This paper, motivated by transport theory, deals with spectral properties of operators G on a complex Hilbert space H such that SG is self-adjoint where S is a nonnegative operator: We give several lower bounds of the spectral radius of G and determine the latter in some cases. We derive the whole spectrum for power compact G by means of Lagrange multiplier theory. We find out spectral connections between G and SG. We give a (spectral) stability estimate for symmetrizable operators in terms of the spectral radius of the perturbation.

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Nonlinear equations, Optimization problems, Quasi-Newton methods, Rate of convergence, Linear convergence, Superlinear convergence, Hilbert space, Matrix equations, Algebraic Riccati equation,

We consider an abstract optimal control problem with additional equality and inequality state and control constraints, we use the exterior penalty function to transform the constrained optimal control problem into a sequence of unconstrained optimal control problems, under conditions in control lie in L 1 , the sequence of the solution to the unconstrained problem contains a subsequence converging of the solution of constrained problem, this convergence is strong when the problem is non convex, and is weak if the problem is convex in control. This generalizes the results of P.Nepomiastcthy [4] where he considered the control in the Hilbert space L 2 (I, R m ).

No abstract