Probabilistic Solutions to Optimal Control Problems

Let ( x 1 ( t ), x 2 ( t )) be a controlled two-dimensional diffusion process. The problem of minimizing, or maximizing, the time spent by ( x 1 ( t ), x 2 ( t )) in a given subset of 2 is solved, in two particular instances, by transforming the optimal control problems into purely probabilistic problems. In Section 2 , ( x 1 ( t ), x 2 ( t )) is a two-dimensional Wiener process and the optimal control is obtained by transforming a nonlinear dynamic programming equation into the Kolmogorov backward equation for a two-dimensional geometric Brownian motion. In Section 3 , the converse problem is solved. The problem of finding the maximal instantaneous reward that we can give for survival in the continuation region is also treated.     

Existence and Structure of Optimal Solutions of Infinite-Dimensional Control Problems

In this work we analyze the structure of optimal solutions for a class of infinite-dimensional control systems. We are concerned with the existence of an overtaking optimal trajectory over an infinite horizon. The existence result that we obtain extends the result of Carlson, Haurie, and Jabrane to a situation where the trajectories are not necessarily bounded. Also, we show that an optimal trajectory defined on an interval [0,τ] is contained in a small neighborhood of the optimal steady-state in the weak topology for all t ∈ [0,τ] \backslash E , where E \subset [0,τ] is a measurable set such that the Lebesgue measure of E does not exceed a constant which depends only on the neighborhood of the optimal steady-state and does not depend on τ . Accepted 26 July 2000. Online publication 13 November 2000.     

Solutions to Constrained Optimal Control Problems with Linear Systems

This paper is devoted to present solutions to constrained finite-horizon optimal control problems with linear systems, and the cost functional of the problem is in a general form. According to the Pontryagin’s maximum principle, the extremal control of such problem is a function of the costate trajectory, but an implicit function. We here develop the canonical backward differential flows method and then give the extremal control explicitly with the costate trajectory by canonical backward differential flows. Moreover, there exists an optimal control if and only if there exists a unique extremal control. We give the proof of the existence of the optimal solution for this optimal control problem with Green functions.     

随机线性二次最优控制:从离散到连续时间模型

在一般情形下,分析了离散时间LQ问题与连续时间情形两者之间的自然联系.首先回顾了连续时间和离散时间随机LQ问题及对应Riccati微分/差分方程的相关结论.接下来在假设Riccati微分方程有解的前提下,证明了离散化步长足够小时,Riccati差分方程有解.然后针对连续和离散时间模型,采用配对问题最优控制的反馈形式,分别构造了一个辅助反馈控制,并证明该控制可驱使对应模型的性能指标逼近于配对问题的值函数,以此得到了关于两个模型之间联系的初步结论.最后藉由前述结论以及控制问题的特性,揭晓了连续时间和离散时间模型之间的自然联系,并给出了Riccati差分方程和微分方程的解之间的误差估计.由此联系,可构造相应离散系统和LQ问题,以适当的阶估计连续时间LQ问题的解,抑或为离散时间模型构造一个近似最优控制.无论哪种思路,都旨在降低直接求解原问题的难度和复杂性.     

Discrete Approximations to Continuum Optimal Flow Problems

Problems in partial differential equations with inequality constraints can be used to describe a continuum analog to various optimal flow/cut problems. While general concepts from convex optimization (like duality) carry over into continuum problems, the application of ideas and algorithms from linear programming and network flow problems is challenging. The capacity constraints are nonlinear (but convex).
In this article, we investigate a discretized version of the planar maximum flow problem that preserves the nonlinear capacity constraints of the continuum problem. The resulting finite-dimensional problem can be cast as a second-order cone programming problem or a quadratically constrained program. Good numerical results can be obtained using commercial solvers. These results are in agreement with the continuum theory of a "challenge" problem posed by Strang.     

2-Regularity and 2-Normality Conditions in Discrete Optimal Control Problems

The paper is devoted to general optimal control problems for discrete-time systems with equality type of constraints on end points and control. We derive first-order necessary optimality conditions that are meaningful under the new nontriviality condition.     

一类非线性奇异最优控制问题的离散解法

本文研究非线性奇异最优控制问题的离散解法。利用在每个“小”时间区间上的积分形式来刻画奇异最优控制的特征,并构造了求解问题的差分方程,同时建立预估、校正格式给出了最优控制近似解的计算方法。     

Coupled Intervals in the Discrete Optimal Control

In this paper, we investigate the nonnegativity and positivity of a quadratic functional ? with variable (i.e. separable and jointly varying) endpoints in the discrete optimal control setting. We introduce a coupled interval notion, which generalizes (i) the conjugate interval notion known for the fixed right endpoint case and (ii) the coupled interval notion known in the discrete calculus of variations. We prove necessary and sufficient conditions for the nonnegativity and positivity of ? in terms of the nonexistence of such coupled intervals. Furthermore, we characterize the nonnegativity of ? in terms of the (previously known notions of) conjugate intervals, a conjoined basis of the associated linear Hamiltonian system, and the solvability of an implicit Riccati equation. This completes the results for the nonnegativity that are parallel to the known ones on the positivity of ?. Finally, we define partial quadratic functionals associated with ? and a (strong) regularity of ?, which we relate to the positivity and nonnegativity of ?.     

On the Existence of Optimal Solutions to an Optimal Control Problem

In this paper, some results concerning the existence of optimal solutions to an optimal control problem are derived. The problem involves a quasilinear hyperbolic differential equation with boundary condition and a nonlinear integral functional of action. The assumption of convexity, under which the main theorem is proved, is not connected directly with the convexity of the functional of action. In the proof, the implicit function theorem for multimappings is used.Communicated by L. D. Berkovitz     

Exploiting the Special Structure for Real‐Time Control of Optimal Control Problems with Linear Controls: Numerical Experience

Often optimal control problems possess control variables appearing linearly in the dynamics, the objective function and the constraints. The special bang‐bang and singular structure of the optimal control is exploited to formulate a nonlinear programming problem (NLP)w ith variables in the switching points and singular subarcs of the controls. This method has several advantages: The dimension of the resulting NLP problem is considerably reduced compared to usual direct optimization methods, several constraints can be neglected, and a parametric sensitivity analysis and real‐time control with respect to the switching points can be performed.     

Relaxation of Optimal Control Problems Involving Time Dependent Subdifferential Operators

This article provides in the setting of infinite dimensional Hilbert space, a result concerning the existence and uniqueness of solutions for Lipschitz single-valued perturbations of evolution problems associated with time-dependent subdifferential operators. The result is used to extend to optimal control problems associated with such equations the relaxation theorems with Young measures established in Casting et al. [7 C. Castaing , P. Raynaud de Fitte , and M. Valadier , M. ( 2004 ). Young Measures on Topological Spaces with Applications in Control Theory and Probability Theory . Kluwer Academic Publishers , Dordrecht . [Google Scholar], 11 J. F. Edmond and L. Thibault ( 2005 ). Relaxation of an optimal control problem involving a perturbed sweeping process . Math. Program Ser. B 104 : 347 – 373 .[Crossref], [Web of Science ®] , [Google Scholar]] and Edmond and Thibault [11 J. F. Edmond and L. Thibault ( 2005 ). Relaxation of an optimal control problem involving a perturbed sweeping process . Math. Program Ser. B 104 : 347 – 373 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Time Optimal Control for Some Ordinary Differential Equations with Multiple Solutions

This paper studies a time optimal control problem for a class of ordinary differential equations. The control systems may have multiple solutions. Based on the properties fulfilled by the solutions of the concerned equations, we get both the existence and the Pontryagin maximum principle for optimal controls.     

Exact Solutions of Some Discrete Deconvolution Problems

In this work, we study the existence of solutions of the deconvolution problems in the discrete setting. More precisely, we prove the existence of solutions of the discrete multichannel deconvolution problems DMDP with convolvers being the characteristic functions of finite sets of positive integers. Also, we provide the reader with a simple method and a fast algorithm for finding the closed forms of the discrete deconvolvers with minimal supports that constitute exact solutions of the DMDP. Moreover, we show that unlike the singular value decomposition scheme, the multichannel deconvolution scheme based on the use of these discrete deconvolvers is not very sensitive to small 2-norm perturbation of the data. Finally, we show how to generalize our method for solving the 2-D version of the DMDP.     

A Characterization of Approximate Solutions of Multiobjective Stochastic Optimal Control Problems

We introduce several concepts of approximate solutions of multiobjective optimization problems, prove existence results and an k -minimum principle for multiobjective stochastic optimal control problems.     

Numerical Solutions to the Bellman Equation of Optimal Control

In this paper, we present a numerical algorithm to compute high-order approximate solutions to Bellman’s dynamic programming equation that arises in the optimal stabilization of discrete-time nonlinear control systems. The method uses a patchy technique to build local Taylor polynomial approximations defined on small domains, which are then patched together to create a piecewise smooth approximation. The numerical domain is dynamically computed as the level sets of the value function are propagated in reverse time under the closed-loop dynamics. The patch domains are constructed such that their radial boundaries are contained in the level sets of the value function and their lateral boundaries are constructed as invariant sets of the closed-loop dynamics. To minimize the computational effort, an adaptive subdivision algorithm is used to determine the number of patches on each level set depending on the relative error in the dynamic programming equation. Numerical tests in 2D and 3D are given to illustrate the accuracy of the method.     

Periodic Solutions of a Discrete Time Predator-Prey System

In this paper, we discuss a discrete predator-prey system with a non-monotonic functional response, which models the dynamics of the prey and the predator having non-overlapping generations. By using the coincidence degree theory, sufficient conditions are obtained for the existence of positive periodic solutions.     

Discontinuous Solutions to Unbounded Differential Inclusions under State Constraints. Applications to Optimal Control Problems

We consider a nonconvex and unbounded differential inclusion derived from a control system whose control sets are time and space-dependent. We extend the inclusion in order to allow discontinuous trajectories. We prove that the set of solutions of the original inclusion is dense in the set of solutions of the extended inclusion and, moreover, these last solutions are stable with respect to the initial data. Both of these results are also proven in the presence of state and integral constraints (assuming suitable conditions at the boundary of the constraining set). As an application, the value function of a Mayer problem is shown to be continuous and the unique viscosity solution of a Hamilton–Jacobi equation with suitable boundary conditions.     

Convergence in Distribution of Optimal Solutions to Stochastic Optimization Problems

1.IntroductionDistributionproblemsareofgreatimportanceinstochasticoptimizationandstatis-tics.Usuallythiskindofproblemscanbedescribedinthefollowingform:wheref(x,w)isafuncti0ndefinedonR"xflandSisasubsetin'R".Because0fc0mplexityoftheproblems,ingeneral,onecangetonlytheirapproximatesolutions.Thefollowingtypeofapproximationis0ftenused:Letuscall(2)thefirsttype0fapproximation.DenotebyZ(w),A(w)theoptima1valueandoptimalsolutionsetofproblem(1)respectivelyandbyZk(w),Ak(w)thecorrespondingonesofproblem(…     

Optimal Solutions of a Discrete Model of Financial Mathematics

A discrete model of the work of a financial enterprise is analyzed. The problem of optimization consists of finding the conditions that ensure the maximal profit.