Stabilizing the Hamiltonian system for constructing optimal trajectories

An infinite-horizon optimal control problem based on an economic growth model is studied. The goal in the problem is to optimize the mechanisms of investment in basic production assets in order to increase the growth rate of the consumption level. The main output variable-the gross domestic product (GDP)-depends on three production factors: capital stock, human capital, and useful work. The first two factors are endogenous variables of the model, and the useful work is an exogenous factor. The dependence of the GDP on the production factors is described by the Cobb-Douglas power-type production function. The economic system under consideration is assumed to be closed, so the GDP is distributed between consumption and investment in the capital stock and human capital. The optimal control problem consists in determining optimal investment strategies that maximize the integral discounted relative consumption index on an infinite time interval. A solution to the problem is constructed on the basis of the Pontryagin maximum principle adapted to infinite-horizon problems. We examine the questions of existence and uniqueness of a solution, verify necessary and sufficient optimality conditions, and perform a qualitative analysis of Hamiltonian systems on the basis of which we propose an algorithm for constructing optimal trajectories. This algorithm uses information on solutions obtained by means of a nonlinear regulator. Finally, we estimate the accuracy of the algorithm with respect to the integral cost functional of the control process.     

Properties of Hamiltonian systems in the Pontryagin maximum principle for economic growth problems

We consider an optimal control problem with a functional defined by an improper integral. We study the concavity properties of the maximized Hamiltonian and analyze the Hamiltonian systems in the Pontryagin maximum principle. On the basis of this analysis, we propose an algorithm for constructing an optimal trajectory by gluing the dynamics of the Hamiltonian systems. The algorithm is illustrated by calculating an optimal economic growth trajectory for macroeconomic data.     

Classical characteristics of the bellman equation in constructions of grid optimal synthesis

We consider optimal control problems with fixed final time and terminal-integral cost functional, and address the question of constructing a grid optimal synthesis (a universal feedback) on the basis of classical characteristics of the Bellman equation. To construct an optimal synthesis, we propose a numerical algorithm that relies on the necessary optimality conditions (the Pontryagin maximum principle) and sufficient conditions in the Hamiltonian form. We obtain estimates for the efficiency of the numerical method. The method is illustrated by an example of the numerical solution of a nonlinear optimal control problem.     

Studying Dynamical Principles of Human Locomotion using Inverse Optimal Control

Human movement, as for example human gait, can be considered as an optimal realization of some given task. However, the criterion for which the naturally performed human motion is optimal, is generally not known. In this article we formulate an inverse optimal control problem to study the relevance of four different optimization criteria in human locomotion. As a walking model we use an actuated three dimensional spring loaded inverted pendulum (3D-SLIP), which is able to mirror the typical shape of the center of mass trajectory in human gait. Using a direct all-at-once approach, the weighting of the optimization criteria and the position of the footsteps are optimized in such a way, that the center of mass trajectory of the resulting optimal state fits real motion capture data as good as possible. Numerical experiments show, that whereas the so called capture point seems to have a great impact on human walking, minimization of the vertical center of mass movement does not show any relevance at all. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling and identification of emotional aspects of locomotion

The studies of emotional facial expressions and emotional body language are currently receiving a lot of attention in the cognitive sciences. In this project, we study implicit bodily expression of emotions during standard motions, such as walking forwards.An underlying assumption of our work is that all human motion is optimal in some sense and that different emotions induce different objective functions, which result in different deformations of normal motion.We created a 3D rigid-body model of a human of which we use the forward dynamics simulation in an optimal control context. We performed two kinds of optimizations: (i) reconstruction of dynamic quantities, such as joint torques, of pre-recorded data of emotional walking motions and (ii) forward optimization that generates neutral and varied walking motions using different objective functions. Optimizations are performed with the software package MUSCOD-II, which uses a direct multiple-shooting discretization scheme. The results of this work form the foundation for further analysis of emotional motions using inverse optimal control methods.     

Optimal synthesis in a control problem with Lipschitz input data

We propose a numerical algorithm for constructing an optimal synthesis in the control problem for a nonlinear system on a fixed time interval. We estimate the difference between the values of the cost functional on optimal trajectories and on the trajectories constructed according to this algorithm. The operation of the algorithm is illustrated by solving model examples on the plane.     

A Fourier-Coefficient Based Solution of an Optimal Control Problem in Quantum Chemistry

We consider an optimal control problem for the time-dependent Schrödinger equation modeling molecular dynamics. The dynamics can be steered by interactions with a tuned laser field. The problem of designing an optimal field can be posed as an optimal control problem. We reformulate the optimization problem by using a Fourier transform of the electric field, and narrow the frequency band. The resulting problem is less memory intense, and can be solved with a superlinearly convergent quasi-Newton method. We show computational results for a Raman-transition example and give numerical evidence that our method can outperform the standard monotonically convergent algorithm.     

Impulsive optimal control model for the trajectory of horizontal wells

This paper presents an impulsive optimal control model for solving the optimal designing problem of the trajectory of horizontal wells. We take fully into account the effect of unknown disturbances in drilling. The optimal control problem can be converted into a nonlinear parametric optimization by integrating the state equation. We discuss here that the locally optimal solution depends in a continuous way on the parameters (disturbances) and utilize this property to propose a revised Hooke–Jeeves algorithm. The uniform design technique is incorporated into the revised Hooke–Jeeves algorithm to handle the multimodal objective function. The numerical simulation is in accordance with theoretical results. The numerical results illustrate the validity of the model and efficiency of the algorithm.     

Single-machine scheduling of proportionally deteriorating jobs by two agents

We consider a problem of scheduling a set of independent jobs by two agents on a single machine. Every agent has its own subset of jobs to be scheduled and uses its own optimality criterion. The processing time of each job proportionally deteriorates with respect to the starting time of the job. The problem is to find a schedule that minimizes the total tardiness of the first agent, provided that no tardy job is allowed for the second agent. We prove basic properties of the problem and give a lower bound on the optimal value of the total tardiness criterion. On the basis of these results, we propose a branch-and-bound algorithm and an evolutionary algorithm for the problem. Computational experiments show that the exact algorithm solves instances up to 50 jobs in a reasonably short time and that solutions obtained by the metaheuristic are close to optimal ones.     

Optimality of Balanced Proper Orthogonal Decomposition for Data Reconstruction

Proper orthogonal decomposition (POD) finds an orthonormal basis yielding an optimal reconstruction of a given dataset. We consider an optimal data reconstruction problem for two general datasets related to balanced POD, which is an algorithm for balanced truncation model reduction for linear systems. We consider balanced POD outside of the linear systems framework, and prove that it solves the optimal data reconstruction problem. The theoretical result is illustrated with an example.     

Optimal switching control of a fed-batch fermentation process

Considering the hybrid nature in fed-batch culture of glycerol biconversion to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae, we propose a state-based switching dynamical system to describe the fermentation process. To maximize the concentration of 1,3-PD at the terminal time, an optimal switching control model subject to our proposed switching system and constraints of continuous state inequality and control function is presented. Because the number of the switchings is not known a priori, we reformulate the above optimal control problem as a two-level optimization problem. An optimization algorithm is developed to seek the optimal solution on the basis of a heuristic approach and control parametrization technique. Numerical results show that, by employing the obtained optimal control strategy, 1,3-PD concentration at the terminal time can be increased considerably.     

Direct methods with maximal lower bound for mixed-integer optimal control problems

Many practical optimal control problems include discrete decisions. These may be either time-independent parameters or time-dependent control functions as gears or valves that can only take discrete values at any given time. While great progress has been achieved in the solution of optimization problems involving integer variables, in particular mixed-integer linear programs, as well as in continuous optimal control problems, the combination of the two is yet an open field of research. We consider the question of lower bounds that can be obtained by a relaxation of the integer requirements. For general nonlinear mixed-integer programs such lower bounds typically suffer from a huge integer gap. We convexify (with respect to binary controls) and relax the original problem and prove that the optimal solution of this continuous control problem yields the best lower bound for the nonlinear integer problem. Building on this theoretical result we present a novel algorithm to solve mixed-integer optimal control problems, with a focus on discrete-valued control functions. Our algorithm is based on the direct multiple shooting method, an adaptive refinement of the underlying control discretization grid and tailored heuristic integer methods. Its applicability is shown by a challenging application, the energy optimal control of a subway train with discrete gears and velocity limits.      

Asymptotic optimization of nonlinear singularly perturbed systems with control constrained by a hypersphere

We consider a terminal control problem for a nonlinear singularly perturbed system with constraints imposed on the right endpoint of the trajectories. The values of the multidimensional controls are bounded in the Euclidean norm. We suggest an algorithm for an approximate (in the asymptotic sense) solution of this problem. The key advantage of the algorithm is the following: the original problem splits into two optimal control problems of lower dimension, one of which is linear.     

Remarks on the algorithm of sakawa for optimal control problems

A general optimal control problem for ordinary differential equations is considered. For this problem, some improvements of the algorithm of Sakawa are discussed. We avoid any convexity assumption and show with an example that the algorithm is even applicable in cases in which no optimal control exists.     

Approximate solution of a control problem on the basis of the nilpotent approximation

We consider the control problem for deterministic systems described by ordinary differential equations with linear controls. On the basis of the nilpotent approximation method, we construct an algorithm for finding an approximate solution of the control problem for three-dimensional nonlinear systems with two linear controls. The algorithm was implemented in Maple and tested in examples including the control of a mobile robot on a plane and the attitude control of a sphere rolling on a plane.     

Systems of control over set-valued trajectories with terminal quality criterion

We consider the optimal control problem with terminal quality criterion in which the state of a system is described by a set-valued mapping, and an admissible control is a summable function. We describe an algorithm that approximates the admissible control function by a piecewise-constant function and prove theorems on the closeness of the corresponding trajectories and the values of quality criteria.     

On the asymptotics of the optimal value of the performance functional in a linear optimal control problem

We consider the asymptotics of the optimal value of the performance functional in an optimal control problem for a linear system with rapid and slow variables, with convex terminal performance functional depending on the slow variables, and with smooth geometric constraints on the control. We find sufficient conditions for the regularity of asymptotics and construct an algorithm for finding the complete asymptotics of the optimal value of the performance functional.     

A numerical method for an optimal control problem with minimum sensitivity on coefficient variation

In this paper, we consider a class of optimal control problem involving an impulsive systems in which some of its coefficients are subject to variation. We formulate this optimal control problem as a two-stage optimal control problem. We first formulate the optimal impulsive control problem with all its coefficients assigned to their nominal values. This becomes a standard optimal impulsive control problem and it can be solved by many existing optimal control computational techniques, such as the control parameterizations technique used in conjunction with the time scaling transform. The optimal control software package, MISER 3.3, is applicable. Then, we formulate the second optimal impulsive control problem, where the sensitivity of the variation of coefficients is minimized subject to an additional constraint indicating the allowable reduction in the optimal cost. The gradient formulae of the cost functional for the second optimal control problem are obtained. On this basis, a gradient-based computational method is established, and the optimal control software, MISER 3.3, can be applied. For illustration, two numerical examples are solved by using the proposed method.     

New developments in the application of Pontryagin's Principle for the hydrothermal optimization

^** Email: bayon{at}uniovi.es In this paper we have developed a much simpler theory than previousones that resolves the problem of the optimization of hydrothermalsystems. The problem involves non-holonomic inequality constraints.In particular, we have established a necessary condition forthe stationary functions of the functional. We shall use Pontryagin'sMinimum Principle as the basis for proving this theorem, settingout our problem in terms of optimal control in continuous time,with the Lagrange-type functional. This theorem allows us toelaborate the optimization algorithm that leads to the determinationof the optimal solution of the hydrothermal system. We generalizethe problem, taking into account a cost associated with thewater, to then set out and solve the corresponding Bolza's problem.Finally, we present an example employing the algorithm developedfor this purpose with the ‘Mathematica’ package.     

Meta-control of an interacting-particle algorithm for global optimization

A common issue for stochastic global optimization algorithms is how to set the parameters of the sampling distribution (e.g. temperature, mutation/cross-over rates, selection rate, etc.) so that the samplings converge to the optimum effectively and efficiently. We consider an interacting-particle algorithm and develop a meta-control methodology which analytically guides the inverse temperature parameter of the algorithm to achieve desired performance characteristics (e.g. quality of the final outcome, algorithm running time, etc.). The main aspect of our meta-control methodology is to formulate an optimal control problem where the fractional change in the inverse temperature parameter is the control variable. The objectives of the optimal control problem are set according to the desired behavior of the interacting-particle algorithm. The control problem considers particles’ average behavior, rather than treating the behavior of individual particles. The solution to the control problem provides feedback on the inverse temperature parameter of the algorithm.