arXiv subjects

Qi Lu

Publications and source records attributed to Qi Lu.

68 records · Page 4Linked to original sources

arXiv2016

A Mini-Course on Stochastic Control

This note is addressed to giving a short introduction to control theory of stochastic systems, governed by stochastic differential equations in both finite and infinite dimensions. We will mainly explain the new phenomenon and difficulties in the study of controllability and optimal control problems for these sort of equations. In particular, we will show by some examples that both the formulation of stochastic control problems and the tools to solve them may differ considerably from their deterministic counterpart.

math.OC↗

arXiv2016

A Scalable and Adaptable Multiple-Place Foraging Algorithm for Ant-Inspired Robot Swarms

Individual robots are not effective at exploring large unmapped areas. An alternate approach is to use a swarm of simple robots that work together, rather than a single highly capable robot. The central-place foraging algorithm (CPFA) is effective for coordinating robot swarm search and collection tasks. Robots start at a centrally placed location (nest), explore potential targets in the area without global localization or central control, and return the targets to the nest. The scalability of the CPFA is limited because large numbers of robots produce more inter-robot collisions and large search areas result in substantial travel costs. We address these problems with the multiple-place foraging algorithm (MPFA), which uses multiple nests distributed throughout the search area. Robots start from a randomly assigned home nest but return to the closest nest with found targets. We simulate the foraging behavior of robot swarms in the robot simulator ARGoS and employ a genetic algorithm to discover different optimized foraging strategies as swarm sizes and the number of targets are scaled up. In our experiments, the MPFA always produces higher foraging rates, fewer collisions, and lower travel and search time compared to the CPFA for the partially clustered targets distribution. The main contribution of this paper is that we systematically quantify the advantages of the MPFA (reduced travel time and collisions), the potential disadvantages (less communication among robots), and the ability of a genetic algorithm to tune MPFA parameters to mitigate search inefficiency due to less communication.

cs.MA↗

arXiv2016

Characterization of Optimal Feedback for Stochastic Linear Quadratic Control Problems

One of the fundamental issues in Control Theory is to design feedback controls. It is well-known that, the purpose of introducing Riccati equations in the deterministic case is to provide the desired feedback controls for linear quadratic control problems. To date, the same problem in the stochastic setting is only partially well-understood. In this paper, we establish the equivalence between the existence of optimal feedback controls for the stochastic linear quadratic control problems with random coefficients and the solvability of the corresponding backward stochastic Riccati equations in a suitable sense. We also give a counterexample showing the nonexistence of feedback controls to a solvable stochastic linear quadratic control problem. This is a new phenomenon in the stochastic setting, significantly different from its deterministic counterpart.

math.OC↗

arXiv2016

An Internal Observability Estimate for Stochastic Hyperbolic Equations

This paper is addressed to establishing an internal observability estimate for some linear stochastic hyperbolic equations. The key is to establish a new global Carleman estimate for forward stochastic hyperbolic equations in the $L^2$-space. Different from the deterministic case, a delicate analysis of the adaptedness for some stochastic processes is required in the stochastic setting.

math.OC↗

arXiv2015

Stochastic Well-posed Systems and Well-posedness of Some Stochastic Partial Differential Equations with Boundary Control and Observation

We generalize the concept "well-posed linear system" to stochastic linear control systems and study some basic properties of such kind systems. Under our generalized definition, we show the well-posedness of the stochastic heat equation and the stochastic Schrödinger equation with suitable boundary control and observation operators, respectively.

math.OC↗

arXiv2015

Averaged controllability for random evolution partial differential equations

We analyze the averaged controllability properties of random evolution Partial Differential Equations. We mainly consider heat and Schrödinger equations with random parameters, although the problem is also formulated in an abstract frame. We show that the averages of parabolic equations lead to parabolic-like dynamics that enjoy the null-controllability properties of solutions of heat equations in an arbitrarily short time and from arbitrary measurable sets of positive measure. In the case of Schrödinger equations we show that, depending on the probability density governing the random parameter, the average may behave either as a conservative or a parabolic-like evolution, leading to controllability properties, in average, of very different kind.

math.OC↗

arXiv2015

Fredholm Transform and Local Rapid Stabilization for a Kuramoto-Sivashinsky Equation

This paper is devoted to the study of the local rapid exponential stabilization problem for a controlled Kuramoto-Sivashinsky equation on a bounded interval. We build a feedback control law to force the solution of the closed-loop system to decay exponentially to zero with arbitrarily prescribed decay rates, provided that the initial datum is small enough. Our approach uses a method we introduced for the rapid stabilization of a Korteweg-de Vries equation. It relies on the construction of a suitable integral transform and can be applied to many other equations.

math.OC↗

arXiv2014

Transposition Method for Backward Stochastic Evolution Equations Revisited, and Its Application

The main purpose of this paper is to improve our transposition method to solve both vector-valued and operator-valued backward stochastic evolution equations with a general filtration. As its application, we obtain a general Pontryagin-type maximum principle for optimal controls of stochastic evolution equations in infinite dimensions. In 1articular, we drop the technical assumption appeared in [Q. Lü and X. Zhang, Springer Briefs in Mathematics,Springer, New York, 2014, Theorem 9.1].

math.OC↗

arXiv2014

Unique Continuation for Stochastic Heat Equations

We establish a unique continuation property for stochastic heat equations evolving in a bounded domain $G$. Our result shows that the value of the solution can be determined uniquely by means of its value on an arbitrary open subdomain of $G$ at any given positive time constant. Further, when $G$ is convex and bounded, we also give a quantitative version of the unique continuation property. As applications, we get an observability estimate for stochastic heat equations, an approximate result and a null controllability result for a backward stochastic heat equation.

math.AP↗

arXiv2013

Observability Estimate and State Observation Problems for Stochastic Hyperbolic Equations

In this paper, we derive a boundary and an internal observability inequality for stochastic hyperbolic equations with nonsmooth lower order terms. The required inequalities are obtained by global Carleman estimate for stochastic hyperbolic equations. By these inequalities, we study a state observation problem for stochastic hyperbolic equations. As a consequence, we also establish a unique continuation property for stochastic hyperbolic equations.

math.OC↗

arXiv2013

Observability Estimate for Stochastic Schroedinger Equations

In this paper, we establish a boundary observability estimate for stochastic Schrödinger equations by means of the global Carleman estimate. Our Carleman estimate is based on a new fundamental identity for a stochastic Schrödinger-like operator. Applications to the state observation problem for semilinear stochastic Schrödinger equations and the unique continuation problem for stochastic Schrödinger equations are also addressed.

math.OC↗

arXiv2013

Carleman Estimate for Stochastic Parabolic Equations and Inverse Stochastic Parabolic Problems

In this paper, we establish a global Carleman estimate for stochastic parabolic equations. Based on this estimate, we solve two inverse problems for stochastic parabolic equations. One is concerned with a determination problem of the history of a stochastic heat process through the observation at the final time $T$, for which we obtain a conditional stability estimate. The other is an inverse source problem with observation on the lateral boundary. We derive the uniqueness of the source.

math.OC↗

arXiv2013

Exact Controllability for Stochastic Schrodinger Equations

This paper is addressed to studying the exact controllability for stochastic Schrödinger equations by two controls. One is a boundary control in the drift term and the other is an internal control in the diffusion term. By means of the standard duality argument, the control problem is converted into an observability problem for backward stochastic Schrödinger equations, and the desired observability estimate is obtained by a global Carleman estimate. At last, we give a result about the lack of exact controllability, which shows that the action of two controls is necessary.

math.OC↗

arXiv2011

Well-posedness of Backward Stochastic Differential Equations with General Filtration

This paper is addressed to the well-posedness of some linear and semilinear backward stochastic differential equations with general filtration, without using the Martingale Representation Theorem. The point of our approach is to introduce a new notion of solution, i.e., the transposition solution, which coincides with the usual strong solution when the filtration is natural but it is more flexible for the general filtration than the existing notion of solutions. A comparison theorem for transposition solutions is also presented.

math.PR↗