to SDE, our Riccati system coincides with the analogue in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. However, these two systems are in general different from each other due to the conditional mean-field terms in the MF-SDE. Eventually, the comparisons with pre-committed optimal strategies, open-loop equilibrium strategies are given in details.

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