parabolic pde
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Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 201
Author(s):  
Xu Ni ◽  
Kejia Yi ◽  
Yiming Jiang ◽  
Ancai Zhang ◽  
Chengdong Yang

This paper discusses consensus control of nonlinear coupled parabolic PDE-ODE-based multi-agent systems (PDE-ODEMASs). First, a consensus controller of leaderless PDE-ODEMASs is designed. Based on a Lyapunov-based approach, coupling strengths are obtained for leaderless PDE-ODEMASs to achieve leaderless consensus. Furthermore, a consensus controller in the leader-following PDE-ODEMAS is designed and the corresponding coupling strengths are obtained to ensure the leader-following consensus. Two examples show the effectiveness of the proposed methods.


Author(s):  
Yu.V. Averboukh

The paper is concerned with the approximation of the value function of the zero-sum differential game with the minimal cost, i.e., the differential game with the payoff functional determined by the minimization of some quantity along the trajectory by the solutions of continuous-time stochastic games with the stopping governed by one player. Notice that the value function of the auxiliary continuous-time stochastic game is described by the Isaacs–Bellman equation with additional inequality constraints. The Isaacs–Bellman equation is a parabolic PDE for the case of stochastic differential game and it takes a form of system of ODEs for the case of continuous-time Markov game. The approximation developed in the paper is based on the concept of the stochastic guide first proposed by Krasovskii and Kotelnikova.


2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Subhankar Mondal ◽  
M. Thamban Nair

Abstract An inverse problem of identifying the diffusion coefficient in matrix form in a parabolic PDE is considered. Following the idea of natural linearization, considered by Cao and Pereverzev (2006), the nonlinear inverse problem is transformed into a problem of solving an operator equation where the operator involved is linear. Solving the linear operator equation turns out to be an ill-posed problem. The method of Tikhonov regularization is employed for obtaining stable approximations and its finite-dimensional analysis is done based on the Galerkin method, for which an orthogonal projection on the space of matrices with entries from L 2 ⁢ ( Ω ) L^{2}(\Omega) is defined. Since the error estimates in Tikhonov regularization method rely heavily on the adjoint operator, an explicit representation of adjoint of the linear operator involved is obtained. For choosing the regularizing parameter, the adaptive technique is employed in order to obtain order optimal rate of convergence. For the relaxed noisy data, we describe a procedure for obtaining a smoothed version so as to obtain the error estimates. Numerical experiments are carried out for a few illustrative examples.


2021 ◽  
Vol 5 (4) ◽  
pp. 1459-1464
Author(s):  
Jingting Zhang ◽  
Chengzhi Yuan ◽  
Wei Zeng ◽  
Paolo Stegagno ◽  
Cong Wang

2021 ◽  
Author(s):  
Xiao-Heng Chang ◽  
Teng-Fei Li ◽  
Ju H. Park

Abstract In this paper, the finite-time H∞ control problem of nonlinear parabolic partial differential equation (PDE) systems with parametric uncertainties is studied. Firstly, based on the definition of the quantiser, static state feedback controller and dynamic state feedback controller with quantization are presented, respectively. The finite-time H∞ control design strategies are subsequently proposed to analyze the nonlinear parabolic PDE systems with respect to the effect of quantization. And by constructing appropriate Lyapunov functionals for the studied systems, sufficient conditions for the existence of the feedback control gains and the quantizer’s adjusting parameters which guarantee the prescribed attenuation level of H∞ performance are expressed as nonlinear matrix inequalities. Then, by using some inequalities and decomposition technic, the nonlinear matrix inequalities are transformed to standard linear matrix inequalities (LMIs). Moreover, the optimal H∞ control performances are pursued by solving optimization problems subject to the LMIs. Finally, to illustrate the feasibility and effectiveness of the finite-time H∞ control design strategies, an application to the catalytic rod in a reactor is explored.


2021 ◽  
pp. S475-S495
Author(s):  
Md. Masud Rana ◽  
Victoria E. Howle ◽  
Katharine Long ◽  
Ashley Meek ◽  
William Milestone

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