Abstract
Sufficiency conditions for Stackelberg strategies for a class of deterministic differential games are derived when the players have recall of the previous trajectory. Sufficient conditions for Nash strategies when the players have recall of the trajectory are also derived. The state equation is linear, and the cost functional is quadratic. The admissible strategies are restricted to be affine in the information available.
Similar content being viewed by others
References
Starr, A. W., andHo, Y. C.,Nonzero-Sum Differential Games, Journal of Optimization Theory and Applications, Vol. 3, pp. 184–206, 1969.
Halanay, A.,Differential Games with Delay, SIAM Journal on Control, Vol. 6, pp. 579–593, 1968.
Ciletti, M. D.,Results in the Theory of Linear Differential Games with an Information Time Lag, Journal of Optimization Theory and Applications, Vol. 5, pp. 347–362, 1970.
Ciletti, M. D.,New Results in the Theory of Differential Games with Information Time Lag, Journal of Optimization Theory and Applications, Vol. 8, pp. 287–315, 1971.
Cruz, Jr., J. B.,Leader-Follower Strategies for Multilevel Systems, IEEE Transactions on Automatic Control, Vol. AC-23, pp. 244–255, 1978.
Papavassilopoulos, G. P., Medanic, J. V., andCruz, Jr., J. B.,On the Existence of Nash Strategies and Solutions to Coupled Riccati Equations in Linear Quadratic Games, Journal of Optimization Theory and Applications, Vol. 28, pp. 49–76, 1979.
Papavassilopoulos, G. P., andCruz, Jr., J. B.,Nonclassical Control Problems and Stackelberg Games, IEEE Transactions on Automatic Control, Vol. AC-24, pp. 155–166, 1979.
Jameson, A., andKreindler, E.,Inverse Problem of Linear Optimal Control, SIAM Journal on Control, Vol. 11, pp. 1–19, 1973.
Basar, T., andSelbuz, H.,Closed-Loop Stackelberg Strategies with Applications in the Optimal Control of Multilevel Systems, IEEE Transactions on Automatic Control, Vol. AC-24, pp. 166–179, 1979.
Banks, H. T.,Variational Problems Involving Functional Differential Equations, SIAM Journal on Control, Vol. 7, pp. 1–17, 1969.
Banks, H. T., andKent, G. A.,Control of Functional Differential Equations of Retarded and Neutral Type to Target Sets in Function Space, SIAM Journal on Control, Vol. 110, pp. 567–593, 1972.
Cameron, R. H., andMartin, W. T.,An Unsymmetric Fubini Theorem, Bulletin of the American Mathematical Society, Vol. 47, pp. 121–125, 1941.
Riesz, F., andNagy, B. Sz.,Functional Analysis, Frederick Ungar Publishing Company, New York, New York, 1955.
Natanson, I. P.,Theory of Functions of a Real Variable, Frederick Ungar Publishing Company, New York, New York, 1961.
Luenberger, D. G.,Optimization by Vector Space Methods, John Wiley and Sons, New York, New York, 1969.
Author information
Authors and Affiliations
Additional information
Communicated by G. Leitmann
This work was supported in part by the Joint Services Electronics Program under Contract No. N00014-79-C-0424, in part by the National Science Foundation under Grant No. ECS-79-19396, and in part by Department of Energy under Contract No. EX-76-C-01-2088.
Rights and permissions
About this article
Cite this article
Papavassilopoulos, G.P., Cruz, J.B. Sufficient conditions for Stackelberg and Nash strategies with memory. J Optim Theory Appl 31, 233–260 (1980). https://doi.org/10.1007/BF00934113
Issue Date:
DOI: https://doi.org/10.1007/BF00934113