Construction of Solutions to Control Problems for Fractional- Order Linear Systems Based on Approximation Models

We consider an optimal control problem for a dynamical system whose motion is described by a linear differential equation with the Caputo fractional derivative of order α ∈ (0, 1). The time interval of the control process is fixed and finite. The control actions are subject to geometric constraints. The aim of the control is to minimize a given terminal-integral quality index. In order to construct a solution, we develop the following approach. First, from the considered problem, we turn to an auxiliary optimal control problem for a first-order linear system with lumped delays, which approximates the original system. After that, the auxiliary problem is reduced to an optimal control problem for an ordinary differential system. Based on this, we propose a closed-loop scheme of optimal control of the original system that uses the approximating system as a guide. In this scheme, the control in the approximating system is formed with the help of an optimal positional control strategy from the reduced problem. The effectiveness of the developed approach is illustrated by a problem in which the quality index is the norm of the terminal state of the system. © 2020 Krasovskii Institute of Mathematics and Mechanics. All rights reserved.This work was supported by RSF (project no. 19-11-00105)

A model predictive control approach to a class of multiplayer minmax differential games

In this dissertation, we consider a class of two-team adversarial differential games in which there are multiple mobile dynamic agents on each team. We describe such games in terms of semi-infinite minmax Model Predictive Control (MPC) problems, and present a numerical optimization technique for efficiently solving them. We also describe the implementation of the solution method in both indoor and outdoor robotic testbeds. Our solution method requires one to solve a sequence of Quadratic Programs (QPs), which together efficiently solve the original semi-infinite min- max MPC problem. The solution method separates the problem into two subproblems called the inner and outer subproblems, respectively. The inner subproblem is based on a constrained nonlinear numerical optimization technique called the Phase I-Phase II method, and we develop a customized version of this method. The outer subproblem is about judiciously initializing the inner subproblems to achieve overall convergence; our method guarantees exponential convergence. We focus on a specific semi-infinite minmax MPC problem called the harbor defense problem. First, we present foundational work on this problem in a formulation containing a single defender and single intruder. We next extend the basic formulation to various advanced scenarios that include cases in which there are multiple defenders and intruders, and also ones that include varying assumptions about intruder strategies. Another main contribution is that we implemented our solution method for the harbor defense problem on both real-time indoor and outdoor testbeds, and demonstrated its computational effectiveness. The indoor testbed is a custom-built robotic testbed named HoTDeC (Hovercraft Testbed for Decentralized Control). The outdoor testbed involved full-sized US Naval Academy patrol ships, and the experiment was conducted in Chesapeake Bay in collaboration with the US Naval Academy. The scenario used involved one ship (the intruder) being commanded by a human pilot, and the defender ship being controlled automatically by our semi-infinite minmax MPC algorithm.The results of several experiments are presented. Finally, we present an efficient algorithm for solving a class of matrix games, and show how this approach can be directly used to effectively solve our original continuous space semi-infinite minmax problem using an adaptive approximation

A Differential Model for a 2x2-Evolutionary Game Dynamics

A dynamical model for an evolutionary nonantagonistic (nonzero sum) game between two populations is considered. A scheme of a dynamical Nash equilibrium in the class of feedback (discontinuous) controls is proposed. The construction is based on solutions of auxiliary antagonistic (zero-sum) differential games. A method for approximating the corresponding value functions is developed. The method uses approximation schemes for constructing generalized (minimax, viscosity) solutions of first order partial differential equations of Hamilton-Jacobi type. A numerical realization of a grid procedure is described. Questions of convergence of approximate solutions to the generalized one (the value function) are discussed, and estimates of convergence are pointed out. The method provides equilibrium feedbacks in parallel with the value functions. Implementation of grid approximations for feedback control is justified. Coordination of long- and short-term interests of populations and individuals is indicated. A possible relation of the proposed game model to the classical replicator dynamics is outlined

Path-Dependent Hamilton–Jacobi Equations: The Minimax Solutions Revised

Motivated by optimal control problems and differential games for functional differential equations of retarded type, the paper deals with a Cauchy problem for a path-dependent Hamilton–Jacobi equation with a right-end boundary condition. Minimax solutions of this problem are studied. The existence and uniqueness result is obtained under assumptions that are weaker than those considered earlier. In contrast to previous works, on the one hand, we do not require any properties concerning positive homogeneity of the Hamiltonian in the impulse variable, and on the other hand, we suppose that the Hamiltonian satisfies a Lipshitz continuity condition with respect to the path variable in the uniform (supremum) norm. The progress is related to the fact that a suitable Lyapunov–Krasovskii functional is built that allows to prove a comparison principle. This functional is in some sense equivalent to the square of the uniform norm of the path variable and, at the same time, it possesses appropriate smoothness properties. In addition, the paper provides non-local and infinitesimal criteria of minimax solutions, their stability with respect to perturbations of the Hamiltonian and the boundary functional, as well as consistency of the approach with the non-path-dependent case. Connection of the problem statement under consideration with some other possible statements (regarding the choice of path spaces and derivatives used) known in the theory of path-dependent Hamilton–Jacobi equations is discussed. Some remarks concerning viscosity solutions of the studied Cauchy problem are given. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature

Evolutionary Computation

This book presents several recent advances on Evolutionary Computation, specially evolution-based optimization methods and hybrid algorithms for several applications, from optimization and learning to pattern recognition and bioinformatics. This book also presents new algorithms based on several analogies and metafores, where one of them is based on philosophy, specifically on the philosophy of praxis and dialectics. In this book it is also presented interesting applications on bioinformatics, specially the use of particle swarms to discover gene expression patterns in DNA microarrays. Therefore, this book features representative work on the field of evolutionary computation and applied sciences. The intended audience is graduate, undergraduate, researchers, and anyone who wishes to become familiar with the latest research work on this field

Advances in Reinforcement Learning

Reinforcement Learning (RL) is a very dynamic area in terms of theory and application. This book brings together many different aspects of the current research on several fields associated to RL which has been growing rapidly, producing a wide variety of learning algorithms for different applications. Based on 24 Chapters, it covers a very broad variety of topics in RL and their application in autonomous systems. A set of chapters in this book provide a general overview of RL while other chapters focus mostly on the applications of RL paradigms: Game Theory, Multi-Agent Theory, Robotic, Networking Technologies, Vehicular Navigation, Medicine and Industrial Logistic

Modelling and Inverse Problems of Control for Distributed Parameter Systems; Proceedings of IFIP(W.G. 7.2)-IIASA Conference, July 24-28, 1989

The techniques of solving inverse problems that arise in the estimation and control of distributed parameter systems in the face of uncertainty as well as the applications of these to mathematical modelling for problems of applied system analysis (environmental issues, technological processes, biomathematical models, mathematical economy and other fields) are among the major topics of research at the Dynamic Systems Project of the Systems and Decision Sciences (SDS) Program at IIASA. In July 1989 the SDS Program was a coorganizer of a regular IFIP (WG 7.2) conference on Modelling and Inverse Problems of Control for Distributed Parameter Systems that was held at IIASA, and was attended by a number of prominent theorists and practitioners. One of the main purpose of this meeting was to review recent developments and perspectives in this field. The proceedings are presented in this volume