9.3 Convergence Results
355
As in Sect. 9.2.1, consider the change of variables q M i (t) = k q q i (t), i =
1, 2, . . . , n, and nonlinearity (9.7), which is reported below for convenience
ˆ
ϕ(q i (t)) =
2h ap (k q q i (t))
R off k q
.
Omitting dependence on t, and assuming t 0 = 0, we obtain in vector notation
that the memristor NN satisfies in the (ϕ, q)-domain the following system of n
differential equations
τ
dq
dt
= −q + G ˆ
Φ(q) + Q 0
(9.13)
for t ≥ 0, where q = (q 1 , q 2 , . . . , q n ) T , ˆ
Φ(q) = ( ˆ
ϕ(q 1 ), ˆ
ϕ(q 2 ), . . . , ˆ
ϕ(q n )) T ,
G =
R off G
2
(9.14)
and
Q 0 =
Cv C 0
k q
+ q 0 −
GR off
2
ˆ
Φ(q 0 ).
(9.15)
Here we have let v C 0 = (v C 1 (0), v C 2 (0), . . . , v C n (0)) T and q 0 = q(0) = q M 0 /k q ,
where q M 0 = (q M 1 (0), q M 2 (0), . . . , q M n (0)) T . In view of the applications, it is
important to remark that the constant term Q 0 depends upon the initial conditions
Cv C 0 and q M 0 for the state variables in the (v, i)-domain.
Equations (9.13) is the SE representation of a memristor NN in terms of
the n state variables given by the memristor incremental charges. As discussed
in Sect. 9.2.1, if R on R off and x(−∞) 1, the nonlinearity ˆ
ϕ(·) enjoys
properties (9.8)–(9.11) and is a good approximation for not too large |q| of the
nonlinearity s(·) of a SCNN. As a consequence, model (9.13) closely approximates
and is analogous to the SCNN model (9.1). Henceforth, we will use the acronym
M-SCNN for model (9.13).
9.3 Convergence Results
In this section, we give a mathematical foundation to the M-SCNN model (9.13) by
studying the existence, uniqueness, boundedness, and prolongability of solutions.
Thereafter, we give conditions ensuring that M-SCNN is convergent, i.e., any
355
As in Sect. 9.2.1, consider the change of variables q M i (t) = k q q i (t), i =
1, 2, . . . , n, and nonlinearity (9.7), which is reported below for convenience
ˆ
ϕ(q i (t)) =
2h ap (k q q i (t))
R off k q
.
Omitting dependence on t, and assuming t 0 = 0, we obtain in vector notation
that the memristor NN satisfies in the (ϕ, q)-domain the following system of n
differential equations
τ
dq
dt
= −q + G ˆ
Φ(q) + Q 0
(9.13)
for t ≥ 0, where q = (q 1 , q 2 , . . . , q n ) T , ˆ
Φ(q) = ( ˆ
ϕ(q 1 ), ˆ
ϕ(q 2 ), . . . , ˆ
ϕ(q n )) T ,
G =
R off G
2
(9.14)
and
Q 0 =
Cv C 0
k q
+ q 0 −
GR off
2
ˆ
Φ(q 0 ).
(9.15)
Here we have let v C 0 = (v C 1 (0), v C 2 (0), . . . , v C n (0)) T and q 0 = q(0) = q M 0 /k q ,
where q M 0 = (q M 1 (0), q M 2 (0), . . . , q M n (0)) T . In view of the applications, it is
important to remark that the constant term Q 0 depends upon the initial conditions
Cv C 0 and q M 0 for the state variables in the (v, i)-domain.
Equations (9.13) is the SE representation of a memristor NN in terms of
the n state variables given by the memristor incremental charges. As discussed
in Sect. 9.2.1, if R on R off and x(−∞) 1, the nonlinearity ˆ
ϕ(·) enjoys
properties (9.8)–(9.11) and is a good approximation for not too large |q| of the
nonlinearity s(·) of a SCNN. As a consequence, model (9.13) closely approximates
and is analogous to the SCNN model (9.1). Henceforth, we will use the acronym
M-SCNN for model (9.13).
9.3 Convergence Results
In this section, we give a mathematical foundation to the M-SCNN model (9.13) by
studying the existence, uniqueness, boundedness, and prolongability of solutions.
Thereafter, we give conditions ensuring that M-SCNN is convergent, i.e., any