TRANSCENDENTAL AND INTERPOLATION METHODS IN SIMULTANEOUS STABILIZATION AND SIMULTANEOUS PARTIAL POLE PLACEMENT PROBLEMS.

We investigate the existence of a compensator which simultaneously renders a given r-tuple of multiinput multioutput p multiplied by m linear dynamical systems internally stable. In particular we parametrize a set of simultaneously stabilizable r-tuples of plants and show that provided r less than equivalent to max (m,p), the above set is semialgebraic and dense in the space SIGMA of r-tuples of plants. We also consider an extension of the classical pole placement and stabilization problems and investigate the simultaneous partial pole placement problem. Consequently, we consider a suitable topology in SIGMA and obtain a necessary condition and sufficient condition for the generic partial pole placement problem. We parametrize the set of all compensators simultaneously stabilizing a m tuple of 1 multiplied by m plants chosen generically, and we obtain a necessary and sufficient condition for simultaneously stabilizing a m plus 1 tuple of 1 multiplied by m plants chosen generically.