scholarly journals Regularized classical optimality conditions in iterative form for convex optimization problems for distributed Volterra-type systems

2021 ◽  
Vol 31 (2) ◽  
pp. 265-284
Author(s):  
V.I. Sumin ◽  
M.I. Sumin
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimality Conditions ◽  
Pontryagin Maximum Principle ◽  
Optimization Problems ◽  
Approximate Solutions ◽  
Lagrange Principle ◽  
The Pontryagin Maximum Principle

We consider the regularization of the classical optimality conditions (COCs) — the Lagrange principle and the Pontryagin maximum principle — in a convex optimal control problem with functional constraints of equality and inequality type. The system to be controlled is given by a general linear functional-operator equation of the second kind in the space $L^m_2$, the main operator of the right-hand side of the equation is assumed to be quasinilpotent. The objective functional of the problem is strongly convex. Obtaining regularized COCs in iterative form is based on the use of the iterative dual regularization method. The main purpose of the regularized Lagrange principle and the Pontryagin maximum principle obtained in the work in iterative form is stable generation of minimizing approximate solutions in the sense of J. Warga. Regularized COCs in iterative form are formulated as existence theorems in the original problem of minimizing approximate solutions. They “overcome” the ill-posedness properties of the COCs and are regularizing algorithms for solving optimization problems. As an illustrative example, we consider an optimal control problem associated with a hyperbolic system of first-order differential equations.

WHY REGULARIZATION OF LAGRANGE PRINCIPLE AND PONTRYAGIN MAXIMUM PRINCIPLE IS NEEDED AND WHAT IT GIVES

2018 ◽  
pp. 757-775
Author(s):  
Mikhail Iosifovich Sumin
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Mathematical Programming ◽  
Optimality Conditions ◽  
Pontryagin Maximum Principle ◽  
Lagrange Principle ◽  
Convex Problems ◽  
The Pontryagin Maximum Principle

We consider the regularization of the classical Lagrange principle and the Pontryagin maximum principle in convex problems of mathematical programming and optimal control. On example of the “simplest” problems of constrained infinitedimensional optimization, two main questions are discussed: why is regularization of the classical optimality conditions necessary and what does it give?

scholarly journals Optimization of cereal output in presence of locusts

2016 ◽  
Vol 6 (1) ◽  
pp. 53-61 ◽  
Author(s):  
Nacima Moussouni ◽  
Mohamed Aidene
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Pontryagin Maximum Principle ◽  
Shooting Method ◽  
The Pontryagin Maximum Principle

In this paper, we study a modelization of the evolution of cereal output production, controlled by adding fertilizers and in presence of locusts, then by adding insecticides. The aim is to maximize the cereal output and meanwhile minimize pollution caused by adding fertilizers and insecticides.The optimal control problem obtained is solved theoretically by using the Pontryagin Maximum Principle, and then numerically with shooting method.

NECESSARY OPTIMALITY CONDITIONS FOR OPTIMAL CONTROL PROBLEM WITH SECOND ORDER EQUATION

2018 ◽  
pp. 437-440
Author(s):  
Alexandr Mikhailovich Kotyukov ◽  
Mikhail Mikhailovich Kotyukov
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimality Conditions ◽  
Pontryagin Maximum Principle ◽  
Necessary Optimality Conditions ◽  
Order Equation ◽  
Second Order

An optimal control problem with second order equation is considered. For this problem Pontryagin maximum principle is obtained. Also non-triviality condition is proved.

scholarly journals Maximum Principle for Stochastic Recursive Optimal Control Problems Involving Impulse Controls

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Zhen Wu ◽  
Feng Zhang
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimality Conditions ◽  
Control Variable ◽  
Necessary Optimality Conditions ◽  
Sufficient Optimality Conditions ◽  
Linear Quadratic ◽  
Regular Control

We consider a stochastic recursive optimal control problem in which the control variable has two components: the regular control and the impulse control. The control variable does not enter the diffusion coefficient, and the domain of the regular controls is not necessarily convex. We establish necessary optimality conditions, of the Pontryagin maximum principle type, for this stochastic optimal control problem. Sufficient optimality conditions are also given. The optimal control is obtained for an example of linear quadratic optimization problem to illustrate the applications of the theoretical results.

On the regularization of classical optimality conditions in a convex optimal control problem with state constraints

2020 ◽  
pp. 263-273
Author(s):  
Fedor A. Kuterin
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimality Conditions ◽  
State Constraints ◽  
Lagrange Principle ◽  
Strongly Convex ◽  
Pointwise State Constraints ◽  
Integrable Functions ◽  
Square Integrable Functions

We consider the regularization of classical optimality conditions in a convex optimal control problem for a linear system of ordinary differential equations with pointwise state constraints such as equality and inequality, understood as constraints in the Hilbert space of square-integrable functions. The set of admissible task controls is traditionally embedded in the space of square-integrable functions. However, the target functional of the optimization problem is not, generally speaking, strongly convex. Obtaining regularized classical optimality conditions is based on a technique involving the use of two regularization parameters. One of them is used for the regularization of the dual problem, while the other is contained in a strongly convex regularizing addition to the target functional of the original problem. The main purpose of the obtained regularized Lagrange principle and the Pontryagin maximum principle is the stable generation of minimizing approximate solutions in the sense of J. Varga for the purpose of practical solving the considered optimal control problem with pointwise state constraints.

scholarly journals Optimal Control Problem Solution with Phase Constraints for Group of Robots by Pontryagin Maximum Principle and Evolutionary Algorithm

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2105
Author(s):  
Askhat Diveev ◽  
Elena Sofronova ◽  
Ivan Zelinka
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Evolutionary Algorithm ◽  
Pontryagin Maximum Principle ◽  
Initial Conditions ◽  
Problem Solution ◽  
Dynamic Phase ◽  
Phase Constraints

A numerical method based on the Pontryagin maximum principle for solving an optimal control problem with static and dynamic phase constraints for a group of objects is considered. Dynamic phase constraints are introduced to avoid collisions between objects. Phase constraints are included in the functional in the form of smooth penalty functions. Additional parameters for special control modes and the terminal time of the control process were introduced. The search for additional parameters and the initial conditions for the conjugate variables was performed by the modified self-organizing migrating algorithm. An example of using this approach to solve the optimal control problem for the oncoming movement of two mobile robots is given. Simulation and comparison with direct approach showed that the problem is multimodal, and it approves application of the evolutionary algorithm for its solution.

scholarly journals REQUIRED OPTIMALITY CONDITIONS IN ONE OPTIMAL CONTROL PROBLEM WITH MULTIPOINT FUNCTIONAL

2020 ◽  
pp. 7-26
Author(s):  
Shahla Rasulzade ◽  
◽  
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Differential Equations ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimality Conditions ◽  
Optimal Control Problems ◽  
Control Problems ◽  
The Maximum Principle ◽  
Distributed Parameters

One specific optimal control problem with distributed parameters of the Moskalenko type with a multipoint quality functional is considered. To date, the theory of necessary first-order optimality conditions such as the Pontryagin maximum principle or its consequences has been sufficiently developed for various optimal control problems described by ordinary differential equations, i.e. for optimal control problems with lumped parameters. Many controlled processes are described by various partial differential equations (processes with distributed parameters). Some features are inherent in optimal control problems with distributed parameters, and therefore, when studying the optimal control problem with distributed parameters, in particular, when deriving various necessary optimality conditions, non-trivial difficulties arise. In particular, in the study of cases of degeneracy of the established necessary optimality conditions, fundamental difficulties arise. In the present work, we study one optimal control problem described by a system of first-order partial differential equations with a controlled initial condition under the assumption that the initial function is a solution to the Cauchy problem for ordinary differential equations. The objective function (quality criterion) is multi-point. Therefore, it becomes necessary to introduce an unconventional conjugate equation, not in differential (classical), but in integral form. In the work, using one version of the increment method, using the explicit linearization method of the original system, the necessary optimality condition is proved in the form of an analog of the maximum principle of L.S. Pontryagin. It is known that the maximum principle of L.S. Pontryagin for various optimal control problems is the strongest necessary condition for optimality. But the principle of a maximum of L.S. Pontryagin, being a necessary condition of the first order, often degenerates. Such cases are called special, and the corresponding management, special management. Based on these considerations, in the considered problem, we study the case of degeneration of the maximum principle of L.S. Pontryagin for the problem under consideration. For this purpose, a formula for incrementing the quality functional of the second order is constructed. By introducing auxiliary matrix functions, it was possible to obtain a second-order increment formula that is constructive in nature. The necessary optimality condition for special controls in the sense of the maximum principle of L.S. Pontryagin is proved. The proved necessary optimality conditions are explicit.

scholarly journals Optimal Campaign in the Smoking Dynamics

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Gul Zaman
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Control Problem ◽  
Pontryagin Maximum Principle ◽  
Numerical Results ◽  
Optimality System ◽  
Control Variables ◽  
Light Smoker ◽  
The Pontryagin Maximum Principle

We present the optimal campaigns in the smoking dynamics. Assuming that the giving up smoking model is described by the simplified (potential-light-smoker-quit smoker) model, we consider two possible control variables in the form of education and treatment campaigns oriented to decrease the attitude towards smoking. In order to do this we minimize the number of light (occasional) and persistent smokers and maximize the number of quit smokers in a community. We first show the existence of an optimal control for the control problem and then derive the optimality system by using the Pontryagin maximum principle. Finally numerical results of real epidemic are presented to show the applicability and efficiency of this approach.

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