stochastic-control

Linear control with additive noise.

As an example, we consider the control problem for the case of controlled drift-diffuation:

$$dx = f (x, u, t)dt + G(x, t)dw$$ $$f(x,t) = Ax+Bu, G(x,t)=G$$ $$g(x,u,y) = x^T Q x+u^T R u$$

We can show that stochastic control law has the form of:

$$u(t) = −R^{-1} BT P(t) x(t)$$

And the Ricatti equation for P(t) is:

$$−\dot{P} = A^T P+PA+Q−PBR^{−1}BTP, P(t_f)=M$$