The adjoint arc in nonsmooth optimization

Abstract

Abstract. We extend the theory of necessary conditions for nonsmooth problems of Bolza in three ways: first, we incorporate state constraints of the intrinsic type x(t) € X(t) for all t ; second, we make no assumption of calmness or normality; and third, we show that a single adjoint function of bounded variation simultaneously satisfies the Hamiltonian inclusion, the Euler-Lagrange inclusion, and the Weierstrass-Pontryagin maximum condition, along with the usual transversality relations.

Necessary conditions for the neutral problem of Bolza with continuously varying time delay

“…The generalized problem of Bolza was introduced by Rockafellar [14,15], and covers a broad class of control systems by using infinite penalization [1,8,9]. In [6], we considered the problem of Bolza with delay in the state variable.…”

“…Indeed, (7) says that x(t) := u(t) is absolutely continuous, (8) implies that γ is the endpoint x(T ), and (9)-(10) together say that…”

“…A relationship between the calculus of variations and control systems was established by Rockafellar by introducing the generalized problem of Bolza [15] as a means to study a broad class of control systems. The generalized problem of Bolza uses the technique of infinite penalization [1,8,9,17] in order to incorporate constraints. Given the use of infinite penalization, smoothness is no longer a viable assumption for the Lagrangian involved.…”

“…Given the use of infinite penalization, smoothness is no longer a viable assumption for the Lagrangian involved. There are a number of results in the literature concerning necessary conditions for the generalized problem of Bolza in the nondelay case [1,2,8,9,13] and in the state-delay case [6,10]. Other nonsmooth neutral optimal control problems have been studied in recent work by Mordukhovich and Wang [11,12].…”

Regularity Properties of Attainable Sets Under State Constraints

“…2,7,9,11,13 In constrained problems there are singularity effects for the trajectories that touch the boundary of Ω. These effects are due to an additional term (containing a measure) that appears in the Maximum Principle.…”