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Register a new userShape Optimization of Actuators over Banach Spaces for Nonlinear Systems
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In this paper, optimal actuator shape for nonlinear parabolic systems is discussed. The system under study is an abstract differential equation with a locally Lipschitz nonlinear part. A quadratic cost on the state and input of the system is considered. The existence of an optimal actuator shape has been established in the literature. This paper focuses on driving the optimality conditions for actuator shapes belonging to a Banach space. The application of the theory to the optimal actuator shape design for railway track model is considered.
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We consider Cheeger-like shape optimization problems of the form $$minbig{|Omega|^alpha J(Omega) : Omegasubset Dbig}$$ where $D$ is a given bounded domain and $alpha$ is above the natural scaling. We show the existence of a solution and analyze as $J(Omega)$ the particular cases of the compliance functional $C(Omega)$ and of the first eigenvalue $lambda_1(Omega)$ of the Dirichlet Laplacian. We prove that optimal sets are open and we obtain some necessary conditions of optimality.
We consider a shape optimization problem written in the optimal control form: the governing operator is the $p$-Laplacian in the Euclidean space $R^d$, the cost is of an integral type, and the control variable is the domain of the state equation. Conditions that guarantee the existence of an optimal domain will be discussed in various situations. It is proved that the optimal domains have a finite perimeter and, under some suitable assumptions, that they are open sets. A crucial difference is between the case $p>d$, where the existence occurs under very mild conditions, and the case $ple d$, where additional assumptions have to be made on the data.
Greedy algorithms which use only function evaluations are applied to convex optimization in a general Banach space $X$. Along with algorithms that use exact evaluations, algorithms with approximate evaluations are treated. A priori upper bounds for the convergence rate of the proposed algorithms are given. These bounds depend on the smoothness of the objective function and the sparsity or compressibility (with respect to a given dictionary) of a point in $X$ where the minimum is attained.
We study the convergence of an inexact version of the classical Krasnoselskii-Mann iteration for computing fixed points of nonexpansive maps. Our main result establishes a new metric bound for the fixed-point residuals, from which we derive their rate of convergence as well as the convergence of the iterates towards a fixed point. The results are applied to three variants of the basic iteration: infeasible iterations with approximate projections, the Ishikawa iteration, and diagonal Krasnoselskii-Mann schemes. The results are also extended to continuous time in order to study the asymptotics of nonautonomous evolution equations governed by nonexpansive operators.
A renewal system divides the slotted timeline into back to back time periods called renewal frames. At the beginning of each frame, it chooses a policy from a set of options for that frame. The policy determines the duration of the frame, the penalty incurred during the frame (such as energy expenditure), and a vector of performance metrics (such as instantaneous number of jobs served). The starting points of this line of research are Chapter 7 of the book [Nee10a], the seminal work [Nee13a], and Chapter 5 of the PhD thesis of Chih-ping Li [Li11]. These works consider stochastic optimization over a single renewal system. By way of contrast, this thesis considers optimization over multiple parallel renewal systems, which is computationally more challenging and yields much more applications. The goal is to minimize the time average overall penalty subject to time average overall constraints on the corresponding performance metrics. The main difficulty, which is not present in earlier works, is that these systems act asynchronously due to the fact that the renewal frames of different renewal systems are not aligned. The goal of the thesis is to resolve this difficulty head-on via a new asynchronous algorithm and a novel supermartingale stopping time analysis which shows our algorithms not only converge to the optimal solution but also enjoy fast convergence rates. Based on this general theory, we further develop novel algorithms for data center server provision problems with performance guarantees as well as new heuristics for the multi-user file downloading problems.
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