Bolza Problem Research Articles

Necessary Conditions for Singular Extremals Involving Multiple Control Variables

New necessary conditions are obtained from the second variation, via a transformed accessory minimum problem, for an important class of singular Bolza problems, which includes most of the singular optimal control problems that have been studied in recent years. This set of necessary conditions is a generalization of the classical Clebsch (Legendre) necessary condition. It is in a form easily used. For problems with multiple control variables, it is required that a certain matrix be symmetric; and if this symmetry property is satisfied, it then requires another matrix to be positive semidefinite. The positive semidefiniteness of the diagonal terms of the latter matrix imposes the same conditions as those obtained by other authors (Kelley, Kopp and Moyer). Should this generalized Clebsch condition be satisfied only in a semidefinite manner, then another similar set of necessary conditions can deduced, and so on.Three examples are studied. Firstly, we impose new necessary conditions on the variable thrust ar...

The Second Variation for the Singular Bolza Problem

The Second Variation for the Singular Bolza Problem

Lagrange and Mayer problems in optimal control

We develop a geometric interpretation for solutions to the Bolza problem in optimal control for systems subject to hybrid dynamics. In particular, the idea of a hybrid Lagrangian submanifold is introduced. These submanifolds are invariant under the hybrid Hamiltonian flow arising from the extended maximum principle for hybrid systems. Solutions to the hybrid optimal control problem are characterized from backward propagating these submanifolds under the hybrid Hamiltonian flow. This approach side-steps issues of multi-valuedness and singularities. This paper ends with a brief example of the controlled bouncing ball to illustrate the theory.

The equivalence of some necessary conditions for optimal control in problems with bounded state variables

Abstract : The problem of optimal control is considered as earlier studied separately by Gamkrelidze, Berkovitz, and Dreyfus --wherein a constraint is placed on the state variables. The three have studied the problem from different viewpoints: Gamkrelidze accounts for the constraint by modifying Pontryagin's maximum principle arguments; Berkovitz applies directly the classical theory developed for the problem of Bolza; and Dreyfus utilizes a dynamic programming approach. The results deduced by the latter approach, in some respects, appear to be unrelated to the Gamkrelidze and Berkovitz results, which are in agreement. It is demonstrated that the two sets of results are actually related. (Author)

Liapounov's theorem on the range of a vector measure and Pontryagin's maximum principle

Abstract : The necessary condition for the optimal control of a non-linear dynamical system is studied. The system under consideration is described by a canonical system of ordinary differential equations for the state variables. The differential equations depend also on certain parameters called control variables. The problem is to choose the value of these control variables, which can be time dependent, in order to satisfy given initialAND END POINTS CONDITIONS AND TO MAXIMIZE A GIVEN FUNCTIONAL. When there are no restrictions on the values which can be taken by the control variables the problem is relatively simple and equivalent to the Problem of Bolza of the calculus of variations. Such situation however is very unlikely to occur in practice because of the physical limitations of the control devices. (Autho )

Variational methods in problems of control and programming

It is shown how a fairly general control problem, or programming problem, with constraints can be reduced to a special type of classical Bolza problem in the calculus of variations. Necessary conditions from the Bolza problem are translated into necessary conditions for optimal control. It is seen from these conditions that Pontryagin's Maximum Principle is a translation of the usual Weierstrass condition, and is applicable to a wider class of problems than that considered by Pontryagin. The differentiability and continuity properties of the value of the control are established under reasonable hypotheses on the synthesis, and it is shown that the value satisfies the Hamilton-Jacobi equation. As a corollary we obtain a rigorous proof of a functional equation of Bellman that is valid for a much wider class of problems than heretofore. A sufficiency theorem for the synthesis of control is also given.

Dynamic programming and the calculus of variations

Problems in the Calculus of Variations can be viewed as multistage decision problems of a continuous type. It follows that their solutions can be characterized by the functional equation technique of dynamic programming [1]. In this paper, it will be shown that the functional equation approach yields, in simple and intuitive fashion, formal derivations of such classical necessary conditions of the Calculus of Variations as the Euler-Lagrange equations, the Weierstrass and Legendre conditions, natural boundary conditions, a transversality condition and the Erdmann corner conditions. The more general “problem of Bolza” in which the final time is defined implicitly and in which the expression to be extremized is the sum of an integral and a function evaluated at the end point is also considered. The principal necessary condition, usually called the “multiplier rule,” is deduced. We shall also derive necessary conditions for the case where the decision variables are restricted by inequality constraints. Finally, it is shown that the functional equation characterization readily yields the Hamilton-Jacobi partial differential equation of classical mechanics.

A degenerate problem of Bolza

A degenerate problem of Bolza

Isoperimetric Problems in the Calculus of Variations

We are concerned with establishing sufficiency theorems for minima of simple integrals of the parametric type in a class of curves with variable end points and satisfying isoperimetric side conditions. The results which are obtained involve no explicit assumptions of normality. Such results can be derived by transforming our problem to a problem of Bolza and using the latest developments in the theory of that problem. More recently [6] an indirect method of proof has been published. Our object is to present a direct method of proof without transformation of the problem which is based upon a generalization of the classical theory of fields.

A Metric in the Space of Generalized Curves

on the class of curves in question. An important step was made when L. C. Young [1, 2] devised the concept of curve. The author [4] metrized the space of generalized curves in such a way that it was complete and on it each (parametric) integral was continuous; it was still possible to establish a compactness theorem. This led to a powerful existence theorem [6] for Bolza problems in parametric form. However, in investigating critical points other than minima the metric was found unsatisfactory. A re-examination led to the results in this paper, which may be summarized as follows. First we set up a concept of convergence in terms of integrals taken along curves, in such a way that the topology produces our minimum requirements on the behavior of integrals and is the coarsest topology which does so. This topology turns out to be a uniform structure of the space of rectifiable curves; with this structure, the space is incomplete. It can be embedded in a complete space in which it is dense, this embedding being essentially unique. The entities composing the complete space are the generalized curves (actually not exactly identical with those previously considered, but the difference is not very significant.) This space is automatically of uniform structure. We show that it can in fact be metrized in conformity with the uniform structure, becoming thus a complete metric space. This metric is convenient in discussing critical point theory. In the last three sections a theory of generalized curves in form is developed, and it is shown that non-parametric Bolza problems of considerable generality can be proved to have solutions in the space of generalized curves.

Expansion Methods for the Isoperimetric Problem of Bolza in Non-Parametric Form

Expansion Methods for the Isoperimetric Problem of Bolza in Non-Parametric Form

An Indirect Sufficiency Proof for the Problem of Bolza in Nonparametric Form

An Indirect Sufficiency Proof for the Problem of Bolza in Nonparametric Form

An Alternate Sufficiency Proof for the Normal Problem of Bolza

An Alternate Sufficiency Proof for the Normal Problem of Bolza

An alternate sufficiency proof for the normal problem of Bolza

An alternate sufficiency proof for the normal problem of Bolza

The isoperimetric problem of Bolza with finite side conditions

The isoperimetric problem of Bolza with finite side conditions

An indirect sufficiency proof for the problem of Bolza in nonparametric form

An indirect sufficiency proof for the problem of Bolza in nonparametric form

Sufficient conditions for the isoperimetric problem of Bolza in the calculus of variations

Sufficient conditions for the isoperimetric problem of Bolza in the calculus of variations

Sufficient Conditions for a Weak Relative Minimum in the Problem of Bolza

Sufficient Conditions for a Weak Relative Minimum in the Problem of Bolza

Sufficient conditions for a weak relative minimum in the problem of Bolza

Sufficient conditions for a weak relative minimum in the problem of Bolza

The problem of Bolza in the calculus of variations

The problem of Bolza in the calculus of variations