CLOT Norm Minimization for Continuous Hands-off Control

In this paper, we consider hands-off control via minimization of the CLOT (Combined $L$-One and Two) norm. The maximum hands-off control is the $L^0$-optimal (or the sparsest) control among all feasible controls that are bounded by a specified value and transfer the state from a given initial state to the origin within a fixed time duration. In general, the maximum hands-off control is a bang-off-bang control taking values of $\pm 1$ and $0$. For many real applications, such discontinuity in the control is not desirable. To obtain a continuous but still relatively sparse control, we propose to use the CLOT norm, a convex combination of $L^1$ and $L^2$ norms. We show by numerical simulations that the CLOT control is continuous and much sparser (i.e. has longer time duration on which the control takes 0) than the conventional EN (elastic net) control, which is a convex combination of $L^1$ and squared $L^2$ norms. We also prove that the CLOT control is continuous in the sense that, if $O(h)$ denotes the sampling period, then the difference between successive values of the CLOT-optimal control is $O(\sqrt{h})$, which is a form of continuity. Also, the CLOT formulation is extended to encompass constraints on the state variable.Comment: 38 pages, 20 figures. enlarged version of arXiv:1611.0207

The turnpike property in the maximum hands-off control

This paper presents analyses for the maximum hands-off control using the geometric methods developed for the theory of turnpike in optimal control. First, a sufficient condition is proved for the existence of the maximum hands-off control for linear time-invariant systems with arbitrarily fixed initial and terminal points using the relation with $L^1$ optimal control. Next, a sufficient condition is derived for the maximum hands-off control to have the turnpike property, which may be useful for approximate design of the control.Comment: Submitted for IEEE Conference on Decision and Control 202

Queueing Subject To Action-Dependent Server Performance: Utilization Rate Reduction

We consider a discrete-time system comprising a first-come-first-served queue, a non-preemptive server, and a stationary non-work-conserving scheduler. New tasks enter the queue according to a Bernoulli process with a pre-specified arrival rate. At each instant, the server is either busy working on a task or is available. When the server is available, the scheduler either assigns a new task to the server or allows it to remain available (to rest). In addition to the aforementioned availability state, we assume that the server has an integer-valued activity state. The activity state is non-decreasing during work periods, and is non-increasing otherwise. In a typical application of our framework, the server performance (understood as task completion probability) worsens as the activity state increases. In this article, we build on and transcend recent stabilizability results obtained for the same framework. Specifically, we establish methods to design scheduling policies that not only stabilize the queue but also reduce the utilization rate - understood as the infinite-horizon time-averaged portion of time the server is working. This article has a main theorem leading to two key results: (i) We put forth a tractable method to determine, using a finite-dimensional linear program (LP), the infimum of all utilization rates that can be achieved by scheduling policies that are stabilizing, for a given arrival rate. (ii) We propose a design method, also based on finite-dimensional LPs, to obtain stabilizing scheduling policies that can attain a utilization rate arbitrarily close to the aforementioned infimum. We also establish structural and distributional convergence properties, which are used throughout the article, and are significant in their own right.Comment: 16 pages, 2 figure