Abstract
In previous work dating back to the early 1970’s F.H. Clarke and the author had independently derived necessary conditions for minimum including a maximum principle for optimal control problems defined by ordinary differential equations in which the right hand side f(t,·, r) and functions defining side conditions are Lipschitz continuous in their dependence on the state variable. Our results, though not the methods, were similar in the formulation of the maximum principle in which the nonexisting derivative f v (t, v, σ) was replaced by an unknown element of Clarke’s generalized Jacobian but differed in handling some side conditions. In the present paper we exhibit a maximum principle in which the dual variables and the related functions are limits of appropriate subsequences of computable sequences.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Clarke, F.H.: Optimization and nonsmooth analysis. Wiley Interscience, New York (1983)
Friedrichs, K.O.: The identity of weak and strong extensions of differential operators. Trans. Am. Math. Soc. 55(1), 132–151 (1944)
Soboleff, S.L.: Sur un thèoréme d’analyse fonctionnelle. Matematicheskii Sbornik 4(46)(3), 472–497, Zbl 0022.14803 (1938) (in Russian, with French summary)
Warga, J.: Optimal control of differential and functional equations. Academic, New York (1972)
Warga, J.: Necessary conditions without differentiability assumptions in optimal control. J. Differ. Equ. 18, 41–62 (1975)
Warga, J.: Fat homeomorpgisms and unbounded derivate containers. J. Math. Anal. Appl. 81, 545–560 (1981); Errata. 90, 582–583 (1982)
Warga, J.: Controllability, estremality and abnormality in nonsmooth optimal control. J. Optim. Theory Appl. 41, 239–260 (1983)
Author information
Authors and Affiliations
Corresponding author
Additional information
This is the last paper by Professor Warga. It was mathematically completed by him right before his death in June 2011 and prepared for publication by Q. J. Zhu. Please send any correspondence to zhu@wmich.edu.
Rights and permissions
About this article
Cite this article
Warga, J. Relaxed Derivatives and Extremality Conditions in Optimal Control. Set-Valued Var. Anal 20, 467–475 (2012). https://doi.org/10.1007/s11228-011-0196-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11228-011-0196-6