Atle Seierstad

Sufficient Conditions in Optimal Control Theory

During the last ten years or so a large number of papers in professional journals in economics dealing with dynamic optimization problems have been employing the modern version of the calculus of variations called optimal control theory. The central result in the theory is the well known Pontryagin maximum principle providing necessary conditions for optimality in very general dynamic optimization problems. These conditions are not, in general, sufficient for optimality. Of course, if an existence theorem can be applied guaranteeing that a solution exists, then by comparing all the candidates for optimality that the necessary conditions produce, we can in principle pick out an optimal solution to the problem. In several cases, however, there is a more convenient method that can be used. Suppose that a solution candidate suggests itself through an application of the necessary conditions, or possibly also by a process of informed guessing. Then, if we can prove that the solution satisfies suLfficiency conditions of the type considered in this paper, then these conditions will ensure the optimality of the solution. In such a case we need not go through the process of finding all the candidates for optimality, comparing them and finally appealing to all existence thieorem. In ordcle to get all idea of what types of conditions might be involved in such sufficiency theorems, it is natural to look at the corresponding problem in static optimization. Here it is well known that the first order calculus or Kulhn-TLcker conditions are sU11ficient for optimality, provided suitable concavity/convexity conditions are imposed on the functions involved. It is natural to expect that similar conditions might secure sufficiency also in dynamic optimization problems. Growth theorists were early aware of this and proofs of sufficienccy in particular problems were constructed; see, e.g., Uzawas 1964 paper [19]. In the mathematical literature few and only rather special results were available until Mangasarian. in a 1966 paper [10] proved a rather general sufficiency theorem in which he was dealing with a nonlinear system, state and control variable constraints and a fixed time interval. In the maximization case, when there are no state space constraints, his result was, essentially, that the Pontryagin necessary conditions plus concavity of the Hamiltonian function with respect to the state and control variables, were sufficient for optimality. The Mangasarian concavity condition is rather strong and in many economic problems his theorem does not apply. Arrow [1] proposed an interesting partial

Necessary conditions for nonsmooth, infinite-horizon, optimal control problems

Necessary conditions are proved for deterministic nonsmooth optimal control problems involving an infinite horizon and terminal conditions at infinity. The necessary conditions include a complete set of transversality conditions.

The dynamic efficiency of capitalism

Abstract Variants of Lancasters original model on the inefficiency of capitalism have continued to interest scholars. In a slight extension of the original finite-horizon model of Lancaster the possibility of more or less completely efficient outcomes is shown in the present paper. Such results are effected by the introduction of standard trigger strategies and soft trigger strategies defined in the present paper.

Differentiability properties of the optimal value function in control theory

Abstract In normal problems, in which the adjoint function is unique, it is proved that subderivatives of the optimal value function exist with respect to changes in the initial and final conditions. When concavity is added, subderivatives change to derivatives.

The macro production function uniquely determines the capacity distribution of the micro units

Abstract A wellknown approach to the theory of production obtains the macro production function from an integration over a distribution of capacities of micro units. An elementary proof is given for the fact that the macro production function uniquely determines the capacity distribution.

Reservation Prices in Optimal Stopping

In an optimal stopping problem where bids on an asset are received, conditions are given that ensure the so-called reservation price property, namely, if a certain price is accepted, then any higher price would also have been accepted at that point in time. The approach followed in this paper is similar to that pursued by D. B. Rosenfield and R. D. Shapiro in 1981.

An extension to Banach space of Pontryagin's maximum principle

An extension of Pontryagins maximum principle to the case where the state space is infinite dimensional is given. The control process is governed by ordinary nonlinear differential equations. A property of control processes, which is analogous to well-known, nonlinear interior mapping theorems, makes up the basis for the proofs.

Sufficient conditions applied to an optimal control problem of resource management

Abstract The purpose of this paper is twofold. First, by solving an interesting problem in the theory of exhaustible resources, it is exemplified how direct sufficiency conditions should properly be used in optimal control problems. The motivation for this aspect of the paper is the almost complete negligence in the economic literature of dealing properly with sufficiency conditions. Second, an important point in the discussion of J. Aarrestad (Scand. J. Econom., 81 (1979), 522–565) is supplemented.

Sufficient conditions in free final time optimal control problems. A comment

Abstract By an application of sufficient conditions, assume that an optimal pair ( χ τ (·), u τ (·)) and an adjoint function p τ (·) were found in the control problem in question with the final time τ fixed but arbitrary. Then a sufficient condition for one of these pairs, say χ τ ∗ (·) , u τ ∗ (·) to be optimal in the corresponding free final time problem is that the Hamiltonian, with ( χ τ ( t ), u τ ( τ −), p τ ( τ ), τ) inserted, is nonnegative (nonpositive) to the left (right) of τ ∗ .

Directional derivatives and the control of nonsmooth systems

Combining standard results in nonsmooth analysis, for nonsmooth control problems this paper establishes necessary conditions for optimality based on directional derivatives and (more generally) contingent derivatives. It is shown in simple examples taht the application of directional derivatives in certain cases can reduce the set of candidate for optimality as compared to the set of candidates obtained from other necessary conditions