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0 0 -** A, This form is called the normal Jordan form of the matrix. T h e corresponding theorem is stated in the following way: There exists for every matrix A a matrix C such that the matrix C-lAC has the normal Jordan form. I n the particular case when there exist n linearly independent eigenvectors, the normal Jordan form is reduced to the diagonal form. It may happen, without the normal form being diagonal, that some Jordan cells are of first order. T h e diagonal form corresponds to the case where all the cells are of first order.

4. It follows from the hypothesis that there exists an yo > 0 such that for E > 0 there exists a quantity T ( E )> 0 with the property that . choose 6, from y o < qo follows y ( t ; to , yo) < a ( € )for t > to T ( E )We such that b(6,) < yo . (I < y ( t ; t o , V(t,, x,,)), it follows that x(t; t o ,xo)l) < a ( € ) Thus, finally, and the theorem is proved. for t 2 to 4-I"(€). 3. 3. Linear Systems We shall now study the characteristic features of the stability problem in the case of linear systems.

2. V ( t ,x , y ) >, a( 1 x 1) with a ( r ) continuous, monotone-increasing, a ( 0 ) = 0. 3. V [ t , x ( t ) , y ( t ) ]is monotone-decreasing for every solution x ( t ) , y ( t ) of the system with I x(t)l < H . 1. 1 then the stability is untform. + < I, > 0, S > 0 chosen such that V ( t o ,x o , y o ) < a ( € ) , if I xo I + I yo 1 < 6; if Y(t,x,y ) < b(l x 1 + 1 y I), we take S = b-'[a(~)]. Consider the solution x ( t ; to , xo , yo),y ( t ; to , xo , y o ) and the function Let Proof. E V * ( t )= V [ t ,x ( t ; t, , x, , yo),y ( t ; t o , xo , yo)]which is, by hypothesis, monotone-decreasing.

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A density lemma


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