Necessary optimality conditions for implicit control systems with applications to control of differential algebraic equations

TL;DR

This paper develops necessary optimality conditions for nonlinear, nonsmooth implicit control systems, including differential algebraic equations, accommodating singular Jacobians and broadening the scope of optimal control theory.

Contribution

It introduces new necessary optimality conditions for implicit control systems with singular Jacobians, applicable to high-index differential algebraic equations.

Findings

Conditions hold under weak constraint qualifications

Applicable to semi-explicit DAEs with index higher than one

Addresses systems with singular Jacobian matrices

Abstract

In this paper we derive necessary optimality conditions for optimal control problems with nonlinear and nonsmooth implicit control systems. Implicit control systems have wide applications including differential algebraic equations (DAEs). The challenge in the study of implicit control system lies in that the system may be truly implicit, i.e., the Jacobian matrix of the constraint mapping may be singular. Our necessary optimality conditions hold under the so-called weak basic constraint qualification plus the calmness of a perturbed constraint mapping. Such constraint qualifications allow for singularity of the Jacobian and hence is suitable for implicit systems. Specifying these results to control of semi-explicit DAEs we obtain necessary optimality conditions for control of semi-explicit DAEs with index higher than one.