Necessary optimality conditions for optimal distributed and (Neumann) boundary control of Burgers equation

In this paper, we investigate the optimal control of the Burgers equation. For both optimal distributed and (Neumann) boundary control problems, the Dubovitskii and Milyutin functional analytical approach is adopted in investigation of the Pontryagin maximum principles of the systems. The necessary optimality conditions are, respectively, presented for two kinds of optimal control problems in both fixed and free final horizon cases, four extremum problems in all. Moreover, in free final horizon case, the assumptions of admissible control set on convexity and non-empty interior are removed so that it can be any set including an interesting case contains only finite many points. Finally, a remark on how to utilize the obtained results is also made for the illustration

Cloud-assisted Distributed Nonlinear Optimal Control for Dynamics over Graph

Dynamics over graph are large-scale systems in which the dynamic coupling among subsystems is modeled by a graph. Examples arise in spatially distributed systems (as discretized PDEs), multi-agent control systems or social dynamics. In this paper, we propose a cloud-assisted distributed algorithm to solve optimal control problems for nonlinear dynamics over graph. Inspired by the centralized Hauser's projection operator approach for optimal control, our main contribution is the design of a descent method in which at each step agents of a network compute a local descent direction, and then obtain a new system trajectory through a distributed feedback controller. Such a controller, iteratively designed by a cloud, allows agents of the network to use only information from neighboring agents, thus resulting into a distributed projection operator over graph. The main advantages of our globally convergent algorithm are dynamic feasibility at each iteration and numerical robustness (thanks to the closed-loop updates) even for unstable dynamics. In order to show the effectiveness of our strategy, we present numerical computations on a discretized model of the Burgers\u2019 nonlinear partial differential equation

A De Giorgi Iteration-based Approach for the Establishment of ISS Properties for Burgers' Equation with Boundary and In-domain Disturbances

This note addresses input-to-state stability (ISS) properties with respect to (w.r.t.) boundary and in-domain disturbances for Burgers' equation. The developed approach is a combination of the method of De~Giorgi iteration and the technique of Lyapunov functionals by adequately splitting the original problem into two subsystems. The ISS properties in $L^2$-norm for Burgers' equation have been established using this method. Moreover, as an application of De~Giorgi iteration, ISS in $L^\infty$-norm w.r.t. in-domain disturbances and actuation errors in boundary feedback control for a 1-$D$ {linear} {unstable reaction-diffusion equation} have also been established. It is the first time that the method of De~Giorgi iteration is introduced in the ISS theory for infinite dimensional systems, and the developed method can be generalized for tackling some problems on multidimensional spatial domains and to a wider class of nonlinear {partial differential equations (PDEs)Comment: This paper has been accepted for publication by IEEE Trans. on Automatic Control, and is available at http://dx.doi.org/10.1109/TAC.2018.2880160. arXiv admin note: substantial text overlap with arXiv:1710.0991

Global Stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers

In this paper we introduce a finite-parameters feedback control algorithm for stabilizing solutions of the Navier-Stokes-Voigt equations, the strongly damped nonlinear wave equations and the nonlinear wave equation with nonlinear damping term, the Benjamin-Bona-Mahony-Burgers equation and the KdV-Burgers equation. This algorithm capitalizes on the fact that such infinite-dimensional dissipative dynamical systems posses finite-dimensional long-time behavior which is represented by, for instance, the finitely many determining parameters of their long-time dynamics, such as determining Fourier modes, determining volume elements, determining nodes , etc..The algorithm utilizes these finite parameters in the form of feedback control to stabilize the relevant solutions. For the sake of clarity, and in order to fix ideas, we focus in this work on the case of low Fourier modes feedback controller, however, our results and tools are equally valid for using other feedback controllers employing other spatial coarse mesh interpolants