Separation principle in stochastic control

The separation principle is one of the fundamental principles of stochastic control theory, which states that the problems of optimal control and state estimation can be decoupled under certain conditions. In its most basic formulation it deals with a linear stochastic system d x = A ( t ) x ( t ) d t + B 1 ( t ) u ( t ) d t + B 2 ( t ) d w d y = C ( t ) x ( t ) d t + D ( t ) d w {\displaystyle {\begin{aligned}dx&=A(t)x(t)\,dt+B_{1}(t)u(t)\,dt+B_{2}(t)\,dw\\dy&=C(t)x(t)\,dt+D(t)\,dw\end{aligned}}} with a state process x {\displaystyle x} , an output process y {\displaystyle y} and a control u {\displaystyle u} , where w {\displaystyle w} is a vector-valued Wiener process, x ( 0 ) {\displaystyle x(0)} is a zero-mean Gaussian random vector independent of w {\displaystyle w} , y ( 0 ) = 0 {\displaystyle y(0)=0} , and A {\displaystyle A} , B 1 {\displaystyle B_{1}} , B 2 {\displaystyle B_{2}} , C {\displaystyle C} , D {\displaystyle D} are matrix-valued functions which generally are taken to be continuous of bounded variation.

Source: Wikipedia — Separation principle in stochastic control (CC BY-SA 4.0)

Separation principle in stochastic control

The separation principle is one of the fundamental principles of stochastic control theory, which states that the problems of optimal control and state estimation can be decoupled under certain conditions. In its most basic formulation it deals with a linear stochastic system d x = A ( t ) x ( t ) d t + B 1 ( t ) u ( t ) d t + B 2 ( t ) d w d y = C ( t ) x ( t ) d t + D ( t ) d w {\displaystyle {\begin{aligned}dx&=A(t)x(t)\,dt+B_{1}(t)u(t)\,dt+B_{2}(t)\,dw\\dy&=C(t)x(t)\,dt+D(t)\,dw\end{aligned}}} with a state process x {\displaystyle x} , an output process y {\displaystyle y} and a control u {\displaystyle u} , where w {\displaystyle w} is a vector-valued Wiener process, x ( 0 ) {\displaystyle x(0)} is a zero-mean Gaussian random vector independent of w {\displaystyle w} , y ( 0 ) = 0 {\displaystyle y(0)=0} , and A {\displaystyle A} , B 1 {\displaystyle B_{1}} , B 2 {\displaystyle B_{2}} , C {\displaystyle C} , D {\displaystyle D} are matrix-valued functions which generally are taken to be continuous of bounded variation.

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Source: Wikipedia "Separation principle in stochastic control" · CC BY-SA 4.0

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