On the application of Liapunov's direct method to linear dynamical systems

ion of a given formal equation; we are not interested in questions of existence and uniqueness of solutions of the preliminary equation. However, we may use % to define a relatively simple Hilbert space S’ = % x “21 with inner product (h, , ha)> = (si , s&e -+ (ui , u& , where hi = (si , ui) E Z, and then define an artificial operator A: JY -+ &’ by II Y IICY = II s IIY + II 24 !I& t y = (s, u) E %. Considering only bounded linear operators 8: % -+ S? that are of “diagonal” form we have where the linear operators B,,: Y + Q, B,,: S --f 9, are bounded. Attempting to satisfy conditions (i) and (iii) of Theorem 4.1 for w = 0, we now require that ,::B,,s, , s,>* = G&s, , G>S for all s, , s2 E Y, G&P, > u&r = <&A 9 u&a for all ffr , us E 4, @,,s, uh = @,,Ks, uh for all (s, u) E %40 X @. Hence, choose B,, = G, B,, = GK, where G: Q -+ 91 is a bounded linear operator such that for all u1 , ua E S, for all si , s2 E .Y. If for some real number y > 0 we have <Gus u>s Z Y(U, uh , <GKs, )q > Y;S, s>e, for all s E 9, u E a, then Theorem 4.1 defines a linear operator A: @A) C 3) -+ S that is the infinitesimal generator of a linear C,,-semigroup on the Hilbert space obtained by completing Sp x % with respect to II IIx , where Il(s, & = <GKs, sh + (Gu, uh for (s, u) E Y x &. Moreover, since (x, Ax)% < 0 for all x E d(A), Theorem 3.9 implies that V(X) = (x, x).~ is a Liapunov functional that proves stability of the equilibrium upon application of Theorem 3.6.

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