Chapter ix optimal control of systems governed by partial. Optimal control of systems governed by partial differential equations by j. Optimal control of systems governed by partial differential equa tions. Optimal control of dynamical systems governed by partial. The space region supporting the partial differential equations is decomposed and the original global optimal control problem is reduced to a sequence of similar local optimal control problems. It presents examples of elliptic control problems, necessary and sufficient conditions for optimality, boundary control and approximate controllability of elliptic systems, and the control of systems governed by parabolic equations. Even though other similarities occur, there is no full generalization of pontryagins maximum principle to nonlinear pdes.
Optimal control of pdes using occupation measures and sdp relaxations. The complete dynamics of the system is given by a coupled set of ordinary and partial differential equations. In this dissertation optimal filters are derived for three important classes of nonlinear stochastic dynamical systems. The main focus is on existence results for optimal controls as well as on optimality conditions. The solution is then obtained as a linear combination of this reduced set of basis functions by means of galerkin projection. This framework is based on the partial integrodifferential fokkerplanck fp equation that governs the time evolution of the probability density function of this process. In fact, since the end of last century, impulsive evolution equations on infinitedimensional spaces have been investigated by many authors including us. We present an iterative domain decomposition method for the optimal control of systems governed by linear partial differential equations. The development of a theory of optimal control deterministic requires the following initial data. Approximationofdelayeddifferential equations 12 ii. This makes it attractive for optimal control and estimation of systems governed by partial differential equations pdes. The interested reader may learn more on this topic from the references above and those on the bibliography at the end of the article. Optimal control of quasilinear parabolic equations volume 125 issue 3 eduardo casas, luis a.
Bittner, on optimal control of processes governed by abstract functional, integral and hyperbolic di. In this paper, a stateconstrained optimal control problem governed by p laplacian elliptic equations is studied. This brief considers recent results on optimal control and stabilization of systems governed by hyperbolic partial differential equations, specifically those in which the control action takes place at the boundary. Optimal control theory ask two different, but related, questions. Kazemidehkordi department of mathematics, university of north carolina at charlotte, charlotte, nc 28223. Optimal control of systems governed by partial differential equations 1. Mesarovic abstract this paper presents a general and complete duality theory for optimal control of systems governed by linear ordinary differential equations with a convex cost criterion. Optimal control of quasilinear parabolic equations.
Optimal control and partial differential equations core. Nasa ames research center, moffett field, ca 94035 this paper presents an optimal control method for a class of distributedparameter systems governed by. In both cases, the classical lions argument rewriting the problem as an optimal control problem for the control without equality constraints, see for instance lions, j. Abstract this survey article summarizes some ideas of the two principle procedures for solving optimal control problems governed by partial differential algebraic equations. Domain decomposition, optimal control of systems governed. We present a method for optimal control of systems governed by partial differential equations pdes with uncertain parameter fields. Control of systems governed by partial differential. Pdf computational optimization of systems governed by.
Optimal control of systems governed by partial differential equations c by j. Adaptive finite element methods for optimal control of. Optimal control of systems governed by partial differential equations. Inbetween the two courses part i theory of optimal control problems with elliptic partial differential equations including semilinear equations in ws 201112 and ii theory of optimal control problems with parabolic partial differential equations including semilinear equations in ws 2012 there will be a seminar which deals with numerical. Beyond that, the goals are to improve the response in some welldefined manner, such as by solving a linearquadratic optimal con. The purpose of this conference is to examine the control theory of partial differential equations and its application.
Optimal control of systems governed by partial differential equations, springer, 1970 can not be applied. The focus of the course is on regularization, variational formulations. The aim of organizing these two workshops in one is to bring together scientists interested in control of distributed parameter systems, namely those having different points of. It is used here in active control of fluid flows governed by the navierstokes equations. The first class of systems, considered in chapter ii, is that governed by. Optimal control of dynamical systems governed by partial differential equations. On optimal sparsecontrol problems governed by jump.
The control design for systems modeled by partial differential equations hence resides at the intersection of mathematics, systems and control theory, control systems technology, and computer and information science making it essential to provide a joint forum to foster and evolve this important and emerging field of. Optimal control of partial differential equations some. The beam model is given by a static nonlinear fourthorder differential equation with some boundary conditions. This book focuses on optimal control problems where the state equation is an elliptic or parabolic partial differential equation. In this paper we consider the problem of optimal regulation of large space structures in the presence of flexible appendages. Included are topics such as the existence of optimal solutions, necessary optimality conditions and adjoint equations, secondorder sufficient conditions, and main principles of selected numerical techniques.
The theory of optimal control of systems governed by partial differential equations pdes was developed by j. Optimal filtering for systems governed by coupled ordinary. This chapter discusses optimal control of systems governed by partial differential equations. Conjugate convex functions, duality, and optimal control. As mathematicians, studying them for their own sake is an entirely acceptable, and even laudable venture. Control theory of systems governed by partial differential.
Introduction optimal control of determined systems governed by partial di erential equations pdes is currently of much interest. Computational optimization of systems governed by partial differential equations offers readers a combined treatment of pdeconstrained optimization and uncertainties and an extensive discussion of multigrid optimization. Content list of 1st ifac workshop on control of systems governed by partial differential equations technical program for wednesday september 25, 20 wempah amphi hermite controllability of pdes and nonlinearity. Pdf optimal control of dynamical systems governed by. Proceedings of the 48h ieee conference on decision and control cdc held jointly with 2009 28th chinese control conference, 64546459. Model predictive optimal control of a timedelay distributedparameter system nhan nguyen. We consider an objective function that involves the mean and variance of the control objective, leading to a riskaverse optimal control problem. Control theory of systems governed by partial differential equations covers the proceedings of the 1976 conference by the same title, held at the naval surface weapons center, silver spring, maryland.
Optimal control problems ocps governed by convection diffusion partial differential equations pdes arise in environmental modeling, petroleum reservoir simulation and in many other engineering applications 9, 10, 27. It provides a bridge between continuous optimization and pde modeling and focuses on the numerical solution of the. Pdf control of systems governed by control of systems. Lions, optimal control of systems governed by partial differential equa. For simplicity of presentation, we consider a spacecraft consisting of a rigid bus and a flexible beam. Computational optimization of systems governed by partial. Optimal control of partial differential equations american. Optimal boundary control and boundary stabilization of. We show that the solution of this hybrid system is. A priori error estimate of stochastic galerkin method for. Methods, results and open problems plenary session chair. In this paper we shall attempt to suznarize sone of the results of optimal control.
Particularly, ahmed and we considered optimal control problems of systems governed by firstorder impulsive evolution equations and firstorder impulsive integrodifferential equations 47. The recursive estimation of states or parameters of stochastic dynamical systems with partial and imperfect measurements is generally referred to as filtering. Systems research center, case western reserve university cleveland, ohio communicated by mihajlo d. Control theory and pdes dustin connerygrigg december 14, 2012 1 introduction di erential equations are extremely useful tools in modelling all sorts of dynamical systems. Discretization of optimal control problems governed by p. Barbu, analysis and control of nonlinear infinite dimensional systems. The mathematical optimization of processes governed by partial differential equations has. A framework for the optimal sparsecontrol of the probability density function of a jumpdiffusion process is presented. Secondorder nonlinear impulsive integrodifferential. In the present paper a bicriterial optimal control problem governed by a parabolic partial differential equation pde and bilateral control constraints is considered. It presents examples of elliptic control problems, necessary and sufficient conditions for optimality, boundary control and approximate controllability of elliptic systems, and. The feasible control set or the cost functional may be nonconvex, and the purpose is to obtain the convergence of a solution of the discretized control problem to an optimal control of the relaxed continuous problem. Optimal control of systems governed by partial differential equations by jacqueslouis lions, 9783642650260, available at book depository with free delivery worldwide. Minimization of functions and unilateral boundary value problems control of systems governed by elliptic partial differential equations control of systems governed by parabolic partial differential equations control of systems governed by hyperbolic equations or by equations which are well posed in the petrowsky sense regularization.
This lecture is an introduction to the theory of optimal control problems governed by elliptic partial di erential equations. Zuazua, enrique basque center for applied mathematics 09. Optimal control of system governed by the gao beam equation. Control of systems governed by control of systems governed by partial differential equations chapter pdf available december 2010 with 425 reads how we measure reads. Even for systeras governed by linear partial differential equations for exanple of the parabolic type with a ceneral cost function there is no result analogous to the maximal principle. Adaptive discontinuous galerkin approximation of optimal. The beam is here subjected to a vertical load and possibly to an axial tension load as well. Optimal control of partial differential equations in. In this contribution several optimal control problems are mathematically formulated and analyzed for a nonlinear beam which was introduced in 1996 by david y. In the stochastic process and, correspondingly, in the fp model the control function enters as a time. Optimal control of systems governed by partial differential equations jacques louis lions download bok. When dealing with controllability problems, to begin with, one has to distinguish between.
Control of systems governed by partial differential equations. Indeed, the ideas of pontryagins maximum principle are used to aid in characterizing an. These systems are modelled by partial di fferential equations pdes. The equations can be of elliptic, parabolic, or hyperbolic type. The wave equation is used as a typical example of a linear system, through which.
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