Optimal control of general dynamic systems under realistic constraints on input signals and state variables is an important problem area in control theory. Many practical control problems can be formulated as optimization tasks, and this leads toa significant demand for efficient numerical solution algorithms. Several such algorithms have been developed, and they are typically derived from a dynamic programming view point. In this thesis a differentapproach is taken. The discretetime dynamic optimization problem is formulated as a static one, with the inputs as free variables. Newton's approach to solving such a problem with constraints, also known as Wilson's method, is then consistently pursued, anda algorithm is developed that isa true Newton algorithm for the problem, at the same time as the inherent structure is utilized for efficient calculations. An advantage with such an approach is that global and local convergence properties can be studied in a familiar framework. The algorithm is tested on several examples and comparisons to other algorithms are carried out. These show that the Newton algorithm performs well and is competitive with other methods. lt handles state variable constraints in a direct and efficient manner, and its practical convergence properties are robust. A general algorithm for !arge scale static problems is also developed in the thesis, and it is tested on a problem with load distribution in an electrical power network.

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I develop a numerical method that combines functional approximations and dynamic programming to solve high-dimensional discrete-time stochastic control problems under general constraints. The method relies on three building blocks: first, a quasi-random grid and the radial basis function method are used to discretize and interpolate the high-dimensional state space; second, to incorporate constraints, the method of Lagrange multipliers is applied to obtain the first order optimality conditions; third, the conditional expectation of the value function is approximated by a second order polynomial basis, estimated using ordinary least squares regressions. To reduce the approximation error, I introduce the test region iterative contraction (TRIC) method to shrink the approximation region around the optimal solution. I apply the method to two Finance applications: a) dynamic portfolio choice with constraints, a continuous control problem; b) dynamic portfolio choice with capital gain taxation, a high-dimensional singular control problem. / text

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.

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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.

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

Bakgrund och problemdiskussion: Styrsystem är ett brett och komplext begrepp som kan innehålla många olika komponenter, till exempel budgetering. Bankverksamheterna är hårt reglerade utifrån de lagar och regler som gäller i det aktuella landet som de är verksamma inom. Därför är det viktigt att ha tydliga riktlinjer och styra de anställda så att de inte bryter mot några lagar och/eller regler. Dock finns det en viss problematik kring en hårt reglerad verksamhet. Till exempel kan både regler och strikta budgetar leda till att de anställdas motivation påverkas negativt då de är hårt hållna och inte kan arbeta på det sätt de vill på grund av strikta riktlinjer. En annan del kring problematiken med en icke vinstdrivande organisation är hur medarbetarna kan motiveras utan monetär bonus/utdelning, då dessa kan ses som volontärarbetare i denna typ av organisation. Syfte: Syftet med denna studie är att beskriva och analysera hur en icke vinstdrivande bank arbetar med sitt interna styrsystem för att bibehålla medarbetarnas motivation till att genomföra arbetsuppgifterna, trots hårda påtryckningar utifrån i form av bland annat lagar och regler. Denna studie avser även att beskriva hur en medarbetare kan belönas utan monetära ersättningar och se om denna har någon inverkan på medarbetarens motivation. Metod: En kvalitativ metod användes i denna studie då djupgående intervjuer genomfördes och därmed kunde respondenternas uppfattningar redogöras. En fallstudiestrategi valdes för att studien skulle kunna fånga upp respondenternas uppfattningar då fallstudier används för att forskaren ska kunna skaffa sig en djupare insikt i hur det ser ut i till exempel en organisation. Intervjufrågorna formulerades efter de teorier och begrepp som finns i den teoretiska referensramen och alla respondenter utom en intervjuades på sin ordinarie arbetsplats. Intervjuerna spelades in, transkriberades och sorterades efter de teoretiska begrepp som används i studien. Slutsats: Trots hårda yttre regleringar och strikta riktlinjer kan styrsystemet utformas på ett sätt som har en positiv inverkan på medarbetarnas motivation. Att arbeta efter budget kan ha en negativ inverkan på motivationen men föreliggande studie visar att detta kan ha en positiv inverkan på motivationen beroende på hur organisationen utformar sitt interna styrsystem utifrån de tre styrningsperspektiven, resultat-, handlings- och social styrning. Även i en icke vinstdrivande organisation kan medarbetarna belönas på andra sätt än genom monetära ersättningar, vilket har en positiv inverkan på medarbetarnas förväntningar och motivation enligt denna studie. Belöningarna måste vara intressanta för medarbetaren, annars är det troligt att medarbetaren inte uppfattar detta som en belöning. / Background and problems: Management control system is a wide and complex concept that can consist of many different components, for example budgeting. The banks are tightly regulated according to the laws and regulations of the country they are operating within. Therefor is it important to have clear guidelines and to control their employees so they don’t break the law and/or the rules. However, there are some concerns related to a tightly regulated business. For example can both strict rules and budgets affect the employee’s motivation in a negative way since they are supervised and can’t perform their tasks in the way they want because of the strict guidelines. Another part of the problem concerning a non-profit organisation is how the employees can be motivated without monetary compensations, as they can be seen as voluntary workers in this kind of organisation. Purpose: The purpose of this paper is to describe and analyse how a non-profit bank uses its internal control system to maintain the motivation of the employees to perform the job assignments despite strong pressure from the outside, inter alia, due to laws and regulations. This study also intends to describe how an employee can be rewarded without monetary compensation and see if this has any effect on the employee’s motivation. Method: A qualitative approach was used in this study as in-depth interviews were performed and therefore could the respondents’ perceptions be defined. A case study strategy was chosen for this paper to be able to capture the respondents' perceptions, as case studies is used when the scholar should be able to gain a deeper understanding of how it appears in an organisation for example. The interview questions were formulated by the theories and concepts in the theoretical framework and all respondents except one were interviewed at their ordinary workplace. The interviews was recorded, transliterated and was sorted out by the theoretical concepts that was used in the study. Conclusions: Despite tight regulation from the outside and strict guidelines can the internal control system be formulated in a way that has a positive effect on the employee’s motivation. To work after a budget could have a negative effect on the motivation but this paper shows that this can have a positive effect on the motivation depending on how the organisation formulates their internal control system according to the three control perspectives, result-, action- and social control. Even in a non-profit organisation can the employees be rewarded in other ways than monetary compensations, which have a positive effect on the employee’s expectations and motivation according to this study. The rewards must be interesting to the employee, in other way it is most likely that the employee doesn’t perceive this as a reward.

As control systems become more integrated with high-end engineering systems as well as consumer products, they are expected to achieve specifications that may include logic rules, safety constraints, startup procedures, and so forth. Control design for such complex specifications is a relatively unexplored research area. One possible design approach is based on partitioning the state space into polytopic regions, and then formulating a certain control problem on each polytope, with the intention that the set of all controllers so obtained would collectively achieve the specification. The control problem which must be solved for each polytope is called the reach control problem, and it has been identified as turnkey to the further development of this approach. The reach control problem (RCP) is to find a state feedback to make the closed-loop trajectories of an affine (or linear) control system defined on a polytope reach and exit a prescribed facet of the polytope in finite time. This dissertation studies a number of aspects of the reach control problem, and it uses tools from convex analysis, nonsmooth analysis, and computational geometry for this study. The dissertation has three main themes. First, we formulate and solve a variant of RCP in which trajectories exit the polytope in a monotonic sense; this provides a triangulation-independent solution of RCP. Second, we develop a Lyapunov-like theory for verifying if RCP is solved using a given candidate controller. This involves the introduction of the notion of generalized flow functions, a LaSalle Principle for RCP, and several converse theorems on existence of generalized flow functions. Third, we study the relationship between affine feedbacks and continuous state feedbacks for RCP on simplices. Although the two feedback classes have been shown to be equivalent under an assumption on the triangulation of the state space, we show by a counterexample that the equivalence is no longer true under arbitrary triangulations. Then we provide for single-input systems a constructive method for the synthesis of multi-affine feedbacks for RCP on simplices.