A steepest-descent method for optimization of mechanical systems

Steepest-descent optimal control techniques have been used extensively for dynamic systems in one independent variable and with a full set of initial conditions. This paper presents an extension of the steepest-descent technique to mechanical design problems that are described by boundary-value problems with one or more independent variables. The method is illustrated by solving finite-dimensional problems, problems with distribution of design over one space dimension, and problems with distribution of design over two space dimensions.     

SOLVING DYNAMIC WILDLIFE RESOURCE OPTIMIZATION PROBLEMS USING REINFORCEMENT LEARNING

ABSTRACT. An important technical component of natural resource management, particularly in an adaptive management context, is optimization. This is used to select the most appropriate management strategy, given a model of the system and all relevant available information. For dynamic resource systems, dynamic programming has been the de facto standard for deriving optimal state‐specific management strategies. Though effective for small‐dimension problems, dynamic programming is incapable of providing solutions to larger problems, even with modern microcomputing technology. Reinforcement learning is an alternative, related procedure for deriving optimal management strategies, based on stochastic approximation. It is an iterative process that improves estimates of the value of state‐specific actions based in interactions with a system, or model thereof. Applications of reinforcement learning in the field of artificial intelligence have illustrated its ability to yield near‐optimal strategies for very complex model systems, highlighting the potential utility of this method for ecological and natural resource management problems, which tend to be of high dimension. I describe the concept of reinforcement learning and its approach of estimating optimal strategies by temporal difference learning. I then illustrate the application of this method using a simple, well‐known case study of Anderson [1975], and compare the reinforcement learning results with those of dynamic programming. Though a globally‐optimal strategy is not discovered, it performs very well relative to the dynamic programming strategy, based on simulated cumulative objective return. I suggest that reinforcement learning be applied to relatively complex problems where an approximate solution to a realistic model is preferable to an exact answer to an oversimplified model.     

Multiobjective optimal control of a four-body kinematic chain

Optimal control of mechanical systems is an active area of research. However, so far, most contributions are taking only one single objective into account, whereas for many practical problems, one is interested in optimizing several conflicting objectives at the same time. Applying singleobjective optimization to each of them leads to several trajectories each being optimal for one objective, but ignoring all others. In contrast to that, combining all objectives and using multiobjective optimization leads to a variety of trade off solutions taking all objectives into account simultaneously. We use the direct discretization method DMOCC (Discrete Mechanics and Optimal Control for Constrained systems) to approximate trajectories of the underlying optimal control problems, resulting in restricted optimization problems of high dimension. For the multiobjective part, we apply a reference point technique which successively utilizes an auxiliary distance function to gain the trade off solutions. The presented approach is illustrated by the multiobjective optimal control of a constrained multibody system. A four-body kinematic chain is controlled in a rest to rest maneuver, for which minimal control effort and minimal required maneuver time are the conflicting objectives. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotically suboptimal control of weakly interconnected dynamical systems

Optimal control problems for a group of systems with weak dynamical interconnections between its constituent subsystems are considered. A method for decentralized control is proposed which distributes the control actions between several controllers calculating in real time control inputs only for theirs subsystems based on the solution of the local optimal control problem. The local problem is solved by asymptotic methods that employ the representation of the weak interconnection by a small parameter. Combination of decentralized control and asymptotic methods allows to significantly reduce the dimension of the problems that have to be solved in the course of the control process.     

Optimal control for evolution equations with memory

In this paper, we investigate the existence and regularity of solutions for Bolza optimal control problems in infinite dimension governed by a class of semilinear evolution equations. Our results apply to systems exhibiting hereditary properties, as heat propagation in real conductors and isothermal viscoelasticity, described by equations with memory terms which account for the past history of the variables in play.     

An efficient computational method for the optimal control problem for the Burgers equation

Pointwise control of the viscous Burgers equation in one spatial dimension is studied with the objective of minimizing the distance between the final state function and target profile along with the energy of the control. An efficient computational method is proposed for solving such problems, which is based on special orthonormal functions that satisfy the associated boundary conditions. Employing these orthonormal functions as a basis of a modal expansion method, the solution space is limited to the smallest lower subspace that is sufficient to describe the original problem. Consequently, the Burgers equation is reduced to a set of a minimal number of ordinary nonlinear differential equations. Thus, by the modal expansion method, the optimal control of a distributed parameter system described by the Burgers equation is converted to the optimal control of lumped parameter dynamical systems in finite dimension. The time-variant control is approximated by a finite term of the Fourier series whose unknown coefficients and frequencies giving an optimal solution are sought, thereby converting the optimal control problem into a mathematical programming problem. The solution space obtained is based on control parameterization by using the Runge–Kutta method. The efficiency of the proposed method is examined using a numerical example for various target functions.     

Optimal Control of a Class of Phase Change Processes with Terminal State Observation

A class of optimal control problems for nonlinear evolutionary processes governed by two-phase Stefan problems is analyzed. The processes with terminal state observation are considered in the case of one space dimension. Approximate optimal solutions (controls, as well as the corresponding states and adjoint states), referring to the problems with time-averaged state observation are shown to converge to the appropriate solutions for the original problem.     

Isolation of the Trivial Part of a Nonlinear Control System by Factorization: II

The problem of constructing aggregated systems (quotient systems) of the simplest form for nonlinear control systems is considered. This factorization reduces the original control system to a decomposition, which permits one to reduce the dimension of control problems.     

Isolation of the Trivial Part of a Nonlinear Control System by Factorization: I

The problem of constructing aggregated systems (quotient systems) of the simplest kind for nonlinear control systems is considered. With the help of this factorization, the original control system is reduced to a decomposition that allows one to reduce the dimension of control problems.     

Exact slow-fast decomposition of the singularly perturbed matrix differential Riccati equation

In this paper the Hamiltonian matrix formulation of the Riccati equation is used to derive the reduced-order pure-slow and pure-fast matrix differential Riccati equations of singularly perturbed systems. These pure-slow and pure-fast matrix differential Riccati equations are obtained by decoupling the singularly perturbed matrix differential Riccati equation of dimension n₁+n₂ into the pure-slow regular matrix differential Riccati equation of dimension n₁ and the pure-fast stiff matrix differential Riccati equation of dimension n₂. A formula is derived that produces the solution of the original singularly perturbed matrix differential Riccati equation in terms of solutions of the pure-slow and pure-fast reduced-order matrix differential Riccati equations and solutions of two reduced-order initial value problems. In addition to its theoretical importance, the main result of this paper can also be used to implement optimal filtering and control schemes for singularly perturbed linear time-invariant systems independently in pure-slow and pure-fast time scales.     

阻尼Sine-Gordon方程的H1-Galerkin混合元方法数值解

利用H1-Galerkin混合有限元方法讨论阻尼Sine-Gordon方程,得到一维情况下半离散和全离散格式的最优阶误差估计,并且推广应用到二维和三维情况,而且不用验证LBB相容性条件.     

The dimension of the global attractor for dissipative reaction-diffusion systems

Our aim in this note is to construct an exponential attractor of optimal (with respect to the dissipation parameter) fractal dimension for dissipative reaction-diffusion systems without conditions on the growth of the nonlinear term.     

Study of optimal control problems on the half-line by the averaging method

We justify the application of the averaging method to optimal control problems for systems of differential equations on the half-line. For optimal control problems for systems of differential equations linear in the control, we prove the existence of optimal controls for the exact and averaged problems. We show that an optimal control in the averaged problem is ɛ-optimal in the exact problem.     

Coherent control of a qubit is trap-free

There is a strong interest in optimal manipulating of quantum systems by external controls. Traps are controls which are optimal only locally but not globally. If they exist, they can be serious obstacles to the search of globally optimal controls in numerical and laboratory experiments, and for this reason the analysis of traps attracts considerable attention. In this paper we prove that for a wide range of control problems for two-level quantum systems all locally optimal controls are also globally optimal. Hence we conclude that two-level systems in general are trap-free. In particular, manipulating qubits—two-level quantum systems forming a basic building block for quantum computation—is free of traps for fundamental problems such as the state preparation and gate generation.     

An iterative two-step algorithm for linear complementarity problems

Summary. We propose an algorithm for the numerical solution of large-scale symmetric positive-definite linear complementarity problems. Each step of the algorithm combines an application of the successive overrelaxation method with projection (to determine an approximation of the optimal active set) with the preconditioned conjugate gradient method (to solve the reduced residual systems of linear equations). Convergence of the iterates to the solution is proved. In the experimental part we compare the efficiency of the algorithm with several other methods. As test example we consider the obstacle problem with different obstacles. For problems of dimension up to 24\,000 variables, the algorithm finds the solution in less then 7 iterations, where each iteration requires about 10 matrix-vector multiplications. Received July 14, 1993 / Revised version received February 1994     

Extremum principles for optimal control in normed linear spaces

Extremum principles intended for use in optimal control are derived in the form of necessary conditions and sufficient conditions, formulated in general normed linear spaces. The method of application is illustrated by several examples involving optimal control problems, mathematical programming problems, lumped-parameter systems, and distributed-parameter systems. The basic theorems provide a unified approach which is applicable to a wide variety of problems in open-loop optimal control.     

年龄相关的非线性时变种群扩散系统最优分布控制的存在性

讨论了一类非线性时变种群扩散系统的最优分布控制问题,利用LionsJL的偏微控制理论和先验估计,证明了系统最优分布控制的存在性.所得结果可为非线性种群扩散系统中的最优控制问题的实际研究提供必要的理论基础.     

On the Hamming distance in combinatorial optimization problems on hypergraph matchings

In this note we consider the properties of the Hamming distance in combinatorial optimization problems on hypergraph matchings, also known as multidimensional assignment problems. It is shown that the Hamming distance between feasible solutions of hypergraph matching problems can be computed as an optimal value of linear assignment problem. For random hypergraph matching problems, an upper bound on the expected Hamming distance to the optimal solution is derived, and an exact expression is obtained in the special case of multidimensional assignment problems with 2 elements in each dimension.     

Maximum principle for the optimal control of a hyperbolic equation in one space dimension,part 1: Theory

A maximum principle is developed for a class of problems involving the optimal control of a damped-parameter system governed by a linear hyperbolic equation in one space dimension that is not necessarily separable. A convex index of performance is formulated, which consists of functionals of the state variable, its first- and second-order space derivatives, its first-order time derivative, and a penalty functional involving the open-loop control force. The solution of the optimal control problem is shown to be unique. The adjoint operator is determined, and a maximum principle relating the control function to the adjoint variable is stated. The proof of the maximum principle is given with the help of convexity arguments. The maximum principle can be used to compute the optimal control function and is particularly suitable for problems involving the active control of structural elements for vibration suppression.