On the Convexity of the Value Function for a Class of Nonconvex Variational Problems: Existence and Optimality Conditions

Abstract

In this paper we study a class of perturbed constrained nonconvex variational problems depending on either time/state or time/state's derivative variables. Its (optimal) value function is proved to be convex and then several related properties are obtained. Existence, strong duality results, and necessary/sufficient optimality conditions are established. Moreover, via a necessary optimality condition in terms of Mordukhovich's normal cone, it is shown that local minima are global. Such results are given in terms of the Hamiltonian function. Finally various examples are exhibited showing the wide applicability of our main results.

Keywords

  1. nonconvex variational problems
  2. Lyapunov theorem
  3. existence of minima
  4. Hamiltonian
  5. strong duality
  6. local minima

MSC codes

  1. 49J15
  2. 49J52
  3. 49K15
  4. 49N15
  5. 90C46

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Information

Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 3673 - 3693
ISSN (online): 1095-7138

History

Submitted: 12 May 2014
Accepted: 4 September 2014
Published online: 13 November 2014

Keywords

  1. nonconvex variational problems
  2. Lyapunov theorem
  3. existence of minima
  4. Hamiltonian
  5. strong duality
  6. local minima

MSC codes

  1. 49J15
  2. 49J52
  3. 49K15
  4. 49N15
  5. 90C46

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