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.     

Properties of Hamiltonian systems in the Pontryagin maximum principle for economic growth problems

We consider an optimal control problem with a functional defined by an improper integral. We study the concavity properties of the maximized Hamiltonian and analyze the Hamiltonian systems in the Pontryagin maximum principle. On the basis of this analysis, we propose an algorithm for constructing an optimal trajectory by gluing the dynamics of the Hamiltonian systems. The algorithm is illustrated by calculating an optimal economic growth trajectory for macroeconomic data.     

On some properties of the adjoint variable in the relations of the Pontryagin maximum principle for optimal economic growth problems

For a class of infinite-horizon optimal control problems that appear in studies on economic growth processes, the properties of the adjoint variable in the relations of the Pontryagin maximum principle, defined by a formula similar to the Cauchy formula for the solutions to linear differential systems, are studied. It is shown that under a dominating discount type condition the adjoint variable defined in this way satisfies both the core relations of the maximum principle (the adjoint system and the maximum condition) in the normal form and the complementary stationarity condition for the Hamiltonian. Moreover, a new economic interpretation of the adjoint variable based on this formula is presented.     

A new approach to the Pontryagin maximum principle for nonlinear fractional optimal control problems

In this paper, we discuss a new general formulation of fractional optimal control problems whose performance index is in the fractional integral form and the dynamics are given by a set of fractional differential equations in the Caputo sense. The approach we use to prove necessary conditions of optimality in the form of Pontryagin maximum principle for fractional nonlinear optimal control problems is new in this context. Moreover, a new method based on a generalization of the Mittag–Leffler function is used to solving this class of fractional optimal control problems. A simple example is provided to illustrate the effectiveness of our main result. Copyright © 2016 John Wiley & Sons, Ltd.     

Dual formulation of the Pontryagin maximum principle in optimal control

An invariant dual formulation of the Pontryagin maximum principle is given for the time-optimal case.     

A proof of the Pontryagin maximum principle for initial-value problems

Many optimization problems in economic analysis, when cast as optimal control problems, are initial-value problems, not two-point boundary-value problems. While the proof of Pontryagin (Ref. 1) is valid also for initial-value problems, it is desirable to present the potential practitioner with a simple proof specially constructed for initial-value problems. This paper proves the Pontryagin maximum principle for an initial-value problem with bounded controls, using a construction in which all comparison controls remain feasible. The continuity of the Hamiltonian is an immediate corollary. The same construction is also shown to produce the maximum principle for the problem of Bolza.     

Properties of the Lagrange multipliers in the Pontryagin maximum principle for optimal control problems with state constraints

We study the Pontryagin maximum principle for an optimal control problem with state constraints. We analyze the continuity of a vector function µ (which is one of the Lagrange multipliers corresponding to an extremal by virtue of the maximum principle) at the points where the extremal trajectory meets the boundary of the set given by the state constraints. We obtain sufficient conditions for the continuity of µ in terms of the smoothness of the extremal trajectory.     

Pontryagin maximum principle,relaxation, and controllability

The relations between the necessary minimum conditions in an optimal control problem (Pontryagin maximum principle), the minimum conditions in the corresponding relaxation (weakened) problem, and sufficient conditions for the local controllability of the controlled system specifying the constraints in the original formulation are studied. An abstract optimization problem that models the basic properties of the optimal control problem is considered.     

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.     

The Pontryagin maximum principle. Ab ovo usque ad mala

A proof of the Pontryagin maximum principle for a sufficiently general optimal control problem is presented; the proof is based on the implicit function theorem and the theorem on the solvability of a finite-dimensional system of nonlinear equations. The exposition is self-contained: all necessary preliminary facts are proved. These facts are mainly related to the properties of solutions to differential equations with discontinuous right-hand side and are derived as corollaries to the implicit function theorem, which, in turn, is a direct consequence of Newton’s method for solving nonlinear equations.     

The Pontryagin maximum principle and a unified theory of dynamic optimization

The Pontryagin maximum principle is the central result of optimal control theory. In the half-century since its appearance, the underlying theorem has been generalized, strengthened, extended, proved and reinterpreted in a variety of ways. We review in this article one of the principal approaches to obtaining the maximum principle in a powerful and unified context, focusing upon recent results that represent the culmination of over thirty years of progress using the methodology of nonsmooth analysis. We illustrate the novel features of this theory, as well as its versatility, by introducing a far-reaching new theorem that bears upon the currently active subject of mixed constraints in optimal control.     

A maximum principle for optimal control problems with mixed constraints

Necessary conditions in the form of maximum principles are derivedfor optimal control problems with mixed control and state constraints.Traditionally, necessary condtions for problems with mixed constraintshave been proved under hypothesis which include the requirementthat the Jacobian of the mixed constraint functional, with respectto the control variable, have full rank. We show that it canbe replaced by a weaker ‘interiority’ hypothesis.This refinement broadens the scope of the optimality conditions,to cover some optimal control problems involving differentialalgebraic constraints, with index greater than unity.     

Weak maximum principle for optimal control problems of nonsmooth systems

In this paper, by considering vector-valued maximum type functions satisfying Lipschitz condition, and optimal control systems with continuous-time which is governed by systems of ordinary differential equation, we derive results similar to Pontryagin’s maximum principle and properties concerning the generalized Jacobian set for optimal control problems of these systems.     

The maximum principle in optimal control problems with concentrated and distributed delays in controls

In the present work there has been posed and studied a general nonlinear optimal problem and a quasi-linear optimal problem with fixed time and free right end. It contains absolutely continuous monotone delays in phase coordinates and absolutely continuous monotone and distributed delays in controls. For these problems the necessary and, respectively, sufficient conditions of optimality in the form of the maximum principle have been proved.     

Extension of the local maximum principle to abnormal optimal control problems

In this paper, an optimal control problem with terminal data is considered in the so-called abnormal case, i.e., when the classical Pontryagin-type maximum principle has a degenerate form which does not depend on the functional being minimized. An extension of the Dubovitskii-Milyutin method to the nonregular case, obtained by applying Avakov's generalization of the Lusternik theorem, is presented. By using this extension, a local maximum principle which has a nondegenerate form also in the abnormal case is proved. An example which supports the theory is given.The author would like to thank Professors S. Walczak and W. Kotarski for fruitful discussions in the process of writing this paper.This research was supported by a SIUE Research Scholar Award and by NSF Grant DMS-91-009324.     

Minimizing sequences in optimal control with approximately given input data and the regularizing properties of the Pontryagin maximum principle

Results related to the optimal control theory for systems with approximately given input data are presented. The basic (desired) element in the theory is the minimizing sequence of feasible controls rather than the classical optimal control. Necessary and sufficient conditions for minimizing sequences are established. The regularizing properties of the Pontryagin maximum principle and of minimizing sequences are discussed. Three basic regularization levels are singled out that are characteristic of any optimal control problem. The stability of the optimal value in a problem depending on the constraint parameter is discussed. Illustrative examples are considered in detail.     

An existence result for optimal economic growth problems

An existence result for optimal control problems of Lagrange type with unbounded time domain is derived very directly from a corresponding result for problems with bounded time domain. This subsumes the main existence result of R. F. Baum ¦J. Optim. Theory Appl.19 (1976), 89–116¦ and has the existence results for optimal economic growth problems of S.-I. Takekuma ¦J. Math. Econom.7 (1980), 193–208¦ and M. J. P. Magill ¦Econometrica49 (1981), 679–711; J. Math. Anal. Appl.82 (1981), 66–74¦ as simple corollaries. In addition, a new notion of uniform integrability is used, which coincides with the classical notion if the time domain is bounded.