In this dissertation we focus on fractional-order dynamical systems and classify these problems as optimal control of system described by fractional derivative, fractional-order nonlinear differential equations, optimal control of systems described by variable-order differential equations and delay fractional optimal control problems. These problems are solved by using the spectral method and reducing the problem to a system of algebraic equations. In fact for the optimal control problems described by fractional and variable-order equations, the variables are approximated by chosen wavelets with unknown coefficients in the constraint equations, performance index and conditions. Thus, a fractional optimal control problem is converted to an optimization problem, which can be solved numerically. We have applied the new generalized wavelets to approximate the fractional-order nonlinear differential equations such as Riccati and Bagley-Torvik equations. Then, the solution of this kind of problem is found using the collocation method. For solving the fractional optimal control described by fractional delay system, a new set of hybrid functions have been constructed. Also, a general and exact formulation for the fractional-order integral operator of these functions has been achieved. Then we utilized it to solve delay fractional optimal control problems directly. The convergence of the present method is discussed. For all cases, some numerical examples are presented and compared with the existing results, which show the efficiency and accuracy of the present method.

This thesis focuses on the challenges that arise from the scheduling of heat exchanger network maintenance problems which undergo fouling and run continuously over time. The original contributions of the current research consist of the development of novel optimisation methodologies for the scheduling of cleaning actions in heat exchanger network problems, the application of the novel solution methodology developed to other general maintenance scheduling problems, the development of a stochastic programming formulation using this optimisation technique and its application to these scheduling problems with parametric uncertainty. The work presented in this thesis can be divided into three areas. To efficiently solve this non-convex heat exchanger network maintenance scheduling problem, new optimisation strategies are developed. The resulting contributions are outlined below. In the first area, a novel methodology is developed for the solution of the heat exchanger network maintenance scheduling problems, which is attributed towards a key discovery in which it is observed that these problems exhibit bang-bang behaviour. This indicates that when integrality on the binary decision variables is relaxed, the solution will tend to either the lower or the upper bound specified, obviating the need for integer programming solution techniques. Therefore, these problems are in ac- tuality optimal control problems. To suitably solve these problems, a feasible path sequential mixed integer optimal control approach is proposed. This methodology is coupled with a simple heuristic approach and applied to a range of heat exchanger network case studies from crude oil refinery preheat trains. The demonstrated meth- odology is shown to be robust, reliable and efficient. In the second area of this thesis, the aforementioned novel technique is applied to the scheduling of the regeneration of membranes in reverse osmosis networks which undergo fouling and are located in desalination plants. The results show that the developed solution methodology can be generalised to other maintenance scheduling problems with decaying performance characteristics. In the third and final area of this thesis, a stochastic programming version of the feasible path mixed integer optimal control problem technique is established. This is based upon a multiple scenario approach and is applied to two heat exchanger network case studies of varying size and complexity. Results show that this methodology runs automatically with ease without any failures in convergence. More importantly due to the significant impact on economics, it is vital that uncertainty in data is taken into account in the heat exchanger network maintenance scheduling problem, as well as other general maintenance scheduling problems when there is a level of uncertainty in parameter values.

Many real-life applications such as the shape optimization of technological devices, the identification of parameters in environmental processes and flow control problems lead to optimization problems governed by systems of convection diusion partial dierential equations (PDEs). When convection dominates diusion, the solutions of these PDEs typically exhibit layers on small regions where the solution has large gradients. Hence, it requires special numerical techniques, which take into account the structure of the convection. The integration of discretization and optimization is important for the overall eciency of the solution process. Discontinuous Galerkin (DG) methods became recently as an alternative to the finite dierence, finite volume and continuous finite element methods for solving wave dominated problems like convection diusion equations since they possess higher accuracy. This thesis will focus on analysis and application of DG methods for linear-quadratic convection dominated optimal control problems. Because of the inconsistencies of the standard stabilized methods such as streamline upwind Petrov Galerkin (SUPG) on convection diusion optimal control problems, the discretize-then-optimize and the optimize-then-discretize do not commute. However, the upwind symmetric interior penalty Galerkin (SIPG) method leads to the same discrete optimality systems. The other DG methods such as nonsymmetric interior penalty Galerkin (NIPG) and incomplete interior penalty Galerkin (IIPG) method also yield the same discrete optimality systems when penalization constant is taken large enough. We will study a posteriori error estimates of the upwind SIPG method for the distributed unconstrained and control constrained optimal control problems. In convection dominated optimal control problems with boundary and/or interior layers, the oscillations are propagated downwind and upwind direction in the interior domain, due the opposite sign of convection terms in state and adjoint equations. Hence, we will use residual based a posteriori error estimators to reduce these oscillations around the boundary and/or interior layers. Finally, theoretical analysis will be confirmed by several numerical examples with and without control constraints

Cette thèse porte sur l'étude d'équation de Hamilton-Jacobi-Bellman associées à des problèmes de contrôle optimal et de jeux à champ moyen avec la particularité qu'on se place sur un réseau (c'est-à-dire, des ensembles constitués d'arêtes connectées par des jonctions) dans les deux problèmes, pour lesquels on autorise différentes dynamiques et différents coûts dans chaque bord d'un réseau. Dans la première partie de cette thèse, on considère un problème de contrôle optimal sur les réseaux dans l'esprit des travaux d'Achdou, Camilli, Cutrì & Tchou (2013) et Imbert, Moneau & Zidani (2013). La principale nouveauté est qu'on rajoute des coûts d'entrée (ou de sortie) aux sommets du réseau conduisant à une éventuelle discontinuité de la fonction valeur. Celle-ci est caractérisée comme l'unique solution de viscosité d'une équation Hamilton-Jacobi pour laquelle une condition de jonction adéquate est établie. L'unicité est une conséquence d'un principe de comparaison pour lequel nous donnons deux preuves différentes, l'une avec des arguments tirés de la théorie du contrôle optimal, inspirée par Achdou, Oudet & Tchou (2015) et l'autre basée sur les équations aux dérivées partielles, d'après Lions & Souganidis (2017). La deuxième partie concerne les jeux à champ moyen stochastiques sur les réseaux. Dans le cas ergodique, ils sont décrits par un système couplant une équation de Hamilton-Jacobi-Bellman et une équation de Fokker- Planck, dont les inconnues sont la densité m de la mesure invariante qui représente la distribution des joueurs, la fonction valeur v qui provient d'un problème de contrôle optimal "moyen" et la constante ergodique ρ. La fonction valeur v est continue et satisfait dans notre problème des conditions de Kirchhoff aux sommets très générales. La fonction m satisfait deux conditions de transmission aux sommets. En particulier, due à la généralité des conditions de Kirchhoff, m est en général discontinue aux sommets. L'existence et l'unicité d'une solution faible sont prouvées pour des Hamiltoniens sous-quadratiques et des hypothèses très générales sur le couplage. Enfin, dans la dernière partie, nous étudions les jeux à champ moyen stochastiques non stationnaires sur les réseaux. Les conditions de transition pour la fonction de valeur v et la densité m sont similaires à celles données dans la deuxième partie. Là aussi, nous prouvons l'existence et l'unicité d'une solution faible pour des Hamiltoniens sous-linéaires et des couplages et dans le cas d'un couplage non-local régularisant et borné inférieurement. La principale difficulté supplémentaire par rapport au cas stationnaire, qui nous impose des hypothèses plus restrictives, est d'établir la régularité des solutions du système posé sur un réseau. Notre approche consiste à étudier la solution de l'équation de Hamilton-Jacobi dérivée pour gagner de la régularité sur la solution de l'équation initiale. / The dissertation focuses on the study of Hamilton-Jacobi-Bellman equations associated with optimal control problems and mean field games problems in the case when the state space is a network. Different dynamics and running costs are allowed in each edge of the network. In the first part of this thesis, we consider an optimal control on networks in the spirit of the works of Achdou, Camilli, Cutrì & Tchou (2013) and Imbert, Monneau & Zidani (2013). The main new feature is that there are entry (or exit) costs at the edges of the network leading to a possible discontinuous value function. The value function is characterized as the unique viscosity solution of a Hamilton-Jacobi equation for which an adequate junction condition is established. The uniqueness is a consequence of a comparison principle for which we give two different proofs. One uses some arguments from the theory of optimal control and is inspired by Achdou, Oudet & Tchou (2015). The other one is based on partial differential equations techniques and is inspired by a recent work of Lions & Souganidis (2017). The second part is about stochastic mean field games for which the state space is a network. In the ergodic case, they are described by a system coupling a Hamilton- Jacobi-Bellman equation and a Fokker-Planck equation, whose unknowns are the density m of the invariant measure which represents the distribution of the players, the value function v which comes from an "average" optimal control problem and the ergodic constant ρ. The function v is continuous and satisfies general Kirchhoff conditions at the vertices. The density m satisfies dual transmission conditions. In particular, due to the generality of Kirchhoff’s conditions, m is in general discontinuous at the vertices. Existence and uniqueness are proven for subquadratic Hamiltonian and very general assumptions about the coupling term. Finally, in the last part, we study non-stationary stochastic mean field games on networks. The transition conditions for value function v and the density m are similar to the ones given in second part. Here again, we prove the existence and uniqueness of a weak solution for sublinear Hamiltonian and bounded non-local regularizing coupling term. The main additional difficulty compared to the stationary case, which imposes us more restrictive hypotheses, is to establish the regularity of the solutions of the system placed on a network. Our approach is to study the solution of the derived Hamilton-Jacobi equation to gain regularity over the initial equation.

The primary goal of this thesis is to study finite element based a priori and a posteriori error estimates of optimal control problems of various kinds governed by linear elliptic PDEs (partial differential equations) of second and fourth orders. This thesis studies interior and boundary control (Neumann and Dirichlet) problems. The initial chapter is introductory in nature. Some preliminary and fundamental results of finite element methods and optimal control problems which play key roles for the subsequent analysis are reviewed in this chapter. This is followed by a brief literature survey of the finite element based numerical analysis of PDE constrained optimal control problems. We conclude the chapter with a discussion on the outline of the thesis. An abstract framework for the error analysis of discontinuous Galerkin methods for control constrained optimal control problems is developed in the second chapter. The analysis establishes the best approximation result from a priori analysis point of view and delivers a reliable and efficient a posteriori error estimator. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. Subsequently, the applications of p p - interior penalty methods for a boundary control problem as well as a distributed control problem governed by the bi-harmonic equation subject to simply supported boundary conditions are discussed through the abstract analysis. In the third chapter, an alternative energy space based approach is proposed for the Dirichlet boundary control problem and then a finite element based numerical method is designed and analyzed for its numerical approximation. A priori error estimates of optimal order in the energy norm and the m norm are derived. Moreover, a reliable and efficient a posteriori error estimator is derived with the help an auxiliary problem. An energy space based Dirichlet boundary control problem governed by bi-harmonic equation is investigated and subsequently a l y - interior penalty method is proposed and analyzed for it in the fourth chapter. An optimal order a priori error estimate is derived under the minimal regularity conditions. The abstract error estimate guarantees optimal order of convergence whenever the solution has minimum regularity. Further an optimal order l l norm error estimate is derived. The fifth chapter studies a super convergence result for the optimal control of an interior control problem with Dirichlet cost functional and governed by second order linear elliptic PDE. An optimal order a priori error estimate is derived and subsequently a super convergence result for the optimal control is derived. A residual based reliable and efficient error estimators are derived in a posteriori error control for the optimal control. Numerical experiments illustrate the theoretical results at the end of every chapter. We conclude the thesis stating the possible extensions which can be made of the results presented in the thesis with some more problems of future interest in this direction.

Conselho Nacional de Desenvolvimento Científico e Tecnológico / Optimal Control Problems (OCP), also known as Dynamic Optimization Problems, consist of an Objective Function to be maximized or minimized, associated with a set of differential and algebraic equations which include equality and inequality constraints in the state or control variables and characterize a system of Differential-Algebraic Equations (DAE). The differential-algebraic approach of numerical solution widely used in process simulation due the guarantee of attendance of the implicit algebraic constraints in the original formulation and the elimination of the necessary manipulations to transform the original problem into a purely differential system,was extended to OCP characterizing the called Differential-Algebraic Optimal Control Problem (DAOCP). A category of DAOCP of special interest includes inequality constraints, due the necessity of previous knowledge of the activations and deactivations sequence of these constraints along the trajectory and also of the instants where they occur, named Events. This DAOCPs with inequality constraints is equivalent to a class of hybrid dynamic optimization problems, where continuous and discrete behaviors are associated (FEEHERY, 1998). A particular type of hybrid OCP is that one where continuous state does not present jumps in the Events, called Switched OCP, for which Xu e Antsaklis (2004) considers a solution methodology based on the parameterization of Events with a previous specification of active subsystems sequence, resulting in the solution of a two-point boundary value differential-algebraic problem, formed by the state, co-state and stationarity equations, boundary and continuity conditions and its differentiations, called sensitivity equations. In this work, this indirect approach for Switched OCP was extended for DAOCP with inequality constraints, with the objective to estimate the Events, along the control, state and adjoint variables. The developed approach for Switched OCP described by Xu e Antsaklis (2004) was implemented in a specific code using Maple 9.5, called EVENTS, with the objective to symbolically generate the equations based on the parameterization of Events. This code was incorporated in a interface named OpCol, that collect characterization tools of DAE systems and generation of the optimality conditions extended Pontryagin s Principle for PCOAD of different types. The characterization tools are the INDEX of Murata (1996) that symbolically identifies the index, the resolubility and the consistency of initial conditions and the ACIG of Cunha e Murata (1999) that implements the Gear s algorithm for the index reduction and the index 1 equivalent system generation. The OTIMA (GOMES, 2000; LOBATO, 2004) generates the Euler-Lagrange equations for DAOCP. These tools had been implemented initially in different versions of Maple and all had been update to 9.5 version using the Maplets package that allows the data entry through interactive windows with the user, demanding a little knowledge of the Maple syntax. The OpCol interface was tested for four cases and for each tool a example data bank with typical problems of literature was created to assist the user in its use. Moreover, the direct method implemented in DIRCOL code was extended for multi-phases formulation with estimates of Events and the indirect method with Events Parameterization and differential-algebraic approach implemented in a Matlab code had been used for the numerical solution of three cases: a switched OCP and 2 DAOCP of batch reactors where the control variable is the feed rate of the component B - the first one has parallel reactions and selectivity constraints with 3 phases of index 1, 3 and 1 and the second a safety constraint with 2 phases of index 2 and 1 respectively and had been described by Srinivasan et al. (2003). The methodology used by this authors was applied to attained analytical expressions for the control variable in each phase necessary in indirect method, composing the called Switching Functions, from the optimality conditions based in the Pontryagin s Principle - specifically from the stationarity condition and the active constraint identification that will allow the control variable determination - and of the physical analysis of the problem in order to discard not appropriate activations/deactivations sequences. The results obtained by indirect and direct methods are compared for the 3 cited problems, showing the viability as much of the multiphase formulation using the DIRCOL and also the satisfactory performance of the indirect method with estimates of Events, beyond the utility of the tools of characterization of EADs, of attainment of optimality conditions and parameterization of Events available in Opcol interface. / Os Problemas de Controle Ótimo, também chamados Problemas de Otimização Dinâmica, são formados por uma Função Objetivo a ser maximizada ou minimizada, associada a conjuntos de equações algébricas e diferenciais que incluem restrições de igualdade e de desigualdade nas variáveis de estado e de controle que caracterizam um sistema de Equações Algébrico-Diferenciais (EADs). A extensão do ponto de vista algébricodiferencial de solução numérica aos PCOs, já amplamente utilizado na simulação de processos devido à garantia de atendimento às restrições algébricas originais e implícitas na formulação e à eliminação das manipulações necessárias para transformar o problema original num sistema de equações puramente diferenciais, caracteriza o chamado Problema de Controle Ótimo Algébrico-Diferencial (PCOAD). Uma categoria de PCOAD de especial interesse é a dos que incluem restrições de desigualdade, devido à necessidade de conhecimento prévio da seqüência de ativações e desativações destas restrições ao longo da trajetória e também dos instantes em que elas ocorrem, chamados Eventos. As ativações/desativações das restrições causam flutuações no índice diferencial e no número de graus de liberdade dinâmicos do PCOAD, exigindo técnicas especiais de redução deste índice até um e o emprego de métodos numéricos eficientes que garantam a convergência e estabilidade da solução. Estes PCOADs com restrições de desigualdade são equivalentes a uma classe de problemas de otimização dinâmica híbridos, que associam comportamentos contínuos e discretos (FEEHERY, 1998). Um tipo particular de PCO híbrido é aquele cujo estado contínuo não apresenta saltos nos Eventos, chamado PCO Chaveado, para o qual Xu e Antsaklis (2004) propõem uma metodologia de solução baseada na parametrização dos Eventos com a especificação prévia da seqüência de subsistemas ativos, resultando na solução de um problema de valor no contorno algébrico-diferencial em dois pontos, formado pelas equações de estado, co-estado e de estacionariedade, condições de contorno e de continuidade e suas diferenciações, chamadas equações de sensibilidade. Neste trabalho, esta abordagem indireta empregada para PCO Chaveados foi estendida para PCOAD com restrições de desigualdade, com o objetivo de estimar também os Eventos, além das variáveis de controle, de estado e adjuntas. A abordagem desenvolvida por Xu e Antsaklis (2004) para PCO Chaveados foi implementada num código específico utilizando o Maple 9.5, chamado EVENTS, com o objetivo de gerar simbolicamente as equações baseadas na parametrização dos Eventos. Este código foi incorporado a uma interface chamada OpCol, que reúne ferramentas de caracterização de sistemas de EAD e de geração das condições de otimalidade segundo o Princípio de Pontryagin estendidas para PCOAD de diferentes classes. As ferramentas de caracterização são o INDEX de Murata (1996) que identifica simbolicamente o índice, a resolubilidade e a consistência das condições iniciais e o ACIG de Cunha e Murata (1999) que implementa o algoritmo de Gear para a redução do índice e geração do sistema equivalente de índice 1. O OTIMA (GOMES, 2000; LOBATO, 2004) gera as equações de Euler-Lagrange para PCOAD. Estas ferramentas foram inicialmente implementadas em diferentes versões do Maple e todas foram atualizadas para a versão 9.5 utilizando o pacote Maplets que permite a entrada de dados através de janelas interativas com o usuário, exigindo dele pouco conhecimento da sintaxe Maple. A interface OpCol foi testada para quatro casos e para cada ferramenta foi criado um banco de exemplos com problemas típicos da literatura que auxiliam o usuário na sua utilização. Além disto, o método direto implementado no código DIRCOL estendido para formulações multifásicas com estimativa dos Eventos e o método indireto com Parametrização dos Eventos e abordagem algébrico-diferencial implementado num código MATLAB foram utilizados na solução numérica de três estudos de casos: um PCO chaveado e 2 PCOAD de reatores batelada onde a variável de controle é a taxa de alimentação do componente B: o primeiro tem reações paralelas e restrições de seletividade com 3 fases de índices 1, 3 e 1 e o segundo restrições de segurança com 2 fases de índices 2 e 1 e respectivamente e foram descritos por Srinivasan et al. (2003). A mesma metodologia utilizada por estes autores foi empregada na obtenção de expressões analíticas para a variável de controle em cada fase necessárias no método indireto, compondo as chamadas Funções Identificadoras de Fase (FIF), a partir das condições de otimalidade baseadas no Princípio de Pontryagin - especificamente a partir da condição de estacionariedade e da identificação da restrição ativa que permitirá a determinação da variável de controle - e da análise física do problema de modo a descartar seqüências de ativação/desativação não apropriadas. Os resultados obtidos pelo método indireto e pelo método direto são comparados entre si para os 3 problemas citados, mostrando a viabilidade tanto da formulação multifásica empregando o DIRCOL quanto o desempenho satisfatório do método indireto com estimativa de Eventos, além da utilidade das ferramentas de caracterização de EADs, de obtenção das condições de otimalidade e de parametrização dos eventos disponibilizadas na interface Opcol. / Mestre em Engenharia Química

Mathematical theory of partial differential equations (PDEs) is a pretty old classical area with wide range of applications to almost every branch of science and engineering. With the advanced development of functional analysis and operator theory in the last century, it became a topic of analysis. The theory of homogenization of partial differential equations is a relatively new area of research which helps to understand the multi-scale phenomena which has tremendous applications in a variety of physical and engineering models, like in composite materials, porous media, thin structures, rapidly oscillating boundaries and so on. Hence, it has emerged as one of the most interesting and useful subject to study for the last few decades both as a theoretical and applied topic. In this thesis, we study asymptotic analysis (homogenization) of second-order partial differential equations posed on an oscillating domain. We consider a two dimensional oscillating domain (comb shape type) consisting of a fixed bottom region and an oscillatory (rugose) upper region. We introduce optimal control problems for the Laplace equation. There are mainly two types of optimal control problems; namely distributed control and boundary control. For distributed control problems in the oscillating domain, one can apply control on the oscillating part or on the fixed part and similarly for boundary control problem (control on the oscillating boundary or on the fixed part the boundary). We consider all the four cases, namely distributed and boundary controls both on the oscillating part and away from the oscillating part. The present thesis consists of 8 chapters. In Chapter 1, a brief introduction to homogenization and optimal control is given with relevant references. In Chapter 2, we introduce the oscillatory domain and define the basic unfolding operators which will be used throughout the thesis. Summary of the thesis is given in Chapter 3 and future plan in Chapter 8. Our main contribution is contained in Chapters 4-7. In chapters 4 and 5, we study the asymptotic analysis of optimal control problems namely distributed and boundary controls, respectively, where the controls act away from the oscillating part of the domain. We consider both L2 cost functional as well as Dirichlet (gradient type) cost functional. We derive homogenized problem and introduce the limit optimal control problems with appropriate cost functional. Finally, we show convergence of the optimal solution, optimal state and associate adjoint solution. Also convergence of cost-functional. In Chapter 6, we consider the periodic controls on the oscillatory part together with Neumann condition on the oscillating boundary. One of the main contributions is the characterization of the optimal control using unfolding operator. This characterization is new and also will be used to study the limiting analysis of the optimality system. Chapter 7 deals with the boundary optimal control problem, where the control is applied through Neumann boundary condition on the oscillating boundary with a suitable scaling parameter. To characterize the optimal control, we introduce boundary unfolding operators which we consider as a novel approach. This characterization is used in the limiting analysis. In the limit, we obtain two limit problems according to the scaling parameters. In one of the limit optimal control problem, we observe that it contains three controls namely; a distributed control, a boundary control and an interface control.

Mathematical theory of homogenization of partial differential equations is relatively a new area of research (30-40 years or so) though the physical and engineering applications were well known. It has tremendous applications in various branches of engineering and science like : material science ,porous media, study of vibrations of thin structures, composite materials to name a few. There are at present various methods to study homogenization problems (basically asymptotic analysis) and there is a vast amount of literature in various directions. Homogenization arise in problems with oscillatory coefficients, domain with large number of perforations, domain with rough boundary and so on. The latter one has applications in fluid flow which is categorized as oscillating boundaries. In fact ,in this thesis, we consider domains with oscillating boundaries. We plan to study to homogenization of certain optimal control problems with oscillating boundaries. This thesis contains 6 chapters including an introductory Chapter 1 and future proposal Chapter 6. Our main contribution contained in chapters 2-5. The oscillatory domain under consideration is a 3-dimensional cuboid (for simplicity) with a large number of pillars of length O(1) attached on one side, but with a small cross sectional area of order ε2 .As ε0, this gives a geometrical domain with oscillating boundary. We also consider 2-dimensional oscillatory domain which is a cross section of the above 3-dimensional domain. In chapters 2 and 3, we consider the optimal control problem described by the Δ operator with two types of cost functionals, namely L2-cost functional and Dirichlet cost functional. We consider both distributed and boundary controls. The limit analysis was carried by considering the associated optimality system in which the adjoint states are introduced. But the main contribution in all the different cases(L2 and Dirichlet cost functionals, distributed and boundary controls) is the derivation of error estimates what is known as correctors in homogenization literature. Though there is a basic test function, one need to introduce different test functions to obtain correctors. Introducing correctors in homogenization is an important aspect of study which is indeed useful in the analysis, but important in numerical study as well. The setup is the same in Chapter 4 as well. But here we consider Stokes’ Problem and study asymptotic analysis as well as corrector results. We obtain corrector results for velocity and pressure terms and also for its adjoint velocity and adjoint pressure. In Chapter 5, we consider a time dependent Kirchhoff-Love equation with the same domain with oscillating boundaries with a distributed control. The state equation is a fourth order hyperbolic type equation with associated L2-cost functional. We do not have corrector results in this chapter, but the limit cost functional is different and new. In the earlier chapters the limit cost functional were of the same type.

This thesis presents a numerical approach based on generalized fractional-order Chebyshev wavelets for solving fractional-order optimal control problems. The exact value of the Riemann– Liouville fractional integral operator of the generalized fractional-order Chebyshev wavelets is computed by applying the regularized beta function. We apply the given wavelets, the exact formula, and the collocation method to transform the studied problem into a new optimization problem. The convergence analysis of the proposed method is provided. The present method is extended for solving fractional-order, distributed-order, and variable-order optimal control problems. Illustrative examples are considered to show the advantage of this method in comparison with the existing methods in the literature.

<p> This work focuses on solving the general optimal control problems with smart-learning-enabled and theory-supported optimal control (SET-OC) approaches. The proposed SET-OC includes two main directions. Firstly, according to the basic idea of the direct method, the smart-learning-enabled iterative optimization algorithm (SEIOA) is proposed for solving discrete optimal control problems. Via discretization and reformulation, the optimal control problem is converted into a general quadratically constrained quadratic programming (QCQP) problem. Then, the SEIOA is applied to solving QCQPs. To be specific, first, a structure-exploiting decomposition scheme is introduced to reduce the complexity of the original problem. Next, an iterative search, combined with an intersection-cutting plane, is developed to achieve global convergence. Furthermore, considering the implicit relationship between the algorithmic parameters and the convergence rate of the iterative search, deep learning is applied to design the algorithmic parameters from an appropriate amount of training data to improve convergence property. To demonstrate the effectiveness and improved computational performance of the proposed SEIOA, the developed algorithms have been implemented in extensive real-world application problems, including unmanned aerial vehicle path planning problems and general QCQP problems. According to the theoretical analysis of global convergence and the simulation results, the efficiency, robustness, and improved convergence rate of the optimization framework compared to the state-of-the-art optimization methods for solving general QCQP problems are analyzed and verified. Secondly, the onboard learning-based optimal control method (L-OCM) is proposed to solve the optimal control problems. Supported by the optimal control theory, the necessary conditions of optimality for optimal control of the optimal control problem can be derived, which leads to two two-point-boundary-value-problems (TPBVPs). Then, critical parameters are identified to approximate the complete solutions of the TPBVPs. To find the implicit relationship between the initial states and these critical parameters, deep neural networks are constructed to learn the values of these critical parameters in real-time with training data obtained from the offline solutions.  To demonstrate the effectiveness and improved computational performance of the proposed L-OCM approaches, the developed algorithms have been implemented in extensive real-world application problems, including two-dimensional human-Mars entry, powered-descent, landing guidance problems, and fuel-optimal powered descent guidance (PDG) problems. In addition, considering there is no thorough analysis of the properties of the optimal control profile for PDG when considering the state constraints, a rigid theoretical analysis of the fuel-optimal PDG problem with state constraints is further provided. According to the theoretical analysis and simulation results, the optimality, robustness, and real-time performance of the proposed L-OCM are analyzed and verified, which indicates the potential for onboard implementation. </p>