Pontryagin Principle for State-Constrained Control Problems Governed by a First-Order PDE System

This paper considers multidimensional control problems governed by a first-order PDE system and state constraints. After performing the standard Young measure relaxation, we are able to prove the Pontryagin principle by means of an -maximum principle. Generalizing the common setting of one-dimensional control theory, we model piecewise-continuous weak derivatives as functions of the first Baire class and obtain regular measures as corresponding multipliers. In a number of corollaries, we derive necessary optimality conditions for local minimizers of the state-constrained problem as well as for global and local minimizers of the unconstrained problem.     

Necessary optimality conditions for infinite dimensional state constrained control problems

This paper is concerned with first order necessary optimality conditions for state constrained control problems in separable Banach spaces. Assuming inward pointing conditions on the constraint, we give a simple proof of Pontryagin maximum principle, relying on infinite dimensional neighboring feasible trajectories theorems proved in [20]. Further, we provide sufficient conditions guaranteeing normality of the maximum principle. We work in the abstract semigroup setting, but nevertheless we apply our results to several concrete models involving controlled PDEs. Pointwise state constraints (as positivity of the solutions) are allowed.     

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.     

Optimal control: Nonlocal conditions,computational methods,and the variational principle of maximum

This paper surveys theoretical results on the Pontryagin maximum principle (together with its conversion) and nonlocal optimality conditions based on the use of the Lyapunov-type functions (solutions to the Hamilton-Jacobi inequalities). We pay special attention to the conversion of the maximum principle to a sufficient condition for the global and strong minimum without assumptions of the linear convexity, normality, or controllability. We give the survey of computational methods for solving classical optimal control problems and describe nonstandard procedures of nonlocal improvement of admissible processes in linear and quadratic problems. Furthermore, we cite some recent results on the variational principle of maximum in hyperbolic control systems. This principle is the strongest first order necessary optimality condition; it implies the classical maximum principle as a consequence.     

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.     

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.     

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.     

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.     

Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions

The paper deals with first order necessary optimality conditions for a class of infinite-horizon optimal control problems that arise in economic applications. Neither convergence of the integral utility functional nor local boundedness of the optimal control is assumed. Using the classical needle variations technique we develop a normal form version of the Pontryagin maximum principle with an explicitly specified adjoint variable under weak regularity assumptions. The result generalizes some previous results in this direction. An illustrative economical example is presented.     

Pontryagin’s principle for state-constrained evolutionary differential complementarity system

This paper is devoted to the state-constrained optimal control problem of evolutionary variational inequality. In this paper, the control domain is not necessarily convex. Moreover, since our state constraint is quite general and, in many cases, it requires pointwise behavior of the state, the framework of the partial differential equation (instead of the abstract framework) is used. Some optimality conditions (in the form of Pontryagin’s principle) for optimal controls are established.     

Optimal strategies to diminish a pest population via bilinear controls

In this work we model and analyze two control strategies to diminish a pest population using traps. The action of any trap depends on its age. Both problems contain bilinear control rates. The large-time behavior of the model with time-periodic inflow is investigated. The first control strategy deals with a finite horizon problem while the second one is related to a time-periodic problem. We obtain Pontryagin’s principle for both control problems. A special attention is given to the periodic problem. Pontryagin’s principle is used to derive a conceptual gradient-type algorithm to approximate the optimal solution. Numerical tests are given.     

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.     

First- and second-order optimality conditions for a strong local minimum in control problems with pure state constraints

In this paper first- and second-order optimality conditions for a strong local minimum are presented for optimal control problems with pure state set-inclusion constraints. The first-order condition is of Pontryagin type, while the second-order condition is of the form of an accessory problem associated with the strong local minimality. This latter condition contains an extra term reflecting the presence of the pure state constraints.     

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.     

Strong Local Optimality Conditions for State Constrained Control Problems

In this paper first- and second-order optimality conditions for strong local minimum are presented for optimal control problems with pure state set-inclusion constraints. The first-order condition is of Pontryagin type, while the second-order condition is of the form of an accessory problem associated with the strong local minimality. This latter condition contains an extra term reflecting the presence of the pure state constraints.     

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.     

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.     

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.     

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.