John L. Troutman

Necessary Conditions for Optimality

What conditions are necessary for optimal performance in our problems? In Chapter 10 we saw that if a control problem can be formulated on a fixed interval and its defining functions are suitably convex, then the methods of variational calculus can be adapted to suggest sufficient conditions for an optimal control. In particular, the minimum principle of §10.3 and §10.4 can guarantee optimality of a solution to the problem. In §11.1 we will discover that this principle is necessary for optimality whether or not convexity is present, even when the underlying interval is not fixed (Theorem 11.10). Then in §11.2, we examine the simple but important class of linear time-optimal problems for which the time interval itself is being minimized and the adjoint equation (a necessary condition) can be used to suggest sufficient conditions for optimality. Finally, in §11.3, we extend our control-theory approach to more general problems involving Lagrangian inequality constraints, and in Theorem 11.20 we obtain a Lagrangian multiplier rule of the Kuhn-Tucker type.

Variational Principles in Mechanics

The recognition that minimizing an integral function through variational methods (as in the last chapters) leads to the second-order differential equations of Euler-Lagrange for the minimizing function made it natural for mathematicians of the eighteenth century to ask for an integral quantity whose minimization would result in Newton’s equations of motion. With such a quantity, a new principle through which the universe acts would be obtained. The belief that “something” should be minimized was in fact a long-standing conviction of natural philosophers who felt that God had constructed the universe to operate in the most efficient manner—but how that efficiency was to be assessed was subject to interpretation. However, Fermat (1657) had already invoked such a principle successfully in declaring that light travels through a medium along the path of least time of transit. Indeed, it was by recognizing that the brachistochrone should give the least time of transit for light in an appropriate medium that Johann Bernoulli “proved” that it should be a cycloid in 1697. (See Problem 1.1.) And it was Johann Bernoulli who in 1717 suggested that static equilibrium might be characterized through requiring that the work done by the external forces during a small displacement from equilibrium should vanish. This “principle of virtual work” marked a departure from other minimizing principles in that it incorporated stationarity—even local stationarity—(tacitly) in its formulation. Efforts were made by Leibniz, by Euler, and most notably, by Lagrange to define a principle of least action (kinetic energy), but it was not until the last century that a truly satisfactory principle emerged, namely, Hamilton’s principle of stationary action (c. 1835) which was foreshadowed by Poisson (1809) and polished by Jacobi (1848) and his successors into an enduring landmark of human intellect, one, moreover, which has survived transition to both relativity and quantum mechanics. (See [L], [Fu] and Problems 8.11 8.12.)

Segmentally Alternating Series

Abstract We consider real series composed of segments having terms of mixed sign and obtain results generalizing the alternating series theorem. We apply this to prove a result on integration of a series of functions which is a numerical series of this type almost everywhere.

The Lemmas of Lagrange and du Bois-Reymond

In most of the examples in Chapter 3, we examined a real valued function F defined on a domain of functions \(\mathcal{D}\) and obtained for it an integral condition in the form I(y, v) = 0, ∀ v in an auxiliary domain \(\mathcal{D}\)0 which is sufficient to guarantee that each y ∈ \(\mathcal{D}\) which satisfies it must minimize F on \(\mathcal{D}\).

Review of Optimization in ℝd

This chapter presents a brief summary of the standard terminology and basic results related to characterizing the maximal and minimal values of a real valued function f defined on a set D in Euclidean space. With the possible exception of the remarks concerning convexity ((0.8) and (0.9)), this material is covered in texts on multidimensional calculus; the notation is explained at the end of §1.5.

Control Problems and Sufficiency Considerations

The discipline now identified as optimal control emerged during the decade 1940–1950, from the efforts by engineers to design electromechanical apparatus which was efficiently self-correcting, relative to some targeted objective. Such efficiency is clearly desirable in, say, the tracking of an aircraft near a busy airport or in the consumption of its fuel, and other economically desirable objectives suggest themselves. The underlying mathematical problems were attacked systematically in the next decade by Bellman [Be], by Hestenes [He], and by a Russian group under Pontjragin [Po]. Their results were quickly adapted to characterize optimal processes in other fields (including economics itself) and the feasibility of optimal control is now a standard consideration in contemporary design strategy.

Piecewise C1 Extremal Functions

In many problems examined thus far we have required continuous differentiability of the function y (or Y) defining the classes for optimization. Already with the example of the minimal surface of revolution from §1.4(a) we have argued that for some configurations, the minimizing curve (if it exists) should exhibit “corners,” and it is natural to wonder whether cornered curves ŷ and Ŷ such as those shown in Figure 7.1 might not give improved results for other problems. Such curves are represented readily by functions which are piecewise continuously differentiable, or piecewise C1.

Linear Spaces and Gâteaux Variations

Each problem considered previously reduces to that of optimizing (usually minimizing) a real valued function J defined on a subset \(\mathcal{D}\) of a linear space \(\mathcal{Y}\) In the present chapter we shall view problems in this context and introduce the associated directional derivatives (Gâteaux variations) of the functions which will be required for what follows. We begin with a catalogue of standard linear spaces presupposing some familiarity with vector space operations, with continuity, and with differentiability in ℝd.