Ronald J. Stern

Indefinite Trust Region Subproblems and Nonsymmetric Eigenvalue Perturbations

This paper extends the theory of trust region subproblems in two ways: (i) it allows indefinite inner products in the quadratic constraint, and (ii) it uses a two-sided (upper and lower bound) quadratic constraint. Characterizations of optimality are presented that have no gap between necessity and sufficiency. Conditions for the existence of solutions are given in terms of the definiteness of a matrix pencil. A simple dual program is introduced that involves the maximization of a strictly concave function on an interval. This dual program simplifies the theory and algorithms for trust region subproblems. It also illustrates that the trust region subproblems are implicit convex programming problems, and thus explains why they are so tractable.The duality theory also provides connections to eigenvalue perturbation theory. Trust region subproblems with zero linear term in the objective function correspond to eigenvalue problems, and adding a linear term in the objective function is seen to correspond to a p...

Exponential nonnegativity on the ice cream cone

Let

Trust Region Problems and Nonsymmetric Eigenvalue Perturbations

K_n

State Constrained Feedback Stabilization

denote the n-dimensional ice cream cone. This paper investigates the structure of those matrices A such that

Controllability of linear systems with positive controls: Geometric considerations

e^{tA} K_n \subset K_n

Invariant ellipsoidal cones

for all

A note on positively invariant cones

t\geqq 0

Invariance theory for infinite dimensional linear control systems

. The characterizations extend to general ellipsoidal cones.

Lyapunov and feedback characterizations of state constrained controllability and stabilization

A characterization is given for the spectrum of a symmetric matrix to remain real after a nonsymmetric sign-restricted border perturbation, including the case where the perturbation is skew-symmetric. The characterization is in terms of the stationary points of a quadratic function on the unit sphere. This yields interlacing relationships between the eigenvalues of the original matrix and those of the perturbed matrix. As a result of the linkage between the perturbation and stationarity problems, new theoretical insights are gained for each. Applications of the main results include a characterization of those matrices that are exponentially nonnegative with respect to the

Cone reachability for linear differential systems

n