Direct methods with maximal lower bound for mixed-integer optimal control problems

Many practical optimal control problems include discrete decisions. These may be either time-independent parameters or time-dependent control functions as gears or valves that can only take discrete values at any given time. While great progress has been achieved in the solution of optimization problems involving integer variables, in particular mixed-integer linear programs, as well as in continuous optimal control problems, the combination of the two is yet an open field of research. We consider the question of lower bounds that can be obtained by a relaxation of the integer requirements. For general nonlinear mixed-integer programs such lower bounds typically suffer from a huge integer gap. We convexify (with respect to binary controls) and relax the original problem and prove that the optimal solution of this continuous control problem yields the best lower bound for the nonlinear integer problem. Building on this theoretical result we present a novel algorithm to solve mixed-integer optimal control problems, with a focus on discrete-valued control functions. Our algorithm is based on the direct multiple shooting method, an adaptive refinement of the underlying control discretization grid and tailored heuristic integer methods. Its applicability is shown by a challenging application, the energy optimal control of a subway train with discrete gears and velocity limits.      

On the relaxation gap for PDE mixed-integer optimal control problems

Mixed-integer optimal control problems require taking discrete and continuous control decisions for the optimization of a dynamical system. We consider dynamics governed by partial differential equations of evolution type and assess the problem by relaxation and rounding strategies. For this solution approach, we present a priori estimates for semilinear evolutions on Banach spaces concerning the optimality gap. The theoretical results show that the gap can be made arbitrary small. We demonstrate the numerical performance of the approach on benchmark problems of parabolic type motivated from thermal manufacturing and of hyperbolic type motivated from traffic flow control. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Error estimates for the numerical approximation of Neumann control problems governed by a class of quasilinear elliptic equations

We study the numerical approximation of Neumann boundary optimal control problems governed by a class of quasilinear elliptic equations. The coefficients of the main part of the operator depend on the state function, as a consequence the state equation is not monotone. We prove that strict local minima of the control problem can be approximated uniformly by local minima of discrete control problems and we also get an estimate of the rate of this convergence. One of the main issues in this study is the error analysis of the discretization of the state and adjoint state equations. Some difficulties arise due to the lack of uniqueness of solution of the discrete equations. The theoretical results are illustrated by numerical tests.     

Spatiotemporal dynamics and feedforward control of an infinite-dimensional,time-variant,thermal-coating process

In this contribution, a model for the spatiotemporal dynamics of a thermal-coating process is derived from first principles (the conservation of energy). The model is a dynamical system consisting of a system of hyperbolic, partial differential equations (PDEs), describing the evolution of the temperature distribution of the substrate. By studying the C ₀- semigroup generated by the system operator we find that in discrete time the infinite-dimensional plant may conveniently be represented by finite-dimensional operators defined on appropriately chosen Euclidean spaces. This representation provides the basis for numerical efficient solution of several optimal feedforward control problems associated with set point changes and launching of the process. Numerical and experimental studies highlight the value of this approach.     

An approach for an algorithmic solution of discrete optimal control problems and their game-theoretical extension

We consider time discrete systems which are described by a system of difference equations. The related discrete optimal control problems are introduced. Additionally, a gametheoretic extension is derived, which leads to general multicriteria decision problems. The characterization of their optimal behavior is studied. Given starting and final states define the decision process; applying dynamic programming techniques suitable optimal solutions can be gained. We generalize that approach to a special gametheoretic decision procedure on networks. We characterize Nash equilibria and present sufficient conditions for their existence. A constructive algorithm is derived. The sufficient conditions are exploited to get the algorithmic solution. Its complexity analysis is presented and at the end we conclude with an extension to the complementary case of Pareto optima.Dmitrii Lozovanu was Supported by BGP CRDF-MRDA MOM2-3049-CS-03.     

一类离散事件动态系统极点分配的优化问题

应用极大代数作为数学工具,用系统矩阵的特征值法讨论了m × n-型离散事件动态系统极点分配的优化问题.给出了系统取得优化的条件.并证明了至少存在,n个具有最优极点分配的优化系统.     

Temporal logic guided safe model-based reinforcement learning: A hybrid systems approach

This paper studies the problem of synthesizing control policies for uncertain continuous-time nonlinear systems from linear temporal logic (LTL) specifications using model-based reinforcement learning (MBRL). Rather than taking an abstraction-based approach, we view the interaction between the LTL formula’s corresponding Büchi automaton and the nonlinear system as a hybrid automaton whose discrete dynamics match exactly those of the Büchi automaton. To find satisfying control policies, we pose a sequence of optimal control problems associated with states in the accepting run of the automaton and leverage control barrier functions (CBFs) to prevent specification violation. Since solving many optimal control problems for a nonlinear system is computationally intractable, we take a learning-based approach in which the value function of each problem is learned online in real-time. Specifically, we propose a novel off-policy MBRL algorithm that allows one to simultaneously learn the uncertain dynamics of the system and the value function of each optimal control problem online while adhering to CBF-based safety constraints. Unlike related approaches, the MBRL method presented herein decouples convergence, stability, and safety, allowing each aspect to be studied independently, leading to stronger safety guarantees than those developed in related works. Numerical results are presented to validate the efficacy of the proposed method.     

Infinite horizon optimal control problems with multiple thermostatic hybrid dynamics

We study an optimal control problem for a hybrid system exhibiting several internal switching variables whose discrete evolutions are governed by some delayed thermostatic laws. By the dynamic programming technique we prove that the value function is the unique viscosity solution of a system of several Hamilton-Jacobi equations, suitably coupled. The method involves a contraction principle and some suitably adapted results for exit-time problems with discontinuous exit cost.     

Algorithms for solving multiobjective discrete control problems and dynamic c-games on networks

In this paper we study a special class of multiobjective discrete control problems on dynamic networks. We assume that the dynamics of the system is controlled by p actors (players) and each of them intend to minimize his own integral-time cost by a certain trajectory. Applying Nash and Pareto optimality principles we study multiobjective control problems on dynamic networks where the dynamics is described by a directed graph.Polynomial-time algorithms for determining the optimal strategies of the players in the considered multiobjective control problems are proposed exploiting the special structure of the underlying graph. Properties of time-expanded networks are characterized. A constructive scheme which consists of several algorithms is presented.