Shape Sensitivity Analysis of Boundary Optimal Control Problems for Parabolic Systems

In this paper, compressible, stationary Navier–Stokes equations are considered. The model is well-posed, and there exist weak solutions in bounded domains, subject to inhomogeneous boundary conditions. The shape sensitivity analysis is performed for Navier–Stokes boundary value problems in the framework of small perturbations of the so-called approximate solutions. The approximate solutions are determined from the Stokes problem, and the small perturbations are given by the unique solutions to the full nonlinear model. The differentiability of small perturbations of the approximate solutions with respect to the coefficients of differential operators implies the shape differentiability of the drag functional. The shape gradient of the drag functional is derived in a form convenient for computations, and an appropriate adjoint state is introduced to this end. The shape derivatives of solutions to the Navier–Stokes equations are given by smooth functions; however, the shape differentiability of the solutions is shown in a weak norm. The proposed method of shape sensitivity analysis is general. The differentiability of solutions to the Navier–Stokes equations with respect to the data leads to the first order necessary conditions for a broad class of optimization problems. The boundary shape gradient as well as the boundary value problems for the shape derivatives of solutions to state equations and the adjoint state equations are obtained in the form of singular limits of volume integrals. This method of shape sensitivity analysis seems to be new and is appropriate for nonlinear problems. It is an important contribution in the field of numerical methods of shape optimization in fluid mechanics.

The optimization problem consists in finding an element that minimizes or maximizes the goal (or cost) functional on the set of admissible controls. The paper is devoted to the optimization problem for a pseudoparabolic equation with control in coefficients of the variational equality as well as on the right-hand side. We consider optimal control problem (parabolic initial boundary value problem) of a viscoelastic plate. The bending of a plate is described by means of the Kirchhoff model. We assume that a nonhomogeneous and anisotropic viscoelastic plate occupying a domain Ω × (—H, H) of the space ℝ3 is loaded be a transversal distributed force perpendicular to the middle plane of the plate. The viscoelastic reinforced plate with welded bars rests on a unilateral elastic foundation. The role of the control variables is played by the thickness of the viscoelastic plate and the stiffness characteristics of the elastic Winkler medium, respectively. We consider the desired moments field (the bending moments and torque) in time T as the cost functional. The control variables have to belong to a set of Lipschitz-continuous functions and to a set of continuously differentiable functions, respectively. The state problem is modelled by a pseudoparabolic variational equality, where the control variables influence both the coefficients of the linear strictly monotone operator and the Winkler functional. We derive there some apriori estimates of the state function which are useful for the problem of the existence of the optimal thickness function and the Winkler modulus, respectively, and for unicity of the optimal controls. On the basis of a general existence theorem for a class of pseudoparabolic optimization problems with the state variational equation, we prove the existence of at least one solution of the optimal control problem.

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.

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.

This article presents a unified approach to simultaneous solution of optimization and optimal control problems in batch distillation, operating under different modes of operation: variable, constant, or optimal reflux. The simplified, computationally efficient short‐cut method and a novel algorithm to solve the optimal control problems in batch distillation is the basis of this unified approach. The short‐cut method identifies the feasible region of operation essential for optimization and optimal control problems, and provides analytical partial derivatives of the model parameters crucial to the solution.The new algorithm for the solution of optimal combination of the maximum principle and NLP optimization techniques. It circumvents the problems associated with the maximum principle approach (iterative solution of a two‐point boundary value problem, undounded control variables, and inability to handle the simultaneous optimization and optimal control problem), and the coupled ODE discretization‐NLP optimization scheme for nonlinear models (higher system nonlinearities, multiplicity of solutions, sensitivity of convergence to initial guesses). This algorithm reduces the dimensionality of the problem, and the nature of the algorithm allows a common platform to optimal solutions of different operating conditions. This article also shows that different categories of the optimal control problems in batch distillation essentially involve the solution of the maximum distillate problem.

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.

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.

A neural network based optimal control synthesis is presented for solving optimal control problems with control and state constraints. The optimal control problem is transcribed into nonlinear programming problem which is implemented with adaptive critic neural network. The proposed simulation methods is illustrated by the optimal control problem of feeding adaptation of filter feeders of Daphnia. Results show that adaptive critic based systematic approach holds promise for obtaining the optimal control with control and state constraints.

The problem of optimal control with a variable structure depending on the particular way on the parameter is described in the article. In the considered issue, the object’s motion equations are described by two differential equations depending on the small parameter and these equations are bound to each other. The study investigated the solution of the problem depending on the small parameter that gives a certain functional minimum in the range of solutions described by differential equations, with a small parameter and the necessary conditions for the optimization of the solution are given. The validity of the theorems obtained in this study was proved by applying the Lagrangian principle to the Lagrangian functions and the application of the Farm Theorem to the Lagrangian problem. At the same time, the problem of optimal control with periodic boundary conditions is also explored. It can also be used extensively in solving some practical issues, for example, model in oil extraction by gas—lift method for the case when the reciprocal value of well’s depth represents a small parameter is considered. Problem of optimal mode construction (i.e., construction of optimal program trajectories and controls) is reduced to the linear—quadratic optimal control problem with a small parameter. This issue is investigated using the Lagrange coefficient and the Lagrange principle for the Lagrange problem. Although this issue has been studied in numerous works, the specificity of the issue under consideration makes it possible to investigate it differently. The considered method can be applied to the solution of optimal control problems for the problems described by some nonlinear differential equations.

This paper introduces the adjoint state method in non‐steady‐state forming processes, such as forging, whereas this method is more frequently used for steady‐state problems. Some practical details for an efficient implementation are given. The emphasis is put on the accuracy of the sensitivity analysis for problems with remeshings and transfers of variables. The precision of the calculated derivatives is verified by comparing its results with the direct differentiation method. The adjoint state method is then applied to shape optimization problems. Copyright © 2003 John Wiley & Sons, Ltd.

1 A Survey on Computational Optimal Control.- Issues in the Direct Transcription of Optimal Control Problems to Sparse Nonlinear Programs.- Optimization in Control of Robots.- Large-scale SQP Methods and their Application in Trajectory Optimization.- Solving Optimal Control and Pursuit-Evasion Game Problems of High Complexity.- 2 Theoretical Aspects of Optimal Control and Nonlinear Programming.- Continuation Methods In Boundary Value Problems.- Second Order Optimality Conditions for Singular Extremals.- Synthesis of Adaptive Optimal Controls for Linear Dynamic Systems.- Control Applications of Reduced SQP Methods.- Time Optimal Control of Mechanical Systems.- 3 Algorithms for Optimal Control Calculations.- Second Order Algorithm for Time Optimal Control of a Linear System.- An SQP-type Solution Method for Constrained Discrete-Time Optimal Control Problems.- Numerical Methods for Solving Differential Games, Prospective Applications to Technical Problems.- Construction of the Optimal Feedback Controller for Constrained Optimal Control Problems with Unknown Disturbances.- Repetitive Optimization for Predictive Control of Dynamic Systems under Uncertainty.- Optimal Control of Multistage Systems Described by High-Index Differential-Algebraic Equations.- A New Class of a High Order Interior Point Method for the Solution of Convex Semiinfinite Optimization Problems.- A Structured Interior Point SQP Method for Nonlinear Optimal Control Problems.- 4 Software for Optimal Control Calculations.- Automated Approach for Optimizing Dynamic Systems.- ANDECS: A Computation Environment for Control Applications of Optimization.- Application of Automatic Differentiation to Optimal Control Problems.- OCCAL: A mixed symbolic-numeric Optimal Control CALculator.- 5 Applications of Optimal Control.- A Robotic Satellite with Simplified Design.- Nonlinear Control under Constraints of a Biological System.- An Object-Oriented Approach to Optimally Describe and Specify a SCADA System Applied to a Power Network.- Near-Optimal Flight Trajectories Generated by Neural Networks.- Performance of a Feedback Method with Respect to Changes in the Air-Density during the Ascent of a Two-Stage-To-Orbit Vehicle.- Linear Optimal Control for Reentry Flight.- Steady-State Modelling of Turbine Engine with Controllers.- Shortest Paths for Satellite Mounted Robot Manipulators.- Optimal Control of the Industrial Robot Manutec r3.

Optimal control problems in Hilbert spaces are considered in a measure-theoretical framework. Instead of minimizing a functional defined on a class of admissible trajectory-control pairs, we minimize one defined on a set of measures; this set is defined by the boundary conditions and the differential equation of the problem. The new problem is an infinite-dimensionallinear programming problem; it is shown that it is possible to approximate its solution by that of a finite-dimensional linear program of sufficiently high dimensions, while this solution itself can be approximated by a trajectory-control pair. This pair may not be strictly admissible; if the dimensionality of the finite-dimensional linear program and the accuracy of the computations are high enough, the conditions of admissibility can be said to be satisfied up to any given accuracy. The value given by this pair to the functional measuring the performance criterion can be about equal to theglobal infimum associated with the classical problem, or it may be less than this number. It appears that this method may become a useful technique for the computation of optimal controls, provided the approximations involved are acceptable.

This paper describes a novel offline approach to estimate the parameters of a nonlinear system using the optimal control theory. In the conventional methods, only the output is considered in the identification process and the problem is stated as a nonlinear optimization problem. Nevertheless, the proposed method here is based on the least square error of both output and input. To do this, the problem is first stated as a tracking problem with some unknown parameters. The necessary and sufficiency conditions of optimality conditions are derived using the fundamental theorem of variational calculus. The problem is deployed as an effort optimal control problem which finally leads to a two point boundary value problem. To solve the resultant equations, a secondary optimal control problem is deployed to be solved using the homotopy algorithm. So the parameters are gradually converged to the real values. The proposed algorithm is analyzed for a one-link manipulator with the full measurements. Thereafter, the proposed method is applied on a real one-link manipulator and nonlinear model of permanent magnet dc motor. Application of the method for the identification of a few test beds at the AUT proves the validity and efficiency of the proposed approach.

An efficient method is introduced for solving optimal control and optimal parameter selection problems of nonlinear differential-algebraic systems involving general constraints. These infinite dimensional problems are converted into the finite dimensional optimization problems by using the control parametrization technique. Moreover, the transformed time locations for the control variables are included in such problems. As a result, optimal control problems become acausal optimal parameter selection problems. A modified version of differential evolution is introduced to solve these problems. The integration of the penalty functions is used to ensure the solution onto the feasible domain of the problems. Two industrial systems, a chemical reactor and a robotics arm, are used to illustrate the robustness and effectiveness of the method. The control effects can be improved by selecting the proper time locations as observed from the simulated results.

Optimal Design and Control of Cyclically Operated Biological Wastewater Treatment Plants