Problem 1
Asymptotic stability under standard regularity and positive-definiteness conditions.
Faculty use
Lyapunov Functions and Stability Certification
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
It may vanish on a larger set; LaSalle-type invariant-set reasoning may be needed.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind using a scalar energy-like function to prove stability without solving trajectories. Use the formulas V(x*)=0, V>0 nearby; V̇=∇V·f≤0.
Problem 4
A strong error analysis identifies the first step that violates classification, algebra, theorem hypotheses, units, initial/boundary data, or numerical assumptions; then repairs that step and re-verifies the final result.
Problem 5
Application responses vary. Require defined variables, a compatible differential-equation model connected to ecology, nonlinear mechanics, chemical oscillations, control, epidemiology, and coupled feedback systems, a complete computation, interpretation, and limitations.
Problem 6
Accept two independent checks appropriate to the task: differentiation/substitution, initial or boundary conditions, dimensional analysis, phase-line or stability evidence, residuals, step refinement, conservation/energy checks, data comparison, or a second method.
Scoring guidance
Six problems × 10 points: 5 mathematical accuracy, 2 structure/justification, 2 verification/interpretation, 1 notation and communication.