Optimal Control Methods: KPI Lectures 2–6 (2013–2014)

This archive joins the five supplied lecture PDFs without altering their pages. It is the main course-specific source for the MOKU ingestion. The PDF metadata names Andrew; KPI curriculum and staff records associate the course with Andrii Pysarenko, but that identification is an inference rather than embedded authorship evidence, so no author is asserted in the registry.

Physical map #

What carries weight #

The durable teaching sequence is geometric. The maximum principle produces an adjoint trajectory; the adjoint produces a switching function; the sign or threshold of that scalar function selects a vertex of the admissible-control set; integrating each constant-control arc produces a curve in the phase plane; and the set of states from which the final arc reaches the target becomes a switching curve. This is much more useful than memorizing “optimal control is bang-bang,” because it explains when that statement is true and how an open-loop necessary condition turns into state feedback.

The course also treats two easily lost subtleties:

• minimum time and minimum $L^1$ effort are different problems even when both produce saturated segments;
• with free terminal time, the infimum of an $L^1$ cost can exist without any finite-time minimizer. A family of progressively longer controls can approach the lower bound, which motivates $\mu$-optimality.

Corrections and boundaries #

• GTK fixes one Hamiltonian convention and translates every sign into it. A maximum and a minimum formulation are equivalent only after the costate/cost sign is changed consistently.
• The oscillator switching curve is piecewise. A circle equation alone does not identify which arc or switching direction belongs to which state.
• The phrase “control energy” for $\int u^T R u\,dt$ is physical only when the control variable and $R$ have the required units and calibration. Otherwise it is a quadratic design penalty.
• Lecture 6's statement that a weight is proportional to the square of a maximum permitted variable is reversed relative to Bryson's normalization. If $z_{i,\max}$ is the acceptable magnitude, the initial diagonal weight is $1/z_{i,\max}^2$.
• A sign/direction in the lower-bound construction in Lecture 5 is internally inconsistent. GTK re-derives the bound from the state integral instead of carrying that line forward.
• The lecture series does not derive the Riccati equation. GTK keeps its fixed-structure frequency-domain calculation separate from unrestricted state-feedback LQR.

Handwritten-note audit #

The 31 explicitly requested photographs under the supplied MOKU/moku folder were read chronologically. They independently confirm the course order and the intended depth: a theorem for switching counts in linear systems, the double-integrator and oscillator constructions, Euclidean-ball vector control, minimum-fuel dead zones and nonattainment, Parseval/determinant tuning, and the finite-horizon differential Riccati equation. They are the student's notebook, so they are not archived, registered, cited, or used as technical evidence.

Visual audit #

All 69 lecture pages were rendered and inspected at readable resolution. Load-bearing visuals include the double-integrator parabolas, oscillator circular arcs, translated tracking geometry, bang-off-bang phase regions, and a coefficient/determinant construction. GTK independently redraws the first four mechanisms as theme-aware SVGs after re-deriving their signs and domains.

The accompanying 133-page departmental laboratory manual was also read completely as topic-discovery material. Its strongest visual candidates were the moving-endpoint sketch, optimal-trajectory fields, bounded-control HJB geometry, and Riccati coefficient flow. None remained clearer at GTK article width than a focused independent redraw, so the manual is not retained or cited under the narrow visual exception.