Singular mass matrix and redundant constraints in unilaterally constrained Lagrangian and Hamiltonian systems

Abstract

This article deals with the analysis of the contact complementarity problem for Lagrangian systems subjected to unilateral constraints, and with a singular mass matrix and redundant constraints. Previous results by the authors on existence and uniqueness of solutions of some classes of variational inequalities are used to characterize the well-posedness of the contact problem. Criteria involving conditions on the tangent cone and the constraints gradient are given. It is shown that the proposed criteria easily extend to the case where the system is also subjected to a set of bilateral holonomic constraints, in addition to the unilateral ones. In the second part, it is shown how basic convex analysis may be used to show the equivalence between the Lagrangian and the Hamiltonian formalisms when the mass matrix is singular.

Appendices

Appendix A: The Mangasarian–Fromovitz Constraint Qualifications (MFCQ) [19, pp. 17, 252]

Let $K=\{x \in \mathbb {R}^{n} : h(x) \geq0\}$, where $h: \mathbb {R}^{n} \rightarrow \mathbb {R}^{m}$ is continuously differentiable, and K is not necessarily convex. Suppose that there exists a vector $v \in \mathbb {R}^{n}$ such that ∇h _i (x)^T v>0 for all $i \in\mathcal{ I}(x)=\{i : h_{i}(x) =0\}$. Then the tangent cone to K at x, defined as the dual of the normal cone in (14) is equal to the linearization cone $\{z \in \mathbb {R}^{n} : z^{\mathrm{T}}\nabla h_{i}(x) \geq0,\ \mbox{for all} i \in\mathcal{I}(x)\}$.

Appendix B: The chain rule of convex analysis

Theorem 1

[35, Theorem 23.9]

Let f(x)=h(Ax) where h(⋅) is a proper convex function on $\mathbb {R}^{m}$ and A is a linear transformation from $\mathbb {R}^{n}$ to $\mathbb {R}^{m}$. Then if h(⋅) is polyhedral and Im(A) contains a point of dom(h), one has ∂f(x)=A ^T ∂h(Ax), for all x.