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Impulse-to-peak response (I2P) analysis for state-space ordinary differential equation (ODE) systems is a well-studied classical problem. However, the techniques employed for I2P optimal control of ODEs have not been extended to partial differential equation (PDE) systems due to the lack of a universal transfer function and state-space representation. Recently, however, partial integral equation (PIE) representation was proposed as the desired state-space representation of a PDE, and Lyapunov stability theory was used to solve various control problems, such as stability and optimal ${H}_\infty$ control. In this work, we utilize this PIE framework, and associated Lyapunov techniques, to formulate the I2P response analysis problem as a solvable convex optimization and obtain provable bounds for the I2P-norm of linear PDEs. Moreover, by establishing strong duality between primal and dual formulations of the optimization problem, we develop a constructive method for I2P optimal state-feedback control of PDEs and demonstrate the effectiveness of the method on various examples.
ENSAE IP Paris, Center for Research in Economics and Statistics（巴黎高等统计与行政学院，经济与统计研究中心）；UC Berkeley, Department of Industrial engineering and Operations Research（加州大学伯克利分校，工业工程与运筹学系）；Laboratoire Manceau de Mathématiques, Institut du Risque et de l’Assurance, Le Mans Université（勒芒大学，风险与保险研究所，马恩河数学实验室）
A De Giorgi Iteration-based Approach for the Establishment of ISS Properties of a Class of Semi-linear Parabolic PDEs with Boundary and In-domain Disturbances
L0TV: A Sparse Optimization Method for Impulse Noise Image Restoration
School of Data and Computer Science, Sun Yat-sen University（中山大学数据与计算机学院）；Key Laboratory of Machine Intelligence and Advanced Computing, Ministry of Education（教育部机器智能与先进计算重点实验室）；Visual Computing Center, King Abdullah University of Science and Technology（阿卜杜拉国王科技大学视觉计算中心）
H$_1$-ISS Analysis and Boundary Control Design for Coupled Linear ODEs and Hyperbolic PDEs
Universidade Federal de Santa Catarina（圣卡塔琳娜联邦大学）；Université Grenoble Alpes, CNRS, Grenoble-INP, GIPSA-lab（格勒诺布尔阿尔卑斯大学，法国国家科学研究中心，格勒诺布尔国立理工学院，GIPSA实验室）
$H_2$-Optimal Estimation of Linear Delayed and PDE Systems
Center for Nonlinear Studies and Applied Mathematics & Plasma Physics Division of Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室非线性研究与应用数学及等离子体物理分部）；School for Engineering of Matter, Transport and Energy at Arizona State University（亚利桑那州立大学物质、运输与能源工程学院）
A comparative study of structural similarity and regularization for joint inverse problems governed by PDEs
Institute for Computational Engineering & Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校计算工程与科学研究所）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）
