We consider the application of feedback control strategies with point actuators to stabilise desired interface shapes. We take a multidimensional Kuramoto--Sivashinsky equation as a test case; this equation arises in the study of thin liquid films, exhibiting a wide range of dynamics in different parameter regimes, including unbounded growth and full spatiotemporal chaos. In the case of limited observability, we utilise a proportional control strategy where forcing at a point depends only on the local observation. We find that point-actuated controls may inhibit unbounded growth of a solution, if they are sufficient in number and in strength, and can exponentially stabilise the desired state. We investigate actuator arrangements, and find that the equidistant case is optimal, with heavy penalties for poorly arranged actuators. We additionally consider the problem of synchronising two chaotic solutions using proportional controls. In the case when the full interface is observable, we construct feedback gain matrices using the linearised dynamics. Such controls improve on the proportional case, and are applied to stabilise non-trivial steady and travelling wave solutions.
Optimal Structured Static State-Feedback Control Design with Limited Model Information for Fully-Actuated Systems
ACCESS Linnaeus Center, School of Electrical Engineering, KTH-Royal Institute of Technology（ACCESS林奈斯中心，电气工程学院，KTH皇家理工学院）；Department of Aerospace Engineering and the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校航空航天工程系与协同科学实验室）
Multidimensional indefinite stochastic Riccati equations and zero-sum stochastic linear-quadratic differential games with non-Markovian regime switching
General Linear Quadratic Optimal Stochastic Control Problem Driven by a Brownian Motion and a Poisson Random Martingale Measure with Random Coefficients
School of Mathematical Sciences Fudan University（复旦大学数学科学学院）
On classical solutions in the stabilization problem for nonholonomic control systems with time-varying feedback laws
Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）；Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine（乌克兰国家科学院应用数学与力学研究所）；Department of Mathematics, University of Klagenfurt（克拉根福大学数学系）
