Infinite dimensional optimization and control theory book. Optimal control theory for infinite dimensional systems. Control of infinitedimensional systems pdf university of waterloo. Citeseerx infinitedimensional optimization and optimal design. The current director and contact person for the group is professor michael malisoff. Sep 30, 2009 infinite dimensional optimization and control theory by hector o. Fattorini skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. Computational methods for control of infinitedimensional. Approximate controllability of infinite dimensional systems of the nth order.
This course is a rigorous introduction to the classical theory of optimal control. Another notification will be sent when the moderators have processed your submisssion. Since the control variable enters the state equation as a coefficient of the partial integrodifferential operator, the resulting optimization problem is nonconvex. Cambridge core optimization, or and risk infinite dimensional optimization and control theory by hector o. Control and optimization with industrial applications abstract. Traditionally, however, this approach has not come with any guarantees. Treats the theory of optimal control with emphasis on optimality conditions, partial differential equations and relaxed solutions fleming w.
Bracketing zfind 3 points such that a control for systems modeled by partial differential equations and delay equations. As is well known, heat conduction, properties of elastic plastic material, fluid dynamics, diffusionreaction processes, etc. In other words, a finitedimensional controller stabilizes the full infinitedimensional. Infinite dimensional optimization and control theory treats optimal problems for systems described by odes and pdes, using an approach that unifies finite and infinite dimensional nonlinear programming. Relation to maximum principle and optimal synthesis 256 6. Infinite dimensional optimization and control theory by hector o.
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. We apply our results to the linearquadratic control problem with quadratic. Optimal control theory for infinite dimensional systems xungjing. Several disciplines which study infinitedimensional optimization problems are calculus of variations, optimal control and shape optimization. Fattorinis extensive monograph is a fundamental contribution to optimal control theory of evolution finite or infinite dimensional systems, and summarizes and extends his many decades of intensive research in this area. Optimal control problems for ordinary and partial differential equations. Group members are studying control design and analysis, nonsmooth analytic. Duality and infinite dimensional optimization sciencedirect.
An infinite dimensional convex optimization problem with the linearquadratic cost function and linearquadratic constraints is considered. Control theory in infinite dimension for the optimal location. 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. Approximate controllability of infinite dimensional systems.
On optimal sparsecontrol problems governed by jumpdiffusion. Deterministic finitedimensional systems, by eduardo d. Every duality is equivalent to a hausdorff locally convex. All authors will be sent email notification when the system receives the article. Infinite dimensional optimization and control theory, encyclopedia of mathematics and its applications, 62. The rigorous treatment of optimization in an infinite dimensional space requires the use of very advanced mathematics.
Global theory of constrained optimization local theory of constrained optimization iterative methods of optimization endofchapter problems constitute a major component of this book and come in two basic varieties. The main emphasis is on applications to convex optimization and convex optimal control problems in banach spaces. This example demonstrates that infinitedimensional optimization theory can be. Typically one needs to employ methods from partial differential equations to solve such problems. An introduction to infinitedimensional linear systems theory. Given a banach space b, a semigroup on b is a family st. Infinite dimensional linear control systems, volume 201 1st. Infinite dimensional optimization and control theory hector. Infinite dimensional systems can be used to describe many phenomena in the real. 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.
Fattorini, 9780521451253, available at book depository with free delivery worldwide. A finite algorithm for solving infinite dimensional. Optimal control as programming in infinite dimensional spaces. Pdf representation and control of infinite dimensional systems. Fundamental issues in applied and computational mathematics are essential to the development of practical computational algorithms. In this form, this is a nonlinear optimization problem with equality constraints. There are three approaches in the optimal control theory. Optimal control theory for infinite dimensional systems birkhauser boston basel berlin. Infinite dimensional optimization problems can be more challenging than finite dimensional ones. Szzj infinite dimensional optimization and control theory. Infinite dimensional optimization and control theory. The main objective of this paper is to provide new explicit criteria to characterize weak lower semi. Citeseerx document details isaac councill, lee giles, pradeep teregowda. This outstanding monograph should be on the desk of every expert in optimal control theory.
The first consists of miscellaneous mathematical problems and proofs that extend and supplement the theoretical. Infinite dimensional systems can be used to describe many phenomena in the. Optimal control theory is an outcome of the calculus of variations, with a history stretching back over 360 years, but interest in it really mushroomed only with the advent of the computer, launched by the spectacular successes of optimal trajectory prediction in aerospace applications in the early 1960s. Relevant gradient formulae pertaining to this finite. Existence of optimal controls is established for arbitrary control sets by means of a general theory of relaxed controls.
Smith department of industrial and operations engineering, the university of michigan, ann arbor, mi 48109, usa abstract. Optimal control is concerned with control laws that maximize a specified measure of a dynamical systems performance. Control theory is the introduction of an input into a dynamical sys. Schochetman department of mathematics and statistics, oakland university, rochester, mi 48309, usa robert l.
Using duality theory, we derive the optimal control, and show that it can be calculated by solving a finite dimensional optimization problem. Optimization including shape optimization optimal control, game theory and calculus of variations wellposedness, stability and control of coupled systems with an interface. We solve a class of convex infinitedimensional optimization problems using a. The princess and infinitedimensional optimization 263 figure 6. Infinite dimensional optimization is a very active research field motivated by a.
The object that we are studying temperature, displace. Differential equations in banach spaces and semigroup theory. We are able to identify a closedform solution to the induced hamiltonjacobibellman hjb equation in infinite dimension and to prove a verification theorem, also providing the optimal control in closed loop form. Infinite dimensional optimization and optimal design martin burger pdf link optimal control peter thompson pdf an introduction to mathematical optimal control theory lawrence c.
The methodology relies on the employment of the classical dynamic programming tool considered in the infinite dimensional context. Infinite dimensional systems can be used to describe many phenomena in the real world. Control and optimization with industrial applications. This book is on existence and necessary conditions, such as potryagins maximum principle, for optimal control problems described by ordinary and partial differential equations. We generalize the interiorpoint techniques of nesterov. Lectures on finite dimensional optimization theory. Infinite horizon problems 264 remarks 272 chapter 7. Lecture notes, 285j infinitedimensional optimization. The terms programming and mathematical programming refer here to the usual constrained maximization problem of the type. Duality and infinite dimensional optimization 1119 if there exists a feasible a for the above problem with ut 0 a. Fortunately, once proven, the major results are quite simple, and analogous to those in the optimization in a finite dimensional space.
Optimal control theory for infinite dimensional systems springerlink. An updated and revised edition of the 1986 title convexity and optimization in banach spaces, this book provides a selfcontained presentation of basic results of the theory of convex sets and functions in infinite dimensional spaces. We generalize the interiorpoint techniques of nesterovnemirovsky to this infinite dimensional situation. Pdf the princess and infinitedimensional optimization. The complementary implicit assertion of bddm2 is that distributed. This barcode number lets you verify that youre getting exactly the right version or edition of a book. Infinite dimensional optimization and control theory encyclopedia of mathematics and its applications series by hector o.
Please click on their names to find out more about their activities. The state of these systems lies in an infinite dimensional space, but finite dimensional approximations must be used. Convex optimization in infinite dimensional spaces 163 a duality x, x is a pair of vector spaces x, x with a bilinear form. Dido pur chases land for the foundation o f carthage, engraving by mathias merian the elder from historische chronica, f rankfurt a. Nonsmooth lyapunov pairs for infinite dimensional firstorder differential inclusions samir adlyy, abderrahim hantoutez, and michel therax abstract. Infinitedimensional optimization and control theory, encyclopedia of mathematics and its applications, 62. Several disciplines which study infinite dimensional optimization problems are calculus of variations, optimal control and shape optimization. Part i finite dimensional control problems 1 1 calculus of variations and control theory 3 1. These necessary conditions are obtained from kuhntucker theorems for nonlinear. Formulation in the most general form, we can write an optimization problem in a topological space endowed with some topology and j.
Purchase infinite dimensional linear control systems, volume 201 1st edition. This is an original and extensive contribution which is not covered by other recent books in the control theory. The current members of the mathematical control theory and optimization group and their students are listed below. The complexity estimates obtained are similar to finite dimensional ones. The objective of the article is to obtain general conditions for several types of controllability at once for an abstract differential equation of arbitrary order, instead of conditions for a fixed order equation.
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