A theorem of Strang [ 11 is extended to provide the best possible lower eigenvalue bound for a certain positive semidefinite matriu. The results are then applied to several asymptotically stable time-invariant linear systems. Some cases in which the bounds become infinite are discussed.
A decomposition of weakly-coupled Markov chains nto reduced-order aggregate chains and "fast" chains is derived. This decomposition is used t o b r e a k a n average cost per unit time problem into reduced-order subproblems, the solutions to which provide a nearoptimal control. W e consider the control of weaklycoupled Markov chains. W e decompose t h i s type of Markov chain i n t o a reduced order aggregate "slow" chain together with a set of decoupled "fast" chains. This decomposition also separates t h e o r i g i n a l c o s t f u n c t i o n into cost functions associated with each subproblem, and l e a d s t o a set of individual control problems, with o r i g i n a l problem. These r e s u l t s are illustrated through solutions providing a near-optimal control for the an example.
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