Stabilization by a linear feedback under phase constraints

We suggest a new method for solving the stabilization problem under phase constraints with the use of linear matrix inequalities. We consider the cases of complete and incomplete state measurements and the presence of nonstationary parametric perturbations. In the synthesis of linear control laws, the suggested method permits one to cover all possible quadratic Lyapunov functions and indicate a set of initial values for trajectories satisfying the phase constraints.     

The problem of conflicting control with mixed constraints

The control problem for a linear dynamical system is considered at a minimax of the terminal quality index. Feasible controls are simultaneously restricted by geometrical constraints and by integrated momentum constraints, the latter being thought of as a store of control resources. The problem is formalized as a differential game [1–4] using concepts [5–8] developed at Ekaterinburg. Here, because of the geometrical constraints, the momentum formulation and its associated difficulties [2–4] do not appear. On the other hand the presence of the integral restrictions leads to the appearance of additional variables whose evolution describes the dynamics of the expenditure of the control resources. These variables are subject to phase restrictions, which is a peculiarity of the problem. A reasonably informative picture and a class of strategies for which the given game has a value and a saddle point are given. A constructive method for computing the value function of the game and constructing optimal strategies is presented. This method is conceptually related to the construction of a stochastic programming synthesis [5] and is based on the recursive construction of upper-convex envelopes for certain auxiliary functions. The possibility of exchanging the minimum and maximum operations over the resource parameters when calculating the value of the game using these procedure is established.     

Solution of an optimal control problem with phase constraints by the double variation method

An optimal control problem with an integral quality index specified in a finite time interval is formulated for a model of economic growth that leads to emission of greenhouse gases. The controlled system is linear with respect to control. The problem contains phase constraints that abandon emission of greenhouse gases above some predefined time-dependent limit. As is known, optimal control problems with phase constraints fall beyond the sphere of efficient application of the Pontryagin maximum principle because, for such problems, this principle is formulated in a complicated form difficult for analytic treatment in particular situations. In this study, the analytic structure of the optimal control and phase trajectories is constructed using the double variation method.     

A bound for the number of different basic solutions generated by the simplex method

In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems (LP) having optimal solutions. The bound is polynomial of the number of constraints, the number of variables, and the ratio between the minimum and the maximum values of all the positive elements of primal basic feasible solutions. When the problem is primal nondegenerate, it becomes a bound for the number of iterations. The result includes strong polynomiality for Markov Decision Problem by Ye (http://www.stanford.edu/~yyye/simplexmdp1.pdf, 2010) and utilize its analysis. We also apply our result to an LP whose constraint matrix is totally unimodular and a constant vector b of constraints is integral.     

On the convergence of a Jacobi-type algorithm for singly linearly-constrained problems subject to simple bounds

In this work we define a block decomposition Jacobi-type method for nonlinear optimization problems with one linear constraint and bound constraints on the variables. We prove convergence of the method to stationary points of the problem under quite general assumptions.     

带补偿的随机二次规划的近似Lagrange-Newton方法

In this paper, two-stage stochastic quadratic programming problems with equality constraints are considered. By Monte Carlo simulation-based approximations of the objective function and its first (second)derivative,an inexact Lagrange-Newton type method is proposed.It is showed that this method is globally convergent with probability one. In particular, the convergence is local superlinear under an integral approximation error bound condition.Moreover, this method can be easily extended to solve stochastic quadratic programming problems with inequality constraints.     

On the asymptotic and practical stability of Persidskii-type systems with switching

In the paper, one class of differential systems with nonlinearities satisfying sector constraints is considered. We study the case where some of the sector constraints are given by linear inequalities, and some by nonlinear ones. It is assumed that the coefficients in the system can switch from one set of values to another. Sufficient conditions for the asymptotic and practical stability of the zero solution of the system are investigated using the direct Lyapunov method and the theory of differential inequalities. Restrictions on the switching law that provide a given region of attraction and ultimate bound for solutions of the system are obtained. An approach based on the construction of different differential inequalities for the Lyapunov function in different parts of the phase space is proposed, which makes it possible to improve the results obtained. The results are applied to the analysis of one automatic control system.     

An effective branch-and-bound algorithm for convex quadratic integer programming

We present a branch-and-bound algorithm for minimizing a convex quadratic objective function over integer variables subject to convex constraints. In a given node of the enumeration tree, corresponding to the fixing of a subset of the variables, a lower bound is given by the continuous minimum of the restricted objective function. We improve this bound by exploiting the integrality of the variables using suitably-defined lattice-free ellipsoids. Experiments show that our approach is very fast on both unconstrained problems and problems with box constraints. The main reason is that all expensive calculations can be done in a preprocessing phase, while a single node in the enumeration tree can be processed in linear time in the problem dimension.     

Approximate construction of attainability sets of control systems with integral constraints on the controls

The approximate construction of attainability sets of control systems with quadratic integral constraints on the controls is considered. It is assumed that a control system is non-linear with respect to the phase variable and linear with respect to the variable which describes the controlling action. The approximation of the attainability sets of a control system is accomplished in several stages. The latter class of controls generates a finite number of trajectories of the system. The trajectories of the system are then replaced by Euler broken lines. An estimate of the accuracy of the Hausdorff distance between the attainability set and the set which has been approximately constructed is obtained.     

On the optimal control of linear systems with linear performance criterion and constraints

We suggest an analytical-numerical method for solving a boundary value optimal control problem with state, integral, and control constraints. The embedding principle underlying the method is based on the general solution of a Fredholm integral equation of the first kind and its analytic representation; the method permits one to reduce the boundary value optimal control problem with constraints to an optimization problem with free right end of the trajectory.     

Feedback control in LQCP with a terminal inequality constraint

This paper considers the linear-quadratic control problem (LQCP) for systems defined by evolution operators with a terminal state inequality constraint. It is shown that, under suitable assumptions, the optimal control exists, is unique, and has a closed-loop structure. The synthesis of the feedback control requires one to solve the integral Riccati equation for the unconstrainted LQCP and a linear integral equation whose solution depends on a real parameter satisfying an additional condition.This work was completed while the author was visiting the Control Theory Centre, University of Warwick, Coventry, England.     

A finite steps algorithm for solving convex feasibility problems

This paper develops a new variant of the classical alternating projection method for solving convex feasibility problems where the constraints are given by the intersection of two convex cones in a Hilbert space. An extension to the feasibility problem for the intersection of two convex sets is presented as well. It is shown that one can solve such problems in a finite number of steps and an explicit upper bound for the required number of steps is obtained. As an application, we propose a new finite steps algorithm for linear programming with linear matrix inequality constraints. This solution is computed by solving a sequence of a matrix eigenvalue decompositions. Moreover, the proposed procedure takes advantage of the structure of the problem. In particular, it is well adapted for problems with several small size constraints.     

Primal-dual interior-point method for thermodynamic gas-particle partitioning

A mathematical model for the computation of the phase equilibrium and gas-particle partitioning in atmospheric organic aerosols is presented. The thermodynamic equilibrium is determined by the global minimum of the Gibbs free energy under equality and inequality constraints for a system that involves one gas phase and many liquid phases. A primal-dual interior-point algorithm is presented for the efficient solution of the phase equilibrium problem and the determination of the active constraints. The first order optimality conditions are solved with a Newton iteration. Sequential quadratic programming techniques are incorporated to decouple the different scales of the problem. Decomposition methods that control the inertia of the matrices arising in the resolution of the Newton system are proposed. A least-squares initialization of the algorithm is proposed to favor the convergence to a global minimum of the Gibbs free energy. Numerical results show the efficiency of the approach for the prediction of gas-liquid-liquid equilibrium for atmospheric organic aerosol particles.     

Linear differential pursuit games with integral constraints in the presence of delay

Problems involving linear differential pursuit games were studied by many authors; their work served as a basis for studying pursuit problems in linear differential games with integral constraints. In the present paper, we obtain sufficient conditions for the solvability of linear pursuit problems with integral constraints on the control of the players in the presence of delay.     

最小能量控制与Lg样条函数

本文考察了一类带有泛函约束的线性系统的最小能量控制问题与Lg样条插值函数的对应关系.得到了求解最优控制——样条函数的递推算法;并建立了最优控制问题解法Ⅱ,指出文献[1]的解法可作为它的特例.     

Existence theorems for multidimensional Lagrange problems

Existence theorems are proved for multidimensional Lagrange problems of the calculus of variations and optimal control. The unknowns are functions of several independent variables in a fixed bounded domain, the cost functional is a multiple integral, and the side conditions are partial differential equations, not necessarily linear, with assigned boundary conditions. Also, unilateral constraints may be prescribed both on the space and the control variables. These constraints are expressed by requiring that space and control variables take their values in certain fixed or variable sets wich are assumed to be closed but not necessarily compact.This research was partially supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-942-65.     

Optimal Control Problems with Integral Functional and Phase Constraints: Reduction to Optimal Consistency Parameter Problems

An iteration method to solve a class of optimal control problems with integral functional and phase constraints is developed in the paper.     

Thevenin decomposition and large-scale optimization

The Thevenin theorem, one of the most celebrated results of electric circuit theory, provides a two-parameter characterization of the behavior of an arbitrarily large circuit, as seen from two of its terminals. We interpret the theorem as a sensitivity result in an associated minimum energy/network flow problem, and we abstract its main idea to develop a decomposition method for convex quadratic programming problems with linear equality constraints, of the type arising in a variety of contexts such as the Newton method, interior point methods, and least squares estimation. Like the Thevenin theorem, our method is particularly useful in problems involving a system consisting of several subssystems, connected to each other with a small number of coupling variables.This research was supported by NSF under Grant CCR-91-03804.     

The pivot and probe algorithm for solving a linear program

In [7] we defined acandidate constraint as one which, for at least one pivot step, contains a potential pivot, discovered that most constraints are never candidate, and devised a modification of the simplex method in which only constraints which are currently candidates are updated. In this paper we present another way to take advantage of this fact. We begin by solving a relaxed linear program consisting of the constraints of the original problem which are initially candidates. Its solution gives an upper bound to the value of the original problem. We also introduce the idea of a probe, that is, a line segment joining two vectors for the primal problem, one of which is primal feasible, and use it to identify a most violated constraint; at the same time this gives a lower bound to the objective value of the original problem. This violated constraint is added to the relaxed problem which is solved again, which gives a new upper bound etc. We present computational experience indicating that time savings of 50–80% over the simplex method can be obtained by this method, which we call PAPA, the Pivot and Probe Algorithm. This report was prepared as part of the activities of the Management Science Research Group, Carnegie-Mellon University, under Contract No. N00014-75-C-0621 NR 047-048 with the U.S. Office of Naval Research. Reproduction in whole or part is permitted for any purpose of the U.S. Government.     

Synthesis of minimal linear stabilizers

We consider the stabilizer synthesis problem for linear stationary control dynamical systems; attention is mainly focused on the investigation of possibilities to reduce the order of such stabilizers. The problem is considered in two settings. The first setting deals with the construction of stabilizers of given order (in particular, the minimum possible order); no conditions are imposed on the spectrum of the closed system. For scalar (single-input-single-output) control systems, we suggest an approach to the solution of this problem and obtain necessary and sufficient conditions for the existence of a stabilizer of a given order. The second problem is that of synthesizing a stabilizer of minimum order with given dynamic properties (a given spectrum or a given distribution of the spectrum of the closed system, in particular, with a guaranteed convergence rate of the closed system). For this problem, we suggest two approaches that permit one to obtain an upper bound for the dimension of such stabilizers.