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However, it is straightforward to verify that there exist constants u1 > 0 and u2 > 0 such that we can write 01(1111(.)11; + ll.r(.)ll; + ~~llE~(.)ll;, and moreover, c0[11.1-011' + do + E:=, TJ,) 5 n[11.r011' + E,"=, d,]. Hence, by combining these inequalities with (4.14), it follows that 111((.)112' + ll.r(.)[lg + E,"=, llEJ(.)llg 5 m/fl1[11.~011~ + E,"=, d,]. That is, condition iii) of Definition 2.2 is satisfied. Conditions ii) and iv) of Definition 2.2 follows directly from the stability of the matrix P. Thus, we can now conclude that the system (2.1), (2.3) is absolutely stabilizable via the control (4.4). 0 Remark: The above theorem gives a necessary and sufficient condition for absolute stabilizability in terms of the solution to a corresponding H m control problem. It is well known that such an H m control problem can be solved in terms of two algebraic Riccati equations; e.g., see [lo]. The following corollary is an immediate consequence of the above theorem. Corollary 4.1: If the uncertain system (2.1), (2.3) satisfies Assumptions 4.1 t 4. 6) and is absolutely stabilizable via nonlinear control, then it will be absolutely stabilizable via a linear controller of the form (4.4). E;=, llEa(-)ll;) 5 (1 / 2 C l)lI4.)lK + ;ll.4.)ll; + Ilso~.

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