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Unformatted text preview: Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS
by A. Megretski TakeHome Test 11
For each problem, give an answer and provide supporting arguments, not to exceed one
page per problem. Return your test paper by 11.05 am on Friday October 17, in the
classroom. Remember that collaboration is not allowed on test assignments. Problem T1.1
Find all values of µ → R for which the function V : R2 ∞� R, deﬁned by
��
� x1
¯
V
= max{x1 , x2 }
¯¯
x2
¯
is monotonically nonincreasing along solutions of the ODE
�
x1 (t) = µx1 (t) + sin(x2 (t)),
˙
x2 (t) = µx2 (t) − sin(x1 (t)).
˙
Hint:  sin(y ) < y  for all y ≤= 0, and sin(y )/y � 1 as y � 0. 1 Posted October 16, 2003. Due at 11.05 am on October 17, 2003 2
Problem T1.2
Find all values of r → R for which diﬀerential inclusion of the form
x(t) → � (x(t)), x(0) = x0 ,
˙
¯
2 where � : R2 ∞� 2R is deﬁned by
x
� (¯) = {f (¯ x)} for x = 0,
x/ ¯
¯≤
� (0) = {f (y ) : y = [y1 ; y2 ] → R2 , y1  + y2  � r},
has a solution x : [0, ∀) ∞� R2 for every continuous function f : R2 ∞� R2 and for every
initial condition x0 → R2 .
¯
Problem T1.3
Find all values q , r → R for which x0 = 0 is not a (locally) stable equilibrium of the ODE
¯
x(t) = Ax(t) + B (Cx(t))1/3
˙
for every set of matrices A, B , C of dimensions nbyn, nby1, and 1byn respectively,
such that A is a Hurwitz matrix and
Re[(1 + j �q )G(j� )] > r � � → R
for
G(s) = C (sI − A)−1 B. ...
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 Spring '09
 AlexandreMegretski
 Dynamics

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