Optimal endpoints

A complete set of necessary and sufficient conditions for selecting optimal endpoints for extremals obtained from the variational Bolza problem in control notation has been developed. The method used to obtain these conditions is based on a seldom used concept of performing a dichotomy on the general optimization problem. With this concept, the problem of Bolza is decomposed into two problems, the first of which involves the selection of optimal paths with the endpoints considered fixed. The second problem involves the selection of optimal endpoints with the paths between the endpoints taken to be stationary curves. The convenience of the dichotomy in deriving the necessary and sufficient conditions for endpoints lies in its simplicity and elementary character; well-known necessary and sufficient conditions from the theory of ordinary maxima and minima are used.An endpoint necessary condition is first obtained which is simply the well-known transversality condition. An additional condition is then developed which, together with the transversality condition, leads to a set of necessary and sufficient conditions for a given extremal to be locally optimal with respect to endpoint variations. While the second condition presented is akin to the classical focal-point condition, the result is new in form and is directly applicable to the optimal control problem. In addition, it is relatively simple to apply and is easy to implement numerically when an analytical solution is not possible. It should be useful in situations where the transversality conditions yield more than one choice for an optimal endpoint.An analytic solution for a simple geodetics problem is presented to illustrate the theory. A discussion of numerical implementation of the sufficiency conditions and its application to an orbit transfer example is also included.This work was supported in part by the National Aeronautics and Space Administration, Grant No. NGR-03-002-001.     

Value Functions and Transversality Conditions for Infinite-Horizon Optimal Control Problems

This paper investigates a relationship between the maximum principle with an infinite horizon and dynamic programming and sheds new light upon the role of the transversality condition at infinity as necessary and sufficient conditions for optimality with or without convexity assumptions. We first derive the nonsmooth maximum principle and the adjoint inclusion for the value function as necessary conditions for optimality. We then present sufficiency theorems that are consistent with the strengthened maximum principle, employing the adjoint inequalities for the Hamiltonian and the value function. Synthesizing these results, necessary and sufficient conditions for optimality are provided for the convex case. In particular, the role of the transversality conditions at infinity is clarified.     

Necessity of limiting co-state arcs in Bolza-type infinite horizon problem

We investigate necessary conditions of optimality for the Bolza-type infinite horizon problem with free right end. The optimality is understood in the sense of weakly uniformly overtaking optimal control. No previous knowledge in the asymptotic behaviour of trajectories or adjoint variables is necessary. Following Seierstad’s idea, we obtain the necessary boundary condition at infinity in the form of a transversality condition for the maximum principle. Those transversality conditions may be expressed in the integral form through an Aseev–Kryazhimskii-type formulae for co-state arcs. The connection between these formulae and limiting gradients of pay-off function at infinity is identified; several conditions under which it is possible to explicitly specify the co-state arc through those Aseev–Kryazhimskii-type formulae are found. For infinite horizon problem of Bolza type, an example is given to clarify the use of the Aseev–Kryazhimskii formula as an explicit expression of the co-state arc.     

Nonsmooth maximum principle for infinite-horizon problems

In this paper, we consider a class of infinite-horizon discounted optimal control problems with nonsmooth problem data. A maximum principle in terms of differential inclusions with a Michel type transversality condition is given. It is shown that, when the discount rate is sufficiently large, the problem admits normal multipliers and a strong transversality condition holds. A relationship between dynamic programming and the maximum principle is also given.The author is indebted to Francis Clarke for helpful suggestions and discussions.     

Maximum principle in the problem of time optimal response with nonsmooth constraints : PMM vol. 40, n 6, 1976, pp. 1014–1023

The problem of optimal response [1, 2] with nonsmooth (generally speaking, nonfunctional) constraints imposed on the state variables is considered. This problem is used to illustrate the method of proving the necessary conditions of optimality in the problems of optimal control with phase constraints, based on constructive approximation of the initial problem with constraints by a sequence of problems of optimal control with constraint-free state variables. The variational analysis of the approximating problems is carried out by means of a purely algebraic method involving the formulas for the incremental growth of a functional [3, 4] and the theorems of separability of convex sets is not used.Using a passage to the limit, the convergence of the approximating problems to the initial problem with constraints is proved, and for general assumptions the necessary conditions of optimality resembling the Pontriagin maximum principle [1] are derived for the generalized solutions of the initial problem. The conditions of transversality are expressed, in the case of nonsmooth (nonfunctional) constraints by a novel concept of a cone conjugate to an arbitrary closed set of a finite-dimensional space. The concept generalizes the usual notions of the normal and the normal cone for the cases of smooth and convex manifolds.     

A controllability minimum principle

A controllability minimum principle and two associated transversality conditions are presented, dealing with the controllability of nonlinear systems. The theorems represent necessary conditions for a control function to generate a system path which lies in the boundary of the set of points that are controllable to a target. The theorems presented here are controllability counterparts to Pontryagin's maximum principle, and undoubtedly these results will seem familiar or may have occurred to other researchers in the area of optimal control. The purpose of this paper is to make the distinction explicit and to establish the validity of these controllability theorems on their own merits. The theorems are demonstrated using a simple example and the principal result (a controllability minimum principle) is shown to be equivalent to the Kalman controllability criterion for linear systems.     

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.     

Sufficient conditions for the optimal control of a class of systems with continuous lags

Sufficient conditions in the form of a maximum principle are obtained for the optimal control of a system described by integro-differential equations and subject to some specified path constraints. The conditions are relaxed to allow for jumps in the adjoint variables at the junction points, provided a certain convexity hypothesis is satisfied for the constraint set at these points.This research was partially supported at Stanford University by the Office of Naval Research, Contract No. N-00014-67-A-0112-0011, by the National Science Foundation, Grant No. GP-31393, and by the US Atomic Energy Commission, Contract No. AT(04-3)-326-PA-18. It was also supported by the Department of Economics at Rice University.     

A maximum principle for an optimal control problem with integral constraints

In this paper, necessary corditions are obtained for an optimal control problem whose state variables are given in terms of integral equations. The conditions are obtained separately for Volterra equations and Fredholm equations. The main result for each case is the maximum principle and multiplier rule. For the Volterra equations, transversality conditions are obtained.     

Geometry of optimality conditions and constraint qualifications

Certain types of necessary optimality conditions for mathematical programming problems are equivalent to corresponding regularity conditions on the constraint set. For any problem, a certain natural optimality condition, dependent upon the particular constraint set, is always satisfied. This condition can be strengthened in numerous ways by invoking appropriate regularity assumptions on the constraint set. Results are presented for Euclidean spaces and some extensions to Banach spaces are given.This work was supported in part by the Office of Naval Research, Contract No. N00014-67-A-0321-0003 (NR-047-095).     

ON THE SUSTAINABLE GROWTH IN AN ECONOMY WITH PERFECTLY SUBSTITUTABLE EXHAUSTIBLE RESOURCES

Abstract A model of sustainable economic growth in an economy with two types of exhaustible resources is analyzed. The resources are assumed to be perfect substitutes with marginal rate of substitution varying over time. The optimal control framework is used to characterize the optimal paths under the maximin criterion. It is shown that the resource with increasing productivity is not used before the constant productivity resource is depleted. Afterwards the resource with an increasing productivity is asymptotically depleted as well. The results are based on an assumption that transversality conditions hold. A new sufficient condition for the transversality conditions is derived. Finally, an analogue of Hartwick’s rule for this non‐autonomous case is established.     

Sufficient Optimality Conditions for Optimal Control Problems with State Constraints

It is well-known in optimal control theory that the maximum principle, in general, furnishes only necessary optimality conditions for an admissible process to be an optimal one. It is also well-known that if a process satisfies the maximum principle in a problem with convex data, the maximum principle turns to be likewise a sufficient condition. Here an invexity type condition for state constrained optimal control problems is defined and shown to be a sufficient optimality condition. Further, it is demonstrated that all optimal control problems where all extremal processes are optimal necessarily obey this invexity condition. Thus optimal control problems which satisfy such a condition constitute the most general class of problems where the maximum principle becomes automatically a set of sufficient optimality conditions.     

Calculus of variations on time scales: weak local piecewise Crd solutions with variable endpoints

A nonlinear calculus of variations problem on time scales with variable endpoints is considered. The space of functions employed is that of piecewise rd-continuously Δ-differentiable functions (C¹_prd). For this problem, the Euler-Lagrange equation, the transversality condition, and the accessory problem are derived as necessary conditions for weak local optimality. Assuming the coercivity of the second variation, a corresponding second order sufficiency criterion is established.     

A minimax optimal control problem

A control system x=f(t,x,u) is considered, and a cost functional ess supT ₀tT ₁ G(t, x(t),u(t)) is to be minimized. Necessary conditions for optimality (maximum principle and transversality conditions) are derived. It is also shown that an optimal control is optimal for the corresponding problem on a subinterval of [T ₀,T ₁], if a certain controllability condition is satisfied.     

Necessary Optimality Conditions for Higher-Order Infinite Horizon Variational Problems on Time Scales

We obtain Euler?CLagrange and transversality optimality conditions for higher-order infinite horizon variational problems on a time scale. The new necessary optimality conditions improve the classical results both in the continuous and discrete settings: Our results seem new and interesting even in the particular cases when the time scale is the set of real numbers or the set of integers.     

Harvesting of a phytoplankton–zooplankton model

Considering that some phytoplankton and zooplankton are harvested for food, a phytoplankton–zooplankton model with harvesting is proposed and investigated. First, stability conditions of equilibria and existence conditions of a Hopf-bifurcation are established. Our results indicate that over exploitation would result in the extinction of the population and an appropriate harvesting strategy should ensure the sustainability of the population which is in line with reality. Furthermore, the existence of bionomic equilibria and the optimal harvesting policy are discussed. The present value of revenues is maximized by using Pontryagin’s maximum principle subject to the state equations and the control constraints. We discussed the case of optimal equilibrium solution. It is found that the shadow prices remain constant over time in optimal equilibrium when they satisfy the transversality condition. It is established that the zero discounting leads to the maximization of economic revenue and that an infinite discount rate leads to complete dissipation of economic rent. Finally, some numerical simulations are given to illustrate our results.     

A second-order condition that strengthens Pontryagin's maximum principle

We derive a second-order necessary condition for optimal control problems defined by ordinary differential equations with endpoint restrictions. This condition, based on a second-order restricted minimization test, bears a somewhat similar relation to the Weierstrass E-condition (the Pontryagin maximum principle) as the Legendre and Jacobi conditions bear to the Euler-Lagrange equation. Specifically, in the context of relaxed controls, the E-condition for free endpoint problems asserts that if a function achieves its minimum over a convex set Q at some point $\text{q}$ then its one-sided directional derivatives at $\text{q}$ into Q are nonnegative. Our new condition, when applied to the special case of free endpoint problems, corresponds to the observation that if such a one-sided directional derivative at $\text{q}$ is 0 then the corresponding second directional derivative is nonnegative. This new condition effectively supplements the Pontryagin maximum principle over the singular regimes of “weakly” normal extremals that are candidates for either a relaxed or an ordinary restricted minimum. Like some other second-order methods, this condition is global over the control set but, unlike the other tests, it is also global over time. A number of examples illustrate its use and behavior.     

Necessary conditions for problems with higher derivative bounded state variables

This paper combines the separate works of two authors. Tan proves a set of necessary conditions for a control problem with second-order state inequality constraints (see Ref. 1). Russak proves necessary conditions for an extended version of that problem. Specifically, the extended version augments the original problem by including state equality constraints, differential and isopermetric equality and inequality constraints, and endpoint constraints. In addition, Russak (i) relaxes the solvability assumption on the state constraints, (ii) extends the maximum principle to a larger set, (iii) obtains modified forms of the relationH =H _t and of the transversality relation usually obtained in problems of this type, and (iv) proves a condition concerning (t ¹), the derivative of the multiplier functions at the final time.Russak's work was supported by a NPS Foundation Grant.Tan is indebted to his thesis advisor, Professor M. R. Hestenes, for suggesting the topic and for his help and guidance in the development of his work. Tan's work was supported by the Army Research Office, Contract No. DA-ARO-D-31-124-71-G18.     

Optimal interception with time constraint

This paper considers the problem of minimum-fuel interception with time constraint. The maneuver consists of using impulsive thrust to bring the interceptor from its initial orbit into a collision course with a target which is moving on a well-defined trajectory. The intercept time is either prescribed or is restricted to be less than an upper limit.The necessary conditions and the transversality conditions for optimality are discussed. The method of solution amounts to first solving a set of equations to obtain the primer vector for an initial one-impulse solution. Then, based on the information provided by the primer vector, rules are established to search for the optimal solution if the initial one-impulse trajectory is not optimal. The method is general, in the sense that it allows for solving the problem of three-dimensional interception with arbitrary motion for the target.Several numerical examples are presented, including orbital interceptions and interception at hyperbolic speeds of a ballistic missile.This research was supported by US Army Strategic Defense Command, Contract No. DASG60-88-C-0037.