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

Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems

We are interested in high-order linear multistep schemes for time discretization of adjoint equations arising within optimal control problems. First we consider optimal control problems for ordinary differential equations and show loss of accuracy for Adams-Moulton and Adams-Bashford methods, whereas BDF methods preserve high--order accuracy. Subsequently we extend these results to semi--lagrangian discretizations of hyperbolic relaxation systems. Computational results illustrate theoretical findings

Asymptotic Preserving time-discretization of optimal control problems for the Goldstein-Taylor model

We consider the development of implicit-explicit time integration schemes for optimal control problems governed by the Goldstein-Taylor model. In the diffusive scaling this model is a hyperbolic approximation to the heat equation. We investigate the relation of time integration schemes and the formal Chapman-Enskog type limiting procedure. For the class of stiffly accurate implicit-explicit Runge-Kutta methods (IMEX) the discrete optimality system also provides a stable numerical method for optimal control problems governed by the heat equation. Numerical examples illustrate the expected behavior

Optimal control problems constrained by hyperbolic conservation laws.

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.This thesis deals with the solutions of optimal control problems constrained by hyperbolic conservation laws. Such problems pose significant challenges for mathematical analysis and numerical simulations. Those challenges are mainly because of the discontinuities that occur in the solutions of non-linear systems of conservation laws and become more acute when dealing with the multidimensional case. The problem is formulated as the minimisation of a flow matching cost functional constrained by multi-dimensional hyperbolic conservation laws. The control variable is the initial condition of the partial differential equations. In our analysis of the problem, we review extensively the constraints equation and we consider successively the one-dimensional and the multi-dimensional cases. In all the cases, we derive the optimality conditions in the adjoint approach at the continuous level, which are then discretised to arrive at a numerical algorithm for the solution. In the derivation of the optimality conditions, we replace the non-linear conservation laws either by the relaxation equation or the Lattice Boltzmann equation. We illustrate our findings on examples related to the multi-dimensional Burger and the Euler equations

Adjoint-based optimization for optimal control problems governed by nonlinear hyperbolic conservation laws

Research considered investigates the optimal control problem which is constrained by a hyperbolic conservation law (HCL). We carried out a comparative study of the solutions of the optimal control problem subject to each one of the two di erent types of hyperbolic relaxation systems [64, 92]. The objective was to employ the adjoint-based optimization to minimize the cost functional of a matching type between the optimal solution and the target solution. Numerical tests were then carried out and promising results obtained. Finally, an extension was made to the adjoint-based optimization approach to apply second-order schemes for the optimal control problem in which also good numerical results were observed

Conservation laws models in networks and multiscale flow optimization.

Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2011.The flow of fluids in a network is of practical importance in gas, oil and water transport for industrial and domestic use. When the flow dynamics are understood, one may be interested in the control of the flow formulated as follows: given some fluid properties at a final time, can one determine the initial flow properties that lead to the desired flow properties? In this thesis, we first consider the flow of a multiphase gas, described by the drift flux model, in a network of pipes and that of water, modeled by the shallow water equations, in a network of rivers. These two models are systems of partial differential equations of first order generally referred to as systems of conservation laws. In particular, our contribution in this regard can be summed up as follows: For the drift-flux model, we consider the flow in a network of pipes seen mathematically as an oriented graph. We solve the standard Riemann problem and prove a well posedness result for the Riemann problem at a junction. This result is obtained using coupling conditions that describe the dynamics at the intersection of the pipes. Moreover, we present numerical results for standard pipes junctions. The numerical results and the analytical results are in agreement. This is an extension for multiphase flows of some known results for single phase flows. Thereafter, the shallow water equations are considered as a model for the flow of water in a network of canals. We analyze coupling conditions at the confluence of rivers, precisely the conservation of mass and the equality of water height at the intersection, and implement these results for some classical river confluences. We also consider the case of pooled stepped chutes, a geometry frequently utilized by dams to spill floodwater. Here we consider an approach different from the engineering community in the sense that we resolve the dynamics by solving a Riemann problem at the dam for the shallow water equations with some suitable coupling conditions. Secondly, we consider an optimization problem constrained by the Euler equations with a flow-matching objective function. Differently from the existing approaches to this problem, we consider a linear approximation of the flow equation in the form of the microscopic Lattice Boltzmann Equations (LBE). We derive an adjoint calculus and the optimality conditions from the microscopic LBE. Using multiscale analysis, we obtain an equivalent macroscopic result at the hydrodynamic limit. Our numerical results demonstrate the ability of our method to solve challenging problems in fluid mechanics

Error estimation and adaptive moment hierarchies for goal-oriented approximations of the Boltzmann equation

This paper presents an a-posteriori goal-oriented error analysis for a numerical approximation of the steady Boltzmann equation based on a moment-system approximation in velocity dependence and a discontinuous Galerkin finite-element (DGFE) approximation in position dependence. We derive computable error estimates and bounds for general target functionals of solutions of the steady Boltzmann equation based on the DGFE moment approximation. The a-posteriori error estimates and bounds are used to guide a model adaptive algorithm for optimal approximations of the goal functional in question. We present results for one-dimensional heat transfer and shock structure problems where the moment model order is refined locally in space for optimal approximation of the heat flux.Comment: arXiv admin note: text overlap with arXiv:1602.0131

PDE–Based Modelling and Control Strategies for Manufacturing Processes

This work aims to design boundary control strategies to solve demand tracking and backlog problems for manufacturing systems in terms of conservation laws coupled with ODEs in different network topologies. The OCPs are investigated in the dispersing and the merging networks. The problems are optimized utilizing open-loop optimal control based on the direct and the indirect approaches. The proposed approaches enable the solution of the OCPs. All of the approaches, in general, reach a local minima with similar behaviour that leads to the steady-state. The results analysis reveals that each method has its own distinct characteristics. The indirect methodology is characterized by excellent accuracy and minimal processing burden; yet, due to the information necessary to compute the gradient, it is a sensitive method. The ease of use and flexibility to any problem distinguishes the direct method. However, this approach takes substantially longer to achieve a solution when compared to the indirect method. Also, the AMPC was introduced to investigate demand tracking and backlog problems in the context of the complex network of production systems. The addressed network includes structures that are dispersing and merging. Furthermore, the appropriate way to handle the parameters of the AMPC for both control and prediction horizons is addressed. Moreover, the proposed AMPC provides for the solutions of demand tracking and backlog problems. In general, AMPC and traditional MPC attain local minima with similar behaviour that leads to steady-state convergence. When compared to a typical MPC, the AMPC's performance shows a considerable reduction in computational time. Additionally, because it provides a mathematical insight into the method's structure, the AMPC allows for great accuracy of optimal solutions. Finally, the AMPC is characterized by its robustness according to perturbation effects