We consider an optimal control problem for a non-autonomous model of ODEs that describes the evolution of the number of customers in some firm. Namely we study the best marketing strategy. Considering a L2 cost functional, we establish the existence and uniqueness of optimal solutions, using an inductive argument to obtain uniqueness on the whole interval from local uniqueness. We also present some simulation results, based on our model, and compare them with results we obtain for an L1 cost functional. For the L1 cost functional the optimal solutions are of bang-bang type and thus easier to implement, because at every moment possible actions are chosen from a finite set of possibilities. For the autonomous case of L2 problem, we show the effectiveness of the optimal control strategy against other formulations of the problem with simpler controls.

We study an optimal control problem for a non-autonomous SEIRS model with incidence given by a general function of the infective, the susceptible and the total population, and with vaccination and treatment as control variables. We prove existence and uniqueness results for our problem and, for the case of mass-action incidence, we present some simulation results designed to compare an autonomous and corresponding periodic model, as well as the controlled versus uncontrolled models.

This paper proposes a distributed system for analysis and monitoring (DSAM) of vibrations and acoustic noise, which consists of an array of intelligent modules, sensor modules, communication bus and a host PC acting as data center. The main advantages of the DSAM are its modularity, scalability, and flexibility for use of different type of sensors/transducers, with analog or digital outputs, and for signals of different nature. Its final cost is also significantly lower than other available commercial solutions. The system is reconfigurable, can operate either with synchronous or asynchronous modes, with programmable sampling frequencies, 8-bit or 12-bit resolution and a memory buffer of 15 kbyte. It allows real-time data-acquisition for signals of different nature, in applications that require a large number of sensors, thus it is suited for monitoring of vibrations in Linear Switched Reluctance Actuators (LSRAs). The acquired data allows the full characterization of the LSRA in terms of its response to vibrations of structural origins, and the vibrations and acoustic noise emitted under normal operation. The DSAM can also be used for electrical machine condition monitoring, machine fault diagnosis, structural characterization and monitoring, among other applications.

From the economic point of view it is important to process high charges of solid in enzimatic hydrolysis of the lignocellulosic residues for the production of 2G ethanol, but batch operation with high initial charges of substrate can cause inefficient mixture, damaging the reactional process. An attractive alternative is operating the reactors in fed batch. The aim of this work was to determine, applying the theory of optimal control, the optimal feeding profile of substrate in bench reactor. For this, it was tested two kinetic of Michaelis-Menten: pseudo-homogeneous Michaelis-Menten kinetic and Michaelis Menten kinetic with inhibition by the product, in the last kinetic was obtained the best fit to experimental data. The developed algorithm was run in Matlab® software and the simulation results were experimentally validated. Enzymatic hydrolysis assays were performed with hydrothermally pre-treated sugar cane straw followed by the alkaline delignification. It was obtained good fiting of the simulation data with respect to the experimental data, with values of glucose final concentration close to 164.647 g/L (triplicate average) and in simulation the concentration was 174.93 g/L. Proving that the sugar cane straw, besides being a good feedstock cheap and abundant, it has a great energetic potential. Furthermore, the optimal control strategy was very efficient in optimization of the proposed problem.

The use of fossil fuels contributes to the increase in the amount of free carbon in the atmosphere, because most of the released carbon does not return to the Earth's surface, boosting the global temperature increase. Biofuels are a renewable and environmentally safe alternative, inasmuch as in its production depends that the carbon present in the atmosphere be captured through photosynthesis forming a cycle of production and consumption of carbon. The use of cellulosic leavings is a Muito promising alternative for the production of biofuels as 2G ethanol. The main reason is the fact that these leavings do not enter the competition "food vs fuels " what happens with 1G ethanol. For the consolidation of the industrial production of 2G bioethanol, the economic improvement of the process is necessary. For this reason, there are several researches that seek to make feasible the technique, in this project, a way is presented to improve the feeding of bioreactors, in order to increase the efficiency of cellulose hydrolysis by releasing reducing sugars, which could be converted into second generation alcohol. The use of fed-batch reactors reduces production costs and increases yields, enjoying of rational feed policies of substrates and biocatalysts. Then, this project intends to determine by means of a computational study, the optimized feeding profiles of cellulosic substrate and fed batch enzyme. For this, a semi- mechanistic model is developed, and the optimal control theory for the development of feed profiles is used. For the application of the feed profiles, a system was designed and built, able to automate the feed for both bioreactors and reactors, which alBaixo s to operate with different feeding profiles. The prototype used 3D printing technology to manufacture the mechanical devices, the entire control is done by two microcontrollers in half duplex network, in charge of acting on a helical conveyor. Due to the ability to work with nonlinear systems, the fuzzy nebulous logic was established as the base of the system, being responsible for carrying out the operations necessary to keep the feeder fully operational.

Application of computational models has contributed to understanding of different dynamics as well as possible more effective control strategies. Three widely used examples are deterministic formulations of compartmental models, individuals based models, and complex networks based models. An alternative to such models is a stochastic approach, which allows uncertainties insertion to models, providing more realistic results. In this context, this work proposes use of deterministic compartmental models to obtain optimum control policies, and later evaluation of such policy applied in a stochastic scenario using a equivalent individual based model. It also proposes three new control strategies based on dynamics and topology in complex network models. To models validation, a case study based on epidemiological dynamics was done, in which proposed strategies resulted in significant reductions in number of infected individuals, optimizing resource spending. Insertion of uncertainty in models was positive for average behavior analysis of dynamics. In addition, a parallel MBI model was proposed to be processed in graphic cards. With this improvement it was possible to obtain a reduction by a factor of twenty in processing time.

Neste trabalho, será considerado um problema de controle ótimo quadrático para a equação do calor em domínios retangulares com condição de fronteira do tipo Dirichlet é nos quais, a função de controle (dependente apenas no tempo) constitui um termo de fonte. Uma caracterização da solução ótima é obtida na forma de uma equação linear em um espaço de funções reais definidas no intervalo de tempo considerado. Em seguida, utiliza-se uma sequência de projeções em subespaços de dimensão finita para obter aproximações para o controle ótimo, o cada uma das quais pode ser gerada por um sistema linear de dimensão finita. A sequência de soluções aproximadas assim obtidas converge para a solução ótima do problema original. Finalmente, são apresentados resultados numéricos para domínios espaciais de dimensão 1.

A presente dissertação propõe uma abordagem alternativa na simulação matemática de um cenário preocupante em ecologia: o controle de pragas nocivas a uma dada lavoura de soja em uma específica região geográfica. O instrumental teórico empregado é a teoria dos jogos, de forma a acoplar ferramentas da matemática discreta à análise e solução de problemas de valor inicial em equações diferenciais, mais especificamente, as chamadas equações de dinâmica populacional de Lotka-Volterra com competição. Essas equações, que modelam o comportamento predador-presa, possuem, com os parâmetros inicialmente utilizados, um ponto de equilíbrio mais alto que o desejado no contexto agrícola sob exame, resultando na necessidade de utilização da teoria do controle ótimo. O esquema desenvolvido neste trabalho conduz a ferramentas suficientemente simples, de forma a tornar viável o seu uso em situações reais. Os dados utilizados para o tratamento do problema que conduziu a esta pesquisa interdisciplinar foram coletados de material bibliográfico da Empresa Brasileira de Pesquisa Agropecuária EMBRAPA.

The main aim of this study is to analyze a fractional parabolic SIR epidemic model of a reaction-diffusion, by using the nonlocal Caputo fractional time-fractional derivative and employing the $p$-Laplacian operator. The immunity is imposed through the vaccination program, which is regarded as a control variable. Finding the optimal control pair that reduces the number of sick people, the associated vaccination, and treatment expenses across a constrained time and space is our main study. The existence and uniqueness of the nonnegative solution for the spatiotemporal SIR model are established. It is also demonstrated that an optimal control exists. In addition, we obtain a description of the optimal control in terms of state and adjoint functions. Then, the optimality system is resolved by a discrete iterative scheme that converges after an appropriate test, similar to the forward-backward sweep method. Finally, numerical approximations are given to show the effectiveness of the proposed control program, which provides meaningful results using different values of the fractional order and $p$, respectively the order of the Caputo derivative and the $p$-Laplacian operators.

In this Ph.D. thesis, we apply optimal control theory to various mathematical models, including a pharmacokinetic/pharmacodynamic (PK/PD) model, a fractional PK/PD model, and an epidemiological SEIR models. First, we focus on the PK/PD model to control the infusion of propofol. We begin by analyzing the mathematical model of anesthesia and provide an analytical solution to the time-optimal control problem for the induction phase of anesthesia. Our approach aligns closely with results obtained using the standard shooting method. Utilizing the Pontryagin minimum principle, we propose a new analytical method for solving the time-optimal control problem. Our findings reveal that the optimal continuous infusion rate of the anesthetic and the minimum time required to transition from an awake state to an anesthetized state are consistent between both methods. Furthermore, we extend our analysis to a PK/PD anesthesia model using psi-caputo fractional derivatives. The second part of the thesis focuses on the development of an SEIR model. Initially, we explore mathematical models for COVID-19 with discrete time delays and vaccination. Specifically, we introduce a time delay to account for the delayed migration of individuals from susceptible to infected states. We establish sufficient conditions for the local stability of both endemic and disease-free equilibrium points in the presence of positive time delays. To address the COVID-19 pandemic, we propose a generalized SEIR-type control model. Additionally, we introduce three time-dependent controls for the SEIR model and analyze the optimal control problem with respect to real data transmission in Italy. Our results demonstrate the effectiveness of the model, particularly concerning the number of quarantined and recovered individuals. By considering Pontryagin controls, we illustrate the potential for significant reductions in susceptible, exposed, infected, quarantined/hospitalized, and deceased individuals through increased population protection. We also present a model for maintaining the efficacy of COVID-19 vaccines during transportation and distribution, emphasizing the importance of vaccination in controlling the pandemic.