Robust piecewise adaptive control for an uncertain semilinear parabolic distributed parameter systems

In this study, we focus on designing a robust piecewise adaptive controller to globally asymptotically stabilize a semilinear parabolic distributed parameter systems (DPSs) with external disturbance, whose nonlinearities are bounded by unknown functions. Firstly, a robust piecewise adaptive control is designed against the unknown nonlinearity and the external disturbance. Then, by constructing an appropriate Lyapunov–Krasovskii functional candidate (LKFC) and using the Wiritinger’s inequality and a variant of the Agmon’s inequality, it is shown that the proposed robust piecewise adaptive controller not only ensures the globally asymptotic stability of the closed-loop system, but also guarantees a given performance. Finally, two simulation examples are given to verify the validity of the design method

Observation and control of PDE with disturbances

In this Thesis, the problem of controlling and Observing some classes of distributed parameter systems is addressed. The particularity of this work is to consider partial differential equations (PDE) under the effect of external unknown disturbances. We consider generalized forms of two popular parabolic and hyperbolic infinite dimensional dynamics, the heat and wave equations. Sliding-mode control is used to achieve the control goals, exploiting the robustness properties of this robust control technique against persistent disturbances and parameter uncertainties

Observation and control of PDE with disturbances

In this Thesis, the problem of controlling and Observing some classes of distributed parameter systems is addressed. The particularity of this work is to consider partial differential equations (PDE) under the effect of external unknown disturbances. We consider generalized forms of two popular parabolic and hyperbolic infinite dimensional dynamics, the heat and wave equations. Sliding-mode control is used to achieve the control goals, exploiting the robustness properties of this robust control technique against persistent disturbances and parameter uncertainties

Multiscale and High-Dimensional Problems

High-dimensional problems appear naturally in various scientific areas. Two primary examples are PDEs describing complex processes in computational chemistry and physics, and stochastic/ parameter-dependent PDEs arising in uncertainty quantification and optimal control. Other highly visible examples are big data analysis including regression and classification which typically encounters high-dimensional data as input and/or output. High dimensional problems cannot be solved by traditional numerical techniques, because of the so-called curse of dimensionality. Rather, they require the development of novel theoretical and computational approaches to make them tractable and to capture fine resolutions and relevant features. Paradoxically, increasing computational power may even serve to heighten this demand, since the wealth of new computational data itself becomes a major obstruction. Extracting essential information from complex structures and developing rigorous models to quantify the quality of information in a high dimensional setting constitute challenging tasks from both theoretical and numerical perspective. The last decade has seen the emergence of several new computational methodologies which address the obstacles to solving high dimensional problems. These include adaptive methods based on mesh refinement or sparsity, random forests, model reduction, compressed sensing, sparse grid and hyperbolic wavelet approximations, and various new tensor structures. Their common features are the nonlinearity of the solution method that prioritize variables and separate solution characteristics living on different scales. These methods have already drastically advanced the frontiers of computability for certain problem classes. This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computational methods and to promote the exchange of ideas emerging in various disciplines about how to treat multiscale and high-dimensional problems

Snapshot-Based Methods and Algorithms

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science

Model Order Reduction by Proper Orthogonal Decomposition

We provide an introduction to POD-MOR with focus on (nonlinear) parametric PDEs and (nonlinear) time-dependent PDEs, and PDE constrained optimization with POD surrogate models as application. We cover the relation of POD and SVD, POD from the infinite-dimensional perspective, reduction of nonlinearities, certification with a priori and a posteriori error estimates, spatial and temporal adaptivity, input dependency of the POD surrogate model, POD basis update strategies in optimal control with surrogate models, and sketch related algorithmic frameworks. The perspective of the method is demonstrated with several numerical examples.Comment: arXiv admin note: substantial text overlap with arXiv:1701.0505

Structure-Preserving Model Reduction of Physical Network Systems

This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p

Model Order Reduction

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science