Title:

Optimal control and asymptotics of stochastic delay evolution equations

This thesis mainly studies stochastic neutral differential equations with delays, which can be studied in the fields of existence, uniqueness, controllability and stability of mild solutions. In Chapter 1, we give a short introduction for the materials in each chapter. We introduce the new models we developed. In Chapter 2, we begin by introducing some definitions and results. To present the proofs of all the results here would require preparatory background material, which would significantly increase both the size and scope of this dissertation. Although this chapter introduces very important theorems, required proofs are omitted here. However, these related proofs can be found from book in Liu [41] and you can also find most of these basic mathematical concepts and their proofs in many wellknown text books such as Pazy [32] and Da Prato and Zabczyk [22] or to be found in the literature reviews. In Chapter 3, we will generalise the previous theory to consider a stochastic optimal control problem for a class of neutral type stochastic systems, which is very important from both theoretic and practical point of view (see, e.g., [39]). We formulate a stochastic optimal control problem with the aim of maximising the objective functional at a given time horizon T > 0. This chapter is organised as follows. In Section 3.2, we formulate the optimal problem with the objective functional as an optimal problem with neutral type for an SDDE both in state and the control. In Section 3.3, we use a representation result that allows us to "lift" this nonMarkovian optimisation problem to a Markovian control problem on a Hilbert space and deal with the general case of delays in the state and in the control and the verification result is given. In Section 3.4, we construct an example of a controlled SDDE, whose HJB equation admits an integral solution. Therefore, there exists an optimal control form for the control problem. In Section 3.5, we establish a linear delay differential equation to obtain solutions. In Section 3.6, we have a summary to state the contribution and development of the chapter. In Chapter 4, we will concentrate on the existence and uniqueness of the squaremean almost periodic mild solutions. This chapter is organised as follows. In Section 4.2, we review and introduce some concepts, basic properties of squaremean almost periodicity and the proofs of two theorems. In Section 4.3, under some suitable conditions, we prove the existence and uniqueness of squaremean almost periodic mild solutions for some stochastic differential equations driven by Poisson jumps. In Section 4.4, we have a summary to state the contribution and development of the chapter. In Chapter 5, we study the problem of determining the attracting sets of neutral stochastic partial differential equations driven by astable noise with impulses. Therefore, the techniques and methods for the global attracting set and stability for neutral SPDEs driven by astable processes with impulses should be developed. This chapter is organised as follows. In Section 5.2, we review and introduce the concepts and basic properties of astable processes. In Section 5.3, we study the global attracting set and stability of the stochastic neutral differential equations with impulses. In Section 5.4, we have a summary to state the contribution and development of the chapter. In Chapter 6, we have a conclusion to summarise the contribution and development of this thesis.
