Analysis of microstructures and phase transition phenomena in one-dimensional, non-linear elasticity by convex optimization

We propose a general method for determining the theoretical microstructure in one-dimensional elastic bars whose internal deformation energy is given by nonconvex polynomials. We use nonconvex variational principles and Young measure theory to describe the optimal energetic configuration of the body. By using convex analysis and classical characterizations of algebraic moments, we can formulate the problem as a convex optimal control problem. Therefore, we can estimate the microstructure of several models by using nonlinear programming techniques. This method can determine the minimizers or the minimizing sequences of nonconvex, variational problems used in one-dimensional, nonlinear elasticity.     

A novel neural network for nonlinear convex programming

In this paper, we present a neural network for solving the nonlinear convex programming problem in real time by means of the projection method. The main idea is to convert the convex programming problem into a variational inequality problem. Then a dynamical system and a convex energy function are constructed for resulting variational inequality problem. It is shown that the proposed neural network is stable in the sense of Lyapunov and can converge to an exact optimal solution of the original problem. Compared with the existing neural networks for solving the nonlinear convex programming problem, the proposed neural network has no Lipschitz condition, no adjustable parameter, and its structure is simple. The validity and transient behavior of the proposed neural network are demonstrated by some simulation results.     

A variational inequality for measurement feedback almost-dissipative control

In this paper, an optimal measurement feedback control problem that yields an almost- (or practically) dissipative closed loop system is considered. That is, the aim is to consider optimal control problems which, when solved, yield a closed loop system which almost satisfies a dissipation inequality. The main idea is that by weakening the required dissipation inequality, a broader class of open loop systems and controllers are admissible, leading to broader application. In obtaining the main results of this paper, dynamic programming is applied to the optimal control problem of interest to derive a variational inequality that generalizes the information state based partial differential equation associated with measurement feedback nonlinear dissipative control. This variational inequality can in principle be used to derive the optimal controller. In the special case of certainty equivalence, an explicit solution of the variational inequality exists and is a functional of the solution of the corresponding optimal state feedback almost-dissipative control problem.     

Control applications of nonlinear convex programming

Since 1984 there has been a concentrated effort to develop efficient interior-point methods for linear programming (LP). In the last few years researchers have begun to appreciate a very important property of these interior-point methods (beyond their efficiency for LP): they extend gracefully to nonlinear convex optimization problems. New interior-point algorithms for problem classes such as semidefinite programming (SDP) or second-order cone programming (SOCP) are now approaching the extreme efficiency of modern linear programming codes. In this paper we discuss three examples of areas of control where our ability to efficiently solve nonlinear convex optimization problems opens up new applications. In the first example we show how SOCP can be used to solve robust open-loop optimal control problems. In the second example, we show how SOCP can be used to simultaneously design the set-point and feedback gains for a controller, and compare this method with the more standard approach. Our final application concerns analysis and synthesis via linear matrix inequalities and SDP.     

Linear‐Quadratic Optimal Control Problem for Partially Observed Forward‐Backward Stochastic Differential Equations of Mean‐Field Type

This paper is concerned with the linear‐quadratic optimal control problem for partially observed forward‐backward stochastic differential equations (FBSDEs) of mean‐field type. Based on the classical spike variational method, backward separation approach as well as filtering technique, we first derive the necessary and sufficient conditions of the optimal control problem with the non‐convex domain. Nextly, by means of the decoupling technique, we obtain two Riccati equations, which are uniquely solvable under certain conditions. Also, the optimal cost functional is represented by the solutions of the Riccati equations for the special case.     

Optimal Control of Forward‐Backward Stochastic Jump‐Diffusion Differential Systems with Observation Noises: Stochastic Maximum Principle

This paper is concerned with a partially observed optimal control problem for a controlled forward‐backward stochastic system with correlated noises between the system and the observation, which generalizes the result of a previous work to a jump‐diffusion system. Under some convexity assumptions, necessary and sufficient optimality conditions for such an optimal control are established in the form of Pontryagin type maximum principle in a unified way by means of duality analysis and convex variational techniques     

A scalable FETI-DP algorithm for a semi-coercive variational inequality

We develop an optimal algorithm for the numerical solution of semi-coercive variational inequalities by combining dual-primal FETI algorithms with recent results for bound and equality constrained quadratic programming problems. The discretized version of the model problem, obtained by using the FETI-DP methodology, is reduced by the duality theory of convex optimization to a quadratic programming problem with bound and equality constraints, which is solved by a new algorithm with a known rate of convergence given in terms of the spectral condition number of the quadratic problem. We present convergence bounds that guarantee the scalability of the algorithm. These results are confirmed by numerical experiments.     

Linear convex stochastic control problems over an infinite horizon

A stochastic control problem over an infinite horizon which involves a linear system and a convex cost functional is analyzed. We prove the convergence of the dynamic programming algorithm associated with the problem, and we show the existence of a stationary Borel measurable optimal control law. The approach used illustrates how results on infinite time reachability [1] can be used for the analysis of dynamic programming algorithms over an infinite horizon subject to state constraints.     

Controller design with multiple objectives

In this paper we study multi-objective control problems that give rise to equivalent convex optimization problems. We develop a uniform treatment of such problems by showing their equivalence to linear programming problems with equality constraints and an appropriate positive cone. We present some specialized results on duality theory, and we apply them to the study of three multi-objective control problems: the optimal l₁ control with time-domain constraints on the response to some fixed input, the mixed H₂/l₁ -control problem, and the l₁ control with magnitude constraint on the frequency response. What makes these problems complicated is that they are often equivalent to infinite-dimensional optimization problems. The characterization of the duality relationship between the primal and dual problem allows us to derive several results. These results establish connections with special convex problems (linear programming or linear matrix inequality problems), uncover finite-dimensional structures in the optimal solution, when possible, and provide finite-dimensional approximations to any degree of accuracy when the problem does not appear to have a finite-dimensional structure. To illustrate the theory and highlight its potential, several numerical examples are presented     

Global exponential stability of recurrent neural networks forsolving optimization and related problems

Global exponential stability is a desirable property for dynamic systems. The paper studies the global exponential stability of several existing recurrent neural networks for solving linear programming problems, convex programming problems with interval constraints, convex programming problems with nonlinear constraints, and monotone variational inequalities. In contrast to the existing results on global exponential stability, the present results do not require additional conditions on the weight matrices of recurrent neural networks and improve some existing conditions for global exponential stability. Therefore, the stability results in the paper further demonstrate the superior convergence properties of the existing neural networks for optimization.     

Control Liapunov function design of neural networks that solve convex optimization and variational inequality problems

This paper presents two neural networks to find the optimal point in convex optimization problems and variational inequality problems, respectively. The domain of the functions that define the problems is a convex set, which is determined by convex inequality constraints and affine equality constraints. The neural networks are based on gradient descent and exact penalization and the convergence analysis is based on a control Liapunov function analysis, since the dynamical system corresponding to each neural network may be viewed as a so-called variable structure closed loop control system.     

An algorithm for a solution of a stochastic adaptive linear quadratic optimal control problem

A variational approach is taken to derive optimality conditions for a discrete-time linear quadratic adaptive stochastic optimal control problem. These conditions lead to an algorithm for computing optimal control laws which differs from the dynamic programming algorithm.     

A new scheme to learn a kernel in regularization networks

In this paper, we propose a new scheme to learn a kernel function from the convex combination of finite given kernels in regularization networks. We show that the corresponding variational problem is convex and under certain conditions, the variational problem can be approximated by a semidefinite programming problem which coincides with the Micchelli and Pontil's (MP's) Model (Micchelli and Pontil, 2005 [10]).     

Improvement of Discrete Control Processes in Problems with Mixed Constraints

The discrete problem of optimal control with mixed constraints was considered from the standpoint of numerical solution. Computational methods are usually based on solving a series of improvement problems. The nonlinear discrete problem and that of nonlinear functional improvement for the variational system were compared. A procedure for constructing an admissible control process improving the original problem from the solution of the reduced problem was described.     

A characterization of convex problems in decentralized control

We consider the problem of constructing optimal decentralized controllers. We formulate this problem as one of minimizing the closed-loop norm of a feedback system subject to constraints on the controller structure. We define the notion of quadratic invariance of a constraint set with respect to a system, and show that if the constraint set has this property, then the constrained minimum-norm problem may be solved via convex programming. We also show that quadratic invariance is necessary and sufficient for the constraint set to be preserved under feedback. These results are developed in a very general framework, and are shown to hold in both continuous and discrete time, for both stable and unstable systems, and for any norm. This notion unifies many previous results identifying specific tractable decentralized control problems, and delineates the largest known class of convex problems in decentralized control. As an example, we show that optimal stabilizing controllers may be efficiently computed in the case where distributed controllers can communicate faster than their dynamics propagate. We also show that symmetric synthesis is included in this classification, and provide a test for sparsity constraints to be quadratically invariant, and thus amenable to convex synthesis.     

A characterization of convex problems in decentralized Control/sup */

We consider the problem of constructing optimal decentralized controllers. We formulate this problem as one of minimizing the closed-loop norm of a feedback system subject to constraints on the controller structure. We define the notion of quadratic invariance of a constraint set with respect to a system, and show that if the constraint set has this property, then the constrained minimum-norm problem may be solved via convex programming. We also show that quadratic invariance is necessary and sufficient for the constraint set to be preserved under feedback. These results are developed in a very general framework, and are shown to hold in both continuous and discrete time, for both stable and unstable systems, and for any norm. This notion unifies many previous results identifying specific tractable decentralized control problems, and delineates the largest known class of convex problems in decentralized control. As an example, we show that optimal stabilizing controllers may be efficiently computed in the case where distributed controllers can communicate faster than their dynamics propagate. We also show that symmetric synthesis is included in this classification, and provide a test for sparsity constraints to be quadratically invariant, and thus amenable to convex synthesis.     

桥式吊车系统的伪谱最优控制设计

对于桥式吊车系统的最优控制问题,根据实际的工况要求,性能指标有时不一定是标准的二次形式.同时,在实际的控制问题中,状态和控制输入往往会受到一些边界条件和路径过程中的约束.针对这一问题,本文应用Chebyshev伪谱优化算法来处理,它可以处理状态和控制约束的非线性最优化问题以及一个非标准的目标函数.首先对桥式吊车系统模型进行一系列的坐标变换,将其转变为上三角系统形式的误差模型.然后将桥式吊车最优控制问题转化成具有一系列代数约束的参数优化问题,即非线性规划问题.通过求解离散化后的参数优化问题,得到桥式吊车的最优控制律.本文还给出了Chebyshev伪谱最优解的可行性和一致性分析.最后,在仿真研究中验证该控制器的有效性.     

Strong observability as a sufficient condition for non-singularity and lossless convexification in optimal control with mixed constraints

This paper analyzes optimal control problems with linear time-varying dynamics defined on a smooth manifold in addition to mixed constraints and pure control constraints. The main contribution is the identification of sufficient conditions for the optimal controls to be non-singular, which enables exact (or lossless) convex relaxations of the control constraints. The problem is analyzed in a geometric framework using a recent maximum principle on manifolds, and it is shown that strong observability of the dual system on the cotangent space is the key condition. Two minimum time problems are analyzed and solved. A minimum fuel planetary descent problem is then analyzed and relaxed to a convex form. Convexity enables its efficient solution in less than one second without any initial guess.     

一个风险敏感最优控制问题的随机最大值原理及在投资选择中的应用

在本文, 我们主要研究了一类产生于金融市场中投资选择问题的风险敏感最优控制问题. 用经典的凸变分技术, 我们得到了该类问题的最大值原理. 最大值原理的形式相似于风险中性的情形. 但是, 对偶方程和变分不等式明显地依赖于风险敏感参数 γ. 这是与风险中性情形的主要区别之一. 我们用该结果解决一类最优投资选择问题. 在投资者仅投资国内债券和股票的情况下, 前人用贝尔曼动态规划原理所得的最优投资策略仅是我们结果的特殊形式. 我们也给了一些数值算例和图, 他们显式地解释了最大期望效用和模型中参数的关系.     

Optimal codesign of industrial networked control systems with state‐dependent correlated fading channels

This paper examines a codesign problem in industrial networked control systems (NCS) whereby physical systems are controlled over wireless fading channels. The considered wireless channels are assumed to be stochastically dependent on the physical states of moving machineries in the industrial working space. In this paper, the moving machineries are modeled as Markov decision processes whereas the characteristics of the correlated fading channels are modeled as a binary random process whose probability measure depends on both the physical states of moving machineries and the transmission power of communication channels. Under such a state‐dependent fading channel model, sufficient conditions to ensure the stochastic safety of the NCS are first derived. Using the derived safety conditions, a codesign problem is then formulated as a constrained joint optimization problem that seeks for optimal control and transmission power policies which simultaneously minimize an infinite time cost on both communication resource and control effort. This paper shows that such optimal policies can be obtained in a computationally efficient manner using convex programming methods. Simulation results of an autonomous forklift truck and a networked DC motor system are presented to illustrate the advantage and efficacy of the proposed codesign framework for industrial NCS.