本文提出了高精度数值方法，用于求解受具有集值阶跃函数的非光滑微分方程约束的最优控制问题。这类系统中的一个重要子类是Filippov系统。本文将集值阶跃函数表示为线性规划的解映射。利用该问题的最优性条件，我们将初始的非光滑系统改写为等价的动态互补系统（DCS）。我们将最初为通过Stewart重构转化为DCS的Filippov系统而开发的带切换检测有限元（FESD）方法[Nurkanović et al., 2024]扩展到具有集值阶跃函数的非光滑系统类别。关键思想是：首先对得到的DCS采用标准Runge-Kutta方法，并让积分步长成为自由度。接下来，我们引入附加条件，以实现隐式但精确的切换检测，并在没有切换发生时消除可能的虚假自由度。本文研究了该方法的理论性质，并通过数值仿真和最优控制算例展示了其良好的性能。本文提出的所有方法均已在开源软件包NOSNOC中实现。
英文摘要：
This paper develops high-accuracy methods for numerically solving optimal control problems subject to nonsmooth differential equations with set-valued step functions. A notable subclass of these systems are Filippov systems. The set-valued step functions are here written as the solution map of a linear program. Using the optimality conditions of this problem we rewrite the initial nonsmooth system into a equivalent dynamic complementarity systems (DCS). We extend the Finite Elements with Switch Detection (FESD) method [Nurkanović et al., 2024], initially developed for Filippov systems transformed via Stewart's reformulation into DCS [Stewart, 1990], to the class of nonsmooth systems with set-valued step functions. The key ideas are to start with a standard Runge-Kutta method for the obtained DCS and to let the integration step sizes to be degrees of freedom. Next, we introduce additional conditions to enable implicit but exact switch detection and to remove possible spurious degrees of freedom if no switches occur. The theoretical properties of the method are studied. Its favorable properties are illustrated on numerical simulation and optimal control examples. All methods introduced in this paper are implemented in the open-source software package NOSNOC.
This paper is concerned with optimal control problems for parabolic partial differential equations with pointwise in time switching constraints on the control. A standard approach to treat constraints in nonlinear optimization is penalization, in particular using $L^1$-type norms. Applying this approach to the switching constraint leads to a nonsmooth and nonconvex infinite-dimensional minimization problem which is challenging both analytically and numerically. Adding $H^1$ regularization or restricting to a finite-dimensional control space allows showing existence of optimal controls. First-o…
This paper studies switching stabilization problems for general switched nonlinear systems. A piecewise smooth control-Lyapunov function (PSCLF) approach is proposed and a constructive way to design a stabilizing switching law is developed. The switching law is constructed via the directional derivatives of the PSCLF with a careful discussion on various technical issues that may occur on the nonsmooth surfaces. Sufficient conditions are derived to ensure stability of the closed-loop Filippov solutions including possible sliding motions. The proposed PSCLF approach contains many existing result…
This letter introduces the NOnSmooth Numerical Optimal Control (NOSNOC) open-source software package. It is a modular MATLAB tool based on CasADi and IPOPT for numerically solving Optimal Control Problems (OCP) with piecewise smooth systems (PSS). The tool supports: 1) automatic reformulation of systems with state jumps into PSS (via the time-freezing reformulation [Nurkanović et al., 2021]) and of PSS into computationally more convenient forms, 2) automatic discretization of the OCP via, e.g., the recently introduced Finite Elements with Switch Detection [Nurkanović et al., 2022] which enable…
In this paper, we establish that for a wide class of controlled stochastic differential equations (SDEs) with stiff coefficients, the value functions of corresponding zero-sum games can be represented by a deep artificial neural network (DNN), whose complexity grows at most polynomially in both the dimension of the state equation and the reciprocal of the required accuracy. Such nonlinear stiff systems may arise, for example, from Galerkin approximations of controlled stochastic partial differential equations (SPDEs), or controlled PDEs with uncertain initial conditions and source terms. This …
Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院）
We analyze stochastic gradient algorithms for optimizing nonconvex, nonsmooth finite-sum problems. In particular, the objective function is given by the summation of a differentiable (possibly nonconvex) component, together with a possibly non-differentiable but convex component. We propose a proximal stochastic gradient algorithm based on variance reduction, called ProxSVRG+. Our main contribution lies in the analysis of ProxSVRG+. It recovers several existing convergence results and improves/generalizes them (in terms of the number of stochastic gradient oracle calls and proximal oracle call…
This paper aims to develop a Newton-type method to solve a class of nonconvex composite programs. In particular, the nonsmooth part is possibly nonconvex. To tackle the nonconvexity, we develop a notion of strong prox-regularity which is related to the singleton property, Lipschitz continuity, and monotonicity of the associated proximal operator, and we verify it in various classes of functions, including weakly convex functions, indicator functions of proximally smooth sets, and two specific sphere-related nonconvex nonsmooth functions. In this case, the problem class we are concerned with co…
We propose and analyze several stochastic gradient algorithms for finding stationary points or local minimum in nonconvex, possibly with nonsmooth regularizer, finite-sum and online optimization problems. First, we propose a simple proximal stochastic gradient algorithm based on variance reduction called ProxSVRG+. We provide a clean and tight analysis of ProxSVRG+, which shows that it outperforms the deterministic proximal gradient descent (ProxGD) for a wide range of minibatch sizes, hence solves an open problem proposed in Reddi et al. (2016b). Also, ProxSVRG+ uses much less proximal oracle…
A method is developed for solving bang-bang and singular optimal control problems using adaptive Legendre-Gauss-Radau (LGR) collocation. The method is divided into several parts. First, a structure detection method is developed that identifies switch times in the control and analyzes the corresponding switching function for segments where the solution is either bang-bang or singular. Second, after the structure has been detected, the domain is decomposed into multiple domains such that the multiple-domain formulation includes additional decision variables that represent the switch times in the…
This article regards numerical optimal control of a class of hybrid systems with hysteresis using solely techniques from nonlinear optimization, without any integer variables. Hysteresis is a rate independent memory effect which often results in severe nonsmoothness in the dynamics. These systems are not simply Piecewise Smooth Systems (PSS); they are a more complicated form of hybrid systems. We introduce a time-freezing reformulation which transforms these systems into a PSS. From the theoretical side, this reformulation opens the door to study systems with hysteresis via the rich tools deve…
