This book studies existence and necessary conditions, such as Pontryagin's maxi mum principle for optimal control problems described by ordinary and partial dif ferential equations. These necessary conditions are obtained from Kuhn-Tucker theorems for nonlinear programming problems in infinite dimensional spaces. The optimal control problems include control constraints, state constraints, and target conditions. Evolution partial differential equations are studied using semi group theory, abstract differential equations in linear spaces, integral equations, and interpolation theory. Existence of optimal controls is established for arbitrary control sets by means of a general theory of relaxed controls. Applications include nonlinear systems described by partial differential equa tions of hyperbolic and parabolic type; the latter case deals with pointwise con straints on the solution and the gradient. The book also includes results on conver gence of suboptimal controls. H. O. Fattorini is Professor of Mathematics at the University of California, Los Angeles. ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS EDITED BY G.-C. ROTA Editorial Board B. Doran, M. Ismail, T.-Y Lam, E. Lutwak Volume 62 Infinite Dimensional Optimization and Control Theory 6 H. Minc Permanents 18 H. O. Fattorini The Cauchy Problem 19 G. G. Lorentz, K. Jetter, and S. D. Riemenschneider Birkhoff Interpolation 22 J. R. Bastida Field Extensions and Galois Theory 23 J. R. Cannon The One-Dimensional Heat Equation 24 S. Wagon The Banach-Tarski Paradox 25 A. Salomaa Computation and Automata 27 N. H. Bingham, C. M. Goldie, and J. L. Teugels Regular Variation 28 P. P. Petrushev and V. A. Popov Rational Approximation of Real Functions 29 N. White (ed.) Combinatorial Geometries 30 M. Pohst and H. Zassenhaus Algorithmic Algebraic Number Theory 31 J. Aczel and J. Dhombres Functional Equations in Several Variables 32 M. Kuczma, B. Choczewski, and R. Ger Iterative Functional Equations 33 R. V. Ambartzumian Factorization Calculus and Geometric Probability 34 G. Gripenberg, S.-O. Londen, and O. Staffans Volterra Integral and Functional Equations 35 G. Gasper and M. Rahman Basic Hypergeometric Series 36 E. Torgersen Comparison of Statistical Experiments 37 A. Neumaier Inteveral Methods for Systems of Equations 38 N. Komeichuk Exact Constants in Approximation Theory 39 R. Brualdi and H. Ryser Combinatorial Matrix Theory 40 N. White (ed.) Matroid Applications 41 S. Sakai Operator Algebras in Dynamical Systems 42 W. Hodges Basic Model Theory 43 H. Stahl and V. Totik General Orthogonal Polynomials 45 G. Da Prato and J. Zabczyk Stochastic Equations in Infinite Dimensions 46 A. Bjomer et al. Oriented matroids 47 G. Edgar and L. Sucheston Stopping Times and Directed Processes 48 C. Sims Computation with Finitely Presented Groups 49 T. Palmer Banach Algebras and the General Theory of *-Algebras 50 F. Borceux Handbook of Categorical Algebra I 51 F. Borceux Handbook of Categorical Algebra II 52 F. Borceux Handbook of Categorical Algebra III 54 A. Katok and B. Hasselblatt Introduction to the Modern Theory of Dynamical Systems 55 V. N. Sachkov Combinatorial Methods in Discrete Mathematics 56 V. N. Sachkov Probabilistic Methods in Discrete Mathematics 57 P. M. Cohn Skew Fields 58 R. Gardner Geometric Topography 59 G. A. Baker Jr. and P. Graves-Morris Pade Approximants 60 J. Krajicek Bounded Arithmetic, Propositional Logic, and Complexity Theory 61 H. Groemer Geometric Applications of Fourier Series and Spherical Harmonics ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS Infinite Dimensional Optimization and Control Theory H. O. FATTORINI CAMBRIDGE UNIVERSITY PRESS CCAAMMBBRRIIDDGGEE UUNNIIVVEERRSSIITTYY PPRREESSSS CCaammbbrriiddggee,, NNeeww YYoorrkk,, MMeellbboouurrnnee,, MMaaddrriidd,, CCaappee TToowwnn,, SSiinnggaappoorree,, SSaaoo PPaauulloo,, DDeellhhii,, DDuubbaaii,, TTookkyyoo CCaammbbrriiddggee UUnniivveerrssiittyy PPrreessss TThhee EEddiinnbbuurrgghh BBuuiillddiinngg,, CCaammbbrriiddggee CCBB22 88RRUU,, UUKK PPuubblliisshheedd iinn tthhee UUnniitteedd SSttaatteess ooff A Ammeerriiccaa bbyy CCaammbbrriiddggee UUnniivveerrssiittyy PPrreessss,, NNeeww YYoorrkk www.cambridge.org IInnffoorrmmaattiioonn oonn tthhiiss ttiittllee:: www.cambridge.org/9780521451253 ©© CCaammbbrriiddggee UUnniivveerrssiittyy PPrreessss 11999999 TThhiiss ppuubblliiccaattiioonn iiss iinn ccooppyyrriigghhtt.. 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IInnffoorrmmaattiioonn rreeggaarrddiinngg pprriicceess,, ttrraavveell ttiimmeettaabblleess aanndd ootthheerr ffaaccttuuaall iinnffoorrmmaattiioonn ggiivveenn iinn tthhiiss wwoorrkk aarree ccoorrrreecctt aatt tthhee ttiimmee ooff ffiirrsstt pprriinnttiinngg bbuutt CCaammbbrriiddggee UUnniivveerrssiittyy PPrreessss ddooeess nnoott gguuaarraanntteeee tthhee aaccccuurraaccyy ooff ssuucchh iinnffoorrmmaattiioonn tthheerreeaafftteerr.. To Natalia CONTENTS –——————————————————————————— Foreword page xiii Part I Finite Dimensional Control Problems 1 1 Calculus of Variations and Control Theory 3 1.1 Calculus of Variations: Surface of Revolution of Minimum Area 3 1.2 Interpretation of the Results 8 1.3 Mechanics and Calculus of Variations 9 1.4 Optimal Control: Fuel Optimal Landing of a Space Vehicle 11 1.5 Optimal Control Problems Described by Ordinary Differential Equations 12 1.6 Calculus of Variations and Optimal Control. Spike Perturbations 13 1.7 Optimal Control: Minimum Drag Nose Shape in Hypersonic Flow 17 1.8 Control of Functional Differential Equations: Optimal Forest Growth 18 1.9 Control of Partial Differential Equations 20 1.10 Finite Dimensional and Infinite Dimensional Control Problems 25 2 Optimal Control Problems Without Target Conditions 26 2.0 Elements of Measure and Integration Theory 26 2.1 Control Systems Described by Ordinary Differential Equations 42 2.2 Existence Theory for Optimal Control Problems 51 2.3 Trajectories and Spike Perturbations 60 2.4 Cost Functionals and Spike Perturbations 66 2.5 Optimal Control Problems without Target Condition: The Hamiltonian Formalism 67 vii viii Contents 2.6 Invariance of the Hamiltonian 71 2.7 The Linear-Quadratic Problem: Existence and Uniqueness of Optimal Controls 76 2.8 The Unconstrained Linear-Quadratic Problem: Feedback, the Riccati Equation 78 2.9 The Constrained Linear-Quadratic Problem 82 3 Abstract Minimization Problems: The Minimum Principle for the Time Optimal Problem 84 3.1 Abstract Minimization Problems 84 3.2 Ekeland's Variational Principle 87 3.3 The Abstract Time Optimal Problem 92 3.4 The Control Spaces 100 3.5 Continuity of the Solution Map 101 3.6 Continuity of the Solution Operator of the Variational Equation 102 3.7 The Minimum Principle for the Time Optimal Problem 104 3.8 Time Optimal Capture of a Wandering Particle 107 3.9 Time Optimal Stopping of an Oscillator 111 3.10 Higher Dimensional Problems 118 4 The Minimum Principle for General Optimal Control Problems 122 4.1 The Abstract Minimization Problem 122 4.2 The Minimum Principle for Problems with Fixed Terminal Time 125 4.3 Optimal Capture of a Wandering Particle in Fixed Time, I 129 4.4 Singular Intervals and Singular Arcs 137 4.5 Optimal Capture of a Wandering Particle in Fixed Time, II 138 4.6 The Minimum Principle for Problems with Variable Terminal Time 143 4.7 Fuel Optimal Soft Landing of a Space Vehicle 146 4.8 Fuel Optimal Soft Landing of a Space Vehicle 149 4.9 The Linear-Quadratic Problem and the Minimum Drag Nose Shape Problem 151 4.10 Nonlinear Programming Problems: The Kuhn-Tucker Theorem 159 Part II Infinite Dimensional Control Problems 167 5 Differential Equations in Banach Spaces and Semigroup Theory 169 5.0 Banach Spaces and Their Duals. Linear Operators. Integration of Vector Valued Functions 169 5.1 Partial Differential Equations as Ordinary Differential Equations in Banach Spaces 189