A short introduction to the control theory in finite-dimensional spaces

The paper proves existence/uniqueness, Pontryagin-type optimality, and the state-feedback form u*(t) = -B^T P_T(t) x*(t) for the finite-horizon LQR, via a convexity argument, an integral optimality identity (Lemma 9.1), and an adjoint/value-function identification y*(t) = P_T(t) x*(t) (Theorem 9.1) . The candidate solution reaches the same conclusions using a standard direct-method existence proof, a variational Lagrange-multiplier derivation of the adjoint, an explicit dynamic programming “splicing” argument, and an envelope-identity to identify y* with the gradient of the value function, hence y*(t)=P_T(t)x*(t), yielding the same feedback law. The paper and model differ mainly in how y*(t)=P_T(t)x*(t) is shown (inner-product identity vs. envelope/gradient), but the results and normalizations coincide; the paper’s Remark 9.6 explicitly situates the proof at the intersection of Pontryagin and dynamic programming, matching the model’s approach in spirit .