Short communications on the existence of touch points for first-order state inequality constraints

The appearance of touch points in state-constrained optimal control problems with general vector-valued control is studied. Under the assumption that the Hamiltonian is regular, touch points for first-order state inequalities are shown to exist only under very special conditions. In many cases of practical importance these conditions can be used to exclude touch points a priori without solving an optimal control problem. The results are demonstrated on a simple example.     

Optimal control of HVAC systems using DDP and NLP techniques

The objective of this study is to apply the differential dynamic programming (DDP) technique of optimal control to heating, ventilating and air-conditioning (HVAC) systems and to compare its performance with a non-linear programming (NLP) technique using the sequential quadratic programming method. The DDP technique is briefly described and studied. Limitations of the technique are noted. Three cases of a system that has been treated previously in the literature are optimized by the two techniques and the computational times compared. The study shows DDP to be efficient compared with NLP for the example problems. NLP is, however, more robust and general and can treat constraints on the state variables directly. Further investigation is needed for larger-scale problems to fully explore the features of the two methods.     

Optimal multi‐interval control of a cantilever beam by a recursive control algorithm

The optimal‐distributed control of a transversely vibrating cantilever beam is studied with the objective of minimizing the deflection and velocity in a given period of time with the minimum possible expenditure of force. The beam undergoes transient vibrations and is subject to given displacement and velocity initial conditions. The control is exercised by means of a transversely distributed force referred to as the control force. In the present study, a multi‐interval optimal control method is developed with the application of a maximum principle. The method consists of dividing the control duration into several intervals and using the maximum principle to obtain the optimality conditions at each interval. The explicit solutions for a cantilever beam are obtained by a recursive algorithm that takes the final conditions of the last interval as the initial conditions of the next interval. The formulation and the method of solution are suitable and convenient for digital computation. Numerical results are given, which compare the deflections, velocities and the control force under the optimal multi‐interval control with those under the optimal single‐interval control. Copyright © 2008 John Wiley & Sons, Ltd.     

Optimal digital redesign of continuous-time systems using fractional-order hold

This paper proposes a novel optimal digital redesign technique using fractional-order hold (f.o.h.) for finding a dynamic digital control law from the available analog counterpart and for simultaneously minimizing a quadratic performance index. A method for converting the dynamic digital control law into a static one, through a tuning parameter, is also presented. The proposed technique can be applied to a system cascaded with a more general class of reference inputs than simple step inputs, and the developed digital controller can be implemented via low cost microcomputers. The superiority of using the fractional-order hold over the zero-order hold and the first-order hold is exploited. © 1997 John Wiley & Sons, Ltd.     

Optimal control of deterministic epidemics

Various deterministic optimal control models for SIR‐epidemics are investigated in this paper. The epidemics are governed by a rather general interaction, which covers most cases studied in the literature. Vaccination, quarantine, screening or health promotion campaigns as forms of control are considered. In all cases one finds a maximum effort control on some initial time interval. In addition, uniqueness and monotonicity properties of these models are studied. The results are also extended to the infinite time‐horizon situation. Copyright © 2000 John Wiley & Sons, Ltd.     

Optimal control of innate immune response

Treatment of a pathogenic disease process is interpreted as the optimal control of a dynamic system. Evolution of the disease is characterized by a non‐linear, fourth‐order ordinary differential equation that describes concentrations of pathogens, plasma cells, and antibodies, as well as a numerical indication of patient health. Without control, the dynamic model evidences sub‐clinical or clinical decay, chronic stabilization, or unrestrained lethal growth of the pathogen, depending on the initial conditions for the infection. The dynamic equations are controlled by therapeutic agents that affect the rate of change of system variables. Control histories that minimize a quadratic cost function are generated by numerical optimization over a fixed time interval, given otherwise lethal initial conditions. Tradeoffs between cost function weighting of pathogens, organ health, and use of therapeutics are evaluated. Optimal control solutions that defeat the pathogen and preserve organ health are demonstrated for four different approaches to therapy. It is shown that control theory can point the way toward new protocols for treatment and remediation of human diseases. Copyright © 2002 John Wiley & Sons, Ltd.     

Optimal drug control in a four-dimensional HIV infection model

In this paper, we consider a four-dimensional version of a human immunodeficiency virus (HIV) infection model, which is an extension of some previous three-dimensional models. We approach the treatment problem by adding two controls u₁ and u₂ to the system for inhibiting viral production and preventing new infections. In fact, u₁ is added to components of uninfected and infected cells to represent the effect of chemotherapy on the interaction of uninfected CD4⁺ T cells with infected cells. u₂ is considered in the effector immune component as immunotherapy. The purpose of this work is to control the progress of the disease in a steady state. Hence, first, we obtain a relation between the two controls u₁ and u₂ such that a Hopf bifurcation occurs. Next, the Pontryagin minimum principle will be applied to derive the optimal therapy for HIV. At the end, numerical results are presented.     

Optimal control of continuous systems with impulse controls

Impulse control problems, in which a continuously evolving state is modified by discrete control actions, have applications in epidemiology, medicine, and ecology. In this paper, we present a simple method for solving impulse control problems for systems of differential equations. In particular, we show how impulse control problems can be reformulated and solved as discrete optimal control problems. The method is illustrated with two examples. Published 2014. This article has been contributed to by US Government employees and their work is in the public domain in the USA.     

Optimal control of Markovian queueing systems

Dynamic optimization of queueing systems is treated by optimal control theory. This work is based on modelling the queueing problem as a time-varying continuous Markov chain. Necessary and sufficient conditions are given for a broad class of problems which include both scalar and Markovian dynamic programming control structures. Continuity of the switching function is used to characterize optimality near the end points of the horizon. Special properties of the model are exploited to ensure the absence of singular subarcs. Numerical results based on the use of a gradient algorithm report the effect of increasing the system capacity, a comparison of scalar versus Markovian dynamic programming controls, and an application to a multiprogrammed computer system.     

Optimal control of a delayed hepatitis B viral infection model with HBV DNA‐containing capsids and CTL immune response

This paper deals with an optimal control problem of a time‐delayed differential equation model that describes the interactions between hepatitis B virus (HBV) with HBV DNA‐containing capsids, liver cells (hepatocytes), and cytotoxic T‐lymphocyte immune response. Both the treatment and the intracellular delay are incorporated into the model. Furthermore, the existence of the optimal control pair is studied, and Pontryagin's minimum principle is used to characterize these 2 optimal controls. The first of them represents the efficiency of drug treatment in preventing new infections, whereas the second stands for the efficiency of drug treatment in inhibiting viral production. The optimality system is derived and solved numerically using the forward and backward difference approximation. Finally, numerical simulations are established to show the role of optimal therapy in controlling viral replication.     

Neural net control of nonlinear plants through state feedback

A heuristic design method for state feedback fixed (non‐adaptive) neural net controller in nonlinear plants is presented. The design method evolves as a natural extension of the optimal control strategies employed in linear systems. A multi‐layered feed‐forward neural network is used as the feedback controller. The controller is trained to directly minimize a suitable cost function comprised of the plant output, states and the input. The optimization is carried out using a gradient scheme that employs the recently developed concept of block partial derivatives. The applicability of the proposed design method is demonstrated through simulated examples. Simulation studies include a variety of optimal control problems in nonlinear plants such as: minimum energy and minimum fuel problems, state tracking, output servo with integrator, and unconstrained and constrained regulation. Copyright © 2000 John Wiley & Sons, Ltd.     

Optimal and sub‐optimal control in Dengue epidemics

This work concerns the application of the optimal control theory to Dengue epidemics. The dynamics of this insect‐borne disease is modelled as a set of non‐linear ordinary differential equations including the effect of educational campaigns organized to motivate the population to break the reproduction cycle of the mosquitoes by avoiding the accumulation of still water in open‐air recipients. The cost functional is such that it reflects a compromise between actual financial spending (in insecticides and educational campaigns) and the population health (which can be objectively measured in terms of, for instance, treatment costs and loss of productivity). The optimal control problem is solved numerically using a multiple shooting method. However, the optimal control policy is difficult to implement by the health authorities because it is not practical to adjust the investment rate continuously in time. Therefore, a suboptimal control policy is computed assuming, as the admissible set, only those controls which are piecewise constant. The performance achieved by the optimal control and the sub‐optimal control policies are compared with the cases of control using only insecticides when Breteau Index is greater or equal to 5 and the case of no‐control. The results show that the sub‐optimal policy yields a substantial reduction in the cost, in terms of the proposed functional, and is only slightly inferior to the optimal control policy. Copyright © 2001 John Wiley & Sons, Ltd.     

Optimal control of harvesting in a stochastic metapopulation model

We consider a metapopulation model for a single species inhabiting two bounded contiguous regions where movement of the population across the shared boundary is allowed. The population in one of the bounded regions can be harvested. We introduce stochastic growth rates for the two populations in a system of ordinary differential equations that model the population dynamics in these two regions. We derive the resulting stochastic control problem with harvesting in the one region as the control. The existence of an optimal control is established by solving an associated quasi‐linear–quadratic optimal control problem. We present numerical simulations to illustrate several scenarios. Copyright © 2011 John Wiley & Sons, Ltd.     

Optimal control approach for discontinuous dynamical systems

In this work, we consider a wide class of discontinuous dynamical systems, discontinuity of which is based on the sign (for short sgn) function. We propose a smooth optimal control problem to solve the main discontinuous system. By solving some numerical examples in mechanical engineering, we show the efficiency of our approach with respect to 2 smoothing methods for discontinuous systems.     

Optimal control analysis of a mathematical model for unemployment

Unemployment is continuously increasing worldwide because of enormous increase in population. This paper attempts to propose an optimal control policy for a deterministic unemployment model. The model considers three states, namely, unemployment, employment, and newly created vacancies. Factors like retirement and death of employed persons, termination from job, and so forth are also included in the model. The optimal control analysis for proposed unemployment model is performed using Pontryagin's maximum principle. The conditions for optimal control of the unemployment problem with effective use of implemented policies to provide employment to unemployed persons and to create new vacancies are derived and analyzed. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal control of a production system with periodic maintenance

In this paper we consider a production system consisting of one machine for which maintenance is performed on a periodic basis. When the machine is undergoing maintenance, the system is shut down and cannot produce. One part-type is produced and the demand rate is assumed to be constant. In order to make on-time delivery, the objective is to produce following the demand as closely as possible. However, the maintenance disruptions make the production deviate from the demand. We formulate the production flow control problem as an optimal control model and use Pontryagin's minimum principle to solve the special case of one up-down cycle. We then solve the general N-cycle problem based on the one-cycle solution.     

Optimal control of a delayed breast cancer stem cells nonlinear model

In this article, we consider a nonlinear model, which is governed by an ordinary differential equations system with time delays in state and control. The model is used in order to describe the growth of breast cancer cells under therapy. We seek optimal therapies to minimize the number of cancer cells as well as the total quantity of drug used in the treatment. In this way, we formulate an optimal control problem. We prove the existence of an optimal therapy and use Pontryagin's maximum principle in order to find optimality conditions, which characterize such optimal therapy. At last, both numerical results and conclusion are presented. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal control study to enhance oil production in labscale Vapex by varying solvent injection pressure with time

In this study, we determine and validate optimal policies of solvent injection pressure versus time to enhance oil production in labscale Vapex or vapor extraction of heavy oil. The study utilizes propane as solvent for a heavy oil of viscosity 14 500 mPa s. The necessary conditions for maximum oil production are derived using a detailed mass transfer model with interfacial solvent concentration versus time as a control function. Based on these conditions, a computational algorithm is developed and implemented to determine the optimal control, which is then converted into the optimal policy of solvent injection pressure versus time using experimental data. The experiments demonstrate that the optimal policy enhances oil production by 20–37% compared with that using constant solvent injection pressure. Moreover, the average deviation of the optimally calculated oil production from its experimental counterpart, both obtained using the same optimal solvent injection policy, is found to be a low value of 4.8%. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal control of a wind energy conversion system and a wind turbine

Currently, a maximum power point tracking (MPPT) unit implemented in a wind energy conversion system (WECS) extracts the maximum mechanical power from the wind turbine used in the WECS. Therefore, the MPPT unit acts as a maximum mechanical power tracker (MMPT) that performs optimal control of the wind turbine to extract the maximum mechanical power from the wind turbine. In this paper, the basic concept of a maximum electrical power tracker (MEPT) is presented both theoretically and technically. It is demonstrated that an MEPT implemented in a WECS maximizes the output electrical power of the WECS. Thus, in contrast with an MMPT, the proposed MEPT optimally controls the whole of the WECS, rather than the wind turbine, to extract the maximum electrical power from the WECS. Since, in the WECS, the power efficiency of the whole of the WECS, not the wind turbine, should be maximized to extract the maximum output electrical power from the WECS, the conventional MPPT unit acting as an MMPT should be replaced with the proposed MEPT, and this is the superiority of the proposed MEPT to an MMPT or a conventional MPPT unit. To provide experimental verifications, 2 novel MEPT and MMPT with simple structures and better performance compared to the MPPT techniques commonly used in WECSs have been constructed, which are presented in detail.