stiff problems and semi-discretizations of partial differential equations. Due to their high stage order they do not suffer from order reduction and work especially well for high accuracy requirements. Our main motivation is to clarify the potential of implicit **peer** **methods** to overcome the deficiencies of linear multistep **methods** when applied to **optimal** **control** problems. We construct **peer** **methods** with high adjoint consistency in the interior of the integration interval and show that the well-known backward differentiation formulas are **optimal** with respect to the achievable order. We will clearly identify that inappropriate adjoint initialization still remains a crucial issue for implicit **peer** **methods**, which restrict the overall consistency order to one. A fact that is also valid for linear multistep **methods**. Given the low consistency order of the discrete adjoints and therefore of the whole discretization, we have to conclude that implicit **peer** **methods** are not suitable for a first-discretize-then-optimize approach. The content of Chapter 4 has been published in [51].

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• Risk-averse stochastic **optimal** **control**. In this thesis the ran- domness in the cost function is quantified with the expecta- tion operator leading to a so-called risk-neutral formulation. In a pioneering paper Artzner et al. (ADEH99) developed an ax- iomatic framework for risk measures 1 which turned out to be suitable for the formulation of risk-averse multistage **optimal** **control** problems (SDR09; Sha09; GR11; Sha12) and have been used in various applications such as power systems (ZG13) and finance (AMRU01). When these measures are introduced in sto- chastic programming models, this lead to lead to a new class of problems –risk-averse optimisation (Sha09; GR11; Sha12). In two recent papers Asamov and Ruszczy ´nski (AR15) and Col- lado et al. (CPR12) proposed **methods** to solve stochastic linear problems which however do not scale up well with the num- ber of scenarios and the prediction horizon. Risk-averse prob- lems, however, are significantly more complex than their risk- neutral counterparts and there are no algorithms to solve them fast enough for them to qualify for typical **control** applications — even for medium-scale problems. Developing optimisation **methods** for risk-averse optimisation problems in **control** appli- cations is an interesting research direction.

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a feature vector is passed to the inverse **optimal** **control**, the inverse **methods** then try to learn the weight vector. The weight vector indicates the relative importance of the different features present in the feature vector. In this paper, the weight vector is always assumed to be positive. In reality, one will never have access to true cost function so that it can be compared against learned cost function. The best one can do is to learn an underlying cost function which respects the constraints and compare the predicted trajectories from that cost function with the actual trajectories generated by the system of interest. Typically, the expert will have some knowledge about the ways in which a given system is constrained. These constraints should be passed to the inverse **methods** so that a correct cost function can be learned. If the expert fails to account for the constraints that may be present in a system, it is very likely that the learned cost function will perform poorly. Also, it is possible that if enough trajectories are not observed, the cost function learned maybe incorrect. This is because the inverse **methods** may not have been able to sample the feasible state/**control** space completely. Another point to note is that although the feature vectors in the examples provided were chosen to be quadratic for reasons of physical intuition, it is possible to choose the feature vectors to be non-quadratic, if the problem so desires. The inverse **methods** will still be able to provide locally **optimal** solution to the residual minimization problem in (5.11)

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The different types of spectral **methods** vary in the type of basis functionals and the deriva- tion of the coefficients which can be carried out analytically, by sampling techniques or by projection techniques. Our main focus in this thesis is the polynomial chaos (PC) method, a spectral method in which Equation (2.19) represents a generalized global Fourier expansion, called polynomial chaos expansion, in a random polynomial basis. When using more general probability distributions such as the normal distribution, this method is also referred to as generalized polynomial chaos (gPC). First practical use of the gPC fracemwork can be found in sparse and linear system, often complicated by high dimensionality in state or uncertain parameter space originating from a discretization of PDE or stochastic processes. Application areas vary amongst computational fluid dynamics [ 57, 151, 89 ], aerodynamic design [ 134 ], mechanics [ 147 ] and geology [ 114 ]. Current work primarily addresses this setting. Examples for recent applications to stochastic **optimal** **control** of PDEs include [ 37, 155, 85, 84 ] as well as in shape and topology optimization [134, 147]. The first works of gPC in **control** systems with ordinary differential equations empirically studied questions such as convergence and stability on small test problems [ 73, 75, 55 ] or linear systems [ 138, 54 ]. Applications to real- world **optimal** **control** problems can be found in, e.g., [111, 80, 18], and the review article [ 81 ]. For recent work on the use of gPC for solving stochastic model predictive **control**, con- sider [ 76, 105, 106 ]. Fagiano [ 52 ] uses gPC to design a controller that induces convergence of the expected value of the state to the origin and satisfies state constraints in expectation.

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state and **control** variables. Amongst these various constraints, the continuous inequal- ity constraints, often involving only state variables, are very diﬃcult to handle directly. Thus, the constraint transcription technique introduced in [80] is used to convert each of them into an equivalent equality constraint in integral form. However, the integrand of each of these equality constraints is non-smooth. Thus, a local smoothing method is used to approximate the non-smooth functions by smooth functions. Consequently, each of these continuous inequality constraints is approximated by a sequence of inequality constraints in integral form, where the integrands are smooth approximating functions. These inequality constraints are known as the inequality constraints in canonical form, as they appear in the same form as the cost function. Then, by using the idea of the penalty function, these inequality constraints are appended to the cost function, forming an augmented cost function. Thus, the constrained time-delay **optimal** **control** problem is approximated by a sequence of time-delay **optimal** parameter selection problems subject to simple bounds on the **control** parameter vector. Each of them is to be solved as a nonlinear programming problem by using a gradient-based optimization technique, such as the sequential quadratic programming approximation scheme with active set strategy (see, for example, [80,147]). For this, the gradient formula of the augmented cost function with respect to the **control** parameter vector is derived. On this basis, an eﬀective com- putational algorithm is developed for the time-delay constrained **optimal** **control** problem through solving a sequence of **optimal** parameter selection problems subject to simple bounds, each of which is regarded as a nonlinear programming problem.

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In (Kozine and Krymsky 2012) the calculus of variations is used again to construct interval-valued probability measures. New constraints are introduced that have relevance for reliability applications. They are upper and lower bounds on failure rate. This has the double effect: better precision in the results and avoidance of the troublesome parameter – the upper bound on time to failure. Another promising outcome of that study was the prospective to depart from the use of rather general and complicated mathematical tool, the calculus of variations. As it was proven, under the constraints on the failure rate, **optimal** solutions are sought on the class of piecewise exponential distributions. In this way, the problem statement can simply be changed to finding the breaking points at which the probability density function either abruptly jumps up or drops.

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that is, with different number of time steps. It turned out that the complexity of the MLMC algorithm can at best reduced down to order ε −2 . Further interesting approaches that reduce the complexity via deterministic quadrature-based algorithms can be found in [47] and [48]. Our aim is to derive an efficient algorithm with complexity rate better than ε −2 . This is achieved by using regression-type algorithms for the construction of **control** variates. As opposite to the SMC approach, our method takes advantage of the smoothness in µ, σ and f (which is needed for nice convergence properties of regression **methods**) and hence is especially efficient for smooth problems.

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The results from the three sets of simulations highlight both the strengths and limitations of the **optimal** **control** model of infant causal perception. There are two major findings. First, OCM quickly learns a set of **optimal** tracking strategies for following a moving object. Second, when presented with a novel causal event, OCM appropriately anticipates the out- come of partially occluded, but not fully occluded, versions of the event.

It was a pleasure to work in the stimulating academic environment established at the Interdisciplinary Center for Scientific Computing and the Faculty of Mathematics and Computer Science at Heidelberg University. In particular, I would like to thank the members of the Simulation and Optimization, the Model-Based Optimizing **Control** and Experimental Design Group for contributing to this unique atmosphere. For the many cups of coffee we shared together and the interesting discussions, I would like to thank Anja Bettendorf, Lilli Bergner, Dr. Holger Diedam, Dr. Kathrin Hatz, Mar´ıa Elena Su´arez Garc´es, J¨urgen Gutekunst, Dr. Christian Hoffmann, Dr. Dennis Janka, Johannes Herold, Dr. Robert Kircheis, Dr. Tom Kraus, Dr. Huu Chuong La, Conrad Leidereiter, Dr. Simon Lenz, Enrique Guerrero Merino, Andreas Meyer, Nadia Said, Dr. Andreas Schmidt, Dr. Andreas Sommer and Leonard Wirsching.

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Interval **methods** have been introduced over the years [10]. But until 1996, the interval analysis was just summarized in simple propositions [24]. In this paper, a discrete time interval based method is proposed for solving the OCPs with interval uncertainties. Time discretization of the OCPs has been introduced since the 1960s [9]. A survey of some of the earlier works can be found in [27]. For instance, properties of Runge- Kutta **methods** have been analyzed in [19, 30]. Since, Runge-Kutta **methods** have a significant role in the numerical treatment of differential systems [16], the main purpose of our study is to propose an interval version of Runge-Kutta **methods** for solving OCPs with interval uncertainties. The **optimal** **control** problem in this study is

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with randomized estimators to reduce the computational eﬀort. The work [VBV18] considers a risk measure that involves only the mean of the objective functional (hereafter named mean-based risk), with an additional penalty on the variance of the state, and proposes a gradient type method, in which the expectation of the gradient is computed by a multilevel Monte Carlo method. In [BvW11], the authors also consider a mean-based risk problem and propose a reduced basis method on the space of controls to dramatically reduce the computational eﬀort. In the work [Kou12], the author presents a more general type of OCP, using the general notion of a risk measure, and derives the corresponding optimality system of PDEs to be solved. For its numerical solution, a trust-region Newton conjugate gradient algorithm is proposed in [KHRvBW13], combined with an adaptive sparse grid collocation for the discretization of the PDE in the stochastic space. The work [KS16] considers derivative-based optimization **methods** for the robust CVaR risk measure, which are built upon introducing smooth approximations to the CVaR. Finally, in the work [GLL11], the authors consider a boundary OCP where the deterministic **control** appears as a Neumann boundary condition. Using again a mean-based risk, they derive an optimality system of equations and provide a complete error analysis of the ﬁnite element approximation, as well as of the random parameter space approximation.

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Abstract - The given thesis puts forth the application of various **methods** for **optimal** **control** of power flow in a network of microgrids(MGs). The **methods** used for solving the mathematical formulation of Microgrid **control** are Steepest Descent method, Newton method and Differential Evolution method. In the above first three come under a group of gradient descent **methods** where as the differential evolution is an Evolutionary Algorithm. A comparative study is performed by calculating the cost function for each of the three **methods**. We have taken a case of four Microgrids collaborating in a network. In the proposed work, the **optimal** **control** microgrid system with Energy storage systems are considered. The gradient descent **methods** and evolutionary algorithm are applied and an optimized result for energy storage systems are obtained. It is observed that the results obtained are good and comparable.

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1.1.1 Combined Homotopy and Neighboring Extremal **Optimal** **Control** For most OCPs in engineering applications, it is difficult to obtain analytical or closed form solutions using Pontryagin’s maximum principle (PMP) or dynamic pro- gramming (DP). Consequently, iterative/numerical **methods** are utilized for solving such OCPs [8], [78]. Two **methods**, which have been used independently in **optimal** **control** theory are homotopy (see, e.g., [10], [18], [51], [79], [91], [100]) and neighboring extremal **optimal** **control** (NEOC) (see, e.g., [14]). However, the combination of these two techniques has not been investigated. With this motivation, we combine these two techniques and arrive at a method for obtaining sub-**optimal** **control** in OCPs defined on a Euclidean space.

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□ Notice that L : = L J ( ) ˆ 2 represents the smallest value of L such that (4.5) is satisfied. We remark that the discussion that follows is valid for L ≥ L J ( ) ˆ 2 as in Lemma 4.5. However, as we discuss below, the efficiency of our proximal schemes depends on how close is the chosen L to the minimal and **optimal** value L J ( ) ˆ 2 . Now, since this value is usually not available analytically, we discuss and implement below some numerical strategies for determining a sufficiently accurate approximation of L J ( ) ˆ 2 . In particular, we consider a power iteration [21], and the backtracking approach discussed in Remark 5.1.

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Time optimal ontrol problems for a lass of linear multi-input systems are.. onsidered.[r]

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for **Optimal** **Control** Problems Carsten Gr¨aser ?
Abstract We present a new approach for the globalization of the primal-dual ac- tive set or equivalently the nonsmooth Newton method applied to an **optimal** **control** problem. The basic result is the equivalence of this method to a nonsmooth New- ton method applied to the nonlinear Schur complement of the optimality system. Our approach does not require the construction of an additional merrit function or additional descent direction. The nonsmooth Newton directions are naturally appro- priate descent directions for a smooth dual energy and guarantee global convergence if standard damping **methods** are applied.

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Section 4.3 discusses the problem of initializing numerical algorithms for solving **optimal** **control** problems. The idea of using order reduction for initialization was suggested by the author, who worked on it independently in early 2004. Experiments in Matlab and SOCS showed the shortcomings of the approach originally adopted by the author. The section discusses possible solutions. Using monitor functions to initialize SOCS was a joint project of Dr. Campbell and a former student Mr. Kalla. In the spring of 2004, the author debugged, organized and rewrote some of the Matlab code written by Mr. Kalla and noted some numerical phenomena that had not been observed before.

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Therefore, (1.3) - (1.5) is, in fact, a reformulation of the classical Dirichlet principle associated to the biharmonic operator. Consequently, the **control** variational method is a modification of the classical variational approaching, via the use of **optimal** **control** theory. From this point of view, it is also important to notice that our method is essentially different from the **optimal** **control** approaches obtained via the least squares fitting to the data procedure applied in various situations. Moreover, the **control** variational approach allows many variants of such modifications. This flexibility may be very advantageous in certain applications. We shall exemplify it in the next section via some abstract schemes. Section 3 is devoted to applications to unilateral problems where constrained **control** problems have to be used. The last section briefly discusses the case of time-dependent pro- blems, which is more difficult and still under development. The paper ends with a short conclusion.

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