Infinite-time Optimal Control Problem Research Articles

New Economic Model of Eutrophication of Shallow Lakes and the Existence of Equivalent Policies

The search for Skiba points in infinite-time optimal control problems with several steady states is an active research area. In this paper, we apply the backward integration method and show its power and relative simplicity as a tool for this task. We apply the method for the classical model of the economy shallow lake based on the phosphorous dynamics and explicitly compute the optimal trajectories and the Skiba points. We also suggest a more natural welfare function, which takes into account the finite value of any economic resource, and show, again in an explicit manner, that it also possesses Skiba points.

Skiba points in free end-time problems

Since the end of the seventies Skiba points have been studied in infinite time optimal control problems with multiple steady states. At such a Skiba point the decision maker is indifferent between choosing trajectories that approach different steady states. This paper extends this theory towards free end-time optimal control problems, where the decision maker collects a salvage value at the endogenous horizon date. In particular, besides operating forever, the decision maker can choose to stop operations immediately, or to operate during a finite time interval after which the decision maker stops and collects a salvage value.This paper illustrates the new theory with a firm-level capital accumulation problem in which the manager has the option to sell the firm (be acquired) at any time. This situation may be relevant for certain high-tech start-ups that create intellectual property which may be of value to an acquiring firm and which is hard for an outside firm to tap in other ways (e.g., via licensing).

On the use of gradual dense–sparse discretizations in receding horizon control

SUMMARYA key factor to success in implementations of real time optimal control, such as receding horizon control (RHC), is making efficient use of computational resources. The main trade‐off is then between efficiency and accuracy of each RHC iteration, and the resulting overall optimality properties of the concatenated iterations, that is, how closely this represents a solution to the underlying infinite time optimal control problem (OCP). Both these issues can be addressed by adapting the RHC solution strategy to the expected form of the solution. Using gradual dense–sparse (GDS) node distributions in direct transcription formulations of the finite time OCP solved in each RHC iteration is a way of adapting the node distribution of this OCP to the fact that it is actually part of an RHC scheme. We have previously argued that this is reasonable, because the near future plan must be implemented now, but the far future plan can and will be revised later. In this paper, we investigate RHC applications where the asymptotic qualitative behavior of the OCP solution can be analyzed in advance. For some classes of systems, explicit exponential convergence rates of the solutions can be computed. We establish such convergence rates for a class of control affine nonlinear systems with a locally quadratic cost and propose to use versions of GDS node distributions for such systems because they will (eventually) be better adapted to the form of the solution. The advantages of the GDS approach in such settings is illustrated with simulations. Copyright © 2013 John Wiley & Sons, Ltd.

Constrained Optimal Control of Hybrid Systems With a Linear Performance Index

We consider the constrained finite and infinite time optimal control problem for the class of discrete-time linear hybrid systems. When a linear performance index is used the finite and infinite time optimal solution is a piecewise affine state feedback control law. In this paper, we present algorithms that compute the optimal solution to both problems in a computationally efficient manner and with guaranteed convergence and error bounds. Both algorithms combine a dynamic programming exploration strategy with multiparametric linear programming and basic polyhedral manipulation

Periodic Near Optimal Control

We consider the infinite time optimal control problem of minimizing an average cost functional for nonlinear singularly perturbed systems. In particular, we are interested in the existence of near optimal trajectories which are periodic. For this purpose, we investigate nonlinear control systems with three time scales and construct a limiting system for the slowest motion by an iterated averaging procedure. We furthermore show that global tools, invariant control sets, and local tools, Lie algebra ranks of vector fields, are preserved under singular perturbations. The existence of periodic near optimal trajectories is verified under the condition that the fast subsystem and an unperturbed slow subsystem are controllable and that a standard Lie algebra rank condition for local accessibility holds.

Estimate of Periodic Suboptimal Controls

We consider the infinite-time optimal control problem of minimizing an average functional for nonlinear control systems. For controllable systems, we give an explicit estimate for the required period T of ɛ-suboptimal T-periodic orbits. Moreover, we show that controllable systems are characterized by the existence of periodic suboptimal trajectories for any average functional.

Necessary conditions for an infinite time optimal control problem

In this paper, we deal with an optimal control problem with infinite transfer time. Using methods which involve local conditional stability, we obtain necessary conditions for optimality under the assumption that the right-hand side of the state equation is Fréchet-differentiable at every point of the optimal solution, and under some weak assumptions about the asymptotical behaviour of the set of perturbations of the solution. The results are illustrated in a specific case considered by Pontryagain et al.

Criteria of optimality in the infinite-time optimal control problem

The present paper gives a systematic presentation of different definitions of optimality in the infinite-time optimal control problem. Some of these definitions are new, while others have been used throughout the literature, sometimes with different names. The logical implications between them are clearly stated, corresponding comparison criteria for solutions are defined, and other relations as well as two types of equivalence relations are established.

Solution of Finite-Time LQP Optimal Control Problems for Discrete-Time Systems

The deterministic discrete-time optimal control problem for a finite optimization interval is considered. A solution is obtained in the z-domain by embedding the problem within a equivalent infinite time problem. The optimal controller is time-invariant and may be easily implemented. The controller is related to the solution of the usual infinite time optimal control problem due to Wiener. This new controller should be of value in self-tuning control laws where a finite interval controller is particularly important.