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Optimal Control of Unbounded Stochastic Evolution Systems in Hilbert Spaces

Shanjian Tang, Jianjun Zhou

Abstract

Optimal control and the associated second-order Hamilton-Jacobi-Bellman (HJB) equation are studied for unbounded stochastic evolution systems in Hilbert spaces. A new notion of viscosity solution, featured by absence of B-continuity, is introduced for the second-order HJB equation in the sense of Crandall and Lions, and is shown to coincide with the classical solutions and to satisfy a stability property. The value functional is proved to be the unique continuous viscosity solution to the second-order HJB equation, with the coefficients being not necessarily B-continuous. Our result provides a new theory of viscosity solutions to the HJB equation for optimal control of stochastic evolutionary equations-driven by a linear unbounded operator-in a Hilbert space, and removes the B-continuity assumption on the coefficients which is used in the existing literature.

Optimal Control of Unbounded Stochastic Evolution Systems in Hilbert Spaces

Abstract

Optimal control and the associated second-order Hamilton-Jacobi-Bellman (HJB) equation are studied for unbounded stochastic evolution systems in Hilbert spaces. A new notion of viscosity solution, featured by absence of B-continuity, is introduced for the second-order HJB equation in the sense of Crandall and Lions, and is shown to coincide with the classical solutions and to satisfy a stability property. The value functional is proved to be the unique continuous viscosity solution to the second-order HJB equation, with the coefficients being not necessarily B-continuous. Our result provides a new theory of viscosity solutions to the HJB equation for optimal control of stochastic evolutionary equations-driven by a linear unbounded operator-in a Hilbert space, and removes the B-continuity assumption on the coefficients which is used in the existing literature.
Paper Structure (10 sections, 23 theorems, 360 equations)

This paper contains 10 sections, 23 theorems, 360 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some Notations
  4. Borwein-Preiss variational principle
  5. Stochastic optimal control problems
  6. Viscosity solutions to HJB equations: existence
  7. Viscosity solutions to PHJB equations: Crandall-Ishii lemma
  8. Viscosity solutions to HJB equations: uniqueness
  9. Properties of the value functional
  10. Consistency and stability for viscosity solutions

Key Result

Lemma 2.3

Let $t\in[0,T)$ and $f:[t,T]\times H\rightarrow K$ be given. If $f\in C^{0+}([t,T]\times H,K)$ (resp., $USC^{0+}([t,T]\times H,K)$, $LSC^{0+}([t,T]\times H,K)$), then $f\in C^0([t,T]\times H,K)$ (resp., $USC^0([t,T]\times H,K)$, $LSC^0([t,T]\times H,K)$).

Theorems & Definitions (60)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Definition 3.1
  • Lemma 3.3
  • Lemma 3.4
  • ...and 50 more