Study of deterministic and stochastic differential equations with applications in control problems (PhD)

Abstract

The present research work deals with the investigation of various kinds of deterministic and stochastic differential equation sinfiniteaswella sininfinite-dimensional spaces. This work provides insight into the different types of controllability, existence, uniqueness, stability and existence of optimal controls for non-instantaneous impulsive differential equations of order one, two and non-integer. We establish the controllability of non-autonomous nonlinear differential equations with non-instantaneous impulses in the space Rn by using a new piecewise control function. also, some significant results on various kinds of controllability fo fractional differential equation with non-instantaneous impulses and state-dependent delay have been investigated. Further, we establish necessary and sufficient conditions for the existence, uniqueness, stability and controllability of non-instantaneous impulsive stochastic differential equations driven by mixed fractional Brownian motion with Hurst parameter H ∈(1/2,1). Next, we study the approximate controllability for a class of non-instantaneous impulsive fractional stochastic differential equations driven by fractional Brownian motion in a Hilbert space. The existence of mild solutions and optimal controls for a new class of second-order stochastic differential equation driven by mixed fractional Brownian motion with non-instantaneous impulses have been established, which has not been proposed so far to the best of our knowledge. Finally, we discuss the optimal control problem for a system governed by fractional differential equation in a real Hilbert space. The optimal pair is obtained as the limit of the optimal pair sequence of the unconstrained problem and also, we derive some approximation results, which guarantee the convergence of the numerical method to optimal pair sequence. Several examples have been provided in order to make our theoretical analysis more concrete. In this work, the main techniques used are fixed point theorems, semigroup theory, q-resolvent family, fractional calculus and stochastic analysis theory.