Existence, Unicité et Contrôlabilité des équations différentielles stochastiques

Abstract

The research reported in this thesis deals with the problem of fractional stochastic differential equations. We formulated and proved the set of sufficient conditions for the controllability of semilinear fractional stochastic differential systems with nonlocal conditions in Hilbert spaces. We have discussed the existence and uniqueness result for a neutral stochastic fractional differential equations involving nonlocal initial conditions, we also investigated a class of dynamic control systems described by semilinear fractional stochastic differential equations of order 1 < q < 2 and Sobolev-type fractional functional stochastic integro-differential systems. The main results are obtained by means of the theory of operators semi-group, fractional calculus, fixed point technique and stochastic analysis theory. The approximate controllability has also been investigated for this class of fractional stochastic functional differential equations. In all the resuts, a new set of sufficient conditions are derived under the assumption that the corresponding linear system is approximately controllable.