Szzj infinite dimensional optimization and control theory. Fortunately, once proven, the major results are quite simple, and analogous to those in the optimization in a finite dimensional space. Theory and application of optimal control have been widely used in different fields such as biomedicine 1, aircraft systems 2, robotic 3, etc. Optimal control of nonlinear systems using the homotopy. Receding horizon optimal control for infinite dimensional systems 743 and the references given there. Infinite dimensional systems can be used to describe many phenomena in the real world. Optimal control theory for infinite dimensional systems edition 1. This book concerns existence and necessary conditions, such as potryagins maximum principle, for optimal control problems described by ordinary and partial differential equations. Infinite dimensional optimal control theory sciencedirect. Download it once and read it on your kindle device, pc, phones or tablets. As is well known, heat conduction, properties of elastic plastic material, fluid dynamics, diffusionreaction. The objective is to develop a control model for controlling such systems using a control action in an optimum manner without delay or overshoot and ensuring control stabil.
An introduction to mathematical optimal control theory version 0. Stability and stabilizability of infinitedimensional systems. An introduction to optimal control ugo boscain benetto piccoli the aim of these notes is to give an introduction to the theory of optimal control for nite dimensional systems and in particular to the use of the pontryagin maximum principle towards the constructionof an optimal. An introduction to infinitedimensional linear systems theory. The purpose of this book is to introduce optimal control theory for infinite dimensional systems. Bernstein government aerospace systems division, harris corporation, ms 224,848, melbourne, fl 32902, u. Infinite dimensional optimization and control theory volume 54 of cambridge studies in advanced mathematics, issn 09506330 volume 62 of encyclopedia of mathematics and its applications, issn 09534806 infinite dimensional optimization and control theory, hector o. The basic optimal control theory is summarised as an infinite dimensional extension of optimisation theory. Calculus of variations and optimal control theory a concise. Optimal control of infinite dimensional bilinear systems. Widely regarded as a milestone in optimal control theory, the significance of the maximum principle lies in the fact that maximizing the hamiltonian is much easier than the original infinite dimensional control problem. It states that it is necessary for any optimal control along with the optimal state trajectory to solve the socalled hamiltonian system, which is a twopoint. Volume i deals with the theory of time evolution of controlled infinite dimensional systems.
Specifically, problems of quantum feedback control, control of tumor growth dynamics and time optimal control are analyzed for bilinear systems. Optimal control theory for in nite dimensional systems. Optimal control theory for infinite dimensional systems xungjing. In this thesis, the problem of designing finite dimensional controllers for infinite dimensional singleinput singleoutput systems is addressed. In the present paper, we consider the quadratic optimal control problem not necessarily in a feedback form for an infinite dimensional discretetime stochastic bilinear system subject to an. Some recent results and open questions in time optimal. This is an original and extensive contribution which is not covered by other recent books in the control theory. The state of these systems lies in an infinite dimensional space, but finite dimensional approximations must be used.
As is well known, heat conduction, properties of elastic plastic material, fluid dynamics, diffusionreaction processes, etc. Evans department of mathematics university of california, berkeley. Introduction to infinitedimensional systems theory. Optimal control theory for infinite dimensional systems optimal control theory for infinite dimensional systems curtain, ruth f. Springer has kindly allowed me to place a copy on the web, as a reference and for ease of web searches. Optimal feedback control of infinite dimensional linear systems with applications to hereditary problems mark milman jet propulsion laboratory, california institute of technology, pasadena, california 91109 james h. There are three approaches in the optimal control theory. Infinite dimensional optimization and control theory book. Stefani springer berlin heidelberg newyork hongkong london milan paris tokyo. One of the milestones in modern optimal control theory is the, pontryagin maximum principle, which was firstly established by pontryagin and his colleagues in later 50s for finite dimensional optimal control. Pdf optimal control of vehicle systems semantic scholar. Infinite dimensional linear control systems, volume 201 1st. Optimal control theory for infinite dimensional systems by xungjing li, 9780817637224, available at book depository with free delivery worldwide.
Summer school held in cetraro, italy, june 1929, 2004 editors. Method 1 introduction theory and application of optimal control have been widely used in different fields such as biomedicine 1, aircraft systems 2, robotic 3, etc. Representation and control of infinite dimensional systems, volume i. Balastoward a more practical control theory for distributed parameter systems. Pdf the linear quadratic optimal control problem for infinite. Calculus of variations and optimal control theory a concise introduction. Optimal control for a class of infinite dimensional systems. Optimal control theory for infinite dimensional systems springerlink.
Optimal control theory for infinite dimensional systems ieee xplore. Optimal control theory emanuel todorov university of california san diego optimal control theory is a mature mathematical discipline with numerous applications in both science and engineering. Optimal control for infinite dimensional systems springerlink. Nov 14, 2014 reflected bsdes and optimal control and stopping for infinite dimensional systems. Fattorini this book concerns existence and necessary conditions, such as potryagins maximum principle, for optimal control problems described by ordinary and partial differential equations. Receding horizon optimal control for infinite dimensional. Optimal feedback control of infinite dimensional linear. Lqoptimal control of infinitedimensional systems by spectral. Control theory is subfield of mathematics and computer science that deals with the control of continuously operating dynamical systems in engineered processes and machines.
In this context the authors introduce a new concept, recovery in subspaces, where the idea basically is to extract for a given system the properties that actually can. The object that we are studying temperature, displace. Infinite dimensional optimization and control theory. Infinite dimensional systems can be used to describe many physical phenomena in the real world. The quadratic cost optimal control problem for systems described by linear ordinary differential equations occupies a central role in the study of control systems both from the theoretical and design points of view. The essential difficulties for the infinite dimensional theory come from two aspects. Numerical evidence is shown to demonstrate the effectiveness of the feedback law to suppress the. More specifically, it is shown how to systematically obtain near optimal finite dimensional compensators for a large class of scalar infinite dimensional plants. Optimal inputoutput stabilization of infinitedimensional. Optimal control theory for infinite dimensional systems birkhauser boston basel berlin. Optimal feedback control of infinite dimensional linear systems with applications. Wellknown examples are heat conduction, vibration of elastic material, diffusionreaction processes, population systems. There are many challenges and research opportunities associated with developing and deploying computational methodologies for problems of control for systems modeled by partial differential equations and delay equations.
It is emerging as the computational framework of choice for studying the neural control of movement, in much the same way that probabilistic infer. We study the optimal inputoutput stabilization of discrete timeinvariant linear systems in hilbert spaces by output injection. Recent theory of infinite dimensional riccati equations is applied to the linearquadratic optimal control problem for hereditary differential systems, and it is shown that, for most such problems. This is a survey of some of the works on optimal control theory for infinite dimensional systems carried out by the research group of fudan university in recent years. Introduction to infinitedimensional systems theory a. Pdf reflected bsdes and optimal control and stopping for. However, formatting rules can vary widely between applications and fields of interest or study. Optimal control of infinite dimensional bilinear systems 3 this. Introduction to infinitedimensional systems theory a state. Optimal control theory for infinite dimensional systems book. Infinite dimensional systems with unbounded control and. Infinite dimensional linear control systems, volume 201.
One of the milestones in modern optimal control theory is the, pontryagin maximum principle, which was firstly established by pontryagin and his colleagues in later 50s for finite dimensional optimal control problems. Nonlinear and optimal control theory lectures given at the c. Representation and control of infinite dimensional systems. In this paper we consider second order optimality conditions for a bilinear optimal control problem governed by a strongly continuous semigroup. An introduction to infinitedimensional linear systems theory with 29 illustrations. Infinite dimensional optimization and control theory by. An optimal control problem with a timeparameter is considered. We show that a necessary and sufficient condition for this problem to be solvable is that the transfer function has a left factorization over hinfinity. This thesis studies the optimal control of vehicular systems, focusing on the solution of minimumlaptime problems for a formula 1 car. Pdf representation and control of infinite dimensional systems. Because of this collnlercxalnple 1 in 60s and 70s, inany people left the area and s\vitchcd to the discussion of time optimal control problem for linear infinite dimensional systems v. In this example we also allow virtually unconstrained control delays.
The author obtains these necessary conditions from kuhntucker theorems for nonlinear programming problems in infinite dimensional spaces. Providing an introduction to stochastic optimal control in infinite dimension, this book gives a complete account of the theory of secondorder hjb equations in infinite dimensional hilbert spaces, focusing on its applicability to associated stochastic optimal control problems. This book treats optimal problems for systems described by ordinary and partial differential equations, using an approach that unifies finite dimensional and infinite dimensional nonlinear problems include control and state constraints and target conditions. A mathematical framework in terms of semigroups is developed which enables the generalisation of the finite dimensional results to infinite dimensions, and which includes partial differential equations and delay equations as special cases. The present book, in two volumes, is in some sense a selfcontained account of this theory of quadratic cost optimal control for a large class of infinite dimensional systems. Researchers in infinite dimensional control theory.
Optimal control theory for infinite dimensional systems. The linear quadratic optimal control problem for infinite dimensional systems over an infinite horizon survey and examples. Infinite dimensional optimization and control theory hector o. Curtain hans zwart an introduction to infinite dimensional linear systems theory with 29 illustrations springerverlag new york berlin heidelberg london paris. An introduction to mathematical optimal control theory.
Computational methods for control of infinitedimensional. However, optimal control of nonlinear systems is a challenging task which has been studied extensively for decades. Now online version available click on link for pdf file, 544 pages please note. Infinite dimensional systems is a well established area of research with an ever increasing number of applications. Purchase infinite dimensional linear control systems, volume 201 1st edition. Quadratic optimal control for discretetime infinite. Foster lockheed missile and space company, sunnyvale, california 94086 and alan schumitzky. The infinitedimensional riccati equation for systems. The paper considers some control problems for systems described on infinite dimensional spaces. Approximate controllability for semilinear systems 286 4. Infinite dimensional linear systems theory, 252286. Linearquadratic optimal control of hereditary differential. Linear systems and lqr a direct calculation of the lqr controllers b riccati equations and linear optimal control c the maximum principle and the hamilton jacobibellman equations no proofs will be discussed on this topic 8.
The use of control lyapunov functions within the context of receding horizon control is a recent one. The objective is to develop a control model for controlling such systems using a control action in an optimum manner without delay or overshoot and ensuring control stability. Pdf available published results are surveyed for a special class of. Lqoptimal control of infinitedimensional systems by. Infinite dimensional linear control systems, volume 201 1st edition the time optimal and norm optimal problems. Optimal control for a class of infinite dimensional systems involving. Jiongmin yong infinite dimensional systems can be used to describe many phenomena in the real world. Given this trend, there is a need for an introductory text treating system and control theory for this class of systems in detail. The relevant numerical techniques for optimisation and integral approximation are compared in view of the application to vehicle systems. Pontryagins maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls.
An introduction to optimal control ugo boscain benetto piccoli the aim of these notes is to give an introduction to the theory of optimal control for nite dimensional systems and in particular to the use of the pontryagin maximum principle towards the constructionof an optimal synthesis. In 15 control lyapunov functions were utilized as explicit constraints in the auxiliary problems to guarantee that the nal statext. The time optimal and norm optimal problems, northholland mathematics studies, 201, amsterdam, 2005. Infinite dimensional optimization and control theory hector. Recent theory of infinite dimensional riccati equations is applied to the linearquadratic optimal control problem for hereditary differential systems, and it is shown that, for most such problems, the operator solutions of the riccati equations are of trace class i. In the dissertation, optimal control problem for bilinear systems motivated from quantum control theory are studied. Control theory in control systems engineering is a subfield of mathematics that deals with the control of continuously operating dynamical systems in engineered processes and machines. Some recent results and open questions in time optimal control for in nite dimensional systems. Sensor placement for optimal control of infinite dimensional systems article in ieee transactions on control of network systems pp99.
715 5 10 119 358 1299 370 1176 1332 1116 1242 361 666 823 211 679 354 675 1300 136 92 1244 1016 1502 318 789 1492 454 492 313 525 886 465 498 383 1405 406 428 498 94 654 1266 159 830 1350 1304 3 14 548