Normality of generalized Euler-Lagrange conditions for state constrained optimal control problems (Q2814279)

The authors ``consider state constrained optimal control problems in which the cost to minimize comprises both integral and end-point terms, establishing normality of the generalized Euler-Lagrange condition.'' The problem is: NEWLINE\[NEWLINE\text{Minimize }J(x(\cdot))=g(x(S),x(T))+\int_S^TL(t,x(t),\dot x(t))dtNEWLINE\]NEWLINE NEWLINE\[NEWLINE\dot x(t)\in F(x(t))\, \text{ a.e. } t\in [S,T],NEWLINE\]NEWLINE NEWLINE\[NEWLINEx(t)\in A\text{ for all }t\in [S,T],\, x(S)\in E_0,\,\, x(\cdot)\in W^{1,1}([S,T] ;\mathbb R^n).NEWLINE\]NEWLINE ``Simple examples illustrate that the validity of the Euler-Lagrange condition (and related necessary conditions) in normal form depends crucially on the interplay between [the] velocity sets, the left end-point constraint set and the state constraint set. We show that this is actually a common feature for general state constrained optimal control problems, in which the state constraint is represented by closed convex sets and the left end-point constraint is a closed set. [Under] these circumstances, classical constraint qualifications involving the state constraints and the velocity sets cannot be used alone to guarantee normality of the necessary conditions. A key feature of this paper is to prove that the additional information involving tangent vectors to the left end-point and the state constraint sets can be used to establish normality.''NEWLINENEWLINENEWLINE(Quotations taken from the authors' abstract.)