An Approximation Scheme for Uncertain Minimax Optimal Control Problems

In this work, we address an uncertain minimax optimal control problem with linear dynamics where the objective functional is the expected value of the supremum of the running cost over a time interval. By taking an independently drawn random sample, the expected value function is approximated by the corresponding sample average function. We study the epi-convergence of the approximated objective functionals as well as the convergence of their global minimizers. Then we define an Euler discretization in time of the sample average problem and prove that the value of the discrete time problem converges to the value of the sample average approximation. In addition, we show that there exists a sequence of discrete problems such that the accumulation points of their minimizers are optimal solutions of the original problem. Finally, we propose a convergent descent method to solve the discrete time problem, and show some preliminary numerical results for two simple examples.     

Optimal Control of First-Order Hamilton–Jacobi Equations with Linearly Bounded Hamiltonian

We consider the optimal control of solutions of first order Hamilton–Jacobi equations, where the Hamiltonian is convex with linear growth. This models the problem of steering the propagation of a front by constructing an obstacle. We prove existence of minimizers to this optimization problem as in a relaxed setting and characterize the minimizers as weak solutions to a mean field game type system of coupled partial differential equations. Furthermore, we prove existence and partial uniqueness of weak solutions to the PDE system. An interpretation in terms of mean field games is also discussed.     

Global optimization of polynomial-expressed nonlinear optimal control problems with semidefinite programming relaxation

In this paper, we propose a new deterministic global optimization method for solving nonlinear optimal control problems in which the constraint conditions of differential equations and the performance index are expressed as polynomials of the state and control functions. The nonlinear optimal control problem is transformed into a relaxed optimal control problem with linear constraint conditions of differential equations, a linear performance index, and a matrix inequality condition with semidefinite programming relaxation. In the process of introducing the relaxed optimal control problem, we discuss the duality theory of optimal control problems, polynomial expression of the approximated value function, and sum-of-squares representation of a non-negative polynomial. By solving the relaxed optimal control problem, we can obtain the approximated global optimal solutions of the control and state functions based on the degree of relaxation. Finally, the proposed global optimization method is explained, and its efficacy is proved using an example of its application.     

High-order adaptive Gegenbauer integral spectral element method for solving non-linear optimal control problems

In this work, we propose an adaptive spectral element algorithm for solving non-linear optimal control problems. The method employs orthogonal collocation at the shifted Gegenbauer–Gauss points combined with very accurate and stable numerical quadratures to fully discretize the multiple-phase integral form of the optimal control problem. The proposed algorithm relies on exploiting the underlying smoothness properties of the solutions for computing approximate solutions efficiently. In particular, the method brackets discontinuities and ‘points of nonsmoothness’ through a novel local adaptive algorithm, which achieves a desired accuracy on the discrete dynamical system equations by adjusting both the mesh size and the degree of the approximating polynomials. A rigorous error analysis of the developed numerical quadratures is presented. Finally, the efficiency of the proposed method is demonstrated on three test examples from the open literature.     

A Consistent and Stable Approach to Generalized Sampling

We consider the problem of generalized sampling, in which one seeks to obtain reconstructions in arbitrary finite dimensional spaces from a finite number of samples taken with respect to an arbitrary orthonormal basis. Typical approaches to this problem consider solutions obtained via the consistent reconstruction technique or as solutions of an overcomplete linear systems. However, the consistent reconstruction technique is known to be non-convergent and ill-conditioned in important cases, such as the recovery of wavelet coefficients from Fourier samples, and whilst the latter approach presents solutions which are convergent and well-conditioned when the system is sufficiently overcomplete, the solution becomes inconsistent with the original measurements. In this paper, we consider generalized sampling via a non-linear minimization problem and prove that the minimizers present solutions which are convergent, stable and consistent with the original measurements. We also provide analysis in the case of recovering wavelets coefficients from Fourier samples. We show that for compactly supported wavelets of sufficient smoothness, there is a linear relationship between the number of wavelet coefficients which can be accurately recovered and the number of Fourier samples available.     

Structure of minimizers in linear-quadratic Bolza problems of optimal control

In this paper, we study intersections of extremals in a linear-quadratic Bolza problem of optimal control. The structure of the inter-sections is described. We show that this structure implies the semipositive definiteness of the quadratic cost functional. In addition, we derive necessary and sufficient conditions for the existence of minimizers.     

Calculus of variations in Hilbert spaces

The objective of this paper is to discuss existence, uniqueness and regularity issues of minimizers of one dimensional variational problems in Hilbert spaces. We obtain existence of C ² local minimizers and prove that the value function of an optimal control problem solves corresponding Hamilton-Jacobi equation in a viscosity sense.     

An adaptive parameter approach for an approximate solution of optimal control problems

In this paper, we consider a class of nonlinear optimal control problems (Bolza-problems) with constraints of the control vector, initial and boundary conditions of the state vectors. The time interval is fixed. Our approach to parametrize both the state functions and the control functions is described by general piecewise polynomials with unknown coefficients (parameters), where a fixed partition of the time interval is used. Here each of these functions in a suitable way individually will be approximated by such polynomials. The optimal control problem thus is reduced to a mathematical programming problem for these parameters. The existence of an optimal solution is assumed. Convergence properties of this method are not considered in this paper.     

On the link between infinite horizon control and quasi-stationary distributions

We study infinite horizon control of continuous-time non-linear branching processes with almost sure extinction for general (positive or negative) discount. Our main goal is to study the link between infinite horizon control of these processes and an optimization problem involving their quasi-stationary distributions and the corresponding extinction rates. More precisely, we obtain an equivalent of the value function when the discount parameter is close to the threshold where the value function becomes infinite, and we characterize the optimal Markov control in this limit. To achieve this, we present a new proof of the dynamic programming principle based upon a pseudo-Markov property for controlled jump processes. We also prove the convergence to a unique quasi-stationary distribution of non-linear branching processes controlled by a Markov control conditioned on non-extinction.     

基于高斯伪谱的最优控制求解及其应用

研究一种基于高斯伪谱法的具有约束受限的最优控制数值计算问题.方法将状态演化和控制规律用多项式参数化近似,微分方程用正交多项式近似.将最优控制问题求解问题转化为一组有约束的非线性规划求解.详细论述了该种近似方法的有效性.作为该种方法的应用,讨论了一个障碍物环境下的机器人最优路径生成问题.将机器人路径规划问题转化为具有约束条件最优控制问题,然后用基于高斯伪谱的方法求解,并给出了仿真结果.     

A discrete dynamic convexized method for nonlinear integer programming

In this paper, we consider the box constrained nonlinear integer programming problem. We present an auxiliary function, which has the same discrete global minimizers as the problem. The minimization of the function using a discrete local search method can escape successfully from previously converged discrete local minimizers by taking increasing values of a parameter. We propose an algorithm to find a global minimizer of the box constrained nonlinear integer programming problem. The algorithm minimizes the auxiliary function from random initial points. We prove that the algorithm can converge asymptotically with probability one. Numerical experiments on a set of test problems show that the algorithm is efficient and robust.     

Régularité des solutions d'un problème variationnel sous contrainte de convexité

We study the minimizers, in the class of convex functions, of an elliptic functional with nonhomogeneous Dirichlet boundary conditions. We prove C¹ regularity of the minimizers under the assumption that the upper envelope of admissible functions is C¹. This condition is optimal at least when the functional depends only on the gradient [4]. We then extend this result to a problem without boundary conditions arising in an economic model introduced by Rochet and Choné in [5].     

Regularity of solutions for some variational problems subject to a convexity constraint

We first study the minimizers, in the class of convex functions, of an elliptic functional with nonhomogeneous Dirichlet boundary conditions. We prove C¹ regularity of the minimizers under the assumption that the upper envelope of admissible functions is C¹. This condition is optimal at least when the functional depends only on the gradient [3]. We then give various extensions of this result. In Particular, we consider a problem without boundary conditions arising in an economic model introduced by Rochet and Choné in [4]. © 2001 John Wiley & Sons, Inc.     

$$L^1$$ penalization of volumetric dose objectives in optimal control of PDEs

This work is concerned with a class of PDE-constrained optimization problems that are motivated by an application in radiotherapy treatment planning. Here the primary design objective is to minimize the volume where a functional of the state violates a prescribed level, but prescribing these levels in the form of pointwise state constraints leads to infeasible problems. We therefore propose an alternative approach based on \(L^1\) penalization of the violation that is also applicable when state constraints are infeasible. We establish well-posedness of the corresponding optimal control problem, derive first-order optimality conditions, discuss convergence of minimizers as the penalty parameter tends to infinity, and present a semismooth Newton method for their efficient numerical solution. The performance of this method for a model problem is illustrated and contrasted with an alternative approach based on (regularized) state constraints.     

A Simple Accurate Method for Solving Fractional Variational and Optimal Control Problems

We develop a simple and accurate method to solve fractional variational and fractional optimal control problems with dependence on Caputo and Riemann–Liouville operators. Using known formulas for computing fractional derivatives of polynomials, we rewrite the fractional functional dynamical optimization problem as a classical static optimization problem. The method for classical optimal control problems is called Ritz’s method. Examples show that the proposed approach is more accurate than recent methods available in the literature.     

最大割问题和最大平分割问题基于半定规划松弛的近似算法

考虑每条边具有非负权重的无向图, 最大割问题要求将顶点集划分为两个集合使得它们之间的边的权重之和最大. 当最大割问题半定规划松弛的最优解落到二维空间时, Goemans将近似比从0.87856...改进为0.88456. 依赖于半定规划松弛的目标值与总权和的比值的曲线, 此曲线的最低点为0.88456, 当半定规划松弛的目标值与总权和的比值在0.5到0.9044之间时, 利用Gegenbauer多项式舍入技巧, 改进了Zwick的近似比曲线. 进一步, 考虑最大割问题的重要变形------最大平分割问题, 在此问题中增加了划分的两部分的点数相等的要求. 同样考虑了最大平分割问题半定规划松弛的最优解落到二维空间的情形, 并利用前述的Gegenbauer多项式舍入技巧得到0.7091-近似算法.     

Primal Dual Methods for Wasserstein Gradient Flows

Combining the classical theory of optimal transport with modern operator splitting techniques, we develop a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media, materials science, and biological swarming. Our method proceeds as follows: first, we discretize in time, either via the classical JKO scheme or via a novel Crank–Nicolson-type method we introduce. Next, we use the Benamou–Brenier dynamical characterization of the Wasserstein distance to reduce computing the solution of the discrete time equations to solving fully discrete minimization problems, with strictly convex objective functions and linear constraints. Third, we compute the minimizers by applying a recently introduced, provably convergent primal dual splitting scheme for three operators (Yan in J Sci Comput 1–20, 2018). By leveraging the PDEs’ underlying variational structure, our method overcomes stability issues present in previous numerical work built on explicit time discretizations, which suffer due to the equations’ strong nonlinearities and degeneracies. Our method is also naturally positivity and mass preserving and, in the case of the JKO scheme, energy decreasing. We prove that minimizers of the fully discrete problem converge to minimizers of the spatially continuous, discrete time problem as the spatial discretization is refined. We conclude with simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our approach, including higher-order convergence our novel Crank–Nicolson-type method, when compared to the classical JKO method.

     

A non-linear parabolic control problem with non-homogeneous boundary condition—convergence of Galerkin approximation

The paper is concerned with optimal control problem for a non-linear parabolic equation with non-homogenous boundary condition and quadratic cost. The control is acting in a nonlinear equation. We derive some results on the existence of optimal controls. Then we treat optimal control problem by Galerkin method and we prove the convergence of optimal values for approximated control problems to the one for the original problem. Finally, we apply the results to give a simple example. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

An Alternative Method for a Classical Problem in the Calculus of Variations

A numerical technique for determining the solution of the brachistochrone problem is presented. The brachistochrone problem is first formulated as a non-linear optimal control problem. Using Chebyshev nodes, we construct the Mth degree polynomial interpolation to approximate the state and the control variables. Application of this method results in the transformation of differential and integral expressions into some non-linear algebraic equations to which Newton-type methods can be applied. Simulation studies demonstrate computational advantages relative to existing methods in the literature.     

A Product Formula Approach to a Nonhomogeneous Boundary Optimal Control Problem Governed by Nonlinear Phase-field Transition System

In this paper we prove the convergence of an iterative scheme of fractional steps type for a non-homogeneous Cauchy-Neumann boundary optimal control problem governed by non-linear phase-field system, when the boundary control is dependent both on time and spatial variables. Moreover, necessary optimality conditions are established for the approximating process. The advantage of such approach leads to a numerical algorithm in order to approximate the original optimal control problem.