In this paper, the null controllability in any positive time T of the first-order equation (1) x'(t)=e^{iθ}Ax(t)+Bu(t) (|θ|<π/2 fixed) is deduced from the null controllability in some positive time L of the second-order equation (2) z''(t)=Az(t)+Bv(t). The differential equations (1) and (2) are set in a Banach space, B is an admissible unbounded control operator, and A is a generator of cosine operator function. The control transmutation method explicits the input function u of (1) in terms of the input function v of (2): u(t,x)=\int k(t,s)v(s)ds, where the compactly supported kernel k depends on T and L only. It proves that the norm of a u steering the system (1) from an initial state x_{0} to zero grows at most like ||x_{0}||\exp(α_{*}L^{2}/T) as the control time T tends to zero. (The rate α_{*} is characterized independently by a one-dimensional controllability problem.) In the applications to the cost of fast controls for the heat equation, L is the length of the longest ray of geometric optics which does not intersect the control region.
This article concerns the exact controllability of unitary groups on Hilbert spaces with unbounded control operator. It provides a necessary and sufficient condition not involving time which blends a resolvent estimate and an observability inequality. By the transmutation of controls in some time L for the corresponding second order conservative system, it is proved that the cost of controls in time T for the unitary group grows at most like \exp(αL^{2}/T) as T tends to 0. In the application to the cost of fast controls for the Schr{ö}dinger equation, L is the length of the longest ray of geom…
This paper is devoted to study the cost of the null controllability for the Stokes system. Using the control transmutation method we show that the cost of driving the Stokes system to rest at time T is of order e^C/T when T -->0^+,i.e., the same order as for the heat equation. For this to be possible, we are led to study the exact controllability of one hyperbolic system with a resistance term, which will be done under assumptions on the control region.
Given a control region $Ω$ on a compact Riemannian manifold $M$, we consider the heat equation with a source term $g$ localized in $Omega$. It is known that any initial data in $L^2(M)$ can be stirred to 0 in an arbitrarily small time $T$ by applying a suitable control $g$ in $L^2([0,T]xOmega)$, and, as $T$ tends to 0, the norm of $g$ grows like $e^(C/T)$ times the norm of the data. We investigate how $C$ depends on the geometry of $Omega$. We prove $C\geq d^{2}/4$ where $d$ is the largest distance of a point in $M$ from $Ω$. When $M$ is a segment of length $L$ controlled at one end, we prove …
Given a time T>0 and a region Omega on a compact Riemannian manifold M, we consider the best constant, denoted C_{T,Omega}, in the observation inequality for the Schroedinger evolution group of the Laplacian Delta with Dirichlet boundary condition: for all f in L^2(M), ||f||_{L^2(M)} \leq C_{T,Omega} ||exp(itDelta)f||_{L^2((0,T)xOmega)}. We investigate the influence of the geometry of Omega on the growth of C_{T,Omega} as T tends to 0. By duality, C_{T,Omega} is also the controllability cost of the free Schroedinger equation on M with Dirichlet boundary condition in time T by interior controls…
The controllability cost for the heat equation as the control time $T$ goes to 0 is well-known of the order $e^{C/T}$ for some positive constant $C$, depending on the controlled domain and for all initial datum. In this paper, we prove that the constant $C$ can be chosen to be arbitrarily small if the support of the initial data is sufficiently close to the controlled domain, but not necessary inside the controlled domain. The proof is in the spirit on Lebeau and Robbiano's approach in which a new spectral inequality is established. The main ingredient of the proof of the new spectral inequali…
In this paper, we consider the cost of null controllability for a large class of linear equations of parabolic or dispersive type in one space dimension in small time. By extending the work of Tenenbaum and Tucsnak in "New blow-up rates for fast controls of Schrödinger and heat equations`", we are able to give precise upper bounds on the time-dependance of the cost of fast controls when the time of control T tends to 0. We also give a lower bound of the cost of fast controls for the same class of equations, which proves the optimality of the power of T involved in the cost of the control. Thes…
This paper studies an online optimization problem with a finite prediction window of cost functions and additional switching costs on decisions. We propose two gradient-based online algorithms: Receding Horizon Gradient Descent (RHGD), and Receding Horizon Accelerated Gradient (RHAG). Both algorithms only require a finite number of projected gradient evaluations at each stage. We provide upper bounds on the dynamic regrets of the proposed algorithms and show that the regret upper bounds decay exponentially with the length of the prediction window. Moreover, we study the fundamental lower bound…
University of the Basque Country（巴斯克大学）；Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学）；Fundación Deusto（德乌斯托基金会）；Universidad Autónoma de Madrid（马德里自治大学）
This paper presents a new fast and robust algorithm that provides fuel-optimal impulsive control input sequences that drive a linear time-variant system to a desired state at a specified time. This algorithm is applicable to a broad class of problems where the cost is expressed as a time-varying norm-like function of the control input, enabling inclusion of complex operational constraints in the control planning problem. First, it is shown that the reachable sets for this problem have identical properties to those in prior works using constant cost functions, enabling use of existing algorithm…
Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on the real line, and suppose the cost of matching two points satisfies the Monge condition. We introduce a notion of locally optimal transport plan, motivated by the weak KAM (Aubry-Mather) theory, and show that all locally optimal transport plans are conjugate to shifts and that the cost of a locally optimal transport plan is a convex function of a shift parameter. This theory is applied to a transportation problem arising in image processing: for two sets of point masses on the circl…
We study the problem of online learning in a class of Markov decision processes known as linearly solvable MDPs. In the stationary version of this problem, a learner interacts with its environment by directly controlling the state transitions, attempting to balance a fixed state-dependent cost and a certain smooth cost penalizing extreme control inputs. In the current paper, we consider an online setting where the state costs may change arbitrarily between consecutive rounds, and the learner only observes the costs at the end of each respective round. We are interested in constructing algorith…
We present an iterative method to efficiently solve the optimal transportation problem for a class of strictly convex costs which includes quadratic and p-power costs. Given two probability measures supported on a discrete grid with n points, we compute the optimal map using O(n) storage space and O(n log(n)) operations per iteration, with an approximately exponential convergence rate. Our approach allows us to solve optimal transportation problems on spatial grids as large as 4096x4096 and 384x384x384 in a matter of minutes.
Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；Shanghai Jiao Tong University（上海交通大学）
Unit commitment (UC) are essential tools to transmission system operators for finding the most economical and feasible generation schedules and dispatch signals. Constraint screening has been receiving attention as it holds the promise for reducing a number of inactive or redundant constraints in the UC problem, so that the solution process of large scale UC problem can be accelerated by considering the reduced optimization problem. Standard constraint screening approach relies on optimizing over load and generations to find binding line flow constraints, yet the screening is conservative with…
We consider the numerical solution of the discrete multi-marginal optimal transport (MOT) by means of the Sinkhorn algorithm. In general, the Sinkhorn algorithm suffers from the curse of dimensionality with respect to the number of marginals. If the MOT cost function decouples according to a tree or circle, its complexity is linear in the number of marginal measures. In this case, we speed up the convolution with the radial kernel required in the Sinkhorn algorithm by non-uniform fast Fourier methods. Each step of the proposed accelerated Sinkhorn algorithm with a tree-structured cost function…
