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.     

Optimality conditions for reflecting boundary control problems

We consider a control problem with reflecting boundary and obtain necessary optimality conditions in the form of the maximum Pontryagin principle. To derive these results we transform the constrained problem in an unconstrained one or we use penalization techniques of Morreau-Yosida type to approach the original problem by a sequence of optimal control problems with Lipschitz dynamics. Then nonsmooth analysis theory is used to study the convergence of the penalization in order to obtain optimality conditions.     

Classical characteristics of the bellman equation in constructions of grid optimal synthesis

We consider optimal control problems with fixed final time and terminal-integral cost functional, and address the question of constructing a grid optimal synthesis (a universal feedback) on the basis of classical characteristics of the Bellman equation. To construct an optimal synthesis, we propose a numerical algorithm that relies on the necessary optimality conditions (the Pontryagin maximum principle) and sufficient conditions in the Hamiltonian form. We obtain estimates for the efficiency of the numerical method. The method is illustrated by an example of the numerical solution of a nonlinear optimal control problem.     

Necessary first-order conditions for optimal crossing of a given region

We consider a nonlinear optimal control problem with an integral functional in which the integrand is the characteristic function of a closed set in the phase space. An approximation method is applied to prove the necessary conditions of optimality in the form of a Pontryagin maximum principle without any prior assumptions on the behavior of the optimal trajectory. Similarly to phase-constrained problems, we derive conditions of nondegeneracy and pointwise nontriviality of the maximum principle. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 179–204, 2004.     

Necessary optimality conditions for a class of optimal control problems with discontinuous integrand

We consider a nonlinear optimal control problem with an integral functional in which the integrand contains the characteristic function of a given closed subset of the phase space. Using an approximation method, we prove necessary optimality conditions in the form of the Pontryagin maximum principle without any a priori assumptions about the behavior of an optimal trajectory.     

Quadratic convergence algorithms for the problem of capture of a point by a family of convex bodies

We consider the nonlinear optimal control problem with an integral functional in which the integrand function is the characteristic function of a given closed set in the phase space. The approximation method is applied to prove the necessary conditions of optimality in the form of the Pontryagin maximum principle without any prior assumptions on the behavior of the optimal trajectory. Similarly to the case of phase-constrained problems, we derive conditions of nondegeneracy and pointwise nontriviality of the maximum principle. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 241–256, 2004.     

Investigation of regularity conditions in optimal control problems with geometric mixed constraints

We investigate regularity conditions in optimal control problems with mixed constraints of a general geometric type, in which a closed non-convex constraint set appears. A closely related question to this issue concerns the derivation of necessary optimality conditions under some regularity conditions on the constraints. By imposing strong and weak regularity condition on the constraints, we provide necessary optimality conditions in the form of Pontryagin maximum principle for the control problem with mixed constraints. The optimality conditions obtained here turn out to be more general than earlier results even in the case when the constraint set is convex. The proofs of our main results are based on a series of technical lemmas which are gathered in the Appendix.     

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.     

Maximum principle for optimal control problems with intermediate constraints

We consider optimal control problems with constraints at intermediate points of the trajectory. A natural technique (propagation of phase and control variables) is applied to reduce these problems to a standard optimal control problem of Pontryagin type with equality and inequality constraints at the trajectory endpoints. In this way we derive necessary optimality conditions that generalize the Pontryagin classical maximum principle. The same technique is applied to so-called variable structure problems and to some hybrid problems. The new optimality conditions are compared with the results of other authors and five examples illustrating their application are presented.     

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.     

Necessary conditions of the minimum in an impulse optimal control problem

The paper is devoted to studying the impulse optimal control problem with inequality-type state constraints and geometric control constraints defined by a measurable multivalued mapping. The author obtains necessary optimality conditions in the form of the Pontryagin maximum principle and nondegeneracy conditions for the latter. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.     

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.     

Optimality Conditions and the Hamiltonian for a Distributed Optimal Control Problem on Controlled Domain

The paper investigates an optimal control problem for a distributed system arising in the economics of endogenous growth. The problem involves a specific coupled family of controlled ODEs parameterized by a parameter (representing the heterogeneity) running over a domain that may dynamically depend on the control and on the state of the system. Existence of an optimal control is obtained and continuity of any optimal control with respect to the parameter of heterogeneity is proved. The latter allows to substantially strengthen previously obtained necessary optimality conditions and to obtain a Pontryagin’s type maximum principle. The necessary optimality conditions obtained here have a Hamiltonian representation, and stationarity of the Hamiltonian along any optimal trajectory is proved in the case of time-independent data.     

On optimality conditions in optimization problems on solutions of operator equations

We study optimization problems in the presence of connection in the form of operator equations defined in Banach spaces. We prove necessary conditions for optimality of first and second order, for example generalizing the Pontryagin maximal principle for these problems. It is not our purpose to state the most general necessary optimality conditions or to compile a list of all necessary conditions that characterize optimal control in any particular minimization problem. In the present article we describe schemes for obtaining necessary conditions for optimality on solutions of general operator equations defined in Banach spaces, and the scheme discussed here does not require that there be no global functional constraints on the controlling parameters. As an example, in a particular Banach space we prove an optimality condition using the Pontryagin-McShane variation. Bibliography: 20 titles. Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 55–67.     

条件平均场随机微分方程的最优控制问题

作者研究了一个条件平均场随机微分方程的最优控制问题.这种方程和某些部分信息下的随机最优控制问题有关,并且可以看做是平均场随机微分方程的推广.作者以庞特里雅金最大值原理的形式给出最优控制满足的必要和充分条件.此外,文中给出一个线性二次最优控制问题来说明理论结果的应用.     

Closed Euler elasticae

Euler’s classical problem on stationary configurations of an elastic rod in a plane is studied as an optimal control problem on the group of motions of a plane. We show complete integrability of the Hamiltonian system of the Pontryagin maximum principle. We prove that a closed elastica is either a circle or a figure-of-eight elastica, wrapped around itself several times. Finally, we study local and global optimality of closed elasticae: the figure-of-eight elastica is optimal only locally, while the circle is optimal globally.     

Pontryagin’s Maximum Principle for Multidimensional Control Problems in Image Processing

In the present paper, we prove a substantially improved version of the Pontryagin maximum principle for convex multidimensional control problems of Dieudonné-Rashevsky type. Although the range of the operator describing the first-order PDE system involved in this problem has infinite codimension, we obtain first-order necessary conditions in a completely analogous form as in the one-dimensional case. Furthermore, the adjoint variables are subjected to a Weyl decomposition. We reformulate two basic problems of mathematical image processing (determination of optical flow and shape from shading problem) within the framework of optimal control, which gives the possibility to incorporate hard constraints in the problems. In the convex case, we state the necessary optimality conditions for these problems.     

Optimal operator control of systems in a Banach space

We consider control systems in an abstract Banach space with control in the form of an operator function. For such systems, we derive necessary optimality conditions in the form of the Pontryagin maximum principle.     

Positional strengthenings of the maximum principle and sufficient optimality conditions

We derive nonlocal necessary optimality conditions, which efficiently strengthen the classical Pontryagin maximum principle and its modification obtained by B. Ka?kosz and S. ?ojasiewicz as well as our previous result of a similar kind named the “feedback minimum principle.” The strengthening of the feedback minimum principle (and, hence, of the Pontryagin principle) is owing to the employment of two types of feedback controls “compatible” with a reference trajectory (i.e., producing this trajectory as a Carath´eodory solution). In each of the versions, the strengthened feedback minimum principle states that the optimality of a reference process implies the optimality of its trajectory in a certain family of variational problems generated by cotrajectories of the original and compatible controls. The basic construction of the feedback minimum principle—a perturbation of a solution to the adjoint system—is employed to prove an exact formula for the increment of the cost functional. We use this formula to obtain sufficient conditions for the strong and global minimum of Pontryagin’s extremals. These conditions are much milder than their known analogs, which require the convexity in the state variable of the functional and of the lower Hamiltonian. Our study is focused on a nonlinear smooth Mayer problem with free terminal states. All assertions are illustrated by examples.