# Browsing by Subject "Optimal Control"

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Item Open Access Nonlinear Energy Harvesting With Tools From Machine Learning(2020) Wang, XuesheEnergy harvesting is a process where self-powered electronic devices scavenge ambient energy and convert it to electrical power. Traditional linear energy harvesters which operate based on linear resonance work well only when excitation frequency is close to its natural frequency. While various control methods applied to an energy harvester realize resonant frequency tuning, they are either energy-consuming or exhibit low efficiency when operating under multi-frequency excitations. In order to overcome these limitations in a linear energy harvester, researchers recently suggested using "nonlinearity" for broad-band frequency response.

Based on existing investigations of nonlinear energy harvesting, this dissertation introduced a novel type of energy harvester designs for space efficiency and intentional nonlinearity: translational-to-rotational conversion. Two dynamical systems were presented: 1) vertically forced rocking elliptical disks, and 2) non-contact magnetic transmission. Both systems realize the translational-to-rotational conversion and exhibit nonlinear behaviors which are beneficial to broad-band energy harvesting.

This dissertation also explores novel methods to overcome the limitation of nonlinear energy harvesting -- the presence of coexisting attractors. A control method was proposed to render a nonlinear harvesting system operating on the desired attractor. This method is based on reinforcement learning and proved to work with various control constraints and optimized energy consumption.

Apart from investigations of energy harvesting, several techniques were presented to improve the efficiency for analyzing generic linear/nonlinear dynamical systems: 1) an analytical method for stroboscopically sampling general periodic functions with arbitrary frequency sweep rates, and 2) a model-free sampling method for estimating basins of attraction using hybrid active learning.

Item Open Access Optimal Monitoring Schedule in Dynamic Contracts(2017-09-16) Chen, M; Sun, P; Xiao, YItem Open Access Robust, Distributed, and Optimal Control via Dissipativity-Augmentation(2024) LoCicero, Ethan JeffreyDissipativity has been an influential tool for stability analysis of networked systems for several decades. In particular, the Network Dissipativity Theorem provides robust stability guarantees for interconnections based on open-loop input-output properties of subsystems. While this theorem has been applied with great success to analysis and controller design for stability, its application in optimal control has been limited. This is due in large part to two factors. First, overly conservative dissipativity characterizations -- especially for network effects like time-varying delays and for data-based estimations -- greatly restrict the controller design space, resulting in poor performance. Second, a framework for combining optimal control problems with the Network Dissipativity Theorem has yet to be put forward in its full generality. In this dissertation addresses these two limitations. First, stochastic dissipativity is defined and used to tightly characterize input-output properties of systems with probabilistically time-varying delays. This includes a KYP-type lemma for nonlinear systems, and a linear matrix inequality condition for linear systems. It is shown that this method yields tighter dissipativity characterizations than its deterministic counterpart, which results in improved controller performance. Second, dissipativity-augmented multiobjective control is proposed as a means of generating network controllers that attain high performance (in terms of H2, colored-H2, H-infinity, pole placement, and learned objectives) with robust stability guarantees using centralized, decentralized, and distributed control structures. While the problem is NP-hard, in each case a feasible point is constructed, and an algorithm for locally optimizing performance is proposed based on the convex-concave procedure. Numerical experiments demonstrate broad applicability, guaranteed stability, and improved performance over existing methods where applicable. In addition, a proof of the optimal convex-concave decomposition for bilinear matrix inequalities is derived, the Network Dissipativity Theorem is extended to subsets of inputs and outputs, some improvements are made to existing model-free data-based dissipativity estimation, and critical limitations of data-based dissipativity estimation are illustrated through a case study.

Item Open Access Solving Partial Differential Equations Using Artificial Neural Networks(2013) Rudd, KeithThis thesis presents a method for solving partial differential equations (PDEs) using articial neural networks. The method uses a constrained backpropagation (CPROP) approach for preserving prior knowledge during incremental training for solving nonlinear elliptic and parabolic PDEs adaptively, in non-stationary environments. Compared to previous methods that use penalty functions or Lagrange multipliers,

CPROP reduces the dimensionality of the optimization problem by using direct elimination, while satisfying the equality constraints associated with the boundary and initial conditions exactly, at every iteration of the algorithm. The effectiveness of this method is demonstrated through several examples, including nonlinear elliptic

and parabolic PDEs with changing parameters and non-homogeneous terms. The computational complexity analysis shows that CPROP compares favorably to existing methods of solution, and that it leads to considerable computational savings when subject to non-stationary environments.

The CPROP based approach is extended to a constrained integration (CINT) method for solving initial boundary value partial differential equations (PDEs). The CINT method combines classical Galerkin methods with CPROP in order to constrain the ANN to approximately satisfy the boundary condition at each stage of integration. The advantage of the CINT method is that it is readily applicable to PDEs in irregular domains and requires no special modification for domains with complex geometries. Furthermore, the CINT method provides a semi-analytical solution that is infinitely differentiable. The CINT method is demonstrated on two hyperbolic and one parabolic initial boundary value problems (IBVPs). These IBVPs are widely used and have known analytical solutions. When compared with Matlab's nite element (FE) method, the CINT method is shown to achieve significant improvements both in terms of computational time and accuracy. The CINT method is applied to a distributed optimal control (DOC) problem of computing optimal state and control trajectories for a multiscale dynamical system comprised of many interacting dynamical systems, or agents. A generalized reduced gradient (GRG) approach is presented in which the agent dynamics are described by a small system of stochastic dierential equations (SDEs). A set of optimality conditions is derived using calculus of variations, and used to compute the optimal macroscopic state and microscopic control laws. An indirect GRG approach is used to solve the optimality conditions numerically for large systems of agents. By assuming a parametric control law obtained from the superposition of linear basis functions, the agent control laws can be determined via set-point regulation, such

that the macroscopic behavior of the agents is optimized over time, based on multiple, interactive navigation objectives.

Lastly, the CINT method is used to identify optimal root profiles in water limited ecosystems. Knowledge of root depths and distributions is vital in order to accurately model and predict hydrological ecosystem dynamics. Therefore, there is interest in accurately predicting distributions for various vegetation types, soils, and climates. Numerical experiments were were performed that identify root profiles that maximize transpiration over a 10 year period across a transect of the Kalahari. Storm types were varied to show the dependence of the optimal profile on storm frequency and intensity. It is shown that more deeply distributed roots are optimal for regions where

storms are more intense and less frequent, and shallower roots are advantageous in regions where storms are less intense and more frequent.

Item Open Access The Economics of Malaria Vector Control(2011) Brown, Zachary StevenIn recent years, government aid agencies and international organizations have increased their financial commitments to controlling and eliminating malaria from the planet. This renewed emphasis on elimination is reminiscent of a previous worldwide campaign to eradicate malaria in the 1960s, a campaign which ultimately failed. To avoid a repeat of the past, mechanisms must be developed to sustain effective malaria control programs.

A number of sociobehavioral, economic, and biophysical challenges exist for sustainable malaria control, particularly in high-burden areas such as sub-Saharan Africa. Sociobehavioral challenges include maintaining high long-term levels of support for and participation in malaria control programs, at all levels of society. Reasons for the failure of the previous eradication campaign included a decline in donor, governmental, community, and household-level support for control programs, as malaria prevalence ebbed due in part to early successes of these programs.

Biophysical challenges for the sustainability of national malaria control programs (NMCPs) encompass evolutionary challenges in controlling the protozoan parasite and the mosquito vector, as well as volatile transmission dynamics which can lead to epidemics. Evolutionary challenges are particularly daunting due to the rapid generational turnover of both the parasites and the vectors: The reliance on a handful of insecticides and antimalarial drugs in NMCPs has placed significant selection pressures on vectors and parasites respectively, leading to a high prevalence of genetic mutations conferring resistance to these biocides.

The renewed global financing of malaria control makes research into how to effectively surmount these challenges arguably more salient now than ever. Economics has proven useful for addressing the sociobehavioral and biophysical challenges for malaria control. A necessary next step is the careful, detailed, and timely integration of economics with the natural sciences to maximize and sustain the impact of this financing.

In this dissertation, I focus on 4 of the challenges identified above: In the first chapter, I use optimal control and dynamic programming techniques to focus on the problem of insecticide resistance in malaria control, and to understand how different models of mosquito evolution can affect our policy prescriptions for dealing with the problem of insecticide resistance. I identify specific details of the biological model--the mechanisms for so-called "fitness costs" in insecticide-resistant mosquitoes--that affect the qualitative properties of the optimal control path. These qualitative differences carry over to large impacts on the economic costs of a given control plan.

In the 2nd chapter, I consider the interaction of parasite resistance to drugs and mosquito resistance to insecticides, and analyze cost-effective malaria control portfolios that balance these 2 dynamics. I construct a mathematical model of malaria transmission and evolutionary dynamics, and calibrate the model to baseline data from a rural Tanzanian district. Four interventions are jointly considered in the model: Insecticide-spraying, insecticide-treated net distribution, and the distribution of 2 antimalarial drugs--sulfadoxine pyramethamine (SP) and artemisinin-based combination therapies (ACTs). Strategies which coordinate vector controls and treatment protocols should provide significant gains, in part due to the issues of insecticide and drug resistance. In particular, conventional vector control and ACT use should be highly complementary, economically and in terms of disease reductions. The ongoing debate concerning the cost-effectiveness of ACTs should thus consider prevailing (and future) levels of conventional vector control methods, such as ITN and IRS: If the cost-effectiveness of widespread ACT distribution is called into question in a given locale, scaling up IRS and/or ITNs probably tilts the scale in favor of distributing ACTs.

In the 3rd chapter, I analyze results from a survey of northern Ugandan households I oversaw in November 2009. The aim of this survey was to assess respondents' perceptions about malaria risks, and mass indoor residual spraying (IRS) of insecticides that had been done there by government-sponsored health workers. Using stated preference methods--specifically, a discrete choice experiment (DCE)--I evaluate: (a) the elasticity of household participation levels in IRS programs with respect to malaria risk, and (b) households' perceived value of programs aimed at reducing malaria risk, such as IRS. Econometric results imply that the average respondent in the survey would be willing to forego a $10 increase in her assets for a permanent 1% reduction in malaria risk. Participation in previous IRS significantly increased the stated willingness to participate in future IRS programs. However, I also find that at least 20% of households in the region perceive significant transactions costs from IRS. One implication of this finding is that compensation for these transactions costs may be necessary to correct theorized public good aspects of malaria prevention via vector control.

In the 4th chapter, I further study these public goods aspects. To do so, I estimate a welfare-maximizing system of cash incentives. Using the econometric models estimated in the 3rd chapter, in conjunction with a modified version of the malaria transmission models developed and utilized in the first 2 chapters, I calculate village-specific incentives aimed at correcting under-provision of a public good--namely, malaria prevention. This under-provision arises from incentives for individual malaria prevention behavior--in this case the decision whether or not to participate in a given IRS round. The magnitude of this inefficiency is determined by the epidemiological model, which dictates the extent to which households' prevention decisions have spillover effects on neighbors.

I therefore compute the efficient incentives in a number of epidemiological contexts. I find that non-negligible monetary incentives for participating in IRS programs are warranted in situations where policymakers are confident that IRS can effectively reduce the incidence of malaria cases, and not just exposure rates. In these cases, I conclude that the use of economic incentives could reduce the incidence of malaria episodes by 5%--10%. Depending on the costs of implementing a system of incentives for IRS participation, such a system could provide an additional tool in the arsenal of malaria controls.