Flow regulation for water quality restoration in a river section: Modeling and control

This work is devoted to the numerical resolution of an optimal control problem that arises in the management of a reservoir for the remediation of a polluted river section. By using mathematical modeling and optimal control techniques we set the mathematical formulation of the problem (as a hyperbolic optimal control problem with control constraints), and obtain a fully discretized problem. Finally, we propose a gradient-free method to solve it, and present realistic numerical results.     

Stability of critical shapes for the drag minimization problem in Stokes flow

We study the stability of some critical (or equilibrium) shapes in the minimization problem of the energy dissipated by a fluid (i.e. the drag minimization problem) governed by the Stokes equations. We first compute the shape derivative up to the second order, then provide a sufficient condition for the shape Hessian of the energy functional to be coercive at a critical shape. Under this condition, the existence of such a local strict minimum is then proved using a precise upper bound for the variations of the second order shape derivative of the functional with respect to the coercivity and differentiability norms. Finally, for smooth domains, a lower bound of the variations of the drag is obtained in terms of the measure of the symmetric difference of domains.     

Variational approach and optimal control of a PEM fuel cell

The purpose of this paper is to propose and study a mathematical model and a boundary control problem associated to the miscible displacement of hydrogen through the porous anode of a PEM fuel cell. Throughout the paper, we study certain variational problems with a priori regularity properties of the weak solutions. We obtain the existence of less regular solutions and then we prove the desired regularity of these solutions. We consider a control problem that permits to determine the boundary distribution of the pressure which provides an optimal configuration for the temperature and for the concentration, as well. Since the solution of the problem is not unique, the control variable does not appear explicitly in the definition of our cost functional. To overcome this difficulty, we introduce a family of penalized control problems which approximates our boundary control problem. The necessary conditions of optimality are derived by passing to the limit in the penalized optimality conditions.     

A multigrid method for constrained optimal control problems

We consider the fast and efficient numerical solution of linear-quadratic optimal control problems with additional constraints on the control. Discretization of the first-order conditions leads to an indefinite linear system of saddle point type with additional complementarity conditions due to the control constraints. The complementarity conditions are treated by a primal-dual active set strategy that serves as outer iteration. At each iteration step, a KKT system has to be solved. Here, we develop a multigrid method for its fast solution. To this end, we use a smoother which is based on an inexact constraint preconditioner.We present numerical results which show that the proposed multigrid method possesses convergence rates of the same order as for the underlying (elliptic) PDE problem. Furthermore, when combined with a nested iteration, the solver is of optimal complexity and achieves the solution of the optimization problem at only a small multiple of the cost for the PDE solution.     

Stability by the linear approximation for discrete equations

This paper is concerned with exponential stability of solutions of perturbed discrete equations. For a given m>1 we will provide necessary and sufficient conditions for exponential stability of all perturbed systems with perturbation of order m under the assumption that the unperturbed linear system is exponentially stable. Basing on this result we obtained necessary and sufficient conditions for exponential stability of the perturbed system for all perturbations of order m>1 for regular systems. Our results are expressed in terms of regular coefficients of the unperturbed system.     

Stochastic optimal control and algorithm of the trajectory of horizontal wells

This paper presents a nonlinear, multi-phase and stochastic dynamical system according to engineering background. We show that the stochastic dynamical system exists a unique solution for every initial state. A stochastic optimal control model is constructed and the sufficient and necessary conditions for optimality are proved via dynamic programming principle. This model can be converted into a parametric nonlinear stochastic programming by integrating the state equation. It is discussed here that the local optimal solution depends in a continuous way on the parameters. A revised Hooke–Jeeves algorithm based on this property has been developed. Computer simulation is used for this paper, and the numerical results illustrate the validity and efficiency of the algorithm.     

The problem of optimal control with reflection studied through a linear optimization problem stated on occupational measures

We obtain a linear programming characterization for the minimum cost associated with finite dimensional reflected optimal control problems. In order to describe the value functions, we employ an infinite dimensional dual formulation instead of using the characterization via Hamilton-Jacobi partial differential equations. In this paper we consider control problems with both infinite and finite horizons. The reflection is given by the normal cone to a proximal retract set.     

Feedback stabilization of a simplified 1d fluid–particle system

We consider the feedback stabilization of a simplified 1d model for a fluid–structure interaction system. The fluid equation is the viscous Burgers equation whereas the motion of the particle is given by the Newton's laws. We stabilize this system around a stationary state by using feedbacks located at the exterior boundary of the fluid domain. With one input, we obtain a local stabilizability of the system with an exponential decay rate of order σ<_σ0$\sigma <{\sigma }_0$. An arbitrary order for the exponential decay rate can be proved if a unique continuation result holds true or if two inputs are used to stabilize the system. Our method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domains of the stationary state and of the stabilized solution are different.     

Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system

In this paper, we study a scalar conservation law that models a highly re-entrant manufacturing system as encountered in semi-conductor production. As a generalization of Coron et al. (2010) [14], the velocity function possesses both the local and nonlocal character. We prove the existence and uniqueness of the weak solution to the Cauchy problem with initial and boundary data in L^∞. We also obtain the stability (continuous dependence) of both the solution and the out-flux with respect to the initial and boundary data. Finally, we prove the existence of an optimal control that minimizes, in the Lp-sense with 1?p?∞, the difference between the actual out-flux and a forecast demand over a fixed time period.     

Weak maximum principle and accessory problem for control problems on time scales

In this paper we derive the first and second variations for a nonlinear time scale optimal control problem with control and state-endpoints equality constraints. Using the first variation, a first order necessary condition for weak local optimality is obtained under the form of a weak maximum principle generalizing the Dubois–Reymond Lemma to the optimal control setting and time scales. A second order necessary condition in terms of the accessory problem is derived by using the nonnegativity of the second variation at all admissible directions. The control problem is studied under a controllability assumption, and with or without the shift in the state variable. These two forms of the problem are shown to be equivalent.     

On PDE solution in transient optimization of gas networks

Operative planning in gas distribution networks leads to large-scale mixed-integer optimization problems involving a hyperbolic PDE defined on a graph. We consider the NLP obtained under prescribed combinatorial decisions—or as relaxation in a branch-and-bound framework, addressing in particular the KKT systems arising in primal–dual interior methods. We propose a custom solution algorithm using sparse projections locally in time, based on the KKT systems’ structural properties in space as induced by the discretized gas flow equations in combination with the underlying network topology. The numerical efficiency and accuracy of the algorithm are investigated, and detailed computational comparisons with a previously developed control space method and with the multifrontal solver MA27 are provided.     

Distributed control for a class of non-Newtonian fluids

We consider control problems with a general cost functional where the state equations are the stationary, incompressible Navier-Stokes equations with shear-dependent viscosity. The equations are quasi-linear. The control function is given as the inhomogeneity of the momentum equation. In this paper, we study a general class of viscosity functions which correspond to shear-thinning or shear-thickening behavior. The basic results concerning existence, uniqueness, boundedness, and regularity of the solutions of the state equations are reviewed. The main topic of the paper is the proof of Gâteaux differentiability, which extends known results. It is shown that the derivative is the unique solution to a linearized equation. Moreover, necessary first-order optimality conditions are stated, and the existence of a solution of a class of control problems is shown.     

Mathematical modelization of equatorial waves

Our purpose is to calculate waves propagating along the equator in an oceanic domain and the influence of a characteristic mean equatorial circulation on the nature of these waves. Equations satisfied by perturbations of currents and temperature are of the Navier-Stokes type and have been linearized around a stationary solution. Existence and uniqueness of the solution have been proved. Numerical experiments have been carried out and provided us with time-dependent values. The excited waves are exhibited by Fourier analysis of these time series.     

Optimal control for multi-phase fluid Stokes problems

Optimal control for a system consistent of the viscosity dependent Stokes equations coupled with a transport equation for the viscosity is studied. Motivated by a lack of sufficient regularity of the adjoint equations, artificial diffusion is introduced to the transport equation. The asymptotic behavior of the regularized system is investigated. Optimality conditions for the regularized optimal control problems are obtained and again the asymptotic behavior is analyzed. The lack of uniqueness of solutions to the underlying system is another source of difficulties for the problem under investigation.     

High-order discretization and multigrid solution of elliptic nonlinear constrained optimal control problems

High(-mixed)-order finite difference discretization of optimality systems arising from elliptic nonlinear constrained optimal control problems are discussed. For the solution of these systems, an efficient and robust multigrid algorithm is presented. Theoretical and experimental results show the advantages of higher-order discretization and demonstrate that the proposed multigrid scheme is able to solve efficiently constrained optimal control problems also in the limit case of bang-bang control.     

A spatial domain decomposition method for parabolic optimal control problems

We present a non-overlapping spatial domain decomposition method for the solution of linear–quadratic parabolic optimal control problems. The spatial domain is decomposed into non-overlapping subdomains. The original parabolic optimal control problem is decomposed into smaller problems posed on space–time cylinder subdomains with auxiliary state and adjoint variables imposed as Dirichlet boundary conditions on the space–time interface boundary. The subdomain problems are coupled through Robin transmission conditions. This leads to a Schur complement equation in which the unknowns are the auxiliary state adjoint variables on the space-time interface boundary. The Schur complement operator is the sum of space–time subdomain Schur complement operators. The application of these subdomain Schur complement operators is equivalent to the solution of an subdomain parabolic optimal control problem. The subdomain Schur complement operators are shown to be invertible and the application of their inverses is equivalent to the solution of a related subdomain parabolic optimal control problem. We introduce a new family of Neumann–Neumann type preconditioners for the Schur complement system including several different coarse grid corrections. We compare the numerical performance of our preconditioners with an alternative approach recently introduced by Benamou.     

Solving optimal control problems for the unsteady Burgers equation in COMSOL Multiphysics

The optimal control of unsteady Burgers equation without constraints and with control constraints are solved using the high-level modelling and simulation package COMSOL Multiphysics. Using the first-order optimality conditions, projection and semi-smooth Newton methods are applied for solving the optimality system. The optimality system is solved numerically using the classical iterative approach by integrating the state equation forward in time and the adjoint equation backward in time using the gradient method and considering the optimality system in the space-time cylinder as an elliptic equation and solving it adaptively. The equivalence of the optimality system to the elliptic partial differential equation (PDE) is shown by transforming the Burgers equation by the Cole-Hopf transformation to a linear diffusion type equation. Numerical results obtained with adaptive and nonadaptive elliptic solvers of COMSOL Multiphysics are presented both for the unconstrained and the control constrained case.     

Third-order nilpotency, nice reachability and asymptotic stability

We consider an affine control system whose vector fields span a third-order nilpotent Lie algebra. We show that the reachable set at time T using measurable controls is equivalent to the reachable set at time T using piecewise-constant controls with no more than four switches. The bound on the number of switches is uniform over any final time T. As a corollary, we derive a new sufficient condition for stability of nonlinear switched systems under arbitrary switching. This provides a partial solution to an open problem posed in [D. Liberzon, Lie algebras and stability of switched nonlinear systems, in: V. Blondel, A. Megretski (Eds.), Unsolved Problems in Mathematical Systems and Control Theory, Princeton Univ. Press, 2004, pp. 203-207].     

Boundary feedback stabilization of the Boussinesq system with mixed boundary conditions

We study the feedback stabilization of the Boussinesq system in a two dimensional domain, with mixed boundary conditions. After ascertaining the precise loss of regularity of the solution in such models, we prove first Green's formulas for functions belonging to weighted Sobolev spaces and then correctly define the underlying control system. This provides a rigorous mathematical framework for models studied in the engineering literature. We prove the stabilizability by extending to the linearized Boussinesq system a local Carleman estimate already established for the Oseen system. Then we determine a feedback control law able to stabilize the linearized system around the stationary solution, with any prescribed exponential decay rate, and able to stabilize locally the nonlinear system.