Results Of Optimal Control Theory Research Articles

Optimal navigation of microswimmers in complex and noisy environments

We design new navigation strategies for travel time optimization of microscopic self-propelled particles in complex and noisy environments. In contrast to strategies relying on the results of optimal control theory or machine learning approaches, implementation of these protocols can be done in a semi-autonomous fashion, as it does not require control over the microswimmer motion via external feedback loops. Although the strategies we propose rely on simple principles, they show arrival time statistics strikingly close to optimality, as well as performances that are robust to environmental changes and strong fluctuations. These features, as well as their applicability to more general optimization problems, make these strategies promising candidates for the realization of optimized semi-autonomous navigation.

Dynamic analysis and optimal control of a class of SISP respiratory diseases

In this paper, the actual background of the susceptible population being directly patients after inhaling a certain amount of PM is taken into account. The concentration response function of PM is introduced, and the SISP respiratory disease model is proposed. Qualitative theoretical analysis proves that the existence, local stability and global stability of the equilibria are all related to the daily emission of PM and PM pathogenic threshold K. Based on the sensitivity factor analysis and time-varying sensitivity analysis of parameters on the number of patients, it is found that the conversion rate β and the inhalation rate η has the largest positive correlation. The cure rate γ of infected persons has the greatest negative correlation on the number of patients. The control strategy formulated by the analysis results of optimal control theory is as follows: The first step is to improve the clearance rate of PM by reducing the PM emissions and increasing the intensity of dust removal. Moreover, such removal work must be maintained for a long time. The second step is to improve the cure rate of patients by being treated in time. After that, people should be reminded to wear masks and go out less so as to reduce the conversion rate of susceptible people becoming patients.

Optimal control of a perturbed sweeping process via discrete approximations

The paper addresses an optimal control problem for a perturbed sweeping process of the rate-independent hysteresis type described by a controlled ``play-and stop operator with separately controlled perturbations. This problem can be reduced to dynamic optimization of a state-constrained unbounded differential inclusion with highly irregular data that cannot be treated by means of known results in optimal control theory for differential inclusions. We develop the method of discrete approximations, which allows us to adequately replace the original optimal control problem by a sequence of well-posed finite-dimensional optimization problems whose optimal solutions strongly converge to that of the controlled perturbed sweeping process. To solve the discretized control systems, we derive effective necessary optimality conditions by using second-order generalized differential tools of variational analysis that explicitly calculated in terms of the given problem data.

Carath&#233odory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory

The purpose of the present paper is to show that the most prominent results in optimal control theory, the distinction between state and control variables, the maximum principle, and the principle of optimality, resp. Bellman's equation are immediate consequences of Carath&#233odory's achievements published about two decades before optimal control theory saw the light of day.

The maximum principle, Bellman's equation, and Carathéodory's work

One of the most important and deep results in optimal control theory is the maximum principle attributed to Hestenes (1950) and in particular to Boltyanskii, Gamkrelidze, and Pontryagin (1956). Another prominent result is known as the Bellman equation, which is associated with Isaacs' and Bellman's work (later than 1951). However, precursors of both the maximum principle and the Bellman equation can already be found in Caratheodory's book of 1935 (Ref. 1a), the first even in his earlier work of 1926 which is given in Ref. 2. This is not a widely acknowledged fact. The present tutorial paper traces Caratheodory's approach to the calculus of variations, once called the "royal road in the calculus of variations," and shows that famous results in optimal control theory, including the maximum principle and the Bellman equation, are consequences of Caratheodory's earlier results.

A class of nondifferentiable control problems

Optimality conditions and duality results are obtained for a class of control problems having a nondifferentiable term in the integrand of the objective functional. These results generalize many well-known results in optimal control theory involving differentiable functions, and also provide a relationship with certain nondifferentiable mathematical programming problems. Some extensions concerning the unified treatment of optimal control theory and continuous programming are also mentioned. Finally, a control problem containing an arbitrary norm, along with its appropriate norm, is given.

Optimal Control Theory Applied to Fusion Plasma Thermal Stabilization

Many authors have investigated stability characteristics and performance of various burn control schemes. The work presented here represents the first application of optimal control theory to the problem of fusion plasma thermal stabilization. The objectives of this initial investigation were to develop analysis methods, demonstrate tractability, and present some preliminary results of optimal control theory in burn control research.

Active protection of domains of a vibrating structure by using optimal control theory: A model determination

The problem of active protection of continuous structure domains is developed by using the well known results of optimal control theory. A mathematical model of the mechanical system is defined and a spatial approximation process is presented according to the basis theory of functional analysis applied to the discretization problem. In finite dimension spaces, the state analysis is developed, including the models of the structure and the perturbation sources, and the corresponding state equations are presented. Various protection criteria compatible with the state analysis scheme are proposed and after presentation of the observer problem, the solution of the optimal regulator problem is developed. Numerical results simulating the time response of a simply supported beam under control are presented for two different protection criteria. The results allow one to proceed to a qualitative comparison of the relative merits of the chosen criteria.

Renewable resource depletion: the conservation problem

In this article we explore some of the basic concepts of management of renewable resources. The idea of maximum sustainable yield is described and growth models are explored. Harvesting effort is introduced and bionomic equilibrium, where costs are equated to revenue, is obtained for an open access resource. The optimal policy for a sole owner is developed, firstly with no time discounting and secondly with a positive discount rate. The results of optimal control theory are used to find optimal policies for the trade‐off between present and future generations.

Stochastic Control Theory Applied to Fishery Management

The results of optimal control theory applied to deterministic fishery models is extended to the continuous stochastic case. The simple logistic model is generalized to stochastic models by singly considering the logistic parameters and the fishing harvest as random processes. Approximate probability density functions are calculated for one model. The stochastic models are applied to the Atlantic sea-scallop fishery. Dynamic programming is used to obtain control solutions.

Singular optimal control problems

For the last 30 years the optimization of nonsingular control problems has been an Important part of control engineering, and its mathematical theory is well developed and widely known. On the other hand, singular control problems prove more difficult to analyse and—although necessary conditions for optimality of singular controls have been established over the past decade—It is only recently that sufficient, and necessary and sufficient, conditions have been formulated. The purpose of this book Is to collect together all known results in optimal control theory (as well as appropriate computational methods) which can be applied to the singular problems In optimal control and which up to now have been scattered In numerous journals. Complete and self-contained, the volume begins with an historical survey of singular control problems and leads to the presentation of important, recent results in the field. There are specific real-world applications and the authors discuss those avenues of research which require further Investigation. All those involved In the optimization of dynamical systems will welcome the publication of this book. In addition to advanced students, lecturers and research workers in universities, this will include practising mechanical, chemical and electrical engineers, builders, textile technologists, paper scientists and chemists, and many concerned with non-technical fields such as economics and business management Contents An historical survey of singular control problems Introduction. Singular control in space navigation. Method of Mlele via Green's theorem. Linear systems—quadratic cost Necessary conditions for singular optimal control. Sufficient conditions and necessary and sufficient conditions for optimality. References. Fundamental concepts Introduction. The general optimal control problem. The first variation of J. The second variation of J. A singular control problem. References. Necessary conditions for singular optimal control Introduction. The generalized Legendre-Clebsch condition. Jacobson's necessary condition. References. Sufficient conditions and necessary and sufficient conditions tor non-negativity of nonsingular and singular second variations Introduction. Preliminaries. The nonsingular case. Strong positivlty and the totally singular second variation. A general sufficiency theorem for the second variation. Necessary and sufficient conditions for non-negativity of the totally singular second variation. Necessary conditions for optimality. Other necessary and sufficient conditions. Sufficient conditions for a weak local minimum. Existence conditions for the matrix Rlccati differential equation. Conclusion. References. Computational methods for singular control problems Introduction. Computational application of the sufficiency conditions of theorems in the previous chapter. Computation of optimal singular controls. Joining of optimal singular and non-singular sub-arcs. Conclusion. References. Conclusion The Importance of singular optimal control problems. Necessary conditions. Necessary and sufficient conditions. Computational methods. Switching conditions. Outlook for the future Author index. Sublect index.

Optimal Control Models of Regional Economies

Control of socio-economic systems which are partitioned into a set of N spatial regions presents some new and fascinating problems in optimal control. Most existing developments of optimal control theory for spatially distributed systems are based on models described by partial differential and difference equations (see for example (1)). For such models it must be assumed that the space is cortinuously differentiable or at least regularly partitioned (e.g. on a Cartesian lattice) mid that interaction between locations is controlled by a pattern of distance decay. In addition it is usually assumed that at least the structure (but not necessarily the parameters) of the model is known a priori. Spatial models described bypartial differential equations are of little utility for developing control models of regional economies, and this paper is concerned with extending the results of optimal control theory to situations in which the spatial domain of interest (e.g. a national economy) is partitioned into N regions which interact arbitrarily such that no distance metric is necessarily involved. In the first section a suitable mathematical representation of regional systems is developed. In section two the results of control theory are extended to this system description for T-stage strategies under uncertainty. Section three treats some of the major research problems that remain to be answered, especially the interaction between parameter estimation and the derivation of optimal controllers.

Three mode control as an optimal control

Three mode control with Ziegler-Nichols or Cohen-Coon controller settings is shown to be the solution to an optimal control problem for a linear first order system with dead time and a quadratic cost functional. The implication of this result is that in the absence of an economically meaningful cost criterion there is no reason to prefer the results of optimal control theory to traditional design procedures for feedback control systems. The multivariable generalization of the optimal control problem solved here provides the means for extending classical control experience in design of single loop processes to multivariable control systems