Optimal Control of a Nonlinear System with White Noise

In this paper, we propose a control problem for nonlinear stochastic differential equations with noise. The system proposed for the control problem is a nonlinear system perturbed by standard Brownian motion. We write this problem in a coupled form where the control <i>u(t)</i> has a value in a convex space. Here, we propose sufficient minimum conditions on the Hamiltonian function to characterize the mean value of the cost function associated with the optimal control problem. The convexity of the Hamiltonian function is a sufficient condition for the existence of the optimal value of the control function. Under certain regularity assumptions, on the control system functional and on the non-differentiability criterion of Brownian motion, the existence and uniqueness results are established by the Cauchy-Lipschitz criteria. We also analyze the mathematical expectation stability of the system to check whether it will converge to an equilibrium point or not. For the study of this stability, the emphasis has been placed on root-mean-square stability. To highlight the results of our work, we apply this control problem to a SIRS-type epidemiological system for the coronavirus epidemic. To study the stability of this epidemiological system, we construct a Lyaunov function associated with the system and then use the results of Lyapunov's theorem to show the convergence of the system to a stable state.