Stochastic Differential Equations: Controllability and Almost Periodicity

Abstract

This dissertation presents fundamental contributions to the theory of infinite-dimensional stochastic differential equations and their optimal control, with particular emphasis on hyperbolic-type systems and almost periodic phenomena. The research establishes profound connections among functional analysis, stochastic analysis, and control theory, thereby addressing long-standing challenges in the study of systems governed by stochastic partial differential equations. In the first part, we develop a comprehensive framework for second-order neutral stochastic differential equations in Hilbert spaces. We establish the existence, uniqueness, and almost periodicity in distribution of mild solutions for a broad class of such equations over the entire real line. The methodology relies on innovative fixed-point arguments in suitable path spaces and introduces a novel generalisation of Grönwall’s inequality capable of treating convolutions on unbounded temporal domains. The second major contribution provides a complete resolution of the stochastic linear– quadratic optimal control problem for hyperbolic systems with multiplicative noise, representing a significant extension beyond classical theory restricted to deterministic systems. We prove the well-posedness, boundedness, and uniqueness of solutions to the associated operator-valued Riccati equation by means of original techniques that combine chronological calculus with a generalised Grönwall–Bihari inequality. The effectiveness of these theoretical developments is demonstrated through applications to almost periodic second-order stochastic differential equations and stochastic wave equations with random forcing, thereby confirming the relevance of the proposed framework to problems in mathematical physics and engineering. By integrating tools from operator theory, stochastic analysis, and harmonic analysis, this work provides a unified analytical toolkit that advances our understanding of infinite-dimensional stochastic dynamics.