Junction phenomena for optimal control with state-variable inequality constraints of third order

It is known that extremal arcs governed by inequality constraints of third order (constraint relations that must be differentiated three times to generate a control equation) cannot join an unconstrained arc, except in special cases. But a control problem is exhibited, for which every extremal includes a constrained arc of third order. The constrained arc joins the end of an infinite sequence of consecutive unconstrained arcs of finite total duration. Evidence (but not proof) is given that this phenomenon is typical, rather than exceptional. An analogous phenomenon is well known for optimal control problems with singular arcs of second order.     

No-gap optimality conditions for an optimal control problem with pointwise control-state constraints

An optimal control problem with pointwise mixed constraints of the instationary three-dimensional Navier–Stokes–Voigt equations is considered. We derive second-order optimality conditions and show that there is no gap between second-order necessary optimality conditions and second-order sufficient optimality conditions. In addition, the second-order sufficient optimality conditions for the problem where the objective functional does not contain a Tikhonov regularization term are also discussed.     

On necessary extremum conditions for finite-dimensional problems with inequality constraints

The finite-dimensional optimization problem with equality and inequality constraints is examined. The case where the classical regularity condition is violated is analyzed. Necessary second-order extremum conditions are obtained that are stronger versions of some available results.     

Successive approximations for Markov decision processes and Markov games with unbounded rewards

The aim of this paper is to give a survey of recent developments in the area of successive approximations for Markov decision processes and Markov games. We will emphasize two aspects, viz. the conditions under which successive approximations converge in some strong sense and variations of these methods which diminish the amount of computational work to be executed. With respect to the first aspect it will be shown how much unboundedness of the rewards may be allowed without violation of the convergence

With respect to the second aspect we will present four ideas, that can be applied in conjunction, which may diminish the amount of work to be done. These ideas are: 1. the use of the actual convergence of the iterates for the construction of upper and lower bounds (Macqueen bounds), 2. the use of alternative policy improvement procedures (based on stopping times), 3. a better evaluation of the values of actual policies in each iteration step by a value oriented approach, 4. the elimination of suboptimal actions not only permanently, but also temporarily. The general presentation is given for Markov decision processes with a final section devoted to the possibilities of extension to Markov games.     

Second-order necessary optimality conditions for semilinear elliptic optimal control problems

This paper deals with the necessary optimality conditions for semilinear elliptic optimal control problems with a pure pointwise state constraint and mixed pointwise constraints. By computing the so-called ‘sigma-term’, we obtain the second-order necessary optimality conditions for the problems, which is sharper than some previously established results in the literature. Besides, we give a condition which relaxes the Slater condition and guarantees that the Lagrangian is normalized.     

Sufficient optimality conditions for nonconvex control problems with state constraints

The sufficient optimality conditions of Zeidan for optimal control problems (Refs. 1 and 2) are generalized such that they are applicable to problems with pure state-variable inequality constraints. We derive conditions which neither assume the concavity of the Hamiltonian nor the quasiconcavity of the constraints. Global as well as local optimality conditions are presented.     

A transformation technique for optimal control problems with partially linear state inequality constraints

This paper considers optimal control problems involving the minimization of a functional subject to differential constraints, terminal constraints, and a state inequality constraint. The state inequality constraint is of a special type, namely, it is linear in some or all of the components of the state vector.A transformation technique is introduced, by means of which the inequality-constrained problem is converted into an equality-constrained problem involving differential constraints, terminal constraints, and a control equality constraint. The transformation technique takes advantage of the partial linearity of the state inequality constraint so as to yield a transformed problem characterized by a new state vector of minimal size. This concept is important computationally, in that the computer time per iteration increases with the square of the dimension of the state vector.In order to illustrate the advantages of the new transformation technique, several numerical examples are solved by means of the sequential gradient-restoration algorithm for optimal control problems involving nondifferential constraints. The examples show the substantial savings in computer time for convergence, which are associated with the new transformation technique.This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-76-3075, and by the National Science Foundation, Grant No. MCS-76-21657.     

On optimal control problems with general boundary conditions

It is shown that the necessary optimality conditions for optimal control problems with terminal constraints and with given initial state allow also to obtain in a straightforward way the necessary optimality conditions for problems involving parameters and general (mixed) boundary conditions. In a similar manner, the corresponding numerical algorithms can be adapted to handle this class of optimal control problems.This research was supported in part by the Commission on International Relations, National Academy of Sciences, under Exchange Visitor Program No. P-1-4174.The author is indebted to the anonymous reviewer bringing to his attention Ref. 9 and making him aware of the possible use of generalized inverse notation when formulating the optimality conditions.     

New necessary conditions of optimality for control problems with state-variable inequality constraints

Necessary conditions of optimality for state-variable inequality constrained problems are derived which differ from those of Bryson, Denham, and Speyer with regard to the behavior of the adjoint variables at junctions of interior and boundary arcs. In particular, it is shown that the earlier conditions under-specify the behavior of the adjoint variables at the junctions. An example is used to demonstrate that the earlier conditions may yield non-stationary trajectories.     

On some elliptic optimal control problems with state constraints

In this paper optimality conditions will be derived for elliptic optimal control problems with a restriction on the state or on the gradient of the state. Essential tools are the method of transposition and generalized trace theorems and green's formulas from the theory of elliptic differential equations.     

On optimal control problems with state constraints

This paper is concerned with necessary conditions for a general optimal control problem developed by Russak and Tan. It is shown that, in most cases, a further relation between the multipliers holds. This result is of interest in particular for the investigation of perturbations of the state constraint.     

Necessary optimality conditions for minimax programming problems with mathematical constraints

In this paper, necessary optimality conditions in terms of upper and/or lower subdifferentials of both cost and constraint functions are derived for minimax optimization problems with inequality, equality and geometric constraints in the setting of non-differentiatiable and non-Lipschitz functions in Asplund spaces. Necessary optimality conditions in the fuzzy form are also presented. An application of the fuzzy necessary optimality condition is shown by considering minimax fractional programming problem.     

Pontryagin's principle for problems with isoperimetric constraints and for problems with inequality terminal constraints

Necessary conditions are derived for optimal control problems subject to isoperimetric constraints and for optimal control problems with inequality constraints at the terminal time. The conditions are derived by transforming the problem into the standard form of optimal control problems and then using Pontryagin's principle.     

A unified framework for linear control problems with state-variable inequality constraints

This paper briefly reviews the literature on necessary optimality conditions for optimal control problems with state-variable inequality constraints. Then, it attempts to unify the treatment of linear optimal control problems with state-variable inequality constraints in the framework of continuous linear programming. The duality theory in this framework makes it possible to relate the adjoint variables arising in different formulations of a problem; these relationships are illustrated by the use of a simple example. This framework also allows more general problems and admits a simplex-like algorithm to solve these problems.This research was partially supported by Grant No. A4619 from the National Research Council of Canada to the first author. The first author also acknowledges the support provided by the Brookhaven National Laboratory, where he conducted his research.     

Existence of minimizers and necessary conditions in set-valued optimization with equilibrium constraints

In this paper we study set-valued optimization problems with equilibrium constraints (SOPECs) described by parametric generalized equations in the form 0 ∈ G(x) + Q(x), where both G and Q are set-valued mappings between infinite-dimensional spaces. Such models particularly arise from certain optimization-related problems governed by set-valued variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish general results on the existence of optimal solutions under appropriate assumptions of the Palais-Smale type and then derive necessary conditions for optimality in the models under consideration by using advanced tools of variational analysis and generalized differentiation. Dedicated to Jiří V. Outrata on the occasion of his 60th birthday. This research was partly supported by the National Science Foundation under grants DMS-0304989 and DMS-0603846 and by the Australian Research Council under grant DP-0451168.     

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.     

A solution method for regular optimal control problems with state constraints

A method of region analysis is developed for solving a class of optimal control problems with one state and one control variable, including state and control constraints. The performance index is strictly convex with respect to the control variable, while this variable appears only linearly in the state equation. The convexity or linearity assumption of the performance index or the state equation with respect to the state variable is not required.The author would like to express his sincere gratitude to Prof. R. Klötzler, Prof. E. Zeidler, Prof. H. Schumann, Prof. J. Focke, and other colleagues of the Department of Mathematics, Karl Marx University, Leipzig, GDR, for their support during his stay in Leipzig.