scholarly journals State Feedback Regulation Problem to the Reaction-Diffusion Equation

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1983
Author(s):  
Francisco Jurado ◽  
Andrés A. Ramírez
Keyword(s):  
State Feedback ◽  
Feedback Regulation ◽  
Parabolic Type ◽  
Reaction Diffusion ◽  
The State ◽  
Reaction Diffusion Equation ◽  
Finite Dimensional ◽  
Sturm Liouville ◽  
Bounded Input ◽  
Feedback Regulator

In this work, we explore the state feedback regulator problem (SFRP) in order to achieve the goal for trajectory tracking with harmonic disturbance rejection to one-dimensional (1-D) reaction-diffusion (R-D) equation, namely, a partial differential equation of parabolic type, while taking into account bounded input, output, and disturbance operators, a finite-dimensional exosystem (exogenous system), and the state of the exosystem as the state to the feedback law. As is well-known, the SFRP can be solved only if the so-called Francis (regulator) equations have solution. In our work, we try with the solution of the Francis equations from the 1-D R-D equation following given criteria to the eigenvalues from the exosystem and transfer function of the system, but the state operator is here defined in terms of the Sturm–Liouville differential operator (SLDO). Within this framework, the SFRP is then solved for the 1-D R-D equation. The numerical simulation results validate the performance of the regulator.

scholarly journals Characteristic phenomena in combustion

1997 ◽  
Vol 1 (2) ◽  
pp. 147-159
Author(s):  
Dirk Meinköhn
Keyword(s):  
State Space ◽  
Reaction Diffusion ◽  
Stationary Solutions ◽  
The State ◽  
State Variable ◽  
Finite Dimensional ◽  
Dimensional State Space ◽  
Singularity Order ◽  
The Relationship ◽  
Stable Stationary Solutions

For the case of a reaction–diffusion system, the stationary states may be represented by means of a state surface in a finite-dimensional state space. In the simplest example of a single semi-linear model equation given. in terms of a Fredholm operator, and under the assumption of a centre of symmetry, the state space is spanned by a single state variable and a number of independent control parameters, whereby the singularities in the set of stationary solutions are necessarily of the cuspoid type. Certain singularities among them represent critical states in that they form the boundaries of sheets of regular stable stationary solutions. Critical solutions provide ignition and extinction criteria, and thus are of particular physical interest. It is shown how a surface may be derived which is below the state surface at any location in state space. Its contours comprise singularities which correspond to similar singularities in the contours of the state surface, i.e., which are of the same singularity order. The relationship between corresponding singularities is in terms of lower bounds with respect to a certain distinguished control parameter associated with the name of Frank-Kamenetzkii.

A Comparison among Multi-Agent Stochastic Optimization Algorithms for State Feedback Regulator Design of a Twin Rotor MIMO System

2016 ◽  
pp. 409-448 ◽  
Author(s):  
Kaushik Das Sharma
Keyword(s):  
Stochastic Optimization ◽  
State Feedback ◽  
Search Algorithm ◽  
Mimo System ◽  
Single Agent ◽  
The State ◽  
Multi Agent ◽  
Twin Rotor Mimo System ◽  
Twin Rotor ◽  
Feedback Regulator

Multi-agent optimization or population based search techniques are increasingly become popular compared to its single-agent counterpart. The single-agent gradient based search algorithms are very prone to be trapped in local optima and also the computational cost is higher. Multi-Agent Stochastic Optimization (MASO) algorithms are much powerful to overcome most of the drawbacks. This chapter presents a comparison of five MASO algorithms, namely genetic algorithm, particle swarm optimization, differential evolution, harmony search algorithm, and gravitational search algorithm. These MASO algorithms are utilized here to design the state feedback regulator for a Twin Rotor MIMO System (TRMS). TRMS is a multi-modal process and the design of its state feedback regulator is quite difficult using conventional methods available. MASO algorithms are typically suitable for such complex process optimizations. The performances of different MASO algorithms are presented and discussed in light of designing the state regulator for TRMS.

scholarly journals Asymptotic Behavior for a Class of Nonclassical Parabolic Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Yanjun Zhang ◽  
Qiaozhen Ma
Keyword(s):  
Parabolic Equations ◽  
Upper Semicontinuity ◽  
Reaction Diffusion ◽  
Regularity Result ◽  
Global Attractors ◽  
Reaction Diffusion Equation ◽  
Exponential Attractors ◽  
Finite Dimensional ◽  
Fixed Subset ◽  
Two Parameters

This paper is devoted to the qualitative analysis of a class of nonclassical parabolic equations ut-εΔut-ωΔu+f(u)=g(x) with critical nonlinearity, where ε∈[0,1] and ω>0 are two parameters. Firstly, we establish some uniform decay estimates for the solutions of the problem for g(x)∈H-1(Ω), which are independent of the parameter ε. Secondly, some uniformly (with respect to ε∈[0,1]) asymptotic regularity about the solutions has been established for g(x)∈L2(Ω), which shows that the solutions are exponentially approaching a more regular, fixed subset uniformly (with respect to ε∈[0,1]). Finally, as an application of this regularity result, a family {ℰε}ε∈[0,1] of finite dimensional exponential attractors has been constructed. Moreover, to characterize the relation with the reaction diffusion equation (ε=0), the upper semicontinuity, at ε=0, of the global attractors has been proved.

scholarly journals Asymptotic $ H^2$ regularity of a stochastic reaction-diffusion equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Hongyong Cui ◽  
Yangrong Li
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Random Sets ◽  
Reaction Diffusion Equation ◽  
Regular Perturbation ◽  
Finite Dimensional ◽  
Main Technique ◽  
Asymptotic Dynamics ◽  
Bounded Smooth ◽  
Stochastic Reaction

<p style='text-indent:20px;'>In this paper we study the asymptotic dynamics for the weak solutions of the following stochastic reaction-diffusion equation defined on a bounded smooth domain <inline-formula><tex-math id="M5">\begin{document}$ {\mathcal{O}} \subset {\mathbb{R}}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ N \leqslant 3 $\end{document}</tex-math></inline-formula>, with Dirichlet boundary condition:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \nonumber\begin{aligned} { {{\rm{d}}} u } +(-\Delta u + u ^3- \beta u ) {{\rm{d}}} t = g(x) {{\rm{d}}} t+h(x) {{\rm{d}}} W , \quad u|_{t = 0} = u_0\in H: = L^2( {\mathcal{O}}), \end{aligned} \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M7">\begin{document}$ \beta&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ g\in H $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M9">\begin{document}$ W $\end{document}</tex-math></inline-formula> a scalar and two-sided Wiener process with a regular perturbation intensity <inline-formula><tex-math id="M10">\begin{document}$ h $\end{document}</tex-math></inline-formula>. We first construct an <inline-formula><tex-math id="M11">\begin{document}$ H^2 $\end{document}</tex-math></inline-formula> tempered random absorbing set of the system, and then prove an <inline-formula><tex-math id="M12">\begin{document}$ (H,H^2) $\end{document}</tex-math></inline-formula>-smoothing property and conclude that the random attractor of the system is in fact a finite-dimensional tempered random set in <inline-formula><tex-math id="M13">\begin{document}$ H^2 $\end{document}</tex-math></inline-formula> and pullback attracts tempered random sets in <inline-formula><tex-math id="M14">\begin{document}$ H $\end{document}</tex-math></inline-formula> under the topology of <inline-formula><tex-math id="M15">\begin{document}$ H^2 $\end{document}</tex-math></inline-formula>. The main technique we shall employ is comparing the regularity of the stochastic equation to that of the corresponding deterministic equation for which the asymptotic <inline-formula><tex-math id="M16">\begin{document}$ H^2 $\end{document}</tex-math></inline-formula> regularity is already known.</p>

scholarly journals Finite-Dimensionality of Tempered Random Uniform Attractors

2021 ◽  
Vol 32 (1) ◽  
Author(s):  
Hongyong Cui ◽  
Arthur C. Cunha ◽  
José A. Langa
Keyword(s):  
Fractal Dimension ◽  
Reaction Diffusion ◽  
The Other ◽  
Reaction Diffusion Equation ◽  
Uniform Attractor ◽  
Symbol Space ◽  
Finite Dimensional Reduction ◽  
Squeezing Property ◽  
Finite Dimensional ◽  
Finite Dimensionality

AbstractFinite-dimensional attractors play an important role in finite-dimensional reduction of PDEs in mathematical modelization and numerical simulations. For non-autonomous random dynamical systems, Cui and Langa (J Differ Equ, 263:1225–1268, 2017) developed a random uniform attractor as a minimal compact random set which provides a certain description of the forward dynamics of the underlying system by forward attraction in probability. In this paper, we study the conditions that ensure a random uniform attractor to have finite fractal dimension. Two main criteria are given, one by a smoothing property and the other by a squeezing property of the system, and neither of the two implies the other. The upper bound of the fractal dimension consists of two parts: the fractal dimension of the symbol space plus a number arising from the smoothing/squeezing property. As an illustrative application, the random uniform attractor of a stochastic reaction–diffusion equation with scalar additive noise is studied, for which the finite-dimensionality in $$L^2$$ L 2 is established by the squeezing approach and that in $$H_0^1$$ H 0 1 by the smoothing framework. In addition, a random absorbing set that absorbs itself after a deterministic period of time is also constructed.

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