WELL-POSED LINEAR SYSTEMS
CONTENTS INTRODUCTION AND OVERVIEW INTRODUCTION BASIC PROPERTIES OF WELL-POSED LINEAR SYSTEMS MOTIVATION DE?NITIONS AND BASIC PROPERTIES BASIC EXAMPLES OF WELL-POSED LINEAR SYSTEMS TIME DISCRETIZATION THE GROWTH BOUND SHIFT REALIZATIONS THE LAX–PHILLIPS SCATTERING MODEL THE WEISS NOTATION COMMENTS STRONGLY CONTINUOUS SEMIGROUPS NORM CONTINUOUS SEMIGROUPS THE GENERATOR OF A C0 SEMIGROUP THE SPECTRA OF SOME GENERATORS WHICH OPERATORS ARE GENERATORS? THE DUAL SEMIGROUP THE RIGGED SPACES INDUCED BY THE GENERATOR APPROXIMATIONS OF THE SEMIGROUP THE NONHOMOGENEOUS CAUCHY PROBLEM SYMBOLIC CALCULUS AND FRACTIONAL POWERS 3 10 ANALYTIC SEMIGROUPS AND SECTORIAL OPERATORS 3 11 SPECTRUM DETERMINED GROWTH 3 12 THE LAPLACE TRANSFORM AND THE FREQUENCY DOMAIN 3 13 SHIFT SEMIGROUPS IN THE FREQUENCY DOMAIN 3 14 INVARIANT SUBSPACES AND SPECTRAL PROJECTIONS 3 15 COMMENTS THE GENERATORS OF A WELL-POSED LINEAR SYSTEM THE CONTROL OPERATOR DIFFERENTIAL REPRESENTATIONS OF THE STATE THE OBSERVATION OPERATOR THE FEEDTHROUGH OPERATOR THE TRANSFER FUNCTION AND THE SYSTEM NODE OPERATOR NODES EXAMPLES OF GENERATORS DIAGONAL AND NORMAL SYSTEMS 4 10 DECOMPOSITIONS OF SYSTEMS 4 11 COMMENTS COMPATIBLE AND REGULAR SYSTEMS COMPATIBLE SYSTEMS BOUNDARY CONTROL SYSTEMS APPROXIMATIONS OF THE IDENTITY IN THE STATE SPACE EXTENDED OBSERVATION OPERATORS EXTENDED OBSERVATION/FEEDTHROUGH OPERATORS REGULAR SYSTEMS EXAMPLES OF REGULAR SYSTEMS ANTI-CAUSAL, DUAL, AND INVERTED SYSTEMS ANTI-CAUSAL SYSTEMS THE DUAL SYSTEM FLOW-INVERSION TIME-INVERSION TIME-?OW-INVERSION PARTIAL ?OW-INVERSION FEEDBACK STATIC OUTPUT FEEDBACK ADDITIONAL FEEDBACK CONNECTIONS STATE FEEDBACK AND OUTPUT INJECTION THE CLOSED-LOOP GENERATORS REGULARITY OF THE CLOSED-LOOP SYSTEM THE DUAL OF THE CLOSED-LOOP SYSTEM EXAMPLES STABILIZATION AND DETECTION STABILITY STABILIZABILITY AND DETECTABILITY COPRIME FRACTIONS AND FACTORIZATIONS COPRIME STABILIZATION AND DETECTION DYNAMIC STABILIZATION REALIZATIONS MINIMAL REALIZATIONS PSEUDO-SIMILARITY OF MINIMAL REALIZATIONS REALIZATIONS BASED ON FACTORIZATIONS OF THE HANKEL OPERATOR EXACT CONTROLLABILITY AND OBSERVABILITY NORMALIZED AND BALANCED REALIZATIONS RESOLVENT TESTS FOR CONTROLLABILITY AND OBSERVABILITY MODAL CONTROLLABILITY AND OBSERVABILITY SPECTRAL MINIMALITY CONTROLLABILITY AND OBSERVABILITY OF TRANSFORMED SYSTEMS 9 10 TIME DOMAIN TESTS AND DUALITY 9 11 COMMENTS ADMISSIBILITY 10 1 INTRODUCTION TO ADMISSIBILITY 10 2 ADMISSIBILITY AND DUALITY 10 3 THE PALEY–WIENER THEOREM AND H ? 10 4 CONTROLLABILITY AND OBSERVABILITY GRAMIANS 10 5 CARLESON MEASURES 10 6 ADMISSIBLE CONTROL AND OBSERVATION OPERATORS FOR DIAGONAL AND NORMAL SEMIGROUPS 10 7 ADMISSIBLE CONTROL AND OBSERVATION OPERATORS FOR CONTRACTION SEMIGROUPS 10 8 ADMISSIBILITY RESULTS BASED ON THE LAX–PHILLIPS MODEL 10 9 COMMENTS PASSIVE AND CONSERVATIVE SCATTERING SYSTEMS 11 1 PASSIVE SYSTEMS 11 2 ENERGY PRESERVING AND CONSERVATIVE SYSTEMS 11 3 SEMI-LOSSLESS AND LOSSLESS SYSTEMS ISOMETRIC AND UNITARY DILATIONS OF CONTRACTION SEMIGROUPS ENERGY PRESERVING AND CONSERVATIVE EXTENSIONS OF PASSIVE SYSTEMS THE UNIVERSAL MODEL OF A CONTRACTION SEMIGROUP CONSERVATIVE REALIZATIONS ENERGY PRESERVING AND PASSIVE REALIZATIONS THE SPECTRUM OF A CONSERVATIVE SYSTEM 11 10 COMMENTS DISCRETE TIME SYSTEMS THE INTERNAL LINEAR FRACTIONAL TRANSFORM THE CAYLEY AND LAGUERRE TRANSFORMS THE RECIPROCAL TRANSFORM