Theory of Optimal Processes I: The Maximum Principle (1960)
The original 40-page primary paper giving the maximum principle for optimal processes, including the adjoint system, pointwise Hamiltonian maximization, endpoint conditions, and linear time-optimal consequences.
Boltyanskii, Gamkrelidze, and Pontryagin's 1960 paper is the historical primary source for the maximum principle used throughout this branch. Its notation predates today's standard textbook conventions, but its logical structure remains recognizable: attach an adjoint to the state dynamics, maximize the Hamiltonian pointwise over the admissible control set, and impose endpoint conditions.
What GTK takes from it #
- The result is a necessary condition for an optimal process, not an automatic certificate of global optimality.
- The maximization is pointwise in time yet coupled globally through the state and adjoint differential equations.
- In linear time-optimal systems, the switching function is an analytic scalar combination of adjoint components; zeros determine switches unless a singular degeneracy occurs.
- Transversality is part of the theorem, not an optional boundary detail.
GTK states a modern normal/abnormal multiplier form and makes the sign convention explicit. It uses Liberzon's later treatment for pedagogy and this paper for the theorem's primary provenance.
Visual audit #
All 40 scanned pages were inspected. The paper is predominantly mathematical prose and displayed equations; no source figure is retained. Formula legibility was checked against the page render because OCR occasionally loses primes and minus signs.