Vertical slot fishways: Mathematical modeling and optimal management

Fishways are the main type of hydraulic devices currently used to facilitate migration of fish past obstructions (dams, waterfalls, rapids,…$\mathrm{rapids},\dots$) in rivers. In this paper we present a mathematical formulation of an optimal control problem related to the optimal management of a vertical slot fishway, where the state system is given by the shallow water equations, the control is the flux of inflow water, and the cost function reflects the need of rest areas for fish and of a water velocity suitable for fish leaping and swimming capabilities. We give a first-order optimality condition for characterizing the optimal solutions of this problem. From a numerical point of view, we use a characteristic-Galerkin method for solving the shallow water equations, and we use an optimization algorithm for the computation of the optimal control. Finally, we present numerical results obtained for the realistic case of a standard nine pools fishway.     

Analysis of a multistate control problem related to food technology

This paper is concerned with an optimal control problem related to the determination of an optimal profile for the steam temperature into the autoclave along the processing of canned foods. The problem studies a system coupling the evolution Navier-Stokes equations with the heat transfer equation by natural convection (the so-called Boussinesq equations), and with the microorganisms removal equation. The essential difficulties in the study of this multistate control problem arise from the lack of uniqueness for the solution of the state system. Here we obtain—after a careful analysis of the problem mathematical formulation—the uniqueness of part of the state, and the existence of optimal solutions.     

Mathematical analysis and optimal control of heavy metals phytoremediation techniques

In this work we optimize different issues related to phytoremediation techniques for heavy metals removal from shallow water, by means of a combination of mathematical modeling, optimal control of partial differential equations and numerical optimization. We introduce, analyze and solve a 2D mathematical system of nonlinear partial differential equations representing the concentrations of heavy metals, algae and nutrients in large waterbodies. Then, we formulate an optimal control problem related to the optimization of the phytoremediation process. In particular, we determine the minimal quantity of algae to be used in the heavy metals remediation process, and locate the optimal place for such algal mass. We also propose two different full algorithms for computing the numerical solution of the control problem and, finally, we present several numerical results for a realistic case.     

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.     

Mesh optimization for singular axisymmetric harmonic maps from the disc into the sphere

We describe in a mathematical setting the singular energy minimizing axisymmetric harmonic maps from the unit disc into the unit sphere; then, we use this as a test case to compute optimal meshes in presence of sharp boundary layers. For the well-posedness of the continuous minimizing problem, we introduce a lower semicontinuous extension of the energy with respect to weak convergence in BV, and we prove that the extended minimization problem has a unique singular solution. We then show how a moving finite element method, in which the mesh is an unknown of the discrete minimization problem obtained by finite element discretization, mimics this geometric point of view. Finally, we present numerical computations with boundary layers of zero thickness, and we give numerical evidence of the convergence of the method. This last aspect is proved in another paper. This work was supported by the Centre de Mathématiques et de Leurs Applications, Ecole Normale Supérieure de Cachan, 61 av. du Président Wilson, 94235 Cachan Cedex, France     

Optimal operation for a river fishway

In this paper we present an optimal control problem related to the optimal management of a vertical slot fishway. We introduce the mathematical statement of the environmental problem, we derive an optimality condition for characterizing its optimal solutions, and we give an optimization algorithm for their numerical computation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Analysis and numerical simulation of a nonlinear mathematical model for testing the manoeuvrability capabilities of a submarine

The aim of this work is to provide a mathematical and numerical tool for the analysis of the manoeuvrability capabilities of a submarine. To this end, we consider a suitable optimal control problem with constraints in both state and control variables. The state law is composed of a highly coupled and nonlinear system of twelve ordinary differential equations. Control inputs appear in linear and quadratic form and physically are linked to rudders and propeller forces and moments. We consider a nonlinear Bolza type cost function which represents a commitment between reaching a final desired state and a minimal expense of control. In a first part, following recent ideas in [F. Periago, J. Tiago, A local existence result for an optimal control problem modeling the manoeuvring of an underwater vehicle, Nonlinear Anal. RWA 11 (2010) 2573–2583], we prove a local existence result for the above mentioned optimal control problem. In a second part, we address the numerical resolution of the problem by using a descent method with projection and optimal step-size parameter. To illustrate the performance of the method proposed in this paper and to show its application in a real engineering problem we include three different numerical experiments for a standard manoeuvre.     

Optimal control of an obstacle problem

We investigate optimal control problems governed by variational inequalities, and more precisely the obstacle problem. Since we adopt a numerical point of view, we first relax the feasible domain of the problem; then using both mathematical programming methods and penalization methods we get optimality conditions with smooth lagrange multipliers.     

Optimal control of the solidification process in metal casting

The optimal control of the solidification process in metal casting is considered. The mathematical model is based on a three-dimensional two-phase initial-boundary value problem of the Stefan type. The mathematical formulation of the optimal control problem is given. The problem is solved numerically by direct optimization methods. The numerical results are described and analyzed. Some of the results are illustrated by plots.     

An inverse problem of identifying the coefficient of parabolic equation

This paper studies an inverse problem of identifying the coefficient of parabolic equation when the final observation is given, which has important application in a large fields of applied science. Based on the optimal control framework, the existence and necessary condition of the minimum for the control functional are established. Since the optimal control problem is nonconvex, one may not expect a unique solution. However, in this paper the solution is proved to be locally unique. After the necessary condition is transformed into an elliptic bilateral variational inequality, an algorithm and some numerical experiments are proposed in the paper. The numerical results show that the algorithm designed in this paper is stable and that the coefficient is recovered very well.     

Extrapolation discontinuous Galerkin method for ultraparabolic equations

Ultraparabolic equations arise from the characterization of the performance index of stochastic optimal control relative to ultradiffusion processes; they evidence multiple temporal variables and may be regarded as parabolic along characteristic directions. We consider theoretical and approximation aspects of a temporally order and step size adaptive extrapolation discontinuous Galerkin method coupled with a spatial Lagrange second-order finite element approximation for a prototype ultraparabolic problem. As an application, we value a so-called Asian option from mathematical finance.     

Boundary optimal control for nonlinear antiplane problems

We consider a nonlinear antiplane problem which models the deformation of an elastic cylindrical body in frictional contact with a rigid foundation. The contact is modelled with Tresca’s law of dry friction in which the friction bound is slip dependent.The aim of this article is to study an optimal control problem which consists of leading the stress tensor as close as possible to a given target, by acting with a control on the boundary of the body. The existence of at least one optimal control is proved. Next we introduce a regularized problem, depending on a small parameter ρ, and we study the convergence of the optimal controls when ρ tends to zero. An optimality condition is delivered for the regularized problem.     

Optimal control,sensitivity analysis and relaxation of maximal monotone integrodifferential inclusions inR N

In this paper we examine optimal control problems governed by maximal monotone integrodifferential inclusions inR ^N . First we establish the existence of an optimal control. Then we show that the value of the problem depends continuously on a parameter appearing in all the data. Then we introduce the relaxed system, we show that under very general hypotheses it has a solution and that its value equals that of the original problem. Subsequently we show that relaxability and performance stability are equivalent concepts. Finally we specialize our results to the class of controlled differential variational inequalities.Research supported by NSF Grant DMS-8802688     

Numerical method for a class of optimal control problems subject to nonsmooth functional constraints

In this paper, we consider a class of optimal control problems which is governed by nonsmooth functional inequality constraints involving convolution. First, we transform it into an equivalent optimal control problem with smooth functional inequality constraints at the expense of doubling the dimension of the control variables. Then, using the Chebyshev polynomial approximation of the control variables, we obtain an semi-infinite quadratic programming problem. At last, we use the dual parametrization technique to solve the problem.     

Joint optimization traffic signal control for an urban arterial road

This paper considers the optimal traffic signal setting for an urban arterial road. By introducing the concepts of synchronization rate and non-synchronization degree, a mathematical model is constructed and an optimization problem is posed. Then, a new iterative algorithm is developed to solve this optimal traffic control signal setting problem. Convergence properties for this iterative algorithm are established. Finally, a numerical example is solved to illustrate the effectiveness of the method.     

Numerical optimization methods for controlled systems with parameters

First- and second-order numerical methods for optimizing controlled dynamical systems with parameters are discussed. In unconstrained-parameter problems, the control parameters are optimized by applying the conjugate gradient method. A more accurate numerical solution in these problems is produced by Newton’s method based on a second-order functional increment formula. Next, a general optimal control problem with state constraints and parameters involved on the righthand sides of the controlled system and in the initial conditions is considered. This complicated problem is reduced to a mathematical programming one, followed by the search for optimal parameter values and control functions by applying a multimethod algorithm. The performance of the proposed technique is demonstrated by solving application problems.     

Finite element approximation of elliptic control problems with constraints on the gradient

We consider an elliptic optimal control problem with control constraints and pointwise bounds on the gradient of the state. We present a tailored finite element approximation to this optimal control problem, where the cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of the lowest order Raviart–Thomas mixed finite element. Pointwise bounds on the gradient variable are enforced in the elements of the triangulation. Controls are not discretized. Error bounds for control and state are obtained in two and three space dimensions. A numerical example confirms our analytical findings.     

Furtivity and Masking Problems in Time-Dependent Electromagnetic Obstacle Scattering

In this paper, we consider furtivity and masking problems in time-dependent three-dimensional electromagnetic obstacle scattering. That is, we propose a criterion based on a merit function to minimize or to mask the electromagnetic field scattered by a bounded obstacle when hit by an incoming electromagnetic field and, with respect to this criterion, we drive the optimal strategy. These problems are natural generalizations to the context of electromagnetic scattering of the furtivity problem in time-dependent acoustic obstacle scattering presented in Ref. 1. We propose mathematical models of the furtivity and masking time-dependent three-dimensional electromagnetic scattering problems that consist in optimal control problems for systems of partial differential equations derived from the Maxwell equations. These control problems are approached using the Pontryagin maximum principle. We formulate the first-order optimality conditions for the control problems considered as exterior problems defined outside the obstacle for systems of partial differential equations. Moreover, the first-order optimality conditions derived are solved numerically with a highly parallelizable numerical method based on a perturbative series of the type considered in Refs. 2–3. Finally, we assess and validate the mathematical models and the numerical method proposed analyzing the numerical results obtained with a parallel implementation of the numerical method in several experiments on test problems. Impressive speedup factors are obtained executing the algorithms on a parallel machine when the number of processors used in the computation ranges between 1 and 100. Some virtual reality applications and some animations relative to the numerical experiments can be found in the website http://www.econ.unian.it/recchioni/w10/.     

Optimal liquidation under partial information with price impact

We study the optimal liquidation problem in a market model where the bid price follows a geometric pure jump process whose local characteristics are driven by an unobservable finite-state Markov chain and by the liquidation rate. This model is consistent with stylized facts of high frequency data such as the discrete nature of tick data and the clustering in the order flow. We include both temporary and permanent effects into our analysis. We use stochastic filtering to reduce the optimal liquidation problem to an equivalent optimization problem under complete information. This leads to a stochastic control problem for piecewise deterministic Markov processes (PDMPs). We carry out a detailed mathematical analysis of this problem. In particular, we derive the optimality equation for the value function, we characterize the value function as continuous viscosity solution of the associated dynamic programming equation, and we prove a novel comparison result. The paper concludes with numerical results illustrating the impact of partial information and price impact on the value function and on the optimal liquidation rate.