Abstract. Many infinite-horizon optimal control problems in management science and economics have optimal paths that approach some stationary level. Often ...
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 23, No. 4, DECEMBER 1977
Nearest Feasible Paths in Optimal Control Problems: Theory, Examples, and Counterexamples S. P.
SETHI 1
Communicated by J. V. Breakwell Abstract. Many infinite-horizon optimal control problems in management science and economics have optimal paths that approach some stationary level. Often, this path has the property of being the nearest feasible path to the stationary equilibrium. ~Ihis paper obtains a simple multiplicative characterization for a single-state single-control problem to have this property. By using Green's theorem it is shown that the property is observed as long as the stationary level is sustainable by a feasible control. If not, the property is, in general, shown to be false. The paper concludes with an important theorem which states that even in the case of multiple equilibria, the optimal path is a nearest feasible path to one of them.
Key Words. Optimal control~ Green's theorem, infinite horizon, multiplicative problems, optimal stationary equilibrium, economic applications. 1. Introduction
Many infinite-horizon optimal control problems encountered in management science and economics have solutions that approach some stationary value for the state variable. Spence and Starrett (Ref. 1) have delineated a class of problems in which the optimal solution is to approach an optimal stationary level, when it exists, as fast as possible. They term such paths as most rapid approach paths. We note that the results obtained by Spence and Starrett are more or less implicit in what is known as the Green's theorem approach to optimal control problems developed by Miele (Ref. 2); in particular, see Sethi (Ref. 3) for the notion of the nearest feasible path to a singular arc in a Green's theorem application. Unaware of the relation between the Green's theorem approach and their analysis, Spence and Starrett have, nevertheless, a A s s o c i a t e P r o f e s s o r of M a n a g e m e n t S c i e n c e , F a c u l t y of M a n a g e m e n t S t u d i e s , U n i v e r s i t y of T o r o n t o , T o r o n t o , O n t a r i o , C a n a d a . T h e a u t h o r t h a n k s J. B o n a f o r his h e l p f u l c o m m e n t s .
563 "ntis journal is copyrighted by Plenum. Each article is available for $7.50 from Plenum Publishing Corporation, 227 West 17th Street, New York, N.Y. 10011.
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organized their results nicely and for the first time in one place. We do indicate, however, that their treatment of the interval constraint on the control variable is highly superficial. This is in the sense that they assume the control required to sustain the singular optimal stationary level to be feasible, a situation which is almost equivalent to the unconstrained problem, at least with respect to the issues under consideration. In this paper, therefore, we attempt to do the following: (i) we derive the results under consideration as a direct consequence of the Green's theorem approach; (ii) we provide a counterexampte when the control constraint is binding and as a result the optimal stationary level is nonsingular; (iii) finally, we provide a management science example with singular nonunique optimal stationary equilibria and obtain its solution.
2. Optimal Control Problem Consider the following optimal control problem with one state and one control variable:
max{ J = Io°°exp(-rt)F(x, u ) dt} ,
(1)
with the discount rate r > O, subject to
Yc=f(x, u),
x(O) = xo,
(2)
and the control variable constraint u e f~ = [0,
Q]CR a.
(3)
Note that we have assumed the stationarity assumption, i.e., F, f, and Q are explicitly independent of the time t. We also assume that F(x, u) is upperbound for all feasible values of x and u. As in Ref. 3, we define the nearest feasible path x*(t) to a given feasible steady state (£, a) as the feasible path starting from Xo in such a way that
Ix*(t) -~1 ~ f f u*(t) = 0
until x = ~, and then u*(t) = ti thereafter.
Xo < ~
6. Counterexample When f l ( Q ) > 1, however, the value computed in (18) is no longer feasible. It can be shown that the stationary optimal control in this case is =pQ/(pQ+k) x s.
(21)
7 In Ref. 3, the optimality of these paths is also established by the theory of the maximum principle.
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Note that x ~ is always in (0, 1). Now, the optimal control is as follows:
~u*(t) = O, $X a, the optimal path consists of a part of the fastest approach segment, followed by a slowest approach to ~. Note that, in either of the last two cases, the nearest approach segment to ~ exists, but it is not a part of the optimal path. Note further that there does not exist nearest feasible paths to Y in both of these cases. We have used the term counterexample to emphasize the fact that, if the control constraint (16) is really binding, then the nearest (or fastest) approach segment is no longer optimal. The counterexample does not, of course, violate Theorem 3.1. This is because the stationary optimal equilibrium $ in this case does not satisfys condition (i) of Theorem 3.1. We further note, in this regard, that Spence and Starrett (Ref. 1) are careless in their discussion of control constraints. Their discussion, although technically correct, seems to give an impression that the control constraint does not create any problem. Of course, it does not, if the control constraint is assumed, as they do, to be nonbinding. This is a trivial assumption and is almost equivalent to having no control constraints. From the discussion in this section, we can state the following theorem, the proof of which follows easily from the Green's theorem approach. Theorem 6.1. such that
If ~ is a unique optimal stationary equilibrium and is < (>)x
where x ~ represents the unique solution of (13), then, only for x0-->(- Jxoco~. This leads to a contradiction with (23). We have thus proved the lemma noting that the part of the lemma in the parentheses follows similarly. Note that, for the case of more than two equilibria, we must modify the lemma appropriately. For this, we replace x] and x~ by x~ and x] with x] > x~ and replace
Xo> ( r > C.
Proof. Since xoAB is not the best path over $1, then, by Lemma 7.1, the best path must cross 2. We have claimed in the lemma that xoCDE is the best path. Suppose that it is not. Then, let the best path cross 2 at some time z > C (see Fig. 2). It is obvious, as can be easily shown by Green's theorem, that the optimal path from x0 to 2 at time 7 (i.e., to point r in Fig. 2) is the path x oAF7 which minimizes the area between the path and x ]. Note that F r is the fastest approach from F to 2 and note that, for r not sufficiently large, it is possible not to have the segment A F in this path; in this case, the path from Xo to point r will consist of the fastest decline from Xo, followed by the fastest rise to point ~r. Furthermore, once the path has crossed g at ~', the best path from r subject to the condition that it does not cross 2 again (since it has already crossed once) is ~-GE, by Lemma 7.1. Under our supposition, in the preceding paragraph, then, xoAFTGE is the best path over the class of paths considered. Then, JxoAI~G ~ JxoCDG~ or
JCxo ~- JxoAl:'¥ ~ JCDO -{-Jot, or
JCxo-~ JxoAF~- '~ J~c >--JcDG q-Jot + J-rc, or
~CxoAF~C ~" A CDCrrC.
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This implies that, for any ~-~", ~-' > ~-> C, A.rFF,,r, r ~ m.rGGqr'r~
with the consequence that the path xoAF'~"G' obtained by increasing r to ~" is also better than JxoCOO'. Thus, JxoAF','G ~ JxoCDO,
V'I"> C;
therefore, lim J.oAFTG JxoAB>- JxoCDE" =
,'i--.*oo
This leads to a contradiction, which completes both parts of the theorem. We have thus shown that, if JxoAB
