Linear-quadratic differential games: Open and Closed Loop Strategies with or without Singularities
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Bibliographic record
Abstract
The object of this paper is to revisit the results of P. Bernhard (J. Optim. Theory Appl. 27 (1979), 51-69) on two-person zero-sum linear quadratic differential games and generalize them to utility functions without positivity assumptions on the matrices acting on the state variable and to linear dynamics with bounded measurable data matrices. This is done in the light of the recent work of P. Zhang [SIAM J. Control Optim., 43 (2005), pp. 2157-2165] who established the equivalence between the flniteness of the open loop value of a game and the finiteness of its open loop lower and upper values. We consider both open and closed loop strategies. We specialize to state feedback via Lebesgue measurable affine closed loop strategies with possible non L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -integrable singularities. We make the connection with the normality and normalizability of the problem and provide a complete classification of closed loop saddle points in terms of the convexity/concavity properties of the utility function and the open loop lower value, upper value, and value of the game.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it