4.4 The adjoint system
Theorem 4.4.1. If \(x' = A(t)x\) and \(y' = -A(t)^{\mathsf T}y\), then \(y(t)^{\mathsf T}x(t)\) is constant. If \(\Phi \) is a fundamental matrix of the first system, \((\Phi ^{-1})^{\mathsf T}\) is a fundamental matrix of the second, the adjoint system.
Proof. \((y^{\mathsf T}x)' = y'^{\mathsf T}x + y^{\mathsf T}x' = -y^{\mathsf T}Ax + y^{\mathsf T}Ax = 0\). For the second statement, \(\frac {d}{dt}\Phi ^{-1} = -\Phi ^{-1}A\), so \(\Psi = (\Phi ^{-1})^{\mathsf T}\) satisfies \(\Psi ' = -A^{\mathsf T}\Psi \), and it is invertible. □
The adjoint propagates linear functionals, such as the sensitivity of a final quantity \(c^{\mathsf T}x(T)\) to the state at earlier times, backwards in time. It is the basis of the adjoint method for computing gradients in optimal control and data assimilation, and of the Fredholm alternative for boundary value problems.
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