Optimal sampled-data control, and generalizations on time scales

In this paper, we derive a version of the Pontryagin maximum principle for general finite-dimensional nonlinear optimal sampled-data control problems. Our framework is actually much more general, and we treat optimal control problems for which the state variable evolves on a given time scale (arbitrary non-empty closed subset of R), and the control variable evolves on a smaller time scale. Sampled-data systems are then a particular case. Our proof is based on the construction of appropriate needle-like variations and on the Ekeland variational principle.Comment: arXiv admin note: text overlap with arXiv:1302.351

Time Blocks Decomposition of Multistage Stochastic Optimization Problems

Multistage stochastic optimization problems are, by essence, complex because their solutions are indexed both by stages (time) and by uncertainties (scenarios). Their large scale nature makes decomposition methods appealing.The most common approaches are time decomposition --- and state-based resolution methods, like stochastic dynamic programming, in stochastic optimal control --- and scenario decomposition --- like progressive hedging in stochastic programming. We present a method to decompose multistage stochastic optimization problems by time blocks, which covers both stochastic programming and stochastic dynamic programming. Once established a dynamic programming equation with value functions defined on the history space (a history is a sequence of uncertainties and controls), we provide conditions to reduce the history using a compressed "state" variable. This reduction is done by time blocks, that is, at stages that are not necessarily all the original unit stages, and we prove areduced dynamic programming equation. Then, we apply the reduction method by time blocks to \emph{two time-scales} stochastic optimization problems and to a novel class of so-called \emph{decision-hazard-decision} problems, arising in many practical situations, like in stock management. The \emph{time blocks decomposition} scheme is as follows: we use dynamic programming at slow time scale where the slow time scale noises are supposed to be stagewise independent, and we produce slow time scale Bellman functions; then, we use stochastic programming at short time scale, within two consecutive slow time steps, with the final short time scale cost given by the slow time scale Bellman functions, and without assuming stagewise independence for the short time scale noises

Noether's Theorem for Control Problems on Time Scales

We prove a generalization of Noether's theorem for optimal control problems defined on time scales. Particularly, our results can be used for discrete-time, quantum, and continuous-time optimal control problems. The generalization involves a one-parameter family of maps which depend also on the control and a Lagrangian which is invariant up to an addition of an exact delta differential. We apply our results to some concrete optimal control problems on an arbitrary time scale.Comment: This is a preprint of a paper whose final and definite form is published in International Journal of Difference Equations ISSN 0973-6069, Vol. 9 (2014), no. 1, 87--10

A survey on fuzzy fractional differential and optimal control nonlocal evolution equations

We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Applied Mathematics', ISSN: 0377-0427. Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication 20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515

Quantum Control Landscapes

Numerous lines of experimental, numerical and analytical evidence indicate that it is surprisingly easy to locate optimal controls steering quantum dynamical systems to desired objectives. This has enabled the control of complex quantum systems despite the expense of solving the Schrodinger equation in simulations and the complicating effects of environmental decoherence in the laboratory. Recent work indicates that this simplicity originates in universal properties of the solution sets to quantum control problems that are fundamentally different from their classical counterparts. Here, we review studies that aim to systematically characterize these properties, enabling the classification of quantum control mechanisms and the design of globally efficient quantum control algorithms.Comment: 45 pages, 15 figures; International Reviews in Physical Chemistry, Vol. 26, Iss. 4, pp. 671-735 (2007

Controlled diffusion processes

This article gives an overview of the developments in controlled diffusion processes, emphasizing key results regarding existence of optimal controls and their characterization via dynamic programming for a variety of cost criteria and structural assumptions. Stochastic maximum principle and control under partial observations (equivalently, control of nonlinear filters) are also discussed. Several other related topics are briefly sketched.Comment: Published at http://dx.doi.org/10.1214/154957805100000131 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Buildings-to-Grid Integration Framework

This paper puts forth a mathematical framework for Buildings-to-Grid (BtG) integration in smart cities. The framework explicitly couples power grid and building's control actions and operational decisions, and can be utilized by buildings and power grids operators to simultaneously optimize their performance. Simplified dynamics of building clusters and building-integrated power networks with algebraic equations are presented---both operating at different time-scales. A model predictive control (MPC)-based algorithm that formulates the BtG integration and accounts for the time-scale discrepancy is developed. The formulation captures dynamic and algebraic power flow constraints of power networks and is shown to be numerically advantageous. The paper analytically establishes that the BtG integration yields a reduced total system cost in comparison with decoupled designs where grid and building operators determine their controls separately. The developed framework is tested on standard power networks that include thousands of buildings modeled using industrial data. Case studies demonstrate building energy savings and significant frequency regulation, while these findings carry over in network simulations with nonlinear power flows and mismatch in building model parameters. Finally, simulations indicate that the performance does not significantly worsen when there is uncertainty in the forecasted weather and base load conditions.Comment: In Press, IEEE Transactions on Smart Gri

Optimal Control of the Thermistor Problem in Three Spatial Dimensions

This paper is concerned with the state-constrained optimal control of the three-dimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system are derived by employing maximal parabolic regularity in the fundamental theorem of Pr\"uss. Global solutions are addressed, which includes analysis of the linearized state system via maximal parabolic regularity, and existence of optimal controls is shown if the temperature gradient is under control. The adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem in form of a qualified optimality system. The theoretical findings are illustrated by numerical results