Optimal control of infinite dimensional stochastic systems.

Abstract

In this thesis we study a Hamilton-Jacobi-Bellman equation arising from the stochastic optimal control problem. More precisely, we study the following second order parabolic partial differential equation$$(P)\left\{\eqalign{&\phi\sb{t}(t,x)={1\over 2}Tr(S\phi\sb{xx}(t,x))+(Bx + \int(x),\phi\sb{x}(t,x))\cr&\qquad\qquad + F(t,x,\phi(t,x),\phi\sb{x}(t,x))\cr&\phi(0,x)=\phi\sb0(x)\right. \cr}$$Where $\phi\sb0,F$ are given functions, B is the infinitesimal generator of a strongly continuous semigroup, and S is a positive, self-adjoint nuclear operator in a Banach space X (Chapter 3) or an identity operator in ${\cal L}(X\sp\*,X)$ (Chapter 4). The main contributions of this thesis include: (1) A direct method suggested recently by Da Prato (26, 28) has been further developed. This method, which is different from ordinary perturbation theorem in semigroup theory, is more constructive, and can be applied to general non-linear parabolic partial differential equations. (2) Many authors have studied semilinear HJB equations (P), with non-linear term of the form$$F(x,\phi\sb{x})={1\over 2}\vert\phi\sb{x}\vert\sp2 + g(x).$$Da Prato (31) has also studied a HJB equation with nonlinear term $F(S\phi\sb{x}).$ In this thesis we will study the more general nonlinear term $F(t,x,\phi,\phi\sb{x}),$ and without coercivity or convexity assumptions. (3) Most of the results about HJB equations obtained before were in $R\sp{n},$ or in a separable Hilbert space. In this thesis we study HJB equations in a separable reflexive Banach space, hence our results are more general.