A numerical algorithm for singular optimal control synthesis using continuation methods

This paper presents an efficient computational method for the synthesis of singular optimal control problems. The proposed numerical procedure consists of two phases. In the first phase the original singular optimal control problem is converted into a non-singular one by adding to the performance index a perturbed (or weighted) energy term. The resultant boundary value problem can easily be solved for an appropriately large value of the perturbation parameter. In the second phase the solution obtained from the first phase is refined in a systematic manner based on continuation methods (imbedding methods or homotopy methods) until the optimal (or suboptimal) solution to the original problem is achieved. One of the major advantages of the proposed algorithm is that the resultant two-point boundary value problem need be solved just once for a properly large perturbation parameter and the refinement of the solution is accomplished by solving a set of initial value problems sequentially and/or in parallel as the perturbation parameter goes to zero. The proposed algorithm is therefore computationally efficient and applicable to a large class of optimal control problems with various boundary conditions (e.g. fixed and free terminal time). The practicability of the method is demonstrated by computer simulations on an example problem.     

A Least-Squares/Fictitious Domain Method for Linear Elliptic Problems with Robin Boundary Conditions

In this article, we discuss a least-squares/fictitious domain method for the solution of linear elliptic boundary value problems with Robin boundary conditions. Let Ω and ω be two bounded domains of R^dsuch that ω⊂Ω. For a linear elliptic problem in Ω\ω with Robin boundary condition on the boundary γ of ω, our goal here is to develop a fictitious domain method where one solves a variant of the original problem on the full Ω, followed by a well-chosen correction over ω. This method is of the virtual control type and relies on a least-squares formulation making the problem solvable by a conjugate gradient algorithm operating in a well chosen control space. Numerical results obtained when applying our method to the solution of two-dimensional elliptic and parabolic problems are given; they suggest optimal order of convergence.     

A Numerical Thermal-Hydraulic Model to Simulate the Fast Transients in a Supercritical Water Channel Subjected to Sharp Pressure Variations

The present work demonstrates the extension of a thermal-hydraulic model, THRUST, with an objective to simulate the fast transient flow dynamics in a supercritical water channel of circular cross section. THRUST is a 1-D model which solves the nonlinearly coupled mass, axial momentum and energy conservation equations in time domain based on a characteristics-dependent fully implicit finite difference scheme using an Eulerian approach. The model developed accounts for the compressibility of the supercritical flow by considering the finite value of acoustic speed in the solution algorithm and treats the boundary conditions naturally. A supercritical water channel of circular cross section, for which the experimental data is available at steady state operating conditions, is chosen for the transient simulations to start with. Two different case studies are undertaken with a purpose to assess the capability of the model to analyze the fast transient processes caused by the large reduction in system pressure. The first transient case study is where the initial exit pressure is reduced by 1 MPa exponentially in a time span of 5 s. In the second case study, the transient is initiated with a sudden step decrease in the exit pressure by the same amount. Results obtained for both the case studies show the desired performance from the model developed.     

Optimal shape design and unilateral boundary value problems: Part I

In the first part we give a general existence theorem and a regularization method for an optimal control problem where the control is a domain in R″ and where the system is governed by a state relation which includes differential equations as well as inequalities. In the second part applications for optimal shape design problems governed by the Dirichlet-Signorini boundary value problem are presented. Several numerical examples are included.     

Gradient flow approach to LQ cost improvement for simultaneous stabilization problem

In this paper we consider LQ cost optimization for the simultaneous stabilization problem. The objective is to find a single simultaneously stabilizing feedback gain matrix such that all closed-loop systems exhibit good transient behaviour. The cost function used is a quadratic function of the system states and the control vector. This paper proposes to seek an optimization solution by solving an ordinary differential equation which is a gradient flow associated with the cost function. Two examples are presented to illustrate the effectiveness of the proposed procedure.     

Solution techniques for periodic control problems: a case study in production planning

Two numerical techniques for solving optimal periodic control problems with a free period are developed. The first method uses shooting techniques for solving an appropriate boundary value problem associated with the necessary conditions of the minimum principle. A convenient form of the transversality condition for the free period is incorporated. The second method is a direct optimization method that applies non-linear programming techniques to a discretized version of the control problem. Both numerical methods are illustrated in detail by a non-convex economic production planning problem. In this model, the π-test reveals that the steady-state operation is not optimal. The optimal periodic control is computed such that a complete set of necessary conditions is verified. The solution techniques are extended to obtain the optimal periodic control under various state constraints. A sensitivity analysis of the optimal solution is performed with respect to a specific parameter in the model. © 1998 John Wiley & Sons, Ltd.     

Numerical solution of a drug displacement problem with bounded state variables

The interaction of the two drugs warfarin and phenylbutazone has previously been considered as a time-optimal control problem with state inequality constraints. We include bounds for the control and show that necessary optimality conditions and junction conditions for bounded state variables lead to boundary value problems with switching and junction conditions. A special version of the multiple-shooting algorithm is employed for solving the different types of boundary value problems. The switching structure of the optimal control is determined for a range of parameters in the state constraint. Owing to the special structure of the control, a state space solution is obtained in a first step which reduces the numerical complexity of the problem. It is shown how the numerical results can be used to compute the generalized gradient of the optimal value function explicitly.     

Analysis of High-Order Absorbing Boundary Conditions for the Schrödinger Equation

The paper is concerned with the numerical solution of Schrödinger equations on an unbounded spatial domain. High-order absorbing boundary conditions for one-dimensional domain are derived, and the stability of the reduced initial boundary value problem in the computational interval is proved by energy estimate. Then a second order finite difference scheme is proposed, and the convergence of the scheme is established as well. Finally, numerical examples are reported to confirm our error estimates of the numerical methods.     

Sequential conjugate gradient-restoration algorithm for optimal control problems with non-differential constraints and general boundary conditions,part I

In this paper, a new member of the family of sequential gradient-restoration algorithms for the solution of optimal control problems is presented. This is an algorithm of the conjugate gradient type, which is designed to solve two classes of optimal control problems, called Problem P1 and Problem P2 for easy indentification. Problem P1 involves minimizing a functional I subject to differential constraints and general boundary conditions. It consists of finding the state x (t), the control u (t), and the parameter pi so that the functional I is minimized, while the constraints and the boundary conditions are satisfied to a predetermined accuracy. Problem P2 extends Problem P1 to include non-differential constraints to be satisfied everywhere along the interval of integration. The approach taken is a sequence of two-phase cycles, composed of a conjugate gradient phase and a restoration phase. The conjugate gradient phase involves one iteration and is designed to decrease the value of the functional, while the constraints are satisfied to first order. The restoration phase involves one or more iterations; each restorative iteration is designed to force constraint satisfaction to first order, while the norm squared of the variations of the control, the parameter, and the missing components of the initial state is minimized. The resulting algorithm has several properties: (i) it produces a sequence of feasible solutions; (ii) each feasible solution is characterized by a value of the functional I which is smaller than that associated with any previous feasible solution; and (iii) for the special case of a quadratic functional subject to linear constraints, the variations of the state, control, and parameter produced by the sequence of conjugate gradient phases satisfy various orthogonality and conjugacy conditions. The algorithm presented here differs from those of References 1-4, in that it is not required that the state vector be given at the initial point. Instead, the initial conditions can be absolutely general. In analogy with References 1-4, the present algorithm is capable of handling general final conditions; therefore, it is suitable for the solution of optimal control problems with general boundary conditions. The importance of the present algorithm lies in that many optimal control problems either arise naturally in the present format or can be brought to such a format by means of suitable transformations.⁵ Therefore, a great variety of optimal control problems can be handled. This includes: (i) problems with control equality constraints, (ii) problems with state equality constraints, (iii) problems with state-derivative equality constraints, (iv) problems with-control inequality constraints, (v) problems with state inequality constraints, (vi) problems with state-derivative inequality constraints, and (vii) Chebyshev minimax problems. Several numerical examples are presented in Part 2 (Reference 6) in order to illustrate the performance of the algorithm associated with Problem P1 and Problem P2. The numerical results show the feasibility as well as the convergence characteristics of the present algorithm.     

A Non-Ersatz Material Approach for the Topology Optimization of Elastic Structures Based on Piecewise Constant Level Set Method

The topology optimization of a linearized elasticity system with the area (volume) constraint is investigated. A non-ersatz material approach is proposed. By introducing a fixed background domain, the linearized elasticity system is extended into the background domain by a characteristic function. The piecewise constant level set (PCLS) method is applied to represent the original material region and the void region. A quadratic function of PCLS function is proposed to replace the characteristic function. The functional derivative of the objective functional with respect to PCLS function is derived, which is zero in the void region and nonzero in the original material region. A penalty gradient algorithm is proposed. Four numerical experiments of 2D and 3D elastic structures with different boundary conditions are presented, illustrating the validity of the proposed algorithm.     

Shape sensitivities in a Navier‐Stokes flow with convective and grey bodies radiative thermal transfer

We study a shape optimal design problem for a forced convection flow: the steady‐state Navier–Stokes equations coupled with an integro‐differential thermal model. The thermal transfers are convective, diffusive and radiative with multiple reflections (model of grey bodies, radiosity equation). The inverse problem consists in minimizing a smooth cost function which depends on the solution, with respect to the domain of the equations. We prove the differentiability of the solution with respect to the domain. It follows the cost function differentiability. We introduce the adjoint state equation and obtain the exact differential of the cost function. The computational method of shape sensitivities and the optimization process are presented too. Copyright © 2003 John Wiley & Sons, Ltd.     

An Edge-Based Smoothed Finite Element Method with TBC for the Elastic Wave Scattering by an Obstacle

Elastic wave scattering has received ever-increasing attention in military and medical fields due to its high-precision solution. In this paper, an edge-based smoothed finite element method (ES-FEM) combined with the transparent boundary condition (TBC) is proposed to solve the elastic wave scattering problem by a rigid obstacle with smooth surface, which is embedded in an isotropic and homogeneous elastic medium in two dimensions. The elastic wave scattering problem satisfies Helmholtz equations with coupled boundary conditions obtained by Helmholtz decomposition. Firstly, the TBC of the elastic wave scattering is constructed by using the analytical solution to Helmholtz equations, which can truncate the boundary value problem (BVP) in an unbounded domain into the BVP in a bounded domain. Then the formulations of ES-FEM with the TBC are derived for Helmholtz equations with coupled boundary conditions. Finally, several numerical examples illustrate that the proposed ES-FEM with the TBC (ES-FEM-TBC) can work effectively and obtain more stable and accurate solution than the standard FEM with the TBC (FEM-TBC) for the elastic wave scattering problem.     

A Fast Local Level Set Method for Inverse Gravimetry

We propose a fast local level set method for the inverse problem of gravimetry. The theoretical foundation for our approach is based on the following uniqueness result: if an open set D is star-shaped or x₃-convex with respect to its center of gravity, then its exterior potential uniquely determines the open set D. To achieve this purpose constructively, the first challenge is how to parametrize this open set D as its boundary may have a variety of possible shapes. To describe those different shapes we propose to use a level-set function to parametrize the unknown boundary of this open set. The second challenge is how to deal with the issue of partial data as gravimetric measurements are only made on a part of a given reference domain Ω. To overcome this difficulty, we propose a linear numerical continuation approach based on the single layer representation to find potentials on the boundary of some artificial domain containing the unknown set D. The third challenge is how to speed up the level set inversion process. Based on some features of the underlying inverse gravimetry problem such as the potential density being constant inside the unknown domain, we propose a novel numerical approach which is able to take advantage of these features so that the computational speed is accelerated by an order of magnitude. We carry out numerical experiments for both two- and three-dimensional cases to demonstrate the effectiveness of the new algorithm.     

A new analytical method for solving a class of nonlinear optimal control problems

In this paper, we introduce a new analytic technique for a class of nonlinear optimal control problems and present a theorem of convergence of the method. In this scheme, first, the original optimal control problem is transformed into a nonlinear two‐point boundary value problem via the Pontryagin's maximum principle, and then, we apply a new method for solving two‐point boundary value problem. The proposed modification is made by introducing He's polynomials in the correction functional. The suggested algorithm is quite efficient and is practically well suited for using in these problems. The proposed iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimal boundary control problem for ill‐posed elliptic equation in domains with rugous boundary. Existence result and optimality conditions

The main purpose is the study of optimal control problem in a domain with rough boundary for the mixed Dirichlet‐Neumann boundary value problem for the strongly nonlinear elliptic equation with exponential nonlinearity. A density of surface traction u acting on a part of rough boundary is taken as a control. The optimal control problem is to minimize the discrepancy between a given distribution and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for a given control. After having defined a suitable functional class in which we look for solutions, we prove the consistency of the original optimal control problem and show that it admits a unique optimal solution. Then we derive a first‐order optimality system assuming the optimal solution is slightly more regular.     

On the solution of the linear-quadratic optimal control problem by extended invariant imbedding

The solution of the linear-quadratic output regulator problem by Pontryagin's maximum principle results in an unstable linear boundary value problem. The method of invariant imbedding, which requires the integration of a matrix Riccati differential equation, solves certain boundary value problems in a stable manner. In order to make this well known method applicable, an extension algorithm is defined, which maps the boundary value problem into a problem of double dimension. The resulting algorithm of extended invariant imbedding does not depend on the boundary values. So the matrix Riccati equation has to be integrated only once ‘offline’: The unknown boundary values, the state, and the control are calculated by solution of systems of linear equations (‘online’). So, if the boundary values change, only systems of linear equations have to be solved once more. The algorithm has two variants with usually contrary stability behaviour.     

A multiobjective optimization approach to filter tuning applied to coupled hyperbolic PDEs describing gas flow dynamics

In this work, we introduce a multiobjective optimization approach that seeks the optimal process noise statistics in the extended Kalman filter (EKF). The bi‐objective Mesh Adaptive Direct Search (Bi MADS) algorithm was used to minimize a performance index based on state estimate errors. The EKF estimated the gas flow dynamics in a pipeline system. Simulations were conducted with outflow boundary conditions for the flow model that contain gradual changes and discontinuities. To ensure shock‐capturing properties, the model was approximated with a semidiscrete finite volume scheme using Roe's SUPERBEE limiter. The knee point in the Pareto front was based on normal boundary intersection approach and selected to compute the flow estimates. Numerical experiments demonstrated that Bi MADS is suitable for tuning the EKF and, compared to the normalized weighted sum method and nondominated sorting genetic algorithm, it showed to be superior in terms of computation time and most effective in finding Pareto optimal solutions.     

Fractional Chebyshev functional link neural network-optimization method for solving delay fractional optimal control problems with Atangana-Baleanu derivative

In this article, we propose a higher order neural network, namely the functional link neural network (FLNN), for the model of linear and nonlinear delay fractional optimal control problems (DFOCPs) with mixed control-state constraints. We consider DFOCPs using a new fractional derivative with nonlocal and nonsingular kernel that was recently proposed by Atangana and Baleanu. The derivative possesses more important characteristics that are very useful in modelling. In the proposed method, a fractional Chebyshev FLNN is developed. At the first step, the delay problem is transformed to a nondelay problem, using a Padé approximation. The necessary optimality condition is stated in a form of fractional two-point boundary value problem. By applying the fractional integration by parts and by constructing an error function, we then define an unconstrained minimization problem. In the optimization problem, trial solutions for state, co-state and control functions are utilized where these trial solutions are constructed by using single-layer fractional Chebyshev neural network model. We then minimize the error function using an unconstrained optimization scheme based on the gradient descent algorithm for updating the network parameters (weights and bias) associated with all neurons. To show the effectiveness of the proposed neural network, some numerical results are provided.     

A Penalty Optimization Algorithm for Eigenmode Optimization Problem Using Sensitivity Analysis

This paper investigates the eigenmode optimization problem governed by the scalar Helmholtz equation in continuum system in which the computed eigenmode approaches the prescribed eigenmode in the whole domain. The first variation for the eigenmode optimization problem is evaluated by the quadratic penalty method, the adjoint variable method, and the formula based on sensitivity analysis. A penalty optimization algorithm is proposed, in which the density evolution is accomplished by introducing an artificial time term and solving an additional ordinary differential equation. The validity of the presented algorithm is confirmed by numerical results of the first and second eigenmode optimizations in 1D and 2D problems.