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Optimal control problems with free terminal time present many challenges including nonsmooth and discontinuous control laws, irregular value functions, many local optima, and the curse of dimensionality. To overcome these issues, we propose an adaptation of the model-based actor-critic paradigm from the field of Reinforcement Learning via an exponential transformation to learn an approximate feedback control and value function pair. We demonstrate the algorithm's effectiveness on prototypical examples featuring each of the main pathological issues present in problems of this type.

Recent research shows that supervised learning can be an effective tool for designing near-optimal feedback controllers for high-dimensional nonlinear dynamic systems. But the behavior of neural network controllers is still not well understood. In particular, some neural networks with high test accuracy can fail to even locally stabilize the dynamic system. To address this challenge we propose several novel neural network architectures, which we show guarantee local asymptotic stability while retaining the approximation capacity to learn the optimal feedback policy semi-globally. The proposed architectures are compared against standard neural network feedback controllers through numerical simulations of two high-dimensional nonlinear optimal control problems: stabilization of an unstable Burgers-type partial differential equation, and altitude and course tracking for an unmanned aerial vehicle. The simulations demonstrate that standard neural networks can fail to stabilize the dynamics even when trained well, while the proposed architectures are always at least locally stabilizing and can achieve near-optimal performance., Comment: arXiv admin note: text overlap with arXiv:2109.07466

Recent research has shown that supervised learning can be an effective tool for designing optimal feedback controllers for high-dimensional nonlinear dynamic systems. But the behavior of these neural network (NN) controllers is still not well understood. In this paper we use numerical simulations to demonstrate that typical test accuracy metrics do not effectively capture the ability of an NN controller to stabilize a system. In particular, some NNs with high test accuracy can fail to stabilize the dynamics. To address this we propose two NN architectures which locally approximate a linear quadratic regulator (LQR). Numerical simulations confirm our intuition that the proposed architectures reliably produce stabilizing feedback controllers without sacrificing optimality. In addition, we introduce a preliminary theoretical result describing some stability properties of such NN-controlled systems.

This paper addresses the problem of optimal defense of a High Value Unit against a large-scale swarm attack. We show that the problem can be cast in the framework of uncertain parameter optimal control and derive a consistency result for the dual problem of this framework. We show that the dual can be computed numerically and apply these numerical results to derive optimal defender strategies against a 100 agent swarm attack.

We theoretically and numerically study the problem of optimal control of large-scale autonomous systems under explicitly adversarial conditions, including probabilistic destruction of agents during the simulation. Large-scale autonomous systems often include an adversarial component, where different agents or groups of agents explicitly compete with one another. An important component of these systems that is not included in current theory or modeling frameworks is random destruction of agents in time. In this case, the modeling and optimal control framework should consider the attrition of agents as well as their position. We propose and test three numerical modeling schemes, where survival probabilities of all agents are smoothly and continuously decreased in time, based on the relative positions of all agents during the simulation. In particular, we apply these schemes to the case of agents defending a high-value unit from an attacking swarm. We show that these models can be successfully used to model this situation, provided that attrition and spatial dynamics are coupled. Our results have relevance to an entire class of adversarial autonomy situations, where the positions of agents and their survival probabilities are both important.

As demonstrated in many areas of real-life applications, neural networks have the capability of dealing with high dimensional data. In the fields of optimal control and dynamical systems, the same capability was studied and verified in many published results in recent years. Towards the goal of revealing the underlying reason why neural networks are capable of solving some high dimensional problems, we develop an algebraic framework and an approximation theory for compositional functions and their neural network approximations. The theoretical foundation is developed in a way so that it supports the error analysis for not only functions as input-output relations, but also numerical algorithms. This capability is critical because it enables the analysis of approximation errors for problems for which analytic solutions are not available, such as differential equations and optimal control. We identify a set of key features of compositional functions and the relationship between the features and the complexity of neural networks. In addition to function approximations, we prove several formulae of error upper bounds for neural networks that approximate the solutions to differential equations, optimization, and optimal control., Comment: 40 pages, 18 figures

In this paper we propose a new computational method for designing optimal regulators for high-dimensional nonlinear systems. The proposed approach leverages physics-informed machine learning to solve high-dimensional Hamilton-Jacobi-Bellman equations arising in optimal feedback control. Concretely, we augment linear quadratic regulators with neural networks to handle nonlinearities. We train the augmented models on data generated without discretizing the state space, enabling application to high-dimensional problems. We use the proposed method to design a candidate optimal regulator for an unstable Burgers' equation, and through this example, demonstrate improved robustness and accuracy compared to existing neural network formulations., Comment: Added IEEE accepted manuscript with copyright notice

The generation and manipulation of strong entanglement and Einstein-Podolsky-Rosen (EPR) steering in macroscopic systems are outstanding challenges in modern physics. Especially, the observation of asymmetric EPR steering is important for both its fundamental role in interpreting the nature of quantum mechanics and its application as resource for the tasks where the levels of trust at different parties are highly asymmetric. Here, we study the entanglement and EPR steering between two macroscopic magnons in a hybrid ferrimagnet-light system. In the absence of light, the two types of magnons on the two sublattices can be entangled, but no quantum steering occurs when they are damped with the same rates. In the presence of the cavity field, the entanglement can be significantly enhanced, and strong two-way asymmetric quantum steering appears between two magnons with equal dispassion. This is very different from the conventional protocols to produce asymmetric steering by imposing additional unbalanced losses or noises on the two parties at the cost of reducing steerability. The essential physics is well understood by the unbalanced population of acoustic and optical magnons under the cooling effect of cavity photons. Our finding may provide a novel platform to manipulate the quantum steering and the detection of bi-party steering provides a knob to probe the magnetic damping on each sublattice of a magnet.

Recent research reveals that deep learning is an effective way of solving high dimensional Hamilton-Jacobi-Bellman equations. The resulting feedback control law in the form of a neural network is computationally efficient for real-time applications of optimal control. A critical part of this design method is to generate data for training the neural network and validating its accuracy. In this paper, we provide a survey of existing algorithms that can be used to generate data. All the algorithms surveyed in this paper are causality-free, i.e., the solution at a point is computed without using the value of the function at any other points. At the end of the paper, an illustrative example of optimal feedback design using deep learning is given., Comment: 15 pages, 1 figure

Uncertainty propagation in nonlinear dynamic systems remains an outstanding problem in scientific computing and control. Numerous approaches have been developed, but are limited in their capability to tackle problems with more than a few uncertain variables or require large amounts of simulation data. In this paper, we propose a data-driven method for approximating joint probability density functions (PDFs) of nonlinear dynamic systems with initial condition and parameter uncertainty. Our approach leverages on the power of deep learning to deal with high-dimensional inputs, but we overcome the need for huge quantities of training data by encoding PDF evolution equations directly into the optimization problem. We demonstrate the potential of the proposed method by applying it to evaluate the robustness of a feedback controller for a six-dimensional rigid body with parameter uncertainty., Comment: This work has been submitted to IFAC for possible publication

We address the design and synthesis of optimal control strategies for high-dimensional stochastic dynamical systems. Such systems may be deterministic nonlinear systems evolving from random initial states, or systems driven by random parameters or processes. The objective is to provide a validated new computational capability for optimal control which will be achieved more efficiently than current state-of-the-art methods. The new framework utilizes direct single or multi-shooting discretization, and is based on efficient vectorized gradient computation with adaptable memory management. The algorithm is demonstrated to be scalable to high-dimensional nonlinear control systems with random initial condition and unknown parameters., Comment: 23 pages, 17 figures

Computing optimal feedback controls for nonlinear systems generally requires solving Hamilton-Jacobi-Bellman (HJB) equations, which are notoriously difficult when the state dimension is large. Existing strategies for high-dimensional problems often rely on specific, restrictive problem structures, or are valid only locally around some nominal trajectory. In this paper, we propose a data-driven method to approximate semi-global solutions to HJB equations for general high-dimensional nonlinear systems and compute candidate optimal feedback controls in real-time. To accomplish this, we model solutions to HJB equations with neural networks (NNs) trained on data generated without discretizing the state space. Training is made more effective and data-efficient by leveraging the known physics of the problem and using the partially-trained NN to aid in adaptive data generation. We demonstrate the effectiveness of our method by learning solutions to HJB equations corresponding to the attitude control of a six-dimensional nonlinear rigid body, and nonlinear systems of dimension up to 30 arising from the stabilization of a Burgers'-type partial differential equation. The trained NNs are then used for real-time feedback control of these systems., Comment: Added section on validation error computation. Updated convergence test formula and associated results

Historically, solving optimal control problems with high index differential algebraic equations (DAEs) has been considered extremely hard. Computational experience with Runge-Kutta (RK) methods confirms the difficulties. High index DAE problems occur quite naturally in many practical engineering applications. Over the last two decades, a vast number of real-world problems have been solved routinely using pseudospectral (PS) optimal control techniques. In view of this, we solve a "provably hard," index-three problem using the PS method implemented in DIDO, a state-of-the-art MATLAB optimal control toolbox. In contrast to RK-type solution techniques, no laborious index-reduction process was used to generate the PS solution. The PS solution is independently verified and validated using standard industry practices. It turns out that proper PS methods can indeed be used to "directly" solve high index DAE optimal control problems. In view of this, it is proposed that a new theory of difficulty for DAEs be put forth., Comment: 14 pages, 9 figures