In this note we contrast two transformation-based methods to deduce absolute extrema and the corresponding extremizers. Unlike variation-based methods, the transformation-based ones of Carlson and Leitmann and the recent one of Silva and Torres are direct in that they permit obtaining solutions by inspection.
Deep Dive into Contrasting Two Transformation-Based Methods for Obtaining Absolute Extrema.
In this note we contrast two transformation-based methods to deduce absolute extrema and the corresponding extremizers. Unlike variation-based methods, the transformation-based ones of Carlson and Leitmann and the recent one of Silva and Torres are direct in that they permit obtaining solutions by inspection.
In the mid 1960's a direct method for the problems of the calculus of variations, which permits one to obtain absolute extremizers directly, without using variational methods, was introduced by Leitmann (Ref. 9). Since then, this direct method has been extended and applied to a variety of problems (see e.g. Refs. 1,3,4,10). A different but related direct approach to problems of optimal control, based on the variational symmetries of the problem (cf. Refs. 7,8), was recently introduced by Silva and Torres (Ref. 11). The emphasis in Ref. 11 has been on showing the differences and similarities between the proposed method and that suggested by Leitmann. In order to illustrate the relation between these two methods, only examples capable of treatment by both methods were presented in Ref. 11. In this note, we discuss some differences between the method of Carlson and Leitmann (C/L) and Silva and Torres (S/T). In particular, we show how one succeeds when the other does not.
Let us consider the problem of optimal control in Lagrange form: minimize an integral
subject to a control system
together with appropriate boundary conditions and constraint on the values of the control variables:
The Lagrangian The application of the invariant transformation method (Ref. 11) depends on the existence of a sufficiently rich family of invariance transformations (variational symmetries). The reader interested on the study of variational symmetries is referred to Refs. 8, 12, 13 and references therein. Definition 2.1 Let h s be a s-parameter family of C 1 mappings satisfying:
for all admissible pairs (x(•), u(•)), then (1)-( 2) is said to be invariant under the transformations h s (t, x, u) up to Φ s (t, x, u); and the transformations h s (t, x, u) are said to be a variational symmetry of (1)-( 2).
The method proposed in Ref. 11 is based on a very simple idea. Given an optimal control problem, one begins by determining its invariance transformations according to Definition 2.1. With respect to this, the tools developed in Refs. 7, 8 are useful. Applying the parameter-invariance transformations, we embed our problem into a parameter-family of optimal control problems. Given the invariance properties, if we are able to solve one of the problems of this family, we also get the solution to our original problem (or to any other problem of the same family) from the invariant transformations. In section 4 we give an example which shows that the Invariant Transformation Method (Ref. 11) is more general than the earlier C/L transformation method in the case of optimal control problems.
Since this method is fully discussed in readily available references, e.g. Refs. 1, 3, 4, 9, 10, many in this journal, we shall only recall that the C/L transformation based method is applicable to problems in the Calculus of Variations format: minimize an integral
with given end conditions
If one wishes to solve an optimal control problem (1)-(3), the “elimination” of u(t) in favor of a function of t, x(t), ẋ(t) must be possible. As illustrated in section 4, this may fail even if the Implicit Function Theorem is satisfied.
Both the S/T and the C/L methods are predicated on posing a problem “equivalent” to the original problems (1)-( 3) and ( 6)- (7), respectively. Thus, these methods are useful only if the solution of the “equivalent” problem is directly obtainable, i.e., by inspection. There is, at present, no result assuring that this can be done in general for the S/T method. However, for the C/L method, at least in the scalar x case, it has been shown in Ref. 5 and generalized to open-loop differential games in Ref. 2, that the “equivalent” problem always has a minimizing solution obtained by inspection. The conditions sufficient for this result are convexity of integrand F (t, x(t), ẋ(t)) with respect to ẋ(t), and existence of a so-called “field of extremals”. Indeed, no matter what the integrand of the original problem is, provided the conditions above are met, the absolute minimizer of the equivalent problem is always a constant.
The advantage of the invariant transformation method when compared with the earlier transformation method is that one can apply it directly to control systems whereas the method of C/L requires that the control u(t) can be expressed as a sufficiently smooth function of t, x(t), ẋ(t), e.g. such that the integrand be continuous in x(t) and ẋ(t). Here we use the invariant transformation method of S/T to solve a simple optimal control problem that is not covered by the classical theory of the Calculus of Variations and which can not be solved by the previous transformation method.
Consider the global minimum problem
We apply the procedure introduced in Ref. 11 and briefly described in section 2. First we notice that problem (8) is variationally invariant according to Definition 2.1 under the one-parameter transformations4
x s
To prove this, we need to show that both the functional integral I[•] and the control sy
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