Bounded State Space Research Articles

Boundedly optimal control of piecewise deterministic systems

This paper deals with a class of undiscounted piecewise deterministic control problems with bounded state space and possibly undiscounted cost. The concept of boundedly optimal control is introduced for undiscounted cost problems. This optimality concept is stronger than the average cost optimality. Via a limit argument for a family of associated problems with discounted cost, we show that a dynamic programming formalism can be developed for the characterization of boundedly optimal solutions.

Optimal Control Theory and Static Optimization in Economics

Optimal control theory is a technique being used increasingly by academic economists to study problems involving optimal decisions in a multi-period framework. This textbook is designed to make the difficult subject of optimal control theory easily accessible to economists while at the same time maintaining rigour. Economic intuitions are emphasized, and examples and problem sets covering a wide range of applications in economics are provided to assist in the learning process. Theorems are clearly stated and their proofs are carefully explained. The development of the text is gradual and fully integrated, beginning with simple formulations and progressing to advanced topics such as control parameters, jumps in state variables, and bounded state space. For greater economy and elegance, optimal control theory is introduced directly, without recourse to the calculus of variations. The connection with the latter and with dynamic programming is explained in a separate chapter. A second purpose of the book is to draw the parallel between optimal control theory and static optimization. Chapter 1 provides an extensive treatment of constrained and unconstrained maximization, with emphasis on economic insight and applications. Starting from basic concepts, it derives and explains important results, including the envelope theorem and the method of comparative statics. This chapter may be used for a course in static optimization. The book is largely self-contained. No previous knowledge of differential equations is required.

Robust control for linear systems with structured state space uncertainty

Abstract A recent sufficient condition for the stability robustness of linear systems with time-varying norm bounded state space uncertainty is extended to include the structure of the uncertainty. Our new result requires that the nominal eigenvalues lie to the left of a vertical line in the complex plane which is determined by a norm involving the structure of the uncertainty and the nominal closed-loop eigenvector matrix. Therefore, this robustness result is especially well suited to the design of control systems using eigenstructure assignment. When the uncertainty is time-invariant, our norm is also an upper bound on the incremental eigenvalue perturbations. We also consider the use of Perron weightings to reduce conservatism and the extension of the results to discrete time systems.

Invariant probabilities for systems in a random environment — With applications to the brusselator

In this paper we are concerned with problems of the long-term behavior for nonlinear systems in random environment. The general model is assumed to be given by an ordinary differential equation with random parameters or random input. The disturbance process can be taken from a fairly general class of Markov processes having a bounded state space. In terms of the system’s dynamics we give sufficient conditions for the existence and uniqueness of invariant probabilities. Finally, we apply these results to the two-dimensional biochemical model which is known as the Brusselator.

Optimal programming problems with a bounded state space.

A new set of necessary conditions are presented for solutions to optimal programming problems with a state variable inequality constraint (SVIC). On a constrained arc, the dimension of the state space is reduced, and the influence functions associated with this reduced state space are shown to be unique and continuous across junctures with unconstrained arcs. It is also shown that unconstrained arcs must satisfy certain constraints at both ends of a constrained arc and that explicit use of this fact must be made if the SVIC is adjoined directly to the performance index. ^EVERAL investigators have discussed necessary condi^ tions for solutions to optimal programming problems with a state variable inequality constraint.15 Most of these investigators recognized that, at a point where the system enters onto a constrained arc (an entry point), all feasible unconstrained arcs passing through this point must satisfy certain tangency constraints, namely, these arcs must have zero values of the state variable constraint function and all of its time derivatives that do not involve the control variable (say p-1 of them). Since control of the constraint function is realized only by changing its pth time derivative, no finite control will keep the system on the constraint boundary if the path reaching the constraint boundary does not meet these tangency constraints. In this paper, we point out that the tangency constraints also apply to the unconstrained arc at a point where the system leaves a constrained arc (an exit point). These exitpoint tangency constraints are satisfied automatically if the necessary conditions of Ref. 3 are used. However, if one uses the direct adjoining approach of Ref. 4, we shall show that explicit use must be made of the tangency constraints at both ends.

Optimal programming problems with a bounded state space

Optimal programming problems with a bounded state space

Application of the maximum principle for bounded state space to the hydro-steam dispatch problem

Application of the maximum principle for bounded state space to the hydro-steam dispatch problem