Optimal control of convection-cooling and numerical implementation

An optimal control problem for the stationary Navier–Stokes equations with variable density is studied. A bilinear control is applied on the flow domain, while Dirichlet and Navier boundary conditions for the velocity are assumed on the boundary. As a first step, we enunciate a result on the existence of weak solutions of the dynamical equation; this is done by firstly expressing the fluid density in terms of the stream-function. Then, the bilinear optimal control problem is analyzed, and the existence of optimal solutions are proved; their corresponding characterization regarding the first-order optimality conditions are obtained. Such optimality conditions are rigorously derived by using a penalty argument since the weak solutions are not necessarily unique neither isolated, and so standard methods cannot be applied.

In this paper, we study a bilinear optimal control problem associated to a chemo-repulsion model with linear production term in a bidimensional domain. The existence, uniqueness and regularity of strong solutions of this model are deduced, proving the existence of a global optimal solution. Afterwards, we derive first-order optimality conditions by using a Lagrange multipliers theorem.

In this paper, we consider an optimal distributed control problem for a reaction-diffusion-based SIR epidemic model with human behavioural effects. We develop a model wherein non-pharmaceutical intervention methods are implemented, but a portion of the population does not comply with them, and this non-compliance affects the spread of the disease. Drawing from social contagion theory, our model allows for the spread of non-compliance parallel to the spread of the disease. The quantities of interest for control are the reduction in infection rate among the compliant population, the rate of spread of non-compliance and the rate at which non-compliant individuals become compliant after, e.g., receiving more or better information about the underlying disease. We prove the existence of global-in-time solutions for fixed controls and study the regularity properties of the resulting control-to-state map. The existence of optimal control is then established in an abstract framework for a fairly general class of objective functions. Necessary first–order optimality conditions are obtained via a Lagrangian-based stationarity system. We conclude with a discussion regarding minimisation of the size of infected and non-compliant populations and present simulations with various parameters values to demonstrate the behaviour of the model.

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.

An optimal birth control problem for a dynamical population model with size-structure

We consider optimal control problems for the flow of gas in a pipe network. The equations of motions are taken to be represented by a nonlinear model derived from a semi-linear approximation of the fully nonlinear isothermal Euler gas equations. We formulate an optimal control problem on a given network and introduce a time discretization thereof. We then study the well-posedness of the corresponding time-discrete optimal control problem. In order to further reduce the complexity, we consider an instantaneous control strategy. This involves a p-Laplace-type problem on the graph with $$p=\frac {3}{2}$$ . We prove well-posedness, existence of optimal controls and derive a first order optimality condition.

This paper is concerned with an optimal control problem of steady-state electrorheological fluids based on an extended Bingham model. Our control parameters are given by finite real numbers representing applied direct voltages, which enter in the viscosity of the electrorheological fluid via an electrostatic potential. The corresponding optimization problem belongs to a class of nonlinear optimal control problems of variational inequalities with control in the coefficients. We analyze the associated variational inequality model and the optimal control problem. Thereafter, we introduce a family of Huber-regularized optimal control problems for the approximation of the original one and verify the convergence of the regularized solutions. Differentiability of the solution operator is proved and an optimality system for each regularized problem is established. In the last part of the paper, an algorithm for the numerical solution of the regularized problem is constructed and numerical experiments are carried out.

In general, optimal control problems rely on numerically rather than analytically solving methods, due to their nonlinearities. The direct method, one of the numerically solving methods, is mainly to transform the optimal control problem into a nonlinear optimization problem with finite dimensions, via discretizing the objective functional and the forced dynamical equations directly. However, in the procedure of the direct method, the classical discretizations of the forced equations will reduce or affect the accuracy of the resulting optimization problem as well as the discrete optimal control. In view of this fact, more accurate and efficient numerical algorithms should be employed to approximate the forced dynamical equations. As verified, the discrete variational difference schemes for forced Birkhoffian systems exhibit excellent numerical behaviors in terms of high accuracy, long-time stability and precise energy prediction. Thus, the forced dynamical equations in optimal control problems, after being represented as forced Birkhoffian equations, can be discretized according to the discrete variational difference schemes for forced Birkhoffian systems. Compared with the method of employing traditional difference schemes to discretize the forced dynamical equations, this way yields faithful nonlinear optimization problems and consequently gives accurate and efficient discrete optimal control. Subsequently, in the paper we are to apply the proposed method of numerically solving optimal control problems to the rendezvous and docking problem of spacecrafts. First, we make a reasonable simplification, i.e., the rendezvous and docking process of two spacecrafts is reduced to the problem of optimally transferring the chaser spacecraft with a continuously acting force from one circular orbit around the Earth to another one. During this transfer, the goal is to minimize the control effort. Second, the dynamical equations of the chaser spacecraft are represented as the form of the forced Birkhoffian equation. Then in this case, the discrete variational difference scheme for forced Birkhoffian system can be employed to discretize the chaser spacecraft's equations of motion. With further discretizing the control effort and the boundary conditions, the resulting nonlinear optimization problem is obtained. Finally, the optimization problem is solved directly by the nonlinear programming method and then the discrete optimal control is achieved. The obtained optimal control is efficient enough to realize the rendezvous and docking process, even though it is only an approximation of the continuous one. Simulation results fully verify the efficiency of the proposed method for numerically solving optimal control problems, if the fact that the time step is chosen to be very large to limit the dimension of the optimization problem is noted.

Turnpike results hold a central position in the area of applied mathematics, offering profound insights into the long-time behavior of optimal control systems. They play an important role in guiding decision-making processes in dynamic systems. Often dynamic systems depend on uncertain data and/or parameters. In this paper, optimal boundary control problems for the transport equation with random initial data and random source term are considered. The paper begins with the numerical analysis of the turnpike phenomenon for an optimal boundary control problem with nonlinear time-varying feedback control, where the initial data are perturbed by randomized Fourier series and the source term is perturbed by a random variable. Further, an integral turnpike result is presented for the optimal controls and the corresponding optimal states of an optimal boundary control problem with random initial data. A turnpike result is also presented for an optimal boundary control problem governed by a transport equation with random source term, where optimal control, optimal expected state, and variance satisfy an integral turnpike inequality. The turnpike constant, the expected value, and the variance are further specified for Gaussian and uniformly distributed random variables.

In this paper, we apply the classical control theory to a variable order fractional differential system in a bounded domain. The Fractional Optimal Control Problem (FOCP) for variable order differential system is considered. The fractional time derivative is considered in a Caputo sense. We first study by using the Lax-Milgram Theorem, the existence and the uniqueness of the solution of the variable order fractional differential system in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. The performance index of a (FOCP) is considered as a function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE) with variable order. The time horizon is fixed. Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of right fractional Caputo derivative, we obtain an optimality system for the optimal control. Some examples are analyzed in details.

A specific elliptic linear-quadratic optimal control problem with Neumann boundary control is investigated. The control has to fulfil inequality constraints. The domain is assumed to be polygonal with reentrant corners. The asymptotic behaviour of two approaches to compute the optimal control is discussed. In the first the piecewise constant approximations of the optimal control are improved by a postprocessing step. In the second the control is not discretized; instead the first order optimality condition is used to determine an approximation of the optimal control. Although the quality of both approximations is in general affected by corner singularities a convergence order of 3/2 can be proven provided that the mesh is sufficiently graded.

This article tackles the optimal control design problem for second order nonlinear affine systems using Krotov sufficient conditions. The computation of optimal control law(s) for nonlinear systems is usually done using the iterative methods based on the standard tools of optimal control theory which viz. Calculus of Variations (CoV), Hamilton-Jacobi-Bellman equation, Pontryagin's principle, etc. This work utilizes the Krotov sufficient conditions of global optimality to obtain non-iterative solutions. These conditions are derived by transforming the optimal control problem into another equivalent optimization problem. This translation is done via an ad-hoc selection of the so-called Krotov function. In this article, the Krotov function is chosen such that the equivalent optimization problem is solved non-iteratively to obtain optimal and sub-optimal control laws for the original optimal control problem. The proposed methodology is demonstrated by a numerical example.

The field of the mathematical and numerical analysis of systems of nonlinear pdes involving interfaces and free boundaries is a burgeoning area of research. Many such systems arise from mathematical models in ma- terial science and fluid dynamics such as phase separation in alloys, crystal growth, dynamics of multiphase fluids and epitaxial growth. In applications of these mathematical models, suitable performance indices and appropriate control actions have to be specified. Mathematically this leads to optimiza- tion problems with pde constraints including free boundaries. It is now timely to consider such control problems because of the maturity of the field of com- putational free boundary problems. The aim of the mini-workshop was to bring together leading experts and young researchers from the separate fields of numerical free boundary problems and optimal control in order to estab- lish links and to identify suitable model problems to serve as paradigms for progressing knowledge of optimal control of free boundaries.

In this work, a new method for solving an optimal semi-active control problem is formulated. It extends previously published results by addressing a continuous-time bilinear quadratic regulator control problem, defined for a structure that is controlled by multiple controlled hydraulic dampers and subjected to external, a priori known deterministic excitation input. The dynamic system is modeled by a linear state-space model, accompanied by constraints on the control forces that reflect the semi-active nature of the prescribed dampers. Next, the control force is written in an equivalent bilinear form, thereby leading to a bilinear state-space model. The optimal control problem is defined by this dynamic bilinear representation, control force constraints, and a performance index that is quadratic in the states and the equivalent damping gains. Namely, as each device has only two allowable operation modes, its control signal’s range is a two-object set. Krotov’s method is used for solving this problem. This method requires to formulate a function’s sequence with special properties and a suitable minimizing feedback. By doing so, a process improvement becomes possible and an improving sequence can be formulated. This sequence of improving functions and the suitable minimizing feedback, which enable the use of Krotov’s method in this case, are formulated in the context of the addressed problem. The results are encapsulated in a convergent algorithm whose outcome is an approximation of a candidate optimal feedback. Numerical example is given to demonstrate the use of the suggested results for a CBQR control design of a structure, subjected to seismic input.

In this paper we model the role of a government of a large population as a mean field optimal control problem. Such control problems are constrained by a PDE of continuity-type, governing the dynamics of the probability distribution of the agent population. We show the existence of mean field optimal controls both in the stochastic and deterministic setting. We derive rigorously the first order optimality conditions useful for numerical computation of mean field optimal controls. We introduce a novel approximating hierarchy of sub-optimal controls based on a Boltzmann approach, whose computation requires a very moderate numerical complexity with respect to the one of the optimal control. We provide numerical experiments for models in opinion formation comparing the behavior of the control hierarchy.