Almost Sure Exponential Stabilization of Stochastic Non-Linear Optimal Control Delay Integro -Differential Equations

This study explores the application of multiplicative Ito-type noise in stabilizing nonlinear optimal control delay differential equations (OCDDES ) that are generally unstable in their deterministic form. The technique applied involves the use of Lyapunov sample exponent and noise perturbation. The equation is perturbed by a multiplicative Ito- type noise to form a stochastic optimal control delay differential equation. The noise scaling parameter in the comparable stochastic optimal control system is replaced with finite integral expression by making it sufficiently as large as possible to stochastically self stabilized the resulting stochastic integro - differential system in an almost sure exponential sense, under additional conditions and sufficiently small time lag. This phenomenon does not occur in deterministic optimal control delay differential equations where noise is absent, since its solutions still admit instability.