Keywords

Evolution inclusions; viability; path-dependent partial differential equations; contingent solutions; viscosity solutions.

Abstract

In this thesis we first study the necessary and sufficient conditions for differential inclusion driven by locally monotone operator in the sense of Liu and Röckner [J. Funct. Anal., 259 (2010), pp. 2902-2922]. The main result of our work provides a necessary and sufficient condition for a set K being viable, that is, a solution of the differential inclusion stay inside of the set K through out the time interval. We then examine the path-dependent Hamilton-Jacobi-Bellman equations (path-dependent HJB equations) associated to the optimal control problem with states controlled by evolution inclusions. With our results on viability, we prove existence and uniqueness of l.s.c.~viscosity solutions, an appropriate nonsmooth solutions for the path-dependent HJB equation that only is lower semicontinuous. The wellposedness of l.s.c.~viscosity solutions leads to the characterization that the value function below is the unique solution to the path-dependent HJB equation.

Completion Date

2025

Semester

Summer

Committee Chair

Yong, Jiongmin

Degree

Doctor of Philosophy (Ph.D.)

College

College of Sciences

Department

Department of Mathematics

Format

PDF

Identifier

DP0029563

Language

English

Document Type

Thesis

Campus Location

Orlando (Main) Campus

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