The **optimal** **control** design is aimed to obtain a best possible system of a particular type with respect to a cer- tain performance index. In the **optimal** **control** design, the performance index replaces the conventional design criteria. A transfer function for the BLDC drive is derived in this paper. In this paper, for digitally PWM con- trolled BLDC motor drive, the **optimal** **control** system is designed. To achieve this optimization, the state and **control** variables are formulated in this paper. Based on **optimal** **control** **theory**, the BLDC performance charac- teristics close to the **optimal** **control** system are synthesized.

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Navigation modules are capable of driving a robotic platform without direct human participation. However, for some specific contexts, it is preferable to give the **control** to a human driver. The human driver participation in the robotic **control** process when the navigation module is running raises the share **control** issue. This work presents a new ap- proach for two agents collaborative planning using the **optimal** **control** **theory** and the three-layer architecture. In particu- lar, the problem of a human and a navigation module collaborative planning for a trajectory following is analyzed. The collaborative plan executed by the platform is a weighted summation of each agent **control** signal. As a result, the pro- posed architecture could be set to work in autonomous mode, in human direct **control** mode or in any aggregation of these two operating modes. A collaborative obstacle avoidance maneuver is used to validate this approach. The pro- posed collaborative architecture could be used for smart wheelchairs, telerobotics and unmanned vehicle applications. Keywords: Robotic Architecture; Share **Control**; Three-Layer Architecture; Cooperative **Control**;

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Abstract: The aim of the article is the elaboration of parametric resonance **theory** at piecewise constant frequency modulation. The investigation is based on the analogy with optics and **optimal** **control** **theory** (OCT) application. The exact expressions of oscillation frequency, gain/damping coefficients, dependencies of these coefficients on the modulation depth, duty ratio and initial phase are derived. First of all, the results obtained on the basis of the energy behavior analysis (at the conjunction conditions execution) in frictionless systems are presented. The well-known parametric resonance triggering condition is revised and adjusted. The heuristic feedback introduction (based on the energy behavior analysis) in the oscillation equation permits one to prove that the frequency modulation satisfying the parametric resonance condition is not necessary and sufficient condition of the oscillations unlimited increase. Their damping/shaking up formally corresponds by the frequency and duty ratio to the condition of the equality of optical paths to the quarter-wavelength characteristic of the interference filter or mirror. The unity of space-time coordinates shows itself in this specific form of the optical-mechanical analogy due to the general Hill’s equation description. It is marked that this equation **theory** underlies most of metamaterials advantages because all transport phenomena imply different wave – electromagnetic, acoustic, spin etc. propagation one way or another. The question about **control** uniqueness arises that is modulating frequency, duty ratio and signature sign uniqueness. Another question of characteristic index extremum at different controls is tightly bound with the former. The answers to these questions are obtained on the basis of OCT. The similarity of the **optimal** **control** problem solution and the one obtained at the heuristic feedback introduction through fundamental solutions product permits one to introduced the new form named general or mixed Hamiltonian along with the ordinary and OCT Hamiltonians. Besides this mixed Hamiltonian equality to zero together with the Wronskian constancy (almost everywhere) is the useful analogous in form to the Liouville’s theorem equation. The nonlinearity accounting using the OCT formalism is described too.

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The whole topic has been extensively studied since the beginning of the twen- tieth century and has been recently revived by its close links with **optimal** **control** **theory**. It is actually of great interest because of its several applications in a wide range of fields such as Physics, Engineering [24] and Economics [12]. Among others, we mention here the pioneering works of Bolza and Bliss [5], the contribu- tion of Pontryagin [17] and the more recent developments by Sussman, Agrachev, Hsu, Montgomery and Griffiths [35, 1, 27, 15, 9], characterized by a differential geometric approach.

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In **optimal** **control**, as in many other disciplines, individuals developing the **theory** and those applying it to real life problems do not always see eye to eye. Some results developed by theoreticians have very limited practical value, while other useful results may be unknown to practitioners or incorrectly interpreted. This work aims to bridge the gap between these two groups by presenting theoretical results in a way that will be useful to practitioners. We concentrate specifically on convergence results relating to a class of methods known as direct transcription, where the entire **optimal** **control** problem is discretized, in our case using a Runge-Kutta method, to form a nonlinear program.

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We state Pontryagin’s maximum principle for **optimal** **control** problems. Symbols in this section will have similar but probably diﬀerent meanings from other sections. Thus we set this part as a separate section. We will state a result given in []. For simplicity, we only state it in a simple way. In other words, Lemma . below is a special case of Theorem . and Corollary . in Chapter V of [].

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recommended in Figures 2(c) and 2(d)[37]. This finding indicates that the additional complexity of our model is warranted when compared with the SI model used in [37]. Timely and sensitive surveillance systems are key to successful application of the **optimal** **control** method as knowledge of both the pathogen characteristics and the community state are assumed. The surveillance systems should be able to identify the virus quickly and provide accurate estimates for parameters which characterize the severity of an influenza. The effectiveness of the **control** policy depends on the accuracy of these estimates, which include infection rate b, death rate τ and recovery rate g. Once the **control** policy is computed, we also need to track the community state to determine if NPIs should be triggered. As early NPI implementation is found to be much more effective, we do not want to miss the begin- ning stage of the outbreak. Thus, the surveillance system should also estimate the community state, including the size of the infectious and susceptible populations. Sensitivity analysis

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Nipah virus, a member of the genus Henipavirus, a new class of virus in the Paramyxoviridae family, has drawn attention as an emerging zoonotic virus in south-east and south-Asian region [3]. This emerging infec- tious disease has become one of the most alarming threats of the public health mainly due to its periodic out- breaks and the high mortality rate [4]. Epidemiology is the study of the distribution and determinants of health related states or events in specified populations and the application of epidemiology is to **control** of health prob- lems. The crucial point is that epidemiology concerns itself with populations or groups of population in contrast to clinical medicine, which deals with individuals (patients). Therefore, epidemiology describes health and dis- ease in terms of frequencies and distributions of determinants and conditions in a population or in a specific group of a population. Although Nipah virus has caused only a few outbreaks, it infects a wide range of animals and causes severe disease and death in people, making it a public health concern [5]. Treatment is mostly symp- tomatic and supportive as the effect of antiviral drugs is not satisfactory. So the very high case fatality addresses the need for adequate and strict **control** and preventive measures.

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Respiratory variables, including tidal volume and respiratory rate, display significant variability. The probability density function (PDF) of respiratory variables has been shown to contain clinical information and can predict the risk for ex- acerbation in asthma. However, it is uncertain why this PDF plays a major role in predicting the dynamic conditions of the respiratory system. This paper introduces a stochastic **optimal** **control** model for noisy spontaneous breathing, and obtains a Shrödinger’s wave equation as the motion equation that can produce a PDF as a solution. Based on the lob- ules-bronchial tree model of the lung system, the tidal volume variable was expressed by a polar coordinate, by use of which the Shrödinger’s wave equation of inter-breath intervals (IBIs) was obtained. Through the wave equation of IBIs, the respiratory rhythm generator was characterized by the potential function including the PDF and the parameter con- cerning the topographical distribution of regional pulmonary ventilations. The stochastic model in this study was as- sumed to have a common variance parameter in the state variables, which would originate from the variability in meta- bolic energy at the cell level. As a conclusion, the PDF of IBIs would become a marker of neuroplasticity in the respi- ratory rhythm generator through Shrödinger’s wave equation for IBIs.

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University of Warwick institutional repository: http://go.warwick.ac.uk/wrap A Thesis Submitted for the Degree of PhD at the University of Warwick http://go.warwick.ac.uk/wrap/72032.. Th[r]

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Since we want to focus on what is the ethically right response to the dilemma rather than what makes a politically feasible or implementable choice, we will first treat humanity as a whole as formally just one single infinitely-lived decision maker that perfectly knows the system as specified in the formal version of the TE, can make a new choice at every generation, can employ randomization for this if desired, can plan ahead, and has the overall goal of having high welfare in all generations. The natural framework for this kind of problem is the language of **optimal** **control** **theory**. Since it will turn out that **optimal** choices and plans (called “policies” in that language) will very much depend on the evaluation of trajectories (sequences of states) in terms of desirability, we will use concepts such as time preferences, inequality aversion, and risk aversion from decision **theory** and welfare economics to derive candidate intergenerational welfare functions to be used for this evaluation, and will discuss their impact on the **optimal** policy. We will restrict our analysis to a consequentialist point of view that takes into account only the actual and potential consequences of actions and their respective probabilities, and leave the inclusion of nonconsequentialist, e.g. procedural [19], preferences for later work.

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Although the framework uses methods from deterministic **optimal** **control** **theory**, it is ex- tended to cover two classes of stochastic **control** problems in which the variance of idiosyncratic shocks to a state equation appears additively in an agent’s bequest function at the time of a regime switch. It is intended that the set of propositions contained herein may be referred to by researchers solving the aforementioned problems, circumventing the need for them to derive their own closed-form solutions or carry out analysis by simulation in cases where theoretical results are unambiguous, highlighting the importance of closed-form solutions and simulations when they are not. Researchers dealing with the latter scenario are referred to the general methods set out in Caputo and Wilen (1995); researchers dealing with the important case of comparative statics for discount rates, in both deterministic and stochastic settings, are referred to Quah and Strulovici (2013).

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method was used to solve the minimum time handling problem in which the starting point, ending point, and trajectory are not constrained. However, this will not occur so universally in general application. Hendriks et al. [18] studied the **optimal** handling inverse problem by applying an **optimal** **control** **theory** and Pontryagin’s minimum principle, the results of which show that dif- ferent vehicle configurations have different **optimal** con- trol strategies, and their method can be used to evaluate the **optimal** transient handling performance of a vehicle. Their algorithm is rather robust with respect to inaccu- rate starting values, but is the least efficient in terms of speed of convergence. Casanova et al. [19, 20] established a vehicle steering **control** model based on the linear opti- mal discrete time preview **control** **theory**, and obtained the **control** input and **optimal** completion time using an inverse dynamics method. Andreasson et al. [21, 22] used the inverse dynamics method to study the vehicle chas- sis **control**, the results of which show that the method is effective for dealing with the complex problems of chassis **control**. Boyer et al. [23] used a handling inverse dynam- ics method to study the multi-body dynamics of a vehicle and analyzed its maneuverability. Wang et al. [24] used a multiple shooting method to study the emergency col- lision avoidance problem. However, the linearization applied in this method makes the quadratic programming problem deviate significantly from the original model. Liu et al. [25–27] used the Gauss pseudospectral method to study the **optimal** path tracking **control** problem. Their method shows many advantages compared to previous traditional methods. However, all research in this regard was based on the assumption that obstacles are station- ary, and dynamic obstacles were not considered.

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As well as the vertical vibration, it is important to reduce the influence on lat- eral speed change for improving overall ride quality. However, unfortunately, there are few studies on ride discomfort due to longitudinal acceleration/dece- leration. Therefore, in this paper, we develop the speed **control** method by gene- rating the longitudinal speed pattern using the jerk which is time derivative of the acceleration and the acceleration as the evaluation index, for improving the ride comfort against the longitudinal acceleration/deceleration. The method is applying the general **optimal** **control** **theory** and also based on the techniques proposed in [10] [11] [12] [13]. The method aims to contribute to improve the beginner driver’s driving skill from the viewpoint of passenger’s comfortability by showing the ideal running pattern and checking the driving.

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Purpose. The study aims at substantiating the method to determine the **optimal** volume of investments for im- proving basic economic indicators of the enterprise’s performance selected by the company management at random costs at each stage of its development. Methodology. The proposed methodology for determining the **optimal** in- vestment volume is based on simulation modeling methods and **optimal** **control** **theory**, in particular, the dynamic programming procedure, since the controlled process of the enterprise`s development is a multi-step one. Using step-by-step planning with generation of costs for transitions and statistical processing of results, a solution to opti- mization problem was obtained, to which the methods of mathematical analysis cannot be applied. Findings. An algorithm has been developed for calculating the minimal volume of capital investments for improving selected economic indicators and constructing the **optimal** trajectory for the enterprise`s development from the initial eco- nomic state to the final desired state. This takes into account unforeseen intermediate costs in the process of enter- prise development. Originality. It is shown that using the methods of the **theory** of **optimal** **control** and simulation modeling, it is possible to calculate the minimal amount of capital investments to improve the selected economic indicators that determine the efficiency of the enterprise performance, taking into account the random costs of in- termediate transitions by the development stages. Such calculation does not depend on the specific content of eco- nomic indicators. Practical value. The proposed methodology for calculating the minimal volume of capital in- vestments is quite simple, but at the same time it allows, on the one hand, determining the priority areas of the en- terprise’s investment activities. On the other hand, it increases the manageability and transparency of the enter- prise’s economic activity, and increases the manager’s confidence in the correctness of the decisions made.

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This paper concentrates on the application of **optimal** **control** **theory** to highlight some aspects of Turkish economy. First the setup is given for Turkey to grow over the balanced path. Then the **optimal** **control** problem is identified. The **control** and state variables are mentioned. The objective is the maximization of life-time discounted utility of the society through **optimal** choice of consumption which automatically determines investment. We make use of Bellman’s principle to guarantee optimality. We make necessary assumptions (technical assumptions) to make use of calculus techniques for a solution. Some functions to represent utility and production are specified. I used the econometric techniques to estimate some parameters of the functions to decide upon the **optimal** level of investment for steady-state in Turkey over the period including 2001 crisis. The corresponding differential equations are obtained as a result of the Hamiltonian defined. The phase diagram is prepared to analyse different trajectories.

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We develop a two-species prey-predator model in which prey is wildebeest and predator is lion. The threats to wildebeest are poaching and drought while to lion are retaliatory killing and drought. The system is found in the Serengeti ecosystem. **Optimal** **control** **theory** is applied to in- vestigate **optimal** strategies for controlling the threats in the system where anti-poaching patrols are used for poaching, construction of strong bomas for retaliatory killing and construction of dams for drought **control**. The possible impact of using a combination of the three controls either one at a time or two at a time on the threats facing the system is also examined. We observe that the best result is achieved by using all controls at the same time, where a combined approach in tackling threats to yield **optimal** results is a good approach in the management of wildlife popula- tions.

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Our goal in this paper is to extend the **optimal** **control** **theory** in the framework of Lions [] to the hyperbolic equation (.) involving p-Laplacian with a damping term. Let H be a Hilbert space and let U be another Hilbert space of **control** variables, and B be a bounded linear operator from U into L (, T; H), which is called a controller. We formulate our

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Other authors have already studied the same kind of questions. B. Bamieh and its co- workers have developed in [1] a method that produces approximate realizations of operators. It furnishes, when it exists, an approximation under the form of a convolution product between the input and a kernel having a small support. Their method is limited to a class of space invariant operators, which excludes **optimal** **control** on bounded domains (in particular the case of boundary **control** or boundary observation). For the same class of problems, the authors have built in [4] **optimal** approximate realizations based on combination of finite diﬀerences operators. The coeﬀicients of the approximate operator are obtained from numerical resolution of Linear Matrix Inequalities (LMIs), which have not always a solution. The authors announced that their method could be extended to bounded domains and also to boundary **control** or observation. In [5], the authors have built an approximation in the sense of high frequencies, by means of a cascade of partial diﬀerential equations. At the moment, this method is restricted to operational equations whose coeﬀicients are all depending on the same partial diﬀerential operator. It turns out that it cannot handle with the kind of operators that we get in **optimal** **control** **theory** when observations or controls are on the boundary (or concentrated on some particular point in the domain). Let us stress out that both methods developed in [4] and in [5] suﬀer of some strong limitations that we aim to overcome in this paper.

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A **control** problem containing support functions in the integrand of the objective of the functional as well as in the inequality constraint function is considered. For this problem, Fritz John and Ka- rush-Kuhn-Tucker type necessary optimality conditions are derived. Using Karush-Kuhn-Tucker type optimality conditions, Wolfe type dual is formulated and usual duality theorems are estab- lished under generalized convexity conditions. Special cases are generated. It is also shown that our duality results have linkage with those of nonlinear programming problems involving support functions.

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