Some Recent Results on the Maximum Principle of Optimal Control Theory

The maximum principle of optimal control theory (also known as the “Pontryagin maximum principle”) was announced in the 1958 International Congress of Mathematicians, and presented in detail in the book [17]. The results of [17], however, are only a version of the principle, i.e. one of several possible ways of translating the statement of the principle into rigorous, precise mathematics.

[1]  Alberto Bressan,et al.  On the qualitative theory of lower semicontinuous differential inclusions , 1989 .

[2]  Andrzej Fryszowski,et al.  Continuous version of Filippov-Wažewski relaxation theorem , 1991 .

[3]  R. Gamkrelidze,et al.  THE EXPONENTIAL REPRESENTATION OF FLOWS AND THE CHRONOLOGICAL CALCULUS , 1979 .

[4]  J. Warga,et al.  Controllability, extremality, and abnormality in nonsmooth optimal control , 1983 .

[5]  R. Colombo,et al.  Continuous selections of solution sets to Lipschitzian differential inclusions , 1991 .

[6]  F. Clarke.  The Maximum Principle under Minimal Hypotheses , 1976 .

[7]  J. Warga.  Homeomorphisms and local C 1 approximations , 1988 .

[8]  H. Whitney.  Analytic Extensions of Differentiable Functions Defined in Closed Sets , 1934 .

[9]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[10]  George Leitman,et al.  Topics in optimization , 1967 .

[11]  Héctor J. Sussmann,et al.  A strong maximum principle for systems of differential inclusions , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[12]  Héctor J. Sussmann,et al.  A strong version of the maximum principle under weak hypotheses , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[13]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[14]  J. Warga,et al.  Fat homeomorphisms and unbounded derivate containers , 1981 .

[15]  E. Hille.  An existence theorem , 1924 .

[16]  On differential relations with lower continuous right-hand side. An existence theorem☆ , 1980 .

[17]  Sussmann.  Nonlinear Controllability and Optimal Control , 1990 .

[18]  J. Tougeron,et al.  Idéaux de fonctions différentiables. II , 1970 .

[19]  A. Krener.  The High Order Maximal Principle and Its Application to Singular Extremals , 1977 .

[20]  R. Gamkrelidze,et al.  A SECOND ORDER OPTIMALITY PRINCIPLE FOR A TIME-OPTIMAL PROBLEM , 1976 .

[21]  J. Warga,et al.  Optimization and Controllability without Differentiability Assumptions , 1983 .

[22]  A. Bressan.  Solutions of lower semicontinuous differential inclusions on closed sets , 1983 .