Discrete mechanics and optimal control for constrained systems

The equations of motion of a controlled mechanical system subject to holonomic constraints may be formulated in terms of the states and controls by applying a constrained version of the Lagrange‐d'Alembert principle. This paper derives a structure‐preserving scheme for the optimal control of such systems using, as one of the key ingredients, a discrete analogue of that principle. This property is inherited when the system is reduced to its minimal dimension by the discrete null space method. Together with initial and final conditions on the configuration and conjugate momentum, the reduced discrete equations serve as nonlinear equality constraints for the minimization of a given objective functional. The algorithm yields a sequence of discrete configurations together with a sequence of actuating forces, optimally guiding the system from the initial to the desired final state. In particular, for the optimal control of multibody systems, a force formulation consistent with the joint constraints is introduced. This enables one to prove the consistency of the evolution of momentum maps. Using a two‐link pendulum, the method is compared with existing methods. Further, it is applied to a satellite reorientation maneuver and a biomotion problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Optimal control of innate immune response

Treatment of a pathogenic disease process is interpreted as the optimal control of a dynamic system. Evolution of the disease is characterized by a non‐linear, fourth‐order ordinary differential equation that describes concentrations of pathogens, plasma cells, and antibodies, as well as a numerical indication of patient health. Without control, the dynamic model evidences sub‐clinical or clinical decay, chronic stabilization, or unrestrained lethal growth of the pathogen, depending on the initial conditions for the infection. The dynamic equations are controlled by therapeutic agents that affect the rate of change of system variables. Control histories that minimize a quadratic cost function are generated by numerical optimization over a fixed time interval, given otherwise lethal initial conditions. Tradeoffs between cost function weighting of pathogens, organ health, and use of therapeutics are evaluated. Optimal control solutions that defeat the pathogen and preserve organ health are demonstrated for four different approaches to therapy. It is shown that control theory can point the way toward new protocols for treatment and remediation of human diseases. Copyright © 2002 John Wiley & Sons, Ltd.     

A Hodge Decomposition Method for Dynamic Ginzburg–Landau Equations in Nonsmooth Domains — A Second Approach

In a general polygonal domain, possibly nonconvex and multi-connected (with holes), the time-dependent Ginzburg–Landau equation is reformulated into a new system of equations. The magnetic field $B$:=∇×A is introduced as an unknown solution in the new system, while the magnetic potential A is solved implicitly through its Hodge decomposition into divergence-free part, curl-free and harmonic parts, separately. Global well-posedness of the new system and its equivalence to the original problem are proved. A linearized and decoupled Galerkin finite element method is proposed for solving the new system. The convergence of numerical solutions is proved based on a compactness argument by utilizing the maximal $L^p$-regularity of the discretized equations. Compared with the Hodge decomposition method proposed in [27]_,the new method has the advantage of approximating the magnetic field B directly and converging for initial conditions that are incompatible with the external magnetic field. Several numerical examples are provided to illustrate the efficiency of the proposed numerical method in both simply connected and multi-connected nonsmooth domains. We observe that even in simply connected domains, the new method is superior to the method in [27] for approximating the magnetic field.     

Optimal control of differential‐algebraic equations from an ordinary differential equation perspective

We study the optimal control problem (OCP) for regular linear differentialalgebraic systems. To this end, we introduce the input index, which allows, on the one hand, to characterize the space of consistent initial values in terms of a Kalman‐like matrix and, on the other hand, the necessary smoothness properties of the control. The latter is essential to make the problem accessible from a numerical point of view. Moreover, we derive an augmented system as the key to analyze the OCP with tools well known from optimal control of ordinary differential equations. The new concepts of the input index and the augmented system provide easily checkable sufficient conditions, which ensure that the stage costs are consistent with the differential‐algebraic system.     

Calculus of variations approach for state and parameter estimation in switched 1D hyperbolic PDEs

This paper proposes the use of calculus of variations to solve the problem of state and parameter estimation for a class of switched 1‐dimensional hyperbolic partial differential equations coupled with an ordinary differential equation. The term “switched” here refers to a system changing its characteristics according to a switching rule, which may depend on time, parameters of the system, and/or state of the system. The estimation method is based on a smooth approximation of the system dynamics and the use of variational calculus on an augmented Lagrangian cost functional to get the sensitivity with respect to the initial state and some (possibly distributed) parameters of the system. Those sensitivities or variations, together with related adjoint systems, are used as inputs for an optimization algorithm to identify the values of the variables to be estimated. Two examples are provided to demonstrate the effectiveness of the proposed method. The first one is concerned with a switched overland flow model, developed from Saint‐Venant equations and Green‐Ampt law; the second example deals with a switched free traffic flow model based on the Lighthill‐Whitham‐Richards representation, modified by the presence of a relief route.     

Primer vector theory applied to the linear relative-motion equations

Primer vector theory is used in analysing a set of linear relative-motion equations—the Clohessy-Wiltshire (C/W) equations—to determine the criteria and necessary conditions for an optimal N-impulse trajectory. Since the state vector for these equations is defined in terms of a linear system of ordinary differential equations, all fundamental relations defining the solution of the state and costate equations and the necessary conditions for optimality can be expressed in terms of elementary functions. The analysis develops the analytical criteria for improving a solution by (1) moving any dependent or independent variable in the initial and/or final orbit, and (2) adding intermediate impulses. If these criteria are violated, the theory establishes a sufficient number of analytical equations. The subsequent satisfaction of these equations will result in the optimal position vectors and times of an N-impulse trajectory. The solution is examined for the specific boundary conditions of (1) fixed-end conditions, two-impulse, and time-open transfer; (2) an orbit-to-orbit transfer; and (3) a generalized rendezvous problem. A sequence of rendezvous problems is solved to illustrate the analysis and the computational procedure.     

Optimal control of non‐linear chemical reactors via an initial‐value Hamiltonian problem

The problem of designing strategies for optimal feedback control of non‐linear processes, specially for regulation and set‐point changing, is attacked in this paper. A novel procedure based on the Hamiltonian equations associated to a bilinear approximation of the dynamics and a quadratic cost is presented. The usual boundary‐value situation for the coupled state–costate system is transformed into an initial‐value problem through the solution of a generalized algebraic Riccati equation. This allows to integrate the Hamiltonian equations on‐line, and to construct the feedback law by using the costate solution trajectory. Results are shown applied to a classical non‐linear chemical reactor model, and compared against suboptimal bilinear‐quadratic strategies based on power series expansions. Since state variables calculated from Hamiltonian equations may differ from the values of physical states, the proposed control strategy is suboptimal with respect to the original plant. Copyright © 2005 John Wiley & Sons, Ltd.     

Multi-Valued Solution and Level Set Methods in Computational High Frequency Wave Propagation

We review the level set methods for computing multi-valued solutions to a class of nonlinear first order partial differential equations, including Hamilton-Jacobi equations, quasi-linear hyperbolic equations, and conservative transport equations with multi-valued transport speeds. The multivalued solutions are embedded as the zeros of a set of scalar functions that solve the initial value problems of a time dependent partial differential equation in an augmented space. We discuss the essential ideas behind the techniques, the coupling of these techniques to the projection of the interaction of zero level sets and a collection of applications including the computation of the semiclassical limit for Schrödinger equations and the high frequency geometrical optics limits of linear wave equations.     

Globally optimal control of self-adjoint distributed systems

The partial differential equations of motion for an uncontrolled distributed structure can be transformed into a set of independent modal equations in terms of natural co-ordinates. It is common practice to design control forces that recouple the modal equations so that the natural co-ordinates for the open-loop (uncontrolled) system cease to be natural co-ordinates for the closed-loop (controlled) system. This approach is referred to as coupled control. In contrast, the independent modal-space control method is a natural control method, i.e. natural co-ordinates for the open-loop system remain natural co-ordinates for the closed-loop system. Moreover, natural control provides a unique and globally optimal closed-form solution to the linear optimal control problem for the distributed structure. Indeed, discretization is not necessary. The optimal control forces are ideally distributed. The distributed control can be approximated by finite-dimensional control, a process that does not require truncation of the plant. Two numerical examples are presented.     

Exact penalization of terminal constraints for optimal control problems

We study optimal control problems for linear systems with prescribed initial and terminal states. We analyze the exact penalization of the terminal constraints. We show that for systems that are exactly controllable, the norm‐minimal exact control can be computed as the solution of an optimization problem without terminal constraint but with a nonsmooth penalization of the end conditions in the objective function, if the penalty parameter is sufficiently large. We describe the application of the method for hyperbolic and parabolic systems of partial differential equations, considering the wave and heat equations as particular examples. Copyright © 2016 John Wiley & Sons, Ltd.     

On linear quadratic optimal control of discrete-time complex-valued linear systems

We study in this paper the linear quadratic optimal control (linear quadratic regulation, LQR for short) for discrete-time complex-valued linear systems, which have several potential applications in control theory. Firstly, an iterative algorithm was proposed to solve the discrete-time bimatrix Riccati equation associated with the LQR problem. It is shown that the proposed algorithm converges to the unique positive definite solution (bimatrix) to the bimatrix Riccati equation with appropriate initial conditions. With the help of this iterative algorithm, the LQR problem for the antilinear system, which is a special case of complex-valued linear system, was carefully examined and three different Riccati equations–based approaches were provided, namely, bimatrix Riccati equation, anti-Riccati equation, and normal Riccati equation. The established approach is then used to solve the LQR problem for a discrete-time time-delay system with one-step state delay, and a numerical example was used to illustrate the effectiveness of the proposed methods.     

An Iterative Discontinuous Galerkin Method for Solving the Nonlinear Poisson Boltzmann Equation

An iterative discontinuous Galerkin (DG) method is proposed to solve the nonlinear Poisson Boltzmann (PB) equation. We first identify a function space in which the solution of the nonlinear PB equation is iteratively approximated through a series of linear PB equations, while an appropriate initial guess and a suitable iterative parameter are selected so that the solutions of linear PB equations are monotone within the identified solution space. For the spatial discretization we apply the direct discontinuous Galerkin method to those linear PB equations. More precisely, we use one initial guess when the Debye parameter λ=O(1), and a special initial guess for λ≪1 to ensure convergence. The iterative parameter is carefully chosen to guarantee the existence, uniqueness, and convergence of the iteration. In particular, iteration steps can be reduced for a variable iterative parameter. Both one and two-dimensional numerical results are carried out to demonstrate both accuracy and capacity of the iterative DG method for both cases of λ=O(1) and λ≪1. The (m+1)th order of accuracy for L² and mth order of accuracy for H¹for P^m elements are numerically obtained.     

Solution Remapping Method with Lower Bound Preservation for Navier-Stokes Equations in Aerodynamic Shape Optimization

It is found that the solution remapping technique proposed in [Numer. Math. Theor. Meth. Appl., 2020, 13(4)] and [J. Sci. Comput., 2021, 87(3): 1-26] does not work out for the Navier-Stokes equations with a high Reynolds number. The shape deformations usually reach several boundary layer mesh sizes for viscous flow, which far exceed one-layer mesh that the original method can tolerate. The direct application to Navier-Stokes equations can result in the unphysical pressures in remapped solutions, even though the conservative variables are within the reasonable range. In this work, a new solution remapping technique with lower bound preservation is proposed to construct initial values for the new shapes, and the global minimum density and pressure of the current shape which serve as lower bounds of the corresponding variables are used to constrain the remapped solutions. The solution distribution provided by the present method is proven to be acceptable as an initial value for the new shape. Several numerical experiments show that the present technique can substantially accelerate the flow convergence for large deformation problems with 70%-80% CPU time reduction in the viscous airfoil drag minimization.     

Stability and identifiability of parameters in Galerkin approximation of distributed systems under multiple inputs

In this communication we consider conditions for stability and identifiability of parameters in initial value problems governed by systems of ordinary differential equations that are obtained as Galerkin approximations of parabolic initial boundary value problems. Of particular interest are problems in which multiple inputs are allowed to obtain multiple outputs. These input—output relations are used to obtain the desired identifiability results.     

Minimum‐time anti‐swing motion planning of cranes using linear programming

A novel method to solve the minimum‐time anti‐swing motion planning problem for cranes based on the use of linear programming is proposed. It is shown that its solution can be obtained by solving a sequence of fixed‐time maximum‐range linear programming problems. A convergence proof is presented. A classical kinematical model for which the trolley acceleration is the control variable is used. The crane motion equations are discretized in time assuming that the control variable is piecewise constant. Inequality constraints on both trolley speed and acceleration are included to represent physical bounds associated to the electromechanical driving system. The load and the trolley are required to be at rest both at the initial and at the final times. The load cable length as a function of time is assumed given. The ease of both the formulation and the solution of the problem is in contrast with the traditional two‐point boundary value problem associated to the Pontryagin's minimum principle. Copyright © 2012 John Wiley & Sons, Ltd.     

Trajectory optimization involving sloshing media

This paper is concerned with the optimization of the transport motion of an open topped fluid filled container within a warehouse environment. In particular, optimal trajectories of the motion of the driver–container system in two‐dimensional space will be investigated via numerical solutions of the model equations using sequential quadratic programming. The fluid and the mechanical facility that moves the container are subject to several constraints. The objective of the optimization is the time to transport the container from an initial position to its final destination within the warehouse. Optimization criteria are investigated to control the movement of the fluid within the container. The systems of ordinary and partial differential equations, representing the dynamics of the models are solved numerically using a direct shooting method. The resulting non‐linear programming problem is solved using sequential quadratic programming (SQP). Copyright © 2002 John Wiley & Sons, Ltd.     

On the bounded and stabilizing solution of a generalized Riccati differential equation arising in connection with a zero-sum linear quadratic stochastic differential game

We study a class of coupled nonlinear matrix differential equations arising in connection with the solution of a zero-sum two-player linear quadratic (LQ) differential game for a dynamical system modeled by an Itô differential equation subject to random switching of its coefficients. The system of differential equations under consideration contains as special cases the game-theoretic Riccati differential equations arising in the solution of the H_∞ control problem from the deterministic and stochastic cases. A set of sufficient conditions that guarantee the existence of the bounded and stabilizing solution of this kind of Riccati differential equations is provided. We show how such stabilizing solution is involved in the construction of the equilibrium strategy of a zero-sum LQ stochastic differential game on an infinite-time horizon and give as a byproduct the solution of such a control problem.     

Mixed h2 and h∞ optimal robust control design

This paper introduces a design methodology for a dynamic compensator that simultaneously minimizes the upper bound of a quadratic performance index and the H_∞-norm of a disturbance transfer function matrix of a multiple-input/multiple-output system whose model contains parameter uncertainty in the state and input matrices. The real parameter uncertainty is modelled as additional measurement outputs and as additional weights on the existing noise inputs and measurement outputs of the system. The compensator equations are derived by taking the dual of a system with parameter variation in the state and output matrices, for which the compensator equations have previously been derived, and then taking the dual of the compensator equations. An algorithm for applying this theory is given and an example is shown.     

Numerical Resolution Near t=0 of Nonlinear Evolution Equations in the Presence of Corner Singularities in Space Dimension 1

The incompatibilities between the initial and boundary data will cause singularities at the time-space corners, which in turn adversely affect the accuracy of the numerical schemes used to compute the solutions. We study the corner singularity issue for nonlinear evolution equations in 1D, and propose two remedy procedures that effectively recover much of the accuracy of the numerical scheme in use. Applications of the remedy procedures to the 1D viscous Burgers equation, and to the 1D nonlinear reaction-diffusion equation are presented. The remedy procedures are applicable to other nonlinear diffusion equations as well.     

Hamiltonian two‐degrees‐of‐freedom control of chemical reactors

A novel unified approach to two‐degrees‐of‐freedom control is devised and applied to a classical chemical reactor model. The scheme is constructed from the optimal control point of view and along the lines of the Hamiltonian formalism for nonlinear processes. The proposed scheme optimizes both the feedforward and the feedback components of the control variable with respect to the same cost objective. The original Hamiltonian function governs the feedforward dynamics, and its derivatives are part of the gain for the feedback component. The optimal state trajectory is generated online, and is tracked by a combination of deterministic and stochastic optimal tools. The relevant numerical data to manipulate all stages come from a unique off‐line calculation, which provides design information for a whole family of related control problems. This is possible because a new set of PDEs (the variational equations) allow to recover the initial value of the costate variable, and the Hamilton equations can then be solved as an initial‐value problem. Perturbations from the optimal trajectory are abated through an optimal state estimator and a deterministic regulator with a generalized Riccati gain. Both gains are updated online, starting with initial values extracted from the solution to the variational equations. The control strategy is particularly useful in driving nonlinear processes from an equilibrium point to an arbitrary target in a finite‐horizon optimization context. Copyright © 2010 John Wiley & Sons, Ltd.