SEMICONCAVE FUNCTIONS, HAMILTON-JACOBI EQUATIONS, AND OPTIMAL CONTROL

By ELIZABETH SMITH

CONTENTS A MODEL PROBLEM SEMICONCAVE FUNCTIONS A PROBLEM IN THE CALCULUS OF VARIATIONS THE HOPF FORMULA HAMILTON–JACOBI EQUATIONS METHOD OF CHARACTERISTICS SEMICONCAVITY OF HOPF’S SOLUTION SEMICONCAVITY AND ENTROPY SOLUTIONS SEMICONCAVE FUNCTIONS DE?NITION AND BASIC PROPERTIES EXAMPLES SPECIAL PROPERTIES OF SCL (A) A DIFFERENTIAL HARNACK INEQUALITY A GENERALIZED SEMICONCAVITY ESTIMATE GENERALIZED GRADIENTS AND SEMICONCAVITY GENERALIZED DIFFERENTIALS DIRECTIONAL DERIVATIVES SUPERDIFFERENTIAL OF A SEMICONCAVE FUNCTION MARGINAL FUNCTIONS INF-CONVOLUTIONS PROXIMAL ANALYSIS AND SEMICONCAVITY SINGULARITIES OF SEMICONCAVE FUNCTIONS RECTI?ABILITY OF THE SINGULAR SETS PROPAGATION ALONG LIPSCHITZ ARCS SINGULAR SETS OF HIGHER DIMENSION APPLICATION TO THE DISTANCE FUNCTION HAMILTON–JACOBI EQUATIONS VISCOSITY SOLUTIONS SEMICONCAVITY AND VISCOSITY PROPAGATION OF SINGULARITIES GENERALIZED CHARACTERISTICS CALCULUS OF VARIATIONS EXISTENCE OF MINIMIZERS NECESSARY CONDITIONS AND REGULARITY THE PROBLEM WITH ONE FREE ENDPOINT THE VALUE FUNCTION THE SINGULAR SET OF U RECTI?ABILITY OF OPTIMAL CONTROL PROBLEMS THE MAYER PROBLEM OPTIMALITY CONDITIONS THE BOLZA PROBLEM CONTROL PROBLEMS WITH EXIT TIME OPTIMAL CONTROL PROBLEMS WITH EXIT TIME LIPSCHITZ CONTINUITY AND SEMICONCAVITY SEMICONVEXITY RESULTS IN THE LINEAR CASE