High-order relaxation approaches for adjoint-based optimal control problems governed by nonlinear hyperbolic systems of conservation laws

A computational investigation of optimal control problems which are constrained by hyperbolic systems of conservation laws is presented. The general framework is to employ the adjoint-based optimization to minimize the cost functional of matching-type between the optimal and the target solution. Extension of the numerical schemes to second-order accuracy for systems for the forward and backward problem are applied. In addition a comparative study of two relaxation approaches as solvers for hyperbolic systems is undertaken. In particular optimal control of the 1-D Riemann problem of Euler equations of gas dynamics is studied. The initial values are used as control parameters. The numerical ow obtained by optimal initial conditions matches accurately with observations.The African Institute of Mathematical Sciences (AIMS), University of the Witwatersrand (Wits) and supported in part by the National Research Foundation (NRF) of South Africa UID: 74260, UID: 81299 and UID: 85566.http://www.degruyter.com/view/j/jnma2017-03-30am2016Mathematics and Applied Mathematic

A stoichiometric method for reducing simulation cost of chemical kinetic models

Mathematical models for chemically reacting systems have high degrees of freedom (very large) and are computationally expensive to analyse. In this discussion, we present and analyse a model reduction method that is based on stoichiometry and mass balances. This method can significantly reduce the high degrees of freedom of such systems. Numerical simulations are undertaken to validate and establish efficiency of the method. A practical example of acid mine drainage is used as a test case to demonstrate the efficacy of the procedure. Analytical results show that the stoichiometrically-reduced model is consistent with the original large model, and numerical simulations demonstrate that the method can accelerate convergence of the numerical schemes in some cases.AEA acknowledges support from the Pilot Bursary of the University of Pretoria, the African Institute of Mathematical Sciences and University of Stellenbosch. MKB is grateful to the African Institute for Mathematical Sciences (AIMS) for hosting him while finalising this work. This work is also supported in part by the National Research Foundation of South Africa (Grant number: 93099 and 93476).http://www.elsevier.com/locate/com/pchemeng2019-04-06hj2018Mathematics and Applied Mathematic

Input-to-state stability of non-uniform linear hyperbolic systems of balance laws via boundary feedback control

Please read abstract in the article.http://link.springer.com/journal/2452021-10-22hj2021Mathematics and Applied Mathematic

Modelling and simulation of reactive transport phenomena

Mathematical modelling and numerical simulation of chemical transport phenomena are very challenging due to large numbers of species and reactions involved. Reactive transport models for such systems have high degrees of freedom, and therefore, are computationally expensive to solve. In this discussion, we present and numerically analyse stoichiometric decoupling method for reducing the high degrees of freedom and hence, cost of simulation. This method is a model reduction procedure that is based on some key properties of chemical systems. A multi-scale model of a passive treatment method for acidic mine effluents is used to test the efficacy of the reduction procedure. Moreover, reduced models are characteristically non-linear and stiff, thus, we used numerical techniques to study the reduction error in order to establish compatibility/efficiency of the reduction procedure.AEA acknowledges support from the Pilot Bursary of the University of Pretoria, the African Institute of Mathematical Sciences and University of Stellenbosch. This work is also supported in part by the National Research Foundation of South Africa (Grant Number: 93099 and 93476).http://www.elsevier.com/locate/jocs2019-09-01hj2018Mathematics and Applied Mathematic

Numerical discretization of coupling conditions by high-order schemes

We consider numerical schemes for 2 2 hyperbolic conservation laws on graphs. The hyperbolic equations are given on the spatially one{dimensional arcs and are coupled at a single point, the node, by a nonlinear coupling condition. We develop high-order nite volume discretizations for the coupled problem. The reconstruction of the uxes at the node is obtained using derivatives of the parameterized algebraic conditions imposed at the nodal points in the network. Numerical results illustrate the expected theoretical behavior.This work has been partly supported by DFG ‘Integrated Production in High-Wage Countries’ and the BMBF KinOpt project. This work is also based on the research supported in part by the National Research Foundation of South Africa (Grant No. 93476) and the Research Development Programme of the University of Pretoria.http://link.springer.com/journal/109152017-10-30hb2016Mathematics and Applied Mathematic

Modified Newton’s method in the leapfrog method for mobile robot path planning

The problem of determining an optimal trajectory for an autonomous mobile robot in an environment with obstacles is considered. The Leapfrog approach is used to solve the ensuing system of equations derived from the first-order optimality conditions of the Pontryagin’s Minimum Principle. A comparison is made between a case in which the classical Newton Method and the Modified Newton Method are used in the shooting method for solving the two-point boundary value problem in the inner loop of the Leapfrog algorithm. It can be observed that with this modification there is an improvement in the convergence rate of the Leapfrog algorithm in general.http://www.springer.comseries/111562019-03-20hj2018Mathematics and Applied Mathematic

Numerical boundary feedback stabilisation of non-uniform hyperbolic systems of balance laws

In this paper, numerical boundary stabilisation of a non-uniform hyperbolic system of balance laws is studied. For the numerical discretisation of the balance laws, a first order explicit upwind scheme is used for the spatial discretisation; and for the temporal discretisation a splitting technique is employed. A discrete L2−Lyapunov function is employed to investigate conditions for the stability of the system. After constructing discrete numerical Lyapunov functionals, we prove an asymptotic expo-nential stability result for a class of non-uniform linear hyperbolic systems of balance laws. Convergence of the solution to its equilibrium is proved. Further application of the approach to practical problems through concrete examples is presented together with suggestions for numerical implementation. The numerical computations also demonstrate the stability of the numerical scheme with parameters chosen to satisfy the stability requirements.G. Weldegiyorgis thanks Department of Mathematics and Applied Mathematics of University of Pretoria for partial funding for his research. M.K. Banda is also grateful to African Institute for Mathematical Sciences - South Africa and for hosting him while finalising this work. He is also grateful for DAAD funding (ID: 57314019) for a visit at RWTH-Aachen. This work is also supported in part by the National Research Foundation of South Africa (Grant number: 93099 and 93476).http://www.tandfonline.com/loi/tcon202019-08-03hj2018Mathematics and Applied Mathematic

Simultaneous identification of damping coefficient and initial value for PDEs from boundary measurement

In this paper, simultaneous identification of damping or anti-damping coefficient and initial value for some Riesz spectral systems is considered. An identification algorithm is proposed based on the fact that the output of the system happens to be decomposed into a product of an exponential function and a periodic function. The former contains information of the damping coefficient, while the latter does not. The convergence and error analysis are also developed. Three examples, namely an anti-stable wave equation with boundary anti-damping, the Schrödinger equation with internal anti-damping and two connected strings with middle joint anti-damping, are investigated and demonstrated by numerical simulations to show the effectiveness of the proposed algorithm.The National Natural Science Foundation of China [grant number 11326196], the National Research Foundation of South Africa [grant number 93099 and 93476], the Doctoral Fund Program of Tianjin Normal University [grant number 52XB1413], and the Postdoctoral Fellowship of the University of Pretoria.http://www.tandfonline.com/loi/tcon202018-05-03Mathematics and Applied Mathematic

Boundary switch on/off control approach to simultaneous identification of diffusion coefficient and initial state for one-dimensional heat equation

In this paper, we consider simultaneous reconstruction of diffusion coefficient and initial state for a one-dimensional heat equation through boundary control and measurement. The boundary measurement is proposed to make the system approximately observable, and both the coefficient and initial state are shown to be identifiable by this measurement under a boundary switch on/off control. By a Dirichlet series representation for the observation, we can transform the problem into an inverse process of reconstruction of the spectrum and coefficients for the Dirichlet series in terms of observation. This happens to be the reconstruction of spectral data for an exponential sequence with measurement error, and it enables us to develop an algorithm based on the matrix pencil method in signal analysis. A theoretical error analysis for the algorithm concerning the coefficient reconstruction is carried out for the proposed method. The numerical simulations are presented to verify the proposed algorithm.The National Natural Science Foundation of China, the Science & Technology Development Fund of Tianjin Education Commission for Higher Education, China, Tianjin 131 Talents Project, Program for Innovative Research Team in Universities of Tianjin, China, the National Research Foundation of South Africa and the Postdoctoral Fellowship of the University of Pretoria.http://aimsciences.org/journal/1531-34922021-07-01hj2021Mathematics and Applied Mathematic