A Solution to Fuller’s Problem Using Constructions of Pontryagin’s Maximum Principle

The classical two-dimensional Fuller problem is considered. The boundary value problem of Pontryagin’s maximum principle is considered. Based on the central symmetry of solutions to the boundary value problem, the Pontryagin maximum principle as a necessary condition of optimality, and the hypothesis of the form of the switching line, a solution to the boundary value problem is constructed and its optimality is substantiated. Invariant group analysis is in this case not used. The results are of considerable methodological interest.     

Study of a special model of resource allocation over an infinite interval of time

A special model of resource allocation over an infinite interval of time is studied. Using the Pontryagin maximum principle, an extreme solution is constructed whose optimality is proven with the help of a theorem on sufficient conditions, in the form of constructions of Pontryagin??s maximum principle. A concrete example in which the classical maximum principle is inapplicable is considered.     

Resource allocation problem in a two-sector economic model of special form

In the present paper, we study the resource allocation problem for a two-sector economic model of special form, which is of interest in applications. The optimization problem is considered on a given finite time interval. We show that, under certain conditions on the model parameters, the optimal solution contains a singular mode. We construct optimal solutions in closed form. The theoretical basis for the obtained results is provided by necessary optimality conditions (the Pontryagin maximum principle) and sufficient optimality conditions in terms of constructions of the Pontryagin maximum principle.     

Averaging of boundary-value problems of control over a standard system with delay

For the terminal problem of optimal control over systems of standard form with constant delay, according to the Pontryagin maximum principle, we study a boundary-value problem with deviating arguments with delay and anticipation. We justify an averaging method for an asymptotic solution of the boundary-value problem obtained.     

Optimal Control for an Obstruction Problem

A question of flow around an obstacle leads to an optimal control problem. If an optimum path exists, then it is calculable from the Pontryagin principle. The optimum is verified to be reached, using a discretization of the problem.     

Optimal Contract Design of Supplier-Led Outsourcing Based on Pontryagin Maximum Principle

This paper provides a supplier-led outsourcing model to maximize the supplier’s profits based on a principal-agent framework with both asymmetric cost information and uncertain market demand information described by continuous random variables. The salvage value of the unsold product is processing-cost dependent. By converting the proposed model, which is a dynamic optimization problem, to an optimal control problem, we obtain the analytical form of the optimal supplier outsourcing contract composed of the wholesale price and the transfer payment by applying Pontryagin’s maximum principle. It is shown that the optimal contract is directly related to the supplier’s beliefs about the manufacturer’s unit cost and the salvage value function. The Pontryagin’s maximum principle-based solution method serves as a powerful tool to support the decision making for the best sourcing strategy, and it provides analytical insights for outsourcing management. Finally, numerical examples are presented to illustrate the validness of the theoretical results.     

Analysis of a gas field development model with an infinite planning horizon

We consider a nonlinear optimal control problem with an infinite planning horizon, which extends a dynamic gas field development model by taking into account a gas price forecast. (The prices varies in time.) The solution is constructed on the basis of the Pontryagin maximum principle. To prove the optimality of the extremal solution, we use the theorem on sufficient optimality conditions in terms of constructions of the Pontryaginmaximum principle. We discuss the problem of constructing an optimal solution by dynamic programming.     

A note on the classical brachistochrone

It is shown that the classical Bernoulli's brachistochrone problem and the brachistochrone problem in a central force field may be solved by the maximum principle of Pontryagin. According to the optimum control theory these problems are singular.     

Non-degenerate necessary optimality conditions for the optimal control problem with equality-type state constraints

In this work, an optimal control problem with state constraints of equality type is considered. Novelty of the problem formulation is justified. Under various regularity assumptions imposed on the optimal trajectory, a non-degenerate Pontryagin Maximum Principle is proven. As a consequence of the maximum principle, the Euler–Lagrange and Legendre conditions for a variational problem with equality and inequality state constraints are obtained. As an application, the equation of the geodesic curve for a complex domain is derived. In control theory, the Maximum Principle suggests the global maximum condition, also known as the Weierstrass–Pontryagin maximum condition, due to which the optimal control function, at each instant of time, turns out to be a solution to a global finite-dimensional optimization problem.     

Optimal control laws for the model of information diffusion in a social group

The article investigates two models of information diffusion in a social group. The dynamics of the process is described by a one-dimensional controlled Riccati differential equation. Our two models differ from the original model of K. V. Izmodenova and A. P. Mikhailov in the choice of the functional being optimized. Two different choices of the optimand functional are considered. The optimal control problems are solved by the Pontryagin maximum principle. It is shown that the optimal control program is a relay function of time with at most one switching point. Conditions on the problem parameters are proposed that are easy to check and guarantee the existence of an optimal-control switching point. The theoretical analysis leads to a one-dimensional convex minimization problem to find the optimal-control switching point. The article also describes an alternative approach to the construction of the optimal solution, which does not resort to the maximum principle and instead utilizes a special representation of the optimand functional and works with reachability sets that are independent of the functional. For the two models considered in this article optimal feedback controls are derived from the programmed optimal controls.     

Optimal control in a macroeconomic problem

The Pontryagin maximum principle is used to develop an original algorithm for finding an optimal control in a macroeconomic problem. Numerical results are presented for the optimal control and optimal trajectory of the development of a regional economic system. For an optimal control satisfying a certain constraint, an invariant of a macroeconomic system is derived.     

Maximum Principle for a Stochastic Optimal Control Problem and Application to Portfolio/Consumption Choice

We consider mainly an optimal control problem motivated by a portfolio and consumption choice problem in a financial market where the utility of the investor is assumed to have a given homogeneous form. A Pontryagin local maximum principle is obtained by using classical variational methods. We apply the result to make optimal portfolio and consumption decisions for the problem under consideration. The optimal selection coincides with the one obtained in Refs. 1 and 2, where the Bellman dynamic programming principle was used.     

Optimal processes in the model of two-sector economy with an integral utility function

An infinite-horizon two-sector economy model with a Cobb–Douglas production function is studied for different depreciation rates, the utility function being an integral functional with discounting and a logarithmic integrand. The application of the Pontryagin maximum principle leads to a boundary value problem with special conditions at infinity. The presence of singular modes in the optimal solution complicates the search for a solution to the boundary value problem of the maximum principle. To construct the solution to the boundary value problem, the singular modes are written in an analytical form; in addition, a special version of the sweep algorithm in continuous form is proposed. The optimality of the extremal solution is proved.     

Bio-economics of a renewable resource subjected to strong Allee effect

In this article bio-economics of a renewable resource that is subjected to strong Allee effect (multiplicative Allee effect) is investigated from sole owner perspective. The considered optimal harvesting problem has been solved using Pontryagin maximum principle. The control problem admits multiple singular equilibrium solutions in contrast to the case where the growth of the resource is of compensatory nature. Thus the choice of optimal singular solution and the nature of associated approach paths make the problem pertinent and interesting.     

A note on the classical brachistochrone

Summary It is shown that the classical Bernoulli's brachistochrone problem and the brachistochrone problem in a central force field may be solved by the maximum principle of Pontryagin. According to the optimum control theory these problems are singular.

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Zusammenfassung In der Arbeit ist gezeigt, daß das klassische Bernoullische brachistochrone Problem und Brachistochronen im zentralen Schwerefeld durch das Pontryagin Maximum Prinzip gelöst werden können. Gemäss Theorie der optimalen Regelung sind diese Probleme singulär.

     

Optimal control and optimal trajectories of regional macroeconomic dynamics based on the Pontryagin maximum principle

The Pontryagin maximum principle is used to prove a theorem concerning optimal control in regional macroeconomics. A boundary value problem for optimal trajectories of the state and adjoint variables is formulated, and optimal curves are analyzed. An algorithm is proposed for solving the boundary value problem of optimal control. The performance of the algorithm is demonstrated by computing an optimal control and the corresponding optimal trajectories.     

Optimal control in a mathematical model for leukemia therapy with phase constraints

We examine the optimal control problem that arises in the mathematical modeling of leukemia therapy, to solve which the Pontryagin maximum principle and the penalty function method are employed. It is assumed that the drug is capable of killing not only diseased cells, but healthy cells as well. The character of the drug??s interaction with cells is described by appropriate therapy functions.     

Pontryagin maximum principle for fractional ordinary optimal control problems

In the paper, fractional systems with Riemann–Liouville derivatives are studied. A theorem on the existence and uniqueness of a solution of a fractional ordinary Cauchy problem is given. Next, the Pontryagin maximum principle for nonlinear fractional control systems with a nonlinear integral performance index is proved. Copyright © 2013 John Wiley & Sons, Ltd.