A spatial domain decomposition method for parabolic optimal control problems

Efficient iterative solvers for a complex valued two-by-two block linear system with application to parabolic optimal control problems

We derive local error estimates for the discretization of optimal control problems governed by linear advection-diffusion partial differential equations (PDEs) using the streamline upwind/Petrov Galerkin (SUPG) stabilized finite element method. We show that if the SUPG method is used to solve optimization problems governed by an advection-dominated PDE, the convergence properties of the SUPG method is substantially different from the convergence properties of the SUPG method applied for the solution of an advection-dominated PDE. The reason is that the solution of the optimal control problem involves another advection-dominated PDE, the so-called adjoint equation, whose advection field is just the negative of the advection of the governing PDEs. For the solution of the optimal control problem, a coupled system involving both the original governing PDE as well as the adjoint PDE must be solved. We show that in the presence of a boundary layer, the local error between the solution of the SUPG discretized optimal control problem and the solution of the infinite dimensional problem is only of first order even if the error is computed locally in a region away from the boundary layer. In the presence of interior layers, we prove optimal convergence rates for the local error in a region away from the layer between the solution of the SUPG discretized optimal control problems and the solution of the infinite dimensional problem. Numerical examples are presented to illustrate some of the theoretical results.

Three algorithms for efficient solution of optimal control problems for high-dimensional systems are presented. Each bypasses the intermediate (and, unnecessary) step of open-loop model reduction. Each also bypasses the solution of the full Riccati equation corresponding to the LQR problem, which is numerically intractable for large n. Motivation for this effort comes from the field of model-based flow control, where open-loop model reduction often fails to capture the dynamics of interest (governed by the Navier–Stokes equation). Our minimum control energy method is a simplified expression for the well-known minimum-energy stabilizing control feedback that depends only on the left eigenvectors corresponding to the unstable eigenvalues of the system matrix A. Our Adjoint of the Direct-Adjoint method is based on the repeated iterative computation of the adjoint of a forward problem, itself defined to be the direct-adjoint vector pair associated with the LQR problem. Our oppositely-shifted subspace iteration (OSSI, the main new result of the present paper) method is based on our new subspace iteration method for computing the Schur vectors corresponding, notably, to the \(m\ll n\) central eigenvalues (near the imaginary axis) of the Hamiltonian matrix related to the Riccati equation of interest. Prototype OSSI implementations are tested on a low-order control problem to illustrate its behavior.

Mathematical formulation and numerical solutions of an optimal Dirichlet boundary control problem for the Boussinesq equations are considered. The solution of the optimal control problem is obtained by adjusting of the temperature on the boundary. We analyze finite element approximations. A gradient method for the solution of the discrete optimal control problem is presented and analyzed. Finally, the results of some computational experiments are presented.

A simplified approach to the solution of linear optimal control problems with quadratic performance indexes is described in this paper. The phase-variable canonical form is used to develop a new type of optimal system equivalence. This concept leads to a substantial simplification of the matrix Riccati equation. The simplified matrix Riccati equation is of the same form for any problem of a given order, say, n, and contains only n nonzero forcing functions. That is, it always corresponds to a set of constant-coefficient scalar differential equations; in various nth-order problems the n nonzero forcing functions and the terminal conditions simply assume different forms. In a very strong sense, this simplified matrix Riccati equation is the simplest possible Riccati equation arising from optimization problems. The method is developed for general time-varying systems with finite terminal time. It is developed also for the important special case of time-invariant systems with infinite terminal time.

A note on preconditioning methods for time-periodic eddy current optimal control problems

Algorithms for the numerical solution of optimal control problems with bounded state variables are developed. Two main cases are considered: either the control variable appears nonlinearly or the control variable appears linearly. In the first case, an extremal are touching the boundary or containing a boundary arc, is shown to satisfy a suitable two-point boundary value problem. In the second case, a numerical idea for solving the problem in statespace is presented which dispenses with the Lagrange-multipliers. Three numerical examples are discussed illustrating the efficiency of the different algorithms. The encountered two-point boundary value problems are solved with the method of multiple shooting.

SUMMARYThis paper presents an algorithm for the indirect solution of optimal control problems that contain mixed state and control variable inequality constraints. The necessary conditions for optimality lead to an inequality constrained two‐point BVP with index‐1 differential‐algebraic equations (BVP‐DAEs). These BVP‐DAEs are solved using a multiple shooting method where the DAEs are approximated using a single‐step linearly implicit Runge–Kutta (Rosenbrock–Wanner) method. An interior‐point Newton method is used to solve the residual equations associated with the multiple shooting discretization. The elements of the residual equations, and the Jacobian of the residual equations, are constructed in parallel. The search direction for the interior‐point method is computed by solving a sparse bordered almost block diagonal (BABD) linear system. Here, a parallel‐structured orthogonal factorization algorithm is used to solve the BABD system. Examples are presented to illustrate the efficiency of the parallel algorithm. It is shown that an American National Standards Institute C implementation of the parallel algorithm achieves significant speedup with the increase in the number of processors used. Copyright © 2013 John Wiley & Sons, Ltd.

Reduction of modes for the control of viscous flows

It is proved that the linear combinations of shifts and contractions of the Gaussian function can be used for an arbitrarily accurate approximation in the space of continuous functions of one variable on any fixed intervals. On the example of the soft lunar landing problem, a method for the numerical solution of optimal control problems based on this approximation procedure of the control function is described. Within the framework of the same example, the sensitivity of constraint functionals to the specification error of optimal parameters is investigated using three approaches as follows: 1) Pontryagin's maximum principle (both numerically and theoretically); 2) the control parametrization technique in combination with the method of sliding nodes; 3) the newly proposed method. A comparative analysis is performed that confirms the effectiveness of the third method.

This paper is concerned with the shape sensitivity analysis of a class of boundary control, constrained, optimal control problems for parabolic systems. The notion of Euler and Lagrange derivatives of a boundary optimal control in the direction of a vector field is introduced. The derivatives are obtained in the form of optimal solutions of auxiliary optimal control problems. The method of sensitivity analysis used in this paper is based on related results on the differential stability of metric projections in Hilbert space onto a convex, closed subset, combined with the material derivative method of shape sensitivity analysis. Parabolic initial-boundary value problems with Dirichlet and Neumann boundary conditions are considered in this paper.

Solution of nonlinear optimal control problems on analog parallel networks is proposed. Recurrent neural networks whose dynamic equations have a Lyapunov function are developed. Such circuits relax to an equilibrium which is the minimum of the Lyapunov function. Nonlinear optimal control problems are formulated in terms of a Lyapunov function and thus are solved using the recurrent networks. Convergence for linear and nonlinear classes of problems is considered. The method is demonstrated by developing and simulating a network to solve a nonlinear vibration problem. Simulation results demonstrate solution times are accurate and extremely fast. Solution times are shown to be independent of the size of the problem.

Receding horizon methods have become very popular in the last decades for the approximate solution of optimal control problems, model predictive control (MPC) being a popular choice. MPC is to a large extent a model-based feedforward technique and as such its performance is rather sensitive to the model quality. This has prompted much interest in the search of methods to make MPC more robust against deviations such as uncertain parameters. This article presents a way to incorporate additional sensitivity terms into the optimization problem in order to reduce the cost function's sensitivity to the model parameters. For tackling the problem Continuation/GMRES (C/GMRES), a method to solve receding horizon nonlinear optimal control problems in an efficient way, was chosen, however, the general formulation is not restricted to this particular method. The potential performance of the approach is shown by means of simulation examples.

Existing methods for nonlinear robust control often use scenario‐based approaches to formulate the control problem as large nonlinear optimization problems. The optimization problems are challenging to solve due to their size, especially if the control problems include time‐varying uncertainty. This paper draws from local reduction methods used in semi‐infinite optimization to solve robust optimal control problems with parametric and time‐varying uncertainty. By iteratively adding interim worst‐case scenarios to the problem, methods based on local reduction provide a way to manage the total number of scenarios. We show that the local reduction method for optimal control problems consists of solving a series of simplified optimal control problems to find worst‐case constraint violations. In particular, we present examples where local reduction methods find worst‐case scenarios that are not on the boundary of the uncertainty set. We also provide bounds on the error if local solvers are used. The proposed approach is illustrated with two case studies with parametric and additive time‐varying uncertainty. In the first case study, the number of scenarios obtained from local reduction is 101, smaller than in the case when all extreme scenarios are considered. In the second case study, the number of scenarios obtained from the local reduction is two compared to 512 extreme scenarios. Our approach was able to satisfy the constraints both for parametric uncertainty and time‐varying disturbances, whereas approaches from literature either violated the constraints or became computationally expensive.

We present an optimal control approach to freeway traffic congestion control. A Link-Node Cell transmission model (LN-CTM) is used to represent the freeway traffic dynamics. The approach searches for solutions represented by a combination of ramp metering and variable speed control. The optimization problem corresponding to the optimal control problem based on the LN-CTM is non-convex and nonlinear. We relax the problem to a linear optimization problem, and propose an approach to map the solution of the linear optimization algorithm to the solution of the original optimal control problem. We prove that the solution derived from this approach is optimal for the original optimal control algorithm. Finally, we use a model predictive framework to demonstrate the optimal control formulation presented in this paper and discuss its potential use.