OPTIMIZATION OF STOCHASTIC SYSTEMS
CONTENTS INTRODUCTION WHY STOCHASTIC DISTRIBUTED PARAMETER CONTROL SYSTEMS? TWO FUNDAMENTAL ISSUES IN CONTROL THEORY RANGE INCLUSION AND THE DUALITY ARGUMENT TWO BASIC METHODS IN THIS BOOK NOTES AND COMMENTS SOME PRELIMINARIES IN STOCHASTIC CALCULUS MEASURES AND PROBABILITY, MEASURABLE FUNCTIONS AND RANDOM VARIABLES INTEGRALS AND EXPECTATION SIGNED/VECTOR MEASURES, CONDITIONAL EXPECTATION SIGNED MEASURES DISTRIBUTION, DENSITY AND CHARACTERISTIC FUNCTIONS VECTOR MEASURES CONDITIONAL EXPECTATION A RIESZ-TYPE REPRESENTATION THEOREM PROOF OF THE NECESSITY FOR A SPECIAL CASE PROOF OF THE NECESSITY FOR THE GENERAL CASE PROOF OF THE SU?CIENCY A SEQUENTIAL BANACH-ALAOGLU-TYPE THEOREM IN THE OPERATOR VERSION STOCHASTIC PROCESSES STOPPING TIMES MARTINGALES REAL VALUED MARTINGALES VECTOR-VALUED MARTINGALES BROWNIAN MOTIONS BROWNIAN MOTIONS IN FINITE DIMENSIONS CONSTRUCTION OF BROWNIAN MOTIONS IN ONE DIMENSION VECTOR-VALUED BROWNIAN MOTIONS 2 10 STOCHASTIC INTEGRALS 2 10 1 ITˆO’S INTEGRALS W R T BROWNIAN MOTIONS IN FINITE DIMENSIONS 2 10 2 ITˆO’S INTEGRALS W R T VECTOR-VALUED BROWNIAN MOTIONS 2 11 PROPERTIES OF STOCHASTIC INTEGRALS 2 11 1 ITˆO’S FORMULA FOR ITˆO’S PROCESSES (IN A STRONG FORM) 2 11 2 BURKHOLDER-DAVIS-GUNDY INEQUALITY 2 11 3 STOCHASTIC FUBINI THEOREM 2 11 4 ITˆO’S FORMULA FOR ITˆO’S PROCESSES IN A WEAK FORM 2 11 5 MARTINGALE REPRESENTATION THEOREM 2 12 NOTES AND COMMENTS STOCHASTIC EVOLUTION EQUATIONS STOCHASTIC EVOLUTION EQUATIONS IN FINITE DIMENSIONS WELL-POSEDNESS OF STOCHASTIC EVOLUTION EQUATIONS NOTIONS OF SOLUTIONS WELL-POSEDNESS IN THE SENSE OF MILD SOLUTION REGULARITY OF MILD SOLUTIONS TO STOCHASTIC EVOLUTION EQUATIONS BURKHOLDER-DAVIS-GUNDY TYPE INEQUALITY AND TIME REGULARITY SPACE REGULARITY BACKWARD STOCHASTIC EVOLUTION EQUATIONS THE CASE OF FINITE DIMENSIONS AND NATURAL ?LTRATION THE CASE OF IN?NITE DIMENSIONS WELL-POSEDNESS IN THE SENSE OF MILD SOLUTION FOR THE CASE OF NATURAL FILTRATION THE CASE OF GENERAL FILTRATION THE CASE OF NATURAL FILTRATION REVISITED CONTROL PROBLEMS FOR STOCHASTIC DISTRIBUTED PARAMETER SYSTEMS AN EXAMPLE OF CONTROLLED STOCHASTIC DI?ERENTIAL EQUATIONS CONTROL SYSTEMS GOVERNED BY STOCHASTIC PARTIAL DI?ERENTIAL EQUATIONS SOME CONTROL PROBLEMS FOR STOCHASTIC DISTRIBUTED PARAMETER SYSTEMS CONTROLLABILITY FOR STOCHASTIC DI?ERENTIAL EQUATIONS IN FINITE DIMENSIONS THE CONTROL SYSTEMS WITH CONTROLS IN BOTH DRIFT AND DI?USION TERMS CONTROL SYSTEM WITH A CONTROL IN THE DRIFT TERM LACK OF ROBUSTNESS FOR NULL/APPROXIMATE CONTROLLABILITY CONTROLLABILITY FOR STOCHASTIC LINEAR EVOLUTION EQUATIONS FORMULATION OF THE PROBLEMS REDUCTION TO THE OBSERVABILITY OF DUAL PROBLEMS EXPLICIT FORMS OF CONTROLS FOR THE CONTROLLABILITY PROBLEMS RELATIONSHIP BETWEEN THE FORWARD AND THE BACKWARD CONTROLLABILITY THE CASE OF BOUNDED CONTROL OPERATORS THE CASE OF UNBOUNDED CONTROL OPERATORS EXACT CONTROLLABILITY FOR STOCHASTIC TRANSPORT EQUATIONS FORMULATION OF THE PROBLEM AND THE MAIN RESULT HIDDEN REGULARITY AND A WEIGHTED IDENTITY OBSERVABILITY ESTIMATE FOR BACKWARD STOCHASTIC TRANSPORT EQUATIONS CONTROLLABILITY AND OBSERVABILITY OF STOCHASTIC PARABOLIC SYSTEMS CONTROLLABILITY OF A CLASS OF STOCHASTIC PARABOLIC SYSTEMS PRELIMINARIES PROOF OF THE NULL CONTROLLABILITY PROOF OF THE APPROXIMATE CONTROLLABILITY CONTROLLABILITY OF A CLASS OF STOCHASTIC PARABOLIC SYSTEMS BY ONE CONTROL PROOF OF THE NULL CONTROLLABILITY RESULT PROOF OF THE NEGATIVE NULL CONTROLLABILITY RESULT CARLEMAN ESTIMATE FOR A STOCHASTIC PARABOLIC-LIKE OPERATOR OBSERVABILITY ESTIMATE FOR STOCHASTIC PARABOLIC EQUATIONS GLOBAL CARLEMAN ESTIMATE FOR STOCHASTIC PARABOLIC EQUATIONS, I GLOBAL CARLEMAN ESTIMATE FOR STOCHASTIC PARABOLIC EQUATIONS, II PROOF OF THE OBSERVABILITY RESULT NULL AND APPROXIMATE CONTROLLABILITY OF STOCHASTIC PARABOLIC EQUATIONS GLOBAL CARLEMAN ESTIMATE FOR BACKWARD STOCHASTIC PARABOLIC EQUATIONS PROOF OF THE OBSERVABILITY ESTIMATE FOR BACKWARD STOCHASTIC PARABOLIC EQUATIONS EXACT CONTROLLABILITY FOR A RE?NED STOCHASTIC WAVE EQUATION 10 1 FORMULATION OF THE PROBLEM 10 2 WELL-POSEDNESS OF STOCHASTIC WAVE EQUATIONS WITH BOUNDARY CONTROLS 10 3 MAIN CONTROLLABILITY RESULTS 10 4 A REDUCTION OF THE EXACT CONTROLLABILITY PROBLEM 10 5 A FUNDAMENTAL IDENTITY FOR STOCHASTIC HYPERBOLIC-LIKE OPERATORS 10 6 OBSERVABILITY ESTIMATE FOR THE STOCHASTIC WAVE EQUATION 10 7 NOTES AND COMMENTS EXACT CONTROLLABILITY FOR STOCHASTIC SCHR¨ODINGER EQUATIONS 11 1 FORMULATION OF THE PROBLEM AND THE MAIN RESULT 11 2 WELL-POSEDNESS OF THE CONTROL SYSTEM 11 3 A FUNDAMENTAL IDENTITY FOR STOCHASTIC SCHR¨ODINGER-LIKE OPERATORS 11 4 OBSERVABILITY ESTIMATE FOR BACKWARD STOCHASTIC SCHR¨ODINGER EQUATIONS 11 5 NOTES AND COMMENTS PONTRYAGIN-TYPE STOCHASTIC MAXIMUM PRINCIPLE AND BEYOND 12 1 FORMULATION OF THE OPTIMAL CONTROL PROBLEM 12 2 THE CASE OF FINITE DIMENSIONS 12 3 NECESSARY CONDITION FOR OPTIMAL CONTROLS FOR CONVEX CONTROL REGIONS 12 4 OPERATOR-VALUED BACKWARD STOCHASTIC EVOLUTION EQUATIONS 12 4 1 NOTIONS OF SOLUTIONS 12 4 2 PRELIMINARIES 12 4 3 PROOF OF THE UNIQUENESS RESULTS 12 4 4 WELL-POSEDNESS RESULT FOR A SPECIAL CASE 12 4 5 PROOF OF THE EXISTENCE AND STABILITY FOR THE GENERAL CASE 12 4 6 A REGULARITY RESULT 12 5 PONTRYAGIN-TYPE MAXIMUM PRINCIPLE 12 6 SU?CIENT CONDITION FOR OPTIMAL CONTROLS 12 6 1 CLARKE’S GENERALIZED GRADIENT 12 6 2 A SU?CIENT CONDITION FOR OPTIMAL CONTROLS 12 7 SECOND ORDER NECESSARY CONDITION FOR OPTIMAL CONTROLS 12 8 NOTES AND COMMENTS LINEAR QUADRATIC OPTIMAL CONTROL PROBLEMS 13 1 FORMULATION OF THE PROBLEM 13 2 OPTIMAL FEEDBACK FOR DETERMINISTIC LQ PROBLEM IN FINITE DIMENSIONS 13 3 OPTIMAL FEEDBACK FOR STOCHASTIC LQ PROBLEM IN FINITE DIMENSIONS 13 3 1 DI?ERENCES BETWEEN DETERMINISTIC AND STOCHASTIC LQ PROBLEMS IN FINITE DIMENSIONS 13 3 2 CHARACTERIZATION OF OPTIMAL FEEDBACKS FOR STOCHASTIC LQ PROBLEMS IN FINITE DIMENSIONS 13 4 FINITENESS AND SOLVABILITY OF PROBLEM (SLQ) 13 5 PONTRYAGIN-TYPE MAXIMUM PRINCIPLE FOR PROBLEM (SLQ) 13 6 TRANSPOSITION SOLUTIONS TO OPERATOR-VALUED RICCATI EQUATIONS 13 7 EXISTENCE OF OPTIMAL FEEDBACK OPERATOR FOR PROBLEM (SLQ) 13 8 SOLVABILITY OF OPERATOR-VALUED BACKWARD STOCHASTIC RICCATI EQUATIONS 13 8 1 SOME PRELIMINARY RESULTS 13 8 2 PROOF OF THE MAIN SOLVABILITY RESULT 13 9 SOME EXAMPLES 13 9 1 LQ PROBLEMS FOR STOCHASTIC WAVE EQUATIONS 13 9 2 LQ PROBLEMS FOR STOCHASTIC SCHR¨ODINGER EQUATIONS 13 10NOTES AND COMMENTS REFERENCES