Define adjoint state
a(t)=āz(t)āLossā(1) From t to t+ϵ (ϵ change in time) we have
z(t+ϵ)=ā«tt+ϵāf(z(t),t,Īø)āt+z(t)=Tϵā(z(t),t)(2) And because of chain rule ( āxāyā=āuāyāāxāuā )
a(t)=a(t+ϵ)āz(t)āTϵā(z(t),t)ā(3) Take the definition of derivative:
ātāa(t)ā=ϵā0limāϵa(t+ϵ)āa(t)ā(4) Substitue (3) in (4)
ātāa(t)ā=ϵā0limāϵa(t+ϵ)āa(t+ϵ)āz(t)āTϵā(z(t),t)āā(5) ātāa(t)ā=ϵā0limāϵa(t+ϵ)āa(t+ϵ)āz(t)āāTϵā(z(t),t)ā(6) Taylor series around z(t) in (6)
ātāa(t)ā=ϵā0limāϵa(t+ϵ)āa(t+ϵ)āz(t)āā(z(t)+ϵf(z(t),t,Īø)+O(ϵ2))ā(7) aka Tϵā(z(t),t) to z(t)+ϵf(z(t),t,Īø)+O(ϵ2) when limϵā0ā
aka when ϵ change in time is small, take range ϵā0=ϵ and become ϵf(z(t),t,Īø), to make up for the loss add O(ϵ2) at the end (notice it is related to ϵ)
Expand (7)
ātāa(t)ā=ϵā0limāϵa(t+ϵ)āa(t+ϵ)(āz(t)āāz(t)+āz(t)āāϵf(z(t),t,Īø)+O(ϵ2))ā(8) aka āz(t)āāz(t)=I
ātāa(t)ā=ϵā0limāϵa(t+ϵ)āa(t+ϵ)(I+āz(t)āāϵf(z(t),t,Īø)+O(ϵ2))ā(9) ātāa(t)ā=ϵā0limāϵa(t+ϵ)āa(t+ϵ)(I+ϵāz(t)āf(z(t),t,Īø)ā+O(ϵ2))ā(10) ātāa(t)ā=ϵā0limāϵāa(t+ϵ)ϵāz(t)āf(z(t),t,Īø)ā+O(ϵ2)ā(11) aka a(t+ϵ)āa(t+ϵ)I=0
ātāa(t)ā=ϵā0limāāa(t+ϵ)āz(t)āf(z(t),t,Īø)ā+O(ϵ)(12) and because limϵā0ā
ātāa(t)ā=āa(t)āz(t)āf(z(t),t,Īø)ā(13)