For more than forty years, the equation y'(t) = Ay(t) u(t) in Banach spaces has been used as model for optimal control processes described by partial differential equations, in particular heat and diffusion processes. Many of the outstanding open problems, however, have remained open until recently, and some have never been solved. This book is a survey of all results know to the author, with emphasis on very recent results (1999 to date).The book is restricted to linear equations and two particular problems (the time optimal problem, the norm optimal problem) which results in a more focused and concrete treatment. As experience shows, results on linear equations are the basis for the treatment of their semilinear counterparts, and techniques for the time and norm optimal problems can often be generalized to more general cost functionals.The main object of this book is to be a state-of-the-art monograph on the theory of the time and norm optimal controls for y'(t) = Ay(t) u(t) that ends at the very latest frontier of research, with open problems and indications for future research.Key features:· Applications to optimal diffusion processes. · Applications to optimal heat propagation processes. · Modelling of optimal processes governed by partial differential equations. · Complete bibliography. · Includes the latest research on the subject. · Does not assume anything from the reader except basic functional analysis. · Accessible to researchers and advanced graduate students alike· Applications to optimal diffusion processes. · Applications to optimal heat propagation processes. · Modelling of optimal processes governed by partial differential equations. · Complete bibliography. · Includes the latest research on the subject. · Does not assume anything from the reader except basic functional analysis. · Accessible to researchers and advanced graduate students alike
PREFACE CHAPTER 1: INTRODUCTIONP> 1.1 Finite dimensional systems: the maximum principle. 1.2. Finite dimensional systems: existence and uniqueness. 1.3. Infinite dimensional systems. CHAPTER 2: SYSTEMS WITH STRONGLY MEASURABLE CONTROLS, I 2.1. The reachable space and the bang-bang property 2.2. Reversible systems 2.3. The reachable space and its dual, I 2.4. The reachable space and its dual, II 2.5. The maximum principle 2.6. Vanishing of the costate and nonuniqueness for norm optimal controls 2.7. Vanishing of the costate for time optimal controls 2.8. Singular norm optimal controls 2.9. Singular norm optimal controls and singular functionals CHAPTER 3: SYSTEMS WITH STRONGLY MEASURABLE CONTROLS, II 3.1. Existence and uniqueness of optimal controls 3.2. The weak maximum principle and the time optimal problem 3.3. Modeling of parabolic equations 3.4. Weakly singular extremals 3.5. More on the weak maximum principle 3.6. Convergence of minimizing sequences and stability of optimal controls CHAPTER 4: OPTIMAL CONTROL OF HEAT PROPAGATION 4.1. Modeling of parabolic equations 4.2. Adjoints 4.3. Adjoint semigroups 4.4. The reachable space 4.5. The reachable space and its dual, I 4.6. The reachable space and its dual, II 4.7. The maximum principle 4.8. Existence, uniqueness and stability of optimal controls 4.9. Examples and applications CHAPTER 5: OPTIMAL CONTROL OF DIFFUSIONS 5.1. Modeling of parabolic equations 5.2. Dual spaces 5.3. The reachable space and its dual 5.4. The maximum principle 5.5. Existence of optimal controls; uniqueness and stability of supports 5.6. Examples and applications. CHAPTER 6: APPENDIX 6.1 Self adjoint operators, I 6.2 Self adjoint operators, II 6.3 Related research REFERENCES NOTATION AND SUBJECT INDEX.
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