On the Existence of Lipschitz Continuous Optimal Feedback Control

We consider a finite-horizon continuous-time optimal control problem with nonlinear dynamics, an integral cost, control constraints and a time-varying parameter which represents perturbations or uncertainty. After time-discretization of the problem we employ a Model Predictive Control (MPC) algorithm with a shrinking horizon, which uses a "prediction"/forecast for the uncertain parameter and (possibly inexact) measurements of the state vector, and generates a piecewise constant control signal by solving auxiliary open-loop control problems. In our main result we derive an estimate of the difference between the MPC-generated control and the optimal feedback control, both obtained for the same value of the perturbation parameter, in terms of the step-size of the discretization and the measurement error. We also give an estimate for a norm of the difference between the MPC-generated control and the optimal open-loop control in the problem with the "true" value of the uncertain parameter, depending on the prediction error.

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

This thesis aims at developing efficient mathematical methods for solving hierarchical dynamic optimization problems. The main motivation is to model processes in nature, for which there is evidence to assume that they run optimally. We describe models of such processes by optimal control problems (called optimal control models (OCMs)). However, an OCM typically includes unknown parameters that cannot be derived entirely on a theoretical basis, which is in particular the case for the cost function. Therefore, we develop parameter estimation techniques to estimate the unknowns in an OCM from observation data of the process. Mathematically, this leads to a hierarchical dynamic optimization problem with a parameter estimation problem on the upper level and an optimal control problem on the lower level. We focus on multi-stage equality and inequality constrained optimal control problems based on nonlinear ordinary differential equations. The main goal of this thesis is to derive numerically efficient mathematical methods for solving hierarchical dynamic optimization problems, and to use these methods to estimate parameters in high-dimensional OCMs from real-world measurement data. We develop parameter-dependent OCMs for the gait of cerebral palsy patients and able-bodied subjects. The unknown parameters in the OCMs are then estimated from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg by using the mathematical methods developed within this work. The main novelties and contributions of this thesis to the field of hierarchical dynamic optimization are summarized herein. - We establish a novel mathematical method, a so-called direct all-at-once approach, for solving hierarchical dynamic optimization problems based on the direct multiple shooting method and first-order optimality conditions. - Furthermore, we propose an efficient numerical algorithm for large-scale hierarchical dynamic optimization problems, which fully exploits the structures inherited from both the hierarchical setting and the discretization. - Pontryagin's maximum principle is used to analyze solution properties of hierarchical dynamic optimization problems like second-order optimality conditions of the lower-level problem. - In addition, we propose and discuss alternative methods for hierarchical dynamic optimization that are based on derivative-free optimization and a bundle approach. These methods keep the hierarchical problem setting and do not reformulate the lower-level problem using first-order optimality conditions. - We establish a novel lifting method for regularizing mathematical programs with complementarity constraints, which is discussed and numerically investigated by means of a well-known collection of benchmark problems. - Proofs of regularity and convergence results for sequential quadratic programming methods applied to lifted mathematical programs with complementarity constraints are provided. - Efficient state-of-the-art implementations of all mathematical methods derived in this thesis, as well as a benchmark collection of hierarchical dynamic optimization problems are presented. - High-dimensional optimal control gait models for cerebral palsy patients and able-bodied subjects are developed. The mathematical methods derived in this thesis are used to estimate the unknown model parameters from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg. The theoretical and practical results presented in this thesis can be considered an initial motivating step towards answering open questions in current medical research in fields like treatment planning, classification of gaits, or the evaluation of surgeries by means of hierarchical dynamic optimization.

An optimal control problem for controlling the cell volume in dehydration and rehydration process

We study parametric optimal control problems governed by a system of time-dependent partial differential equations (PDE) and subject to additional control and state constraints. An approach is presented to compute the optimal control functions and the so-called sensitivity differentials of the optimal solution with respect to perturbations. This information plays an important role in the analysis of optimal solutions as well as in real-time optimal control. The method of lines is used to transform the perturbed PDE system into a large system of ordinary differential equations. A subsequent discretization then transcribes parametric ODE optimal control problems into perturbed nonlinear programming problems (NLP), which can be solved efficiently by SQP methods. Second-order sufficient conditions can be checked numerically and we propose to apply an NLP-based approach for the robust computation of the sensitivity differentials of the optimal solutions with respect to the perturbation parameters. The numerical method is illustrated by the optimal control and sensitivity analysis of the Burgers equation.

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 the present paper, we are concerned with a fractional wave equation of Sturm–Liouville type in a general star graph. We first give several existence, uniqueness and regularity results of weak solutions for the one-dimensional case using the spectral theory; we prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal control via the Euler–Lagrange first-order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary conditions and controls of the velocity. We show the existence and uniqueness of minimizers, and by using the first-order optimality conditions with the Lagrange multipliers, we are able to characterize the optimal controls.

This paper proposes a solution method for the discrete-time long-term dynamic portfolio optimization problem with state and asset allocation constraints. We use the ideas of Model Predictive Control (MPC) to solve the constrained stochastic control problem. MPC is a solution technique which was developed to solve constrained optimal control problems for deterministic control applications. MPC solves the optimal control problem with a receding horizon where a series of consecutive open-loop optimal control problems is solved. The aim of this paper is to develop an MPC approach to the problem of long-term portfolio optimization when the expected returns of the risky assets are modeled using a factor model based on stochastic Gaussian processes. We prove that MPC is a suboptimal control strategy for stochastic systems which uses the new information advantageously and thus is better than the pure optimal open-loop control. For the open-loop optimal control optimization, we derive the conditional portfolio distribution and the corresponding conditional portfolio mean and variance. The mean and the variance depend on future decision about the asset allocation. For the dynamic portfolio optimization problem, we consider constraints on the asset allocation as well as probabilistic constraints on the attainable values of the portfolio wealth. We discuss two different objectives, a classical mean–variance objective and the objective to maximize the probability of exceeding a predetermined value of the portfolio. The dynamic portfolio optimization problem is stated, and the solution via MPC is explained in detail. The results are then illustrated in a case study.

The control of artificial in-stream aeration of polluted rivers with multiple waste effluent sources is treated. The optimal feedback control law for this distributed parameter system is determined by solving the partial differential equations along characteristic lines. In this process the double integral cost functional of the distributed parameter system is reduced to a single integral cost. Because certain measurements are time consuming, the feedback control law is obtained in the presence of observation delay in some but not all of the system variables. The open loop optimal control is then found, showing explicity the effect of changes in any of the effluent sources on the aeration strategy. It is shown that the optimal strategy for a distribution of sources can be written as an affine transformation upon the optimal controls for sources of unit strength.

We are honored to organize this special issue of Neural Computation and Applications, based on contributions to the Ninth International Symposium on Neural Networks (ISNN 2012) held on July 11–14, 2012, in Shenyang, China. Evaluated by the contributions and the recommendation of ISNN 2012 organizers, 22 papers were selected for the further review process with their extended form. Each paper has been reviewed thoroughly by three independent experts in related research area. We are very grateful for the hard work of the reviewers, which help to facilitate the outcome of this special issue with greatly improved quality of the contributions. Finally, 20 papers are accepted, to be introduced as follows. In brief, the contributions of this special issue can be roughly classified into three major categories: data-based control, data-based optimization, and data-based modeling. In each category, not only the theoretical foundation is analyzed, but also some prospective applications are demonstrated. In this introduction, we like to borrow some ideas from the contributions, such as hierarchical, and cluster as the subsidiary category idea. The first category consisting of 6 papers is devoted to the data-based control and applications area, by which an optimal or near-optimal control can be derived from the input and output data. The first 4 papers focus on the hot research direction, adaptive dynamic programming and reinforcement learning (ADPRL), and the remaining 2 papers relate to iterative learning control and optimal control. Derong Liu et al. develop an online algorithm based on policy iteration for continuous-time optimal control with infinite horizon cost for nonlinear systems. In the proposed method, a discount value function is employed, which is considered to be a more general case for optimal control problems. Meanwhile, without knowledge of the internal system dynamics, the algorithm can converge uniformly online to the optimal solution of the modified Hamilton– Jacobi–Bellman equation. By means of two neural networks, the algorithm is able to find suitable approximations of both the optimal control and the optimal cost. Dual iterative adaptive dynamic programming for a class of discrete-time nonlinear systems with time-delays is investigated by Qinglai Wei et al. Dual iterative ADP algorithm is introduced to obtain the optimal solutions of the optimal performance index function and control, where in each iteration, the performance index function and the system states are both updated. Convergence analysis is presented to prove the performance index function to reach D. Zhao (&) State Key Laboratory of Management and Control for Complex Systems, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China e-mail: Dongbin.zhao@ia.ac.cn

This thesis is concerned with numerical methods for Mixed-Integer Optimal Control Problems with Combinatorial Constraints. We establish an approximation theorem relating a Mixed-Integer Optimal Control Problem with Combinatorial Constraints to a continuous relaxed convexified Optimal Control Problems with Vanishing Constraints that provides the basis for numerical computations. We develop a a Vanishing- Constraint respecting rounding algorithm to exploit this correspondence computationally. Direct Discretization of the Optimal Control Problem with Vanishing Constraints yield a subclass of Mathematical Programs with Equilibrium Constraints. Mathematical Programs with Equilibrium Constraint constitute a class of challenging problems due to their inherent non-convexity and non-smoothness. We develop an active-set algorithm for Mathematical Programs with Equilibrium Constraints and prove global convergence to Bouligand stationary points of this algorithm under suitable technical conditions. For efficient computation of Newton-type steps of Optimal Control Problems, we establish the Generalized Lanczos Method for trust region problems in a Hilbert space context. To ensure real-time feasibility in Online Optimal Control Applications with tracking-type Lagrangian objective, we develop a Gaus-Newton preconditioner for the iterative solution method of the trust region problem. We implement the proposed methods and demonstrate their applicability and efficacy on several benchmark problems.

To consider the energy saving during the robot motion, optimal posture control method for a robot manipulator is proposed. The Chebyshev-Gauss (CG) Pseudo-spectral method is used to discuss the problem with the energy optimal control. The Lagrange interpolation of barycentre is adopted to approximate the state and control variables. The continuous optimal control problem can be converted to a discrete nonlinear programming (NLP) problem. And then it can be solved by Sequential Quadratic Programming (SQP) algorithm. The simulation results of optimal energy control make the robot manipulator from the initial state to the desired terminal posture successfully. The terminal angular velocity of each link is also meet the predetermined value. The control law does not exceed the preset boundary of control.

In this paper, the nonlinear optimal control problem is investigated for spacecraft formation flying with collision avoidance. Based on a nonlinear model of formation flying, two optimal tracking control laws are proposed to ensure the formation pattern converges to the predetermined configuration. The first optimal control law which combines Lyapunov optimizing control with a trajectory-following optimization technique is developed to solve the finite-time nonlinear optimal control problem. For the second controller, an extended [Formula: see text]-[Formula: see text] method is applied to design a closed-form feedback control scheme for nonlinear control problem of spacecraft formation flying with non-standard cost functions. By taking into account the flying safety, a repulsive control scheme is incorporated in the optimal controller to ensure collision avoidance and minimize the performance index. Finally, a numerical example is performed to demonstrate the effectiveness of the proposed approaches.

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.

This paper deals with the problem of optimal control of RD allocation of investments among rival projects under a given total budget; timing of the ultimate selection of new technology (activity) according to the most successful project. The process of data improvement (PDI) is described as a (random) process of reducing the intervals of uncertainty, the rate of data improvement being controlled, in effect, by the investment policy. A stationary dynamic model with infinite horizon of planning and with the criterion of total discounted costs is developed. The model is carefully investigated from the economic point of view, and some new effects are discovered. It is demonstrated that uncertainty may be the source of additional gain; the bigger the initial uncertainty, the bigger the gain may be under optimal control. This effect is based on PDI, the economic background of which is presented as multiplication of the expected gain over time, which is typical for the classical macromodels of economics. Optimal continuous control is constructed at every time moment, and answers to two main questions are given: (1) either to prolong R & D or to make ultimate selection of one project, (2) how to allocate resources among the rival projects in the case of R&D prolongation. The last question is answered on the basis of profitability indices that relate efficiencies of PDI for particular projects to the costs of information.