Lipschitzian regularity of minimizers for optimal control problems with control-affine dynamics. (English) Zbl 0961.49021

Sarychev, A. V.; Torres, D. F. M.

Appl. Math. Optimization 41, No. 2, 237-254 (2000).

The paper deals with the Lagrange optimal control problem with a cost functional \(\int_a^b L(t,x,\dot{x}) dt\) and control-affine dynamics \(\dot{x}=f(t,x)+g(t,x)u\). The main object of study is the boundedness of optimal controls \(u\), corresponding to the Lipschitz regularity of minimizers for the basic problem of the calculus of variations. The main result of the paper is Theorem 1, where boundedness of controls is achieved reducing first the original problem to an autonomous time-optimal control problem, and then applying the Pontryagin maximum principle. This method has been first used, for the purpose of proving existence, by Gamkrelidze. Applications to the regularity of solutions of the basic problem of the calculus of variations (comparing the results to those of Clarke and Vinter) and to variational problems involving higher order derivatives are given.

Reviewer: L.Ambrosio (Pisa)

MSC:

49N60	Regularity of solutions in optimal control
49J15	Existence theories for optimal control problems involving ordinary differential equations

Keywords:

Lipschitz regularity; Pontryagin maximum principle; optimal control