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Sensitivity of optimal control for diffusion Hopfield neural network in the presence of perturbation
Applied Mathematics and Computation 219 (2012) 3793–3808
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Sensitivity of optimal control for diffusion Hopfield neural network in the presence of perturbation q Quan-Fang Wang a,⇑, Shin-ichi Nakagiri b a b
Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong Faculty of Engineering, Kobe University, Nada, Kobe 657-8501, Japan
a r t i c l e
i n f o
Keywords: Optimal control Hopfield neural network Perturbation Sensitivity Diffusion Numerical simulation
a b s t r a c t With the wide investigation of Hopfield neural network (HNN) and its control problems, this paper is to control diffusion HNN system in the presence of perturbation (disturbance, uncertainties) in the control field. In particular, it is try to answer the most interesting question on the sensitivity of these perturbations both in theoretical and computational aspects. Crown Copyright Ó 2012 Published by Elsevier Inc. All rights reserved.
1. Preliminaries A great deal contributed works are reported in the studying Hopfield neural networks (HNNs) respect to theoretically problems, see [2–6,8,7,13,15]. Even inclusive of our proposed diffusion HNN system in [12], the existing research is limited to the theoretic and computational issues, and lost of generality and realizability. How about working on real neural networks? Could our conclusion be simply applied to supposed HNN structure? What will be happened in the meaning of physical viewpoint? Keep these questions in our mind to consider control problem for perturbed HNN system. Actually, numerous factors of disturbances and uncertainties would be involved in our extracted neural network as mathematics model. Furthermore, these perturbations can significantly change the experimental results if proceeding of the investigation on neural networks consist of large-number of biological neurons. How about the sensitivity of each physics factor? All of these questions will attract our attention definitely. The motivation of this paper is to directly manipulate the perturbed HNN with diffusion term. Hopefully, the current research results could provide us the access and direction in continuing investigation. Briefly to review our research papers as follows. Theoretic result reported at [12] in the framework of variational method, numerical solution of diffusion HNN had been handled in [17], 1D numerical approach are shown in [18,19], bang–bang control can be found in [16], pointwise control is published in [20]. Recent published paper [21] is focus on boundary pointwise control of diffusion HNN in two dimension case. The objective of this work is aimed at controlling of diffusion HNN in the presence of disturbances and uncertainties. Meanwhile, it is quite desired to achieve the control in the realistic sense. The article is organized by the following sections. Section 2 is to give perturbed HNN model with necessary notations and mathematical setting in the framework of variational method. Section 3 is to address control theory for such a diffusion HNN in the presence of perturbation arising in control field. Section 4 roughly state the adapted numerical computation algorithm q
This work is related to presentation at 2010 Annual Conference of Japan Society for Industrial and Applied Mathematics.
⇑ Corresponding author.
E-mail addresses: [email protected], [email protected] (Q.-F. Wang), [email protected] (S.-i. Nakagiri). 0096-3003/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.10.008
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based on the minimization of defined quadratic cost function. Section 5 is to show the demonstration results for three neurons for perturbed control field. The comparison with no perturbed case is also presented with simulated graphics for various physical quantities. 2. Perturbed HNN model Diffusion term has been included in our physical model for having a nonlinear neural network model starting in [12]. For simplicity, this article is restricted for perturbation occurred in control field. As to a neuron system, the control might be language, electric current, drug, magnetic source, light source and so on. If these types of control variables are used to control HNN, perturbation or disturbance will be happen without doubt. Let X be an open bounded domain of R3 with a piecewise smooth boundary C ¼ @ X. Denote x ¼ ðx1 ; x2 ; x3 Þ. Let Q ¼ ð0; TÞ X and R ¼ ð0; TÞ C with T > 0. Let yi ðx; tÞ denote the activation potential of the ith neuron for i ¼ 1; 2; . . . ; n. The diffusion HNN model with perturbation is described by simultaneous system of n-numbers neurons
Ci
n @yi ðtÞ y ðtÞ X di Dyi ðtÞ ¼ i þ F ij fj ðuj ðtÞÞ þ ðui ðtÞ þ dui Þ in Q @t Ri j¼1
ð1Þ
@yi ðtÞ ¼ 0 on R, and initial guess yi ðx; 0Þ ¼ yi0 ðxÞ for i ¼ 1; 2; . . . ; n. Here in (1), C i denote the total in@g put capacitance of the amplifier ith and its associated input lead. di > 0 are diffusion constants, F ij are connection weight con1 stants. The magnitude of F ij ¼ , where Rij is the resistor connecting the output of j to the input line i, while the sign of F ij is Rij 1 1 1 , determined by the choice of the positive or negative output of amplifier jth at the connection site. Ri defined as ¼ þ Ri pi Rij where pi is the input resistance of amplifier ith. fj : R ¼ ð1; 1Þ ! ð1; 1Þ are nonlinear sigmoidal activation functions, e.g., fj ðsÞ ¼ tanh s. Particularly, ui ðtÞ denotes external current to i-neuron, i.e. control inputs, and dui represents perturbation term in control field. In general, suppose dui is bounded for well-defined issue, that is, there exist constant C such that with boundary condition
8t 2 ½0; T:
jdui j 6 C;
Unbounded control input ui is exclusive of our research for lost generality, and make non-sense for biological neural network. For virtual artificial neural networks, it needs to set the physical model, and deduce the corresponding theoretical and computational results. Denote uðtÞ ¼ ðu1 ðtÞ; u2 ðtÞ; . . . ; un ðtÞÞ, and its the space U ¼ L2 ð0; TÞn . Let U ad be a closed and convex admissible set of U. In order to find quantum optimal control pairing u ¼ ðu1 ; u2 ; . . . ; un Þ in system (1), define two Hilbert spaces H ¼ L2 ðXÞ and V ¼ H1 ðXÞ corresponding to Neumann boundary condition of systems (1). They are endowed with the usual inner products ð; Þ; ðð; ÞÞ and norms j j; k k, respectively (cf. [1,11]). Then the embeddings in Gelfand triple spaces V,!H,!V 0 are continuous, dense and compact. The associated cost function subject to system (1) is n 2 X n X JðuÞ ¼ yfi ðuÞ ytarget ðui ; ui ÞU ; þ i V
i¼1
8u ¼ ðu1 ; u2 ; . . . ; un Þ 2 U;
ð2Þ
i¼1
where ytarget 2 V; i ¼ 1; 2; . . . ; n are target states, and yfi ðuÞ; i ¼ 1; 2; . . . ; n are final observed states at time t f , respectively. Here i are weighted coefficients for balancing the evaluates of inherent and running costs. Denote dual space of V by V 0 , and the symbol h ; i denotes the dual pairing from V and V 0 . Denote w ¼ ðw1 ; w2 ; . . . ; wn Þt and / ¼ ð/1 ; /2 ; . . . ; /n Þt , then the Hilbert spaces H ¼ L2 ðXÞn and V ¼ H1 ðXÞn with inner products defined by
ðw; /ÞH ¼ ðw; /ÞV ¼
n X ðwi ; /i Þ; i¼1 n X
ðwi ; /i Þ;
w; / 2 H; w; / 2 V;
i¼1
respectively. Then dual space V 0 ¼ ðV 0 Þn and dual pairing between V 0 and V is given by
hw; /iV;V 0 ¼
n X hwi ; /i i;
w 2 V; / 2 V 0 ;
i¼1
where hwi ; /i i denotes the dual pairing between V and V 0 . The norms of H and V are denoted by jwjH and kwkV , respectively. Let us define the following vectors and matrix representations:
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C ¼ ½C 1 ; C 2 ; . . . ; C n t ; 2 3 F 11 F 12 F 1n 6F 7 6 21 F 22 F 2n 7 F ¼6 .. .. .. 7 6 .. 7; 4 . . . . 5 F n1 F n2 F nn 3 2 3 2 y1 f1 ðy1 Þ 6y 7 6 f ðy Þ 7 6 27 6 2 2 7 7 6 7 f ðyÞ ¼ 6 6 .. 7; y ¼ 6 .. 7; 4 . 5 4 . 5 fn ðyn Þ
yn
D ¼ diagfd1 ; d2 ; . . . ; dn g; 1 1 1 1 ; ¼ diag ; ;...; R R1 R2 Rn t y0 ¼ y10 ð0Þ; y20 ð0Þ; . . . ; yn0 ð0Þ ; uðtÞ ¼ ½u1 ðtÞ; u2 ðtÞ; . . . ; un ðtÞt : duðtÞ ¼ ½du1 ðtÞ; du2 ðtÞ; . . . ; dun ðtÞt : ~ ¼ u þ du. Hence, the system (1) is converted to the Here ½. . . t denote the transposition of vector. For simplicity, denote u vectors-matrices form of
8 @y y > ~ ðtÞ in Q ; > > C @t DDy ¼ R þ F f ðyÞ þ u > < @y ¼ 0 on R; > > @ g > > : yðx; 0Þ ¼ y0 ðxÞ in X:
ð3Þ
Definition 1. Define Hilbert space Wð0; T; V; V 0 Þ as the solution space of system (3) by
Wð0; T; V; V 0 Þ ¼ fg j g 2 L2 ð0; T; VÞ; g0 2 L2 ð0; T; V 0 Þg: Its inner product and the norm are defined by
ðg1 ; g2 Þ ¼
Z
0
T
ðg 1 ; g 2 ÞV þ ðg 1 ; g 2 ÞV 0 dt;
12 kgkWð0;T;V;V 0 Þ ¼ kgk2L2 ð0;T;VÞ þ kg0 k2L2 ð0;T;V 0 Þ : Assume fj satisfy the uniform Lipschitz continuity:
9k0 > 0 : jfj ðsÞ fj ðrÞj 6 k0 js rj;
j ¼ 1; 2; . . . ; n
1
for s; r 2 R . Refer [12,20], there exists k such that f satisfies the uniform Lipschitz continuity. RT Due to the unified boundedness jdui j 6 C for all t 2 ½0; T, then there exists M > 0 such that 0 jdui j2 dt 6 M, thus 2 2 ~ 2 U. ~ i ¼ ui þ dui still belongs to L ð0; TÞ, i.e. u dui 2 L ð0; TÞ. Therefore u Definition 2. A function y is called a weak solution of (3) if y 2 Wð0; T; V; V 0 Þ and satisfy
8 RT RT @yðtÞ > > C v ðtÞ dt þ 0 ðDryðtÞ; rv ðtÞÞ dt ; > 0 > @t > V 0 ;V < RT R T yðtÞ ~ ; v ðtÞÞ in Q ; > ; v ðtÞ dt þ 0 ðF f ðyðtÞÞ; v ðtÞÞ dt þ ðu ¼ 0 > > R > > : yðx; 0Þ ¼ y0 ðxÞ; 8v ðtÞ 2 V:
ð4Þ
Theorem 1. Suppose f ð0Þ ¼ 0 and u 2 L2 ð0; TÞ, then the system (3) has a unique weak solution y 2 Wð0; T; V; V 0 Þ and Cð0; T; HÞ, and we get estimate
~ k2L2 ð0;TÞ ; kyk2L1 ð0;T;HÞ þ kyk2L2 ð0;T;VÞ 6 C 0 1 þ jy0 j2H þ ku
where C 0 > 0 just depends on C; D and R. See [17] for the proof of existence and uniqueness.
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3. Control theory for HNN ~ i ðtÞ for i ¼ 1; 2; . . . ; n, respectively. Let U ¼ L2 ð0; TÞn be the HilSuppose L2 ð0; TÞ is the Hilbert spaces of control variables u ~ ~ ~ Þ of system (3) in bert spaces of uðtÞ. For any u 2 U, by virtue of Theorem 1, there exist a unique weak solution y ¼ yðu ~ Þ of U into Wð0; T; V; V 0 Þ is well-defined. Wð0; T; V; V 0 Þ. Furthermore, the solution map y ! yðu Consider terminal observation of activation functions at time T, and quadratic cost function associated with control system (3) is given by
~ Þ ytarget k2L2 ðXÞn þ ðu ~; u ~ ÞU ; JðuÞ ¼ kyf ðu
8u~ 2 U;
~; u ~ ÞU ¼ where ytarget ¼ ðytarget ; ytarget ; . . . ; ytarget Þt 2 L2 ðXÞn is desired values, and ðu n 1 2
ð5Þ Xn i¼1
~i ; u ~ i Þ. ðu
Let U ad be admissible subsets of U. Two fundamental theoretic optimal control problems for (3) with quadratic cost (5) are in here, also see [10] and [14]. ~ ¼ ðu ~ 1 ; u ~ 2 ; . . . ; u ~ n Þ 2 U ad such that (a)Find an element u
~Þ ¼ Jðu ~ Þ; inf Jðu
u2U ad
8u~ ¼ ðu~ 1 ; u~ 2 ; . . . ; u~ n Þ 2 U ad :
~. (b)Characterization of such u ~ is so-called optimal control of perturbed system (3) subject to cost function (5). Such u Theorem 2. Suppose U ad is a non-empty bounded closed convex set of U. Then there exists at least one optimal control ~ ¼ ðu ~1 ; u ~2 ; . . . ; u ~ n Þ for control problem (3) subject to cost function (5). u ~ composed of u and du, if one consider u ~ as a new control input in domain U, then the proof can be found in Although u [20]. ~ is given by the variational inequality The first order optimality condition for u
~ Þðu ~u ~ Þ P 0; J 0 ðu 0
8u~ 2 U ad ;
~
~ Þ in (5) at u ~ . where J ðu Þ denotes the Gâteaux derivative of Jðu ~ 2 U ad for (1) with cost function (5) is Theorem 3. Let all assumptions be satisfied, then the optimal distributed control u characterized by optimality systems
8 ~ Þ ~ Þ @yðu yðu > > ~ ÞÞ þ u ~ Þ ¼ ~ ðtÞ in Q ; > DDyðu þ F f ðyðu C > > @t R < ~ Þ @yðu ¼ 0 on R; > > > @g > > : ~ yðu ; 0Þ ¼ y0 onX; 8 ~ Þ ~ Þ @pðu pðu > > ~ Þ ¼ ~ ÞÞpðu ~ Þ in Q; DDpðu þ f 0 F ðyðu C > > > @t R < ~ Þ @pðu ¼ 0 on R; > > @g > > > : ~ Þ ¼ ðyf ðu ~ Þ ytarget Þ on X; pðt f ; u Z T Z ~ 2 U ad : ~ Þ þ u ~ ðtÞ; u ~ ðtÞ u ~ ðtÞÞ dx dt P 0; 8u ðpðu 0
ð6Þ
ð7Þ
X
~ . Clearly, it is well known that inequality (7) is the necessary optimality condition of u
4. Computational algorithm For meeting the two dimension demonstration, consider x ¼ ðx1 ; x2 Þ 2 X, where X ¼ ð0; LÞ ð0; LÞ in this section. Let n n L ~ h ¼ ðu ~ h1 ; u ~ h2 ; . . . ; u ~ hn Þ and yh ¼ ðyh1 ; yh2 ; . . . ; yhn Þ. be the partition number of domain ½0; L ½0; L, and the interval length h ¼ . Set u n h h Let tf be the final observed time, and 0 6 t f 6 T. Thus for yi;f ¼ yi ðtf Þ, citing (4) to formulate the minimization problem for approximate cost function
Jðuh Þ ¼
n Z X i¼1
X
yhi;f ytarget i
~hi y ~i ðuh Þ 2 V h satisfying with y
2
dx þ
Z 0
T
h 2 ~ i dt u
ð8Þ
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Ci
Z
T
Z
0
for all
h
X
dyi dt
v h dx dt þ di
Z 0
T
Z X
ryhi rv h dx dt ¼
1 Ri
Z 0
T
Z X
yhi v h dx dt þ
n Z X j¼1
0
T
Z X
F ij fj ðyhj Þv h dx dt þ
Z 0
T
~ hi v h dt; u
ð9Þ
v h 2 V h.
Theorem 4. From Theorem 2, there exists at least a minimizer to finite element problems (8)–(9). The Gâteaux derivative for the discrete solution yhi at any direction v h , denotes as yh0 i v h , satisfying (9) for 8v h 2 V h . Let phi pi ðyh Þ, hence, for quadratic cost to have that
J 0 ðyh Þv h ¼ 2
n Z X i¼1
0
T
Z
n Z X pi ðyh Þv h dx dt þ 2
X
i¼1
0
T
~ hi ðtÞv h ðtÞ dt: u
2 ~ h The necessary optimality condition (7) for optimal control pairing ðyh i ; ui Þ 2 V h L ð0; TÞ can be obtained as
n Z X i¼1
T 0
Z X
pi ðyh Þðuhi uh i Þ dx dt þ
n Z X i¼1
0
T h h uh i ðtÞðui ðtÞ ui ðtÞÞ dt P 0;
~ h ¼ ðu ~ h1 ; u ~ h2 ; . . . ; u ~ hn Þ 2 U ad , where U ad is the admissible set of U. for all u In order to solve such a optimization problem, let us to give the following minimization theorem, and omit its proof. ~ hk g be a minimizing sequence to finite element problems (8)–(9). Then each sequence of fu ~ hk g has a Theorem 5. Set fu n 2 hk ~ g, in L ð0; TÞ to minimize approximate cost function (8). subsequence, rewritten as fu Using modified conjugate gradient method to solve above minimization problems by Lasdon in [9]. The convergency in the order of OðhÞ can be proved as in our previous papers [18–20]. 5. Numerical demonstration Set L ¼ 100 and X ¼ ½0; 100 ½0; 100 is spatial domain for two dimensions case. Three neurons P1 ; P 2 ; P 3 are located at ð30; 30Þ; ð70; 30Þ; ð50; 50Þ, corresponding to activities potential functions y1 ; y2 ; y3 respectively. Let iteration number
~ 10 ðx1 ; x2 ; tÞ (red curve), t 2 ½0; 1. (For interpretation of the references Fig. 1. No perturbed initial control u10 ðx1 ; x2 ; tÞ (black curve), perturbed initial control u to color in this figure legend, the reader is referred to the web version of this article.)
~ 20 ðx1 ; x2 ; tÞ (green curve), t 2 ½0; 1. (For interpretation of the Fig. 2. No perturbed initial control u20 ðx1 ; x2 ; tÞ (black curve), perturbed initial control u references to color in this figure legend, the reader is referred to the web version of this article.)
~ 3 ðx1 ; x2 ; tÞ (blue curve), t 2 ½0; 1. (For interpretation of the references Fig. 3. No perturbed initial control u3 ðx1 ; x2 ; tÞ (black curve), perturbed initial control u to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 4. y1 ðx1 ; x2 ; 0Þ; y1 ðx1 ; x2 ; TÞ and their contour plots for ðx1 ; x2 Þ 2 ½0; 100 ½0; 100.
Fig. 5. y2 ðx1 ; x2 ; 0Þ; y2 ðx1 ; x2 ; TÞ and their contour plots for ðx1 ; x2 Þ 2 ½0; 100 ½0; 100.
N ¼ 10, partition number n ¼ 100; h ¼ L=n; t0 ¼ 0:0; T ¼ 10:0; dt ¼ 1:0 and take t f ¼ T. Consider diffusion HNN simultaneously systems for three neurons.
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Fig. 6. y3 ðx1 ; x2 ; 0Þ; y3 ðx1 ; x2 ; TÞ and their contour plots for ðx1 ; x2 Þ 2 ½0; 100 ½0; 100.
Fig. 7. Mexican hat functions at t ¼ 0 : y1 ð30; x2 ; 0Þ in x2 direction (red curve), y1 ðx1 ; 30; 0Þ in x1 direction (blue curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 8. Mexican hat functions at t ¼ 0 : y2 ð70; x2 ; 0Þ in x2 direction (red curve), y2 ðx1 ; 30; 0Þ in x1 direction (blue curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 9. Mexican hat functions at t ¼ 0 : y3 ð50; x2 ; 0Þ in x2 direction (blue curve), y3 ðx1 ; 30; 0Þ in x1 direction (red curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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8 3 X @y1 > > > c1j tanhðxyj Þ þ u1 ðtÞ in Q ; d1 Dy1 ¼ a1 y1 þ > > @t > > j¼1 > > > > 3 X > @y2 > > > c2j tanhðxyj Þ þ u2 ðtÞ in Q ; d2 Dy2 ¼ a2 y2 þ > > @t > j¼1 < 3 X @y3 > > c3j tanhðxyj Þ þ u3 ðtÞ in Q; d3 Dy3 ¼ a3 u3 þ > > > @t > j¼1 > > > > > > y1 ðx1 ; x2 ; 0Þ ¼ y10 ðx1 ; x2 Þ on X; > > > > > y2 ðx1 ; x2 ; 0Þ ¼ y20 ðx1 ; x2 Þ on X; > : y3 ðx1 ; x2 ; 0Þ ¼ y30 ðx1 ; x2 Þ on X:
Here, d1 ¼ d2 ¼ d3 ¼ 0:0001, a1 ¼ a2 ¼ a3 ¼ 1:0; c1j ¼ c2j ¼ c3j ¼
2
pk
and x ¼ p2 ; k ¼ 1:4. Then give the starting distributed
control by using coefficients b ¼ 0:1 102; f ¼ 1:5 102 in term of
u10 ðtÞ ¼ bð2 þ 3 sinðftÞÞ; u20 ðtÞ ¼ 2bð2 þ 3 sinðftÞÞ; u30 ðtÞ ¼ bð2 þ 3 cosðftÞÞ:
~ 1 ðtÞ in t 2 ½0; t f . Fig. 10. Optima control u
~ 2 ðtÞ in t 2 ½0; tf . Fig. 11. Optimal control u
~ 3 ðtÞ in t 2 ½0; tf . Fig. 12. Optimal control u
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~1 ðx1 ; x2 ; tÞ at each iteration n ¼ 1; 2; . . . ; 10 for ðx1 ; x2 Þ 2 ½0; 100 ½0; 100; t 2 ½0; tf . Fig. 13. Activity function y
The perturbations in control variables are given by amplitude b ¼ 0:2b and frequency f ¼ 0:2f . Then perturbed system with 0 the amplitude b ¼ ð1 þ 0:2Þb and frequency f 0 ¼ ð1 þ 0:2Þf in u1 ; u2 ; u3 appeared at each equation of y1 ; y2 ; y3 . See Figs. 1–3 for two starting controls of no perturbed and perturbed cases.
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~2 ðx1 ; x2 ; tÞ at each iteration n ¼ 1; 2; . . . ; 10 for ðx1 ; x2 Þ 2 ½0; 100 ½0; 100; t 2 ½0; tf . Fig. 14. Activity function y
The initial activities potential functions are given by
p y10 ¼ 240sech 0:01ðx1 30Þ2 þ 0:01ðx2 30Þ2 þ ; 5 p y20 ¼ 240sech 0:01ðx1 70Þ2 þ 0:01ðx2 30Þ2 þ ; 5 p y30 ¼ 240sech 0:01ðx1 50Þ2 þ 0:01ðx2 70Þ2 þ : 5
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~3 ðx1 ; x2 ; tÞ at each iteration n ¼ 1; 2; . . . ; 10 for ðx1 ; x2 Þ 2 ½0; 100 ½0; 100; t 2 ½0; tf . Fig. 15. Activity function y
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~1 ð30; x2 ; t f Þ in x2 direction (red curve), y ~1 ðx1 ; 30; t f Þ in x1 direction (blue curve). (For interpretation of the references to Fig. 16. Mexican hat functions at t f : y color in this figure legend, the reader is referred to the web version of this article.)
~2 ð70; x2 ; t f Þ in x2 direction (blue curve), y ~2 ðx1 ; 30; tf Þ in x1 direction (red curve). (For interpretation of the references to Fig. 17. Mexican hat functions at t f : y color in this figure legend, the reader is referred to the web version of this article.)
~3 ð50; x2 ; t f Þ in x2 direction (red curve), y ~3 ðx1 ; 70; t f Þ in x1 direction (blue curve). (For interpretation of the references to Fig. 18. Mexican hat functions at t f : y color in this figure legend, the reader is referred to the web version of this article.)
~1 ð30; 30; tÞ (red curve), y ~2 ð70; 30; tÞ (green curve), y ~3 ð50; 70; tÞ (blue curve) for t 2 ½0; tf . (For interpretation of the references to color in this figure Fig. 19. y legend, the reader is referred to the web version of this article.)
The target activities functions are supposed by
20 ðy þ 0:7y20 Þ; 240 10 90 y ; y2t ¼ 90 þ 240 20 20 ðy þ 0:7y20 Þ: y3t ¼ 240 30 y1t ¼
The start and target activities functions of y1 ; y2 ; y3 and their contour plots are shown in Figs. 4–6. As is well known, the activation potential is like Mexican hat function, before the control processing, their graphics at points P 1 ; P 2 ; P 3 in direction x1 ; x2 are given in Figs. 7–9.
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eJ 2.0 108
J 1.8 108
1.5 108
1.6 108 1.0 108
1.4 108 1.2 108
5.0 107
2
4
6
8
10
n
8.0 107
2
4
6
8
10
n
~ Þ; eeJ ¼ eJðu ~ Þ with JðuÞ; eJðuÞ (green, red points represent no perturbation, black point is perturbed one). (For interpretation Fig. 20. The comparison of eJ ¼ Jðu of the references to color in this figure legend, the reader is referred to the web version of this article.)
~ 1 ðtÞ (purple curve is perturbed one) for t 2 ½0; 1. (For interpretation of the references to color in Fig. 21. Optimal controls u1 ðtÞ (blue curve is original one), u this figure legend, the reader is referred to the web version of this article.)
~ 2 ðtÞ (purple curve is perturbed one) for t 2 ½0; 1. (For interpretation of the references to color in Fig. 22. Optimal controls u2 ðtÞ(blue curve is original one), u this figure legend, the reader is referred to the web version of this article.)
~ 3 ðtÞ (purple curve is perturbed one) for t 2 ½0; 1. (For interpretation of the references to color in Fig. 23. Optimal controls u3 ðtÞ (blue curve is original one), u this figure legend, the reader is referred to the web version of this article.)
Through the limited iterations steps, the optimal controls will be obtained as the form of
~1 ðtÞ ¼ 19:8486 þ 30:0ð2 þ 3 sinð150tÞÞ; u ~2 ðtÞ ¼ 0:275766 þ 59:9981ð2 þ 3 sinð150tÞÞ; u ~3 ðtÞ ¼ 19:8532 þ 30:0ð2 þ 3 sinð150tÞÞ: u The perturbed optimal control inputs are displayed in Figs. 10–12.
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~ 1 ðtÞ for t 2 ½0; 10. Fig. 24. Error function eu
~ 2 ðtÞ for t 2 ½0; 10. Fig. 25. Error function eu
~ 3 ðtÞ for t 2 ½0; 10. Fig. 26. Error function eu
~ 1 ðtÞ (blue line) for t 2 ½0; 1. (For interpretation of the references to color in this figure legend, the reader is Fig. 27. Error functions eu1 ðtÞ (red line), eu referred to the web version of this article.)
~1 ; y ~2 ; y ~3 (corresponding to u ~1 ; u ~2 ; u ~ 3 ) are changed with time at each iterAfter controlling process, the activities function y ation in following. In last iteration, the activities potential in presence of perturbation are computed in grouped graphics, see Figs. 13–15. These Mexican hat functions are changed and simulated at final iteration time tf in Figs. 16–18. The calculated Mexican hat functions verified the Theorem 1 and 2 numerically. At point ð30; 30Þ; ð70; 30Þ and ð50; 70Þ, ~i ðtÞ; i ¼ 1; 2; 3 are changed with time t, their graphics are plotted in Fig. 19. activation potential functions y
Q.-F. Wang, S.-i. Nakagiri / Applied Mathematics and Computation 219 (2012) 3793–3808
3807
~ 2 ðtÞ (blue line) for t 2 ½0; 1. (For interpretation of the references to color in this figure legend, the reader is Fig. 28. Error functions eu2 ðtÞ (red line), eu referred to the web version of this article.)
~ 3 ðtÞ (blue line) for t 2 ½0; 1. (For interpretation of the references to color in this figure legend, the reader is Fig. 29. Error functions eu3 ðtÞ (red line), eu referred to the web version of this article.)
It is accordant with the results in [5]. Optimal cost values attain J½u ¼ 8:33629 107 and eJ½u ¼ 8:70052 107 for no ~ Þ are calculated at each iteration. perturbation case and perturbed case. Cost functions values JðuÞ and eJ ¼ Jðu
J½1 ¼ 1:68294 108 ; J½2 ¼ 8:35024 107 ; J½3 ¼ 8:34863 107 ; 7
J½4 ¼ 8:34938 10 ;
eJ½1 ¼ 1:93494 108 ; eJ½2 ¼ 8:7219 107 ; eJ½3 ¼ 8:72139 107 ; eJ½4 ¼ 8:7156 107 ;
eJ½5 ¼ 8:71734 107 ; J½6 ¼ 8; 34379 107 ; eJ½6 ¼ 8:71167 107 ; J½7 ¼ 8:33859 107 ; eJ½7 ¼ 8:71043 107 ;
J½5 ¼ 8:34332 107 ;
J½8 ¼ 8:33741 107 ; J½9 ¼ 8:33673 107 ; J½10 ¼ 8:33629 107 ;
eJ½8 ¼ 8:70567 107 ; eJ½9 ¼ 8:70669 107 ; eJ½10 ¼ 8:70052 107 :
Two cases error values eJ ¼ J½n þ 1 J½n; eeJ ¼ eJ½n þ 1 eJ½n of cost functions are obtained as
eJ½1 ¼ 1:68294 108 ;
eeJ½1 ¼ 1:93494 108 ; eeJ½2 ¼ 1:06275 108 ;
eJ½2 ¼ 8:47918 107 ; eJ½3 ¼ 16086:6; eeJ½3 ¼ 5054:44; eJ½4 ¼ 7479:55; eJ½5 ¼ 60597:8; eJ½6 ¼ 4670:06; eJ½7 ¼ 51987:6; eJ½8 ¼ 11761:3; eJ½9 ¼ 6818:04; eJ½10 ¼ 4396:42;
eeJ½4 ¼ 57920:0; eeJ½5 ¼ 17437:0; eeJ½6 ¼ 56783:0; eeJ½7 ¼ 12310:0; eeJ½8 ¼ 47619:0; eeJ½9 ¼ 10186:8; eeJ½10 ¼ 61735:7:
The comparison graphics of two cost values and two error values at each iteration see Fig. 20.
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~ i ; i ¼ 1; 2; 3 are given in Figs. 21–23. The comparison of optimal control functions ui ; u ~i ¼ u ~ i ½n þ 1 u ~ ½n; i ¼ 1; 2; 3 at final iteration calculated as: Errors of control functions eu
~ 1 ¼ 0:0000517268 8:17772 109 sinð150tÞ; eu ~ 2 ¼ 0:00883129 4:25091 107 sinð150tÞ; eu ~ 3 ¼ 0:0000505376 8:07544 109 cosð150tÞ: eu Their graphs are simulated in Figs. 24–26. The comparison with error of no perturbed case eui ¼ ui ½n þ 1 ui ½n; i ¼ 1; 2; 3 are displayed in Figs. 27–29 for time interval t 2 ½0; 1, The numerous results directly illustrate Theorem 3. Obviously, the stability and convergence of nonlinear scheme are evidenced. For no perturbation case and perturbed case, the total running CPU times are 1314.6 s, and 1720.77 s, the used maximum memory are 165330.200 bytes, and 167774.368 bytes, respectively. Remark 1. Our results can be extended to three dimensions case clearly. In this work, the control input ui ðtÞ is depended time t, it would be efficient to solve the spatial depended case ui ðt; xÞ, see [20] to render details. It would be working for timedelayed HNNs certainly.
6. Conclusions This work presented the conclusion for control of Hopfield neural network (HNN) with diffusion term in the presence of perturbations occurred in control field. It would be a new consideration in the realization of control HNN. Future perspective research would be connected with the neuroscience and relative fields. Acknowledgments The authors thank 2010 Annual Conference of Japan Society for Industrial and Applied Mathematics (JSIAM) for presentation ‘‘Sensitivity of optimal control for diffusion Hopfield Neural Network in the presence of perturbation’’. References [1] R. Dautray, J.L. Lions, Mathematical analysis and numerical methods for science and technology, Evolution Problems I, vol. 5, Springer-Verlag, Berlin– Heidelberg–New York, 1992. [2] R. Fitz-Hugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophys. 17 (1955) 257–278. [3] A.L. Hodgkin, A.F. Huxley, A qualitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117 (1952) 500–544. [4] J.J. Hopfield, Neural network and physics system with collective computational abilities, Proc. Natl. Acad. Sci. USA 79 (1982) 2554–2558. [5] J.J. Hopfield, Neurons with graded response have collective properties like those of two state neurons, Proc. Natl. Acad. Sci. USA 81 (1984) 3088–3092. [6] J.J. Hopfield, D.W. Tank, Computing with neural circuits: a model, Science 233 (1986) 625–632. [7] D. Ji, J. Koo, S. Won, S. Lee, J. Park, Passivity-based control for Hopfield neural networks using convex representation, Appl. Math. Comput. 217 (2011) 6168–6175. [8] S. Lakshmanan, J. Park, H. Jung, P. Balasubramaniam, Design of state estimator for neural networks with leakage, discrete and distributed delays, Appl. Math. Comput. 218 (2012) 11297–11310. [9] L.S. Lasdon, S.K. Mitter, A.D. Warren, The conjugate gradient method for optimal control problems, IEEE Trans. Autom. Control 12 (1967) 132–138. [10] J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin–Heidelberg–New York, 1971. [11] J.L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I, II, Springer-Verlag, Berlin–Heidelberg–New York, 1972. [12] S. Nakagiri, Q.F. Wang, Optimal control problems for distributed Hopfield-type neural networks, Nonlinear Funct. Anal. Appl. 7 (2) (2002) 167–186. [13] J. Park, O. Kwon, On improved delay-dependent criterion for global stability of bidirectional associative memory neural networks with time-varying delays, Appl. Math. Comput. 2008 (1999) 435–446. [14] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematica Sciences, second ed., 68, Springer-Verlag, Berlin– Heidelberg–New York, 1997. [15] M. Velazco Fontova, A. Oliveira, C. Lyra, Hopfield neural networks in large-scale linear optimization problems, Appl. Math. Comput. 218 (2012) 6851– 6859. [16] Q.F. Wang, Boundary bang–bang control of distributed diffusion Hopfield neural network equations, Int. J. Comput. Numer. Anal. Appl. 6 (1) (2004) 33– 63. [17] Q.F. Wang, S. Nakagiri, Numerical solution of distributed diffusion Hopfield neural network equations, Appl. Math. Comput. 174 (2) (2006) 1343–1362. [18] Q.F. Wang, Theoretical and computational issue of optimal control for distributed Hopfield neural network equations with diffusion term, SIAM J. Sci. Comput. 29 (2) (2007) 890–911. [19] Q.F. Wang, Numerical approximate of optimal control for distributed diffusion Hopfield neural networks, Int. J. Numer. Methods Eng. 69 (3) (2007) 443–468. [20] Q.F. Wang, Computation issues of pointwise control for diffusion Hopfield neural network system, Appl. Math. Comput. 207 (2009) 148–163. [21] Q.F. Wang, Boundary pointwise control for diffusion Hopfield neural network, Int. J. Nanotechnol. Mol. Comput. 2 (1) (2010) 13–29.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Sensitivity of optimal control for diffusion Hopfield neural network in the presence of perturbation q Quan-Fang Wang a,⇑, Shin-ichi Nakagiri b a b
Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong Faculty of Engineering, Kobe University, Nada, Kobe 657-8501, Japan
a r t i c l e
i n f o
Keywords: Optimal control Hopfield neural network Perturbation Sensitivity Diffusion Numerical simulation
a b s t r a c t With the wide investigation of Hopfield neural network (HNN) and its control problems, this paper is to control diffusion HNN system in the presence of perturbation (disturbance, uncertainties) in the control field. In particular, it is try to answer the most interesting question on the sensitivity of these perturbations both in theoretical and computational aspects. Crown Copyright Ó 2012 Published by Elsevier Inc. All rights reserved.
1. Preliminaries A great deal contributed works are reported in the studying Hopfield neural networks (HNNs) respect to theoretically problems, see [2–6,8,7,13,15]. Even inclusive of our proposed diffusion HNN system in [12], the existing research is limited to the theoretic and computational issues, and lost of generality and realizability. How about working on real neural networks? Could our conclusion be simply applied to supposed HNN structure? What will be happened in the meaning of physical viewpoint? Keep these questions in our mind to consider control problem for perturbed HNN system. Actually, numerous factors of disturbances and uncertainties would be involved in our extracted neural network as mathematics model. Furthermore, these perturbations can significantly change the experimental results if proceeding of the investigation on neural networks consist of large-number of biological neurons. How about the sensitivity of each physics factor? All of these questions will attract our attention definitely. The motivation of this paper is to directly manipulate the perturbed HNN with diffusion term. Hopefully, the current research results could provide us the access and direction in continuing investigation. Briefly to review our research papers as follows. Theoretic result reported at [12] in the framework of variational method, numerical solution of diffusion HNN had been handled in [17], 1D numerical approach are shown in [18,19], bang–bang control can be found in [16], pointwise control is published in [20]. Recent published paper [21] is focus on boundary pointwise control of diffusion HNN in two dimension case. The objective of this work is aimed at controlling of diffusion HNN in the presence of disturbances and uncertainties. Meanwhile, it is quite desired to achieve the control in the realistic sense. The article is organized by the following sections. Section 2 is to give perturbed HNN model with necessary notations and mathematical setting in the framework of variational method. Section 3 is to address control theory for such a diffusion HNN in the presence of perturbation arising in control field. Section 4 roughly state the adapted numerical computation algorithm q
This work is related to presentation at 2010 Annual Conference of Japan Society for Industrial and Applied Mathematics.
⇑ Corresponding author.
E-mail addresses: [email protected], [email protected] (Q.-F. Wang), [email protected] (S.-i. Nakagiri). 0096-3003/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.10.008
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based on the minimization of defined quadratic cost function. Section 5 is to show the demonstration results for three neurons for perturbed control field. The comparison with no perturbed case is also presented with simulated graphics for various physical quantities. 2. Perturbed HNN model Diffusion term has been included in our physical model for having a nonlinear neural network model starting in [12]. For simplicity, this article is restricted for perturbation occurred in control field. As to a neuron system, the control might be language, electric current, drug, magnetic source, light source and so on. If these types of control variables are used to control HNN, perturbation or disturbance will be happen without doubt. Let X be an open bounded domain of R3 with a piecewise smooth boundary C ¼ @ X. Denote x ¼ ðx1 ; x2 ; x3 Þ. Let Q ¼ ð0; TÞ X and R ¼ ð0; TÞ C with T > 0. Let yi ðx; tÞ denote the activation potential of the ith neuron for i ¼ 1; 2; . . . ; n. The diffusion HNN model with perturbation is described by simultaneous system of n-numbers neurons
Ci
n @yi ðtÞ y ðtÞ X di Dyi ðtÞ ¼ i þ F ij fj ðuj ðtÞÞ þ ðui ðtÞ þ dui Þ in Q @t Ri j¼1
ð1Þ
@yi ðtÞ ¼ 0 on R, and initial guess yi ðx; 0Þ ¼ yi0 ðxÞ for i ¼ 1; 2; . . . ; n. Here in (1), C i denote the total in@g put capacitance of the amplifier ith and its associated input lead. di > 0 are diffusion constants, F ij are connection weight con1 stants. The magnitude of F ij ¼ , where Rij is the resistor connecting the output of j to the input line i, while the sign of F ij is Rij 1 1 1 , determined by the choice of the positive or negative output of amplifier jth at the connection site. Ri defined as ¼ þ Ri pi Rij where pi is the input resistance of amplifier ith. fj : R ¼ ð1; 1Þ ! ð1; 1Þ are nonlinear sigmoidal activation functions, e.g., fj ðsÞ ¼ tanh s. Particularly, ui ðtÞ denotes external current to i-neuron, i.e. control inputs, and dui represents perturbation term in control field. In general, suppose dui is bounded for well-defined issue, that is, there exist constant C such that with boundary condition
8t 2 ½0; T:
jdui j 6 C;
Unbounded control input ui is exclusive of our research for lost generality, and make non-sense for biological neural network. For virtual artificial neural networks, it needs to set the physical model, and deduce the corresponding theoretical and computational results. Denote uðtÞ ¼ ðu1 ðtÞ; u2 ðtÞ; . . . ; un ðtÞÞ, and its the space U ¼ L2 ð0; TÞn . Let U ad be a closed and convex admissible set of U. In order to find quantum optimal control pairing u ¼ ðu1 ; u2 ; . . . ; un Þ in system (1), define two Hilbert spaces H ¼ L2 ðXÞ and V ¼ H1 ðXÞ corresponding to Neumann boundary condition of systems (1). They are endowed with the usual inner products ð; Þ; ðð; ÞÞ and norms j j; k k, respectively (cf. [1,11]). Then the embeddings in Gelfand triple spaces V,!H,!V 0 are continuous, dense and compact. The associated cost function subject to system (1) is n 2 X n X JðuÞ ¼ yfi ðuÞ ytarget ðui ; ui ÞU ; þ i V
i¼1
8u ¼ ðu1 ; u2 ; . . . ; un Þ 2 U;
ð2Þ
i¼1
where ytarget 2 V; i ¼ 1; 2; . . . ; n are target states, and yfi ðuÞ; i ¼ 1; 2; . . . ; n are final observed states at time t f , respectively. Here i are weighted coefficients for balancing the evaluates of inherent and running costs. Denote dual space of V by V 0 , and the symbol h ; i denotes the dual pairing from V and V 0 . Denote w ¼ ðw1 ; w2 ; . . . ; wn Þt and / ¼ ð/1 ; /2 ; . . . ; /n Þt , then the Hilbert spaces H ¼ L2 ðXÞn and V ¼ H1 ðXÞn with inner products defined by
ðw; /ÞH ¼ ðw; /ÞV ¼
n X ðwi ; /i Þ; i¼1 n X
ðwi ; /i Þ;
w; / 2 H; w; / 2 V;
i¼1
respectively. Then dual space V 0 ¼ ðV 0 Þn and dual pairing between V 0 and V is given by
hw; /iV;V 0 ¼
n X hwi ; /i i;
w 2 V; / 2 V 0 ;
i¼1
where hwi ; /i i denotes the dual pairing between V and V 0 . The norms of H and V are denoted by jwjH and kwkV , respectively. Let us define the following vectors and matrix representations:
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C ¼ ½C 1 ; C 2 ; . . . ; C n t ; 2 3 F 11 F 12 F 1n 6F 7 6 21 F 22 F 2n 7 F ¼6 .. .. .. 7 6 .. 7; 4 . . . . 5 F n1 F n2 F nn 3 2 3 2 y1 f1 ðy1 Þ 6y 7 6 f ðy Þ 7 6 27 6 2 2 7 7 6 7 f ðyÞ ¼ 6 6 .. 7; y ¼ 6 .. 7; 4 . 5 4 . 5 fn ðyn Þ
yn
D ¼ diagfd1 ; d2 ; . . . ; dn g; 1 1 1 1 ; ¼ diag ; ;...; R R1 R2 Rn t y0 ¼ y10 ð0Þ; y20 ð0Þ; . . . ; yn0 ð0Þ ; uðtÞ ¼ ½u1 ðtÞ; u2 ðtÞ; . . . ; un ðtÞt : duðtÞ ¼ ½du1 ðtÞ; du2 ðtÞ; . . . ; dun ðtÞt : ~ ¼ u þ du. Hence, the system (1) is converted to the Here ½. . . t denote the transposition of vector. For simplicity, denote u vectors-matrices form of
8 @y y > ~ ðtÞ in Q ; > > C @t DDy ¼ R þ F f ðyÞ þ u > < @y ¼ 0 on R; > > @ g > > : yðx; 0Þ ¼ y0 ðxÞ in X:
ð3Þ
Definition 1. Define Hilbert space Wð0; T; V; V 0 Þ as the solution space of system (3) by
Wð0; T; V; V 0 Þ ¼ fg j g 2 L2 ð0; T; VÞ; g0 2 L2 ð0; T; V 0 Þg: Its inner product and the norm are defined by
ðg1 ; g2 Þ ¼
Z
0
T
ðg 1 ; g 2 ÞV þ ðg 1 ; g 2 ÞV 0 dt;
12 kgkWð0;T;V;V 0 Þ ¼ kgk2L2 ð0;T;VÞ þ kg0 k2L2 ð0;T;V 0 Þ : Assume fj satisfy the uniform Lipschitz continuity:
9k0 > 0 : jfj ðsÞ fj ðrÞj 6 k0 js rj;
j ¼ 1; 2; . . . ; n
1
for s; r 2 R . Refer [12,20], there exists k such that f satisfies the uniform Lipschitz continuity. RT Due to the unified boundedness jdui j 6 C for all t 2 ½0; T, then there exists M > 0 such that 0 jdui j2 dt 6 M, thus 2 2 ~ 2 U. ~ i ¼ ui þ dui still belongs to L ð0; TÞ, i.e. u dui 2 L ð0; TÞ. Therefore u Definition 2. A function y is called a weak solution of (3) if y 2 Wð0; T; V; V 0 Þ and satisfy
8 RT RT @yðtÞ > > C v ðtÞ dt þ 0 ðDryðtÞ; rv ðtÞÞ dt ; > 0 > @t > V 0 ;V < RT R T yðtÞ ~ ; v ðtÞÞ in Q ; > ; v ðtÞ dt þ 0 ðF f ðyðtÞÞ; v ðtÞÞ dt þ ðu ¼ 0 > > R > > : yðx; 0Þ ¼ y0 ðxÞ; 8v ðtÞ 2 V:
ð4Þ
Theorem 1. Suppose f ð0Þ ¼ 0 and u 2 L2 ð0; TÞ, then the system (3) has a unique weak solution y 2 Wð0; T; V; V 0 Þ and Cð0; T; HÞ, and we get estimate
~ k2L2 ð0;TÞ ; kyk2L1 ð0;T;HÞ þ kyk2L2 ð0;T;VÞ 6 C 0 1 þ jy0 j2H þ ku
where C 0 > 0 just depends on C; D and R. See [17] for the proof of existence and uniqueness.
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3. Control theory for HNN ~ i ðtÞ for i ¼ 1; 2; . . . ; n, respectively. Let U ¼ L2 ð0; TÞn be the HilSuppose L2 ð0; TÞ is the Hilbert spaces of control variables u ~ ~ ~ Þ of system (3) in bert spaces of uðtÞ. For any u 2 U, by virtue of Theorem 1, there exist a unique weak solution y ¼ yðu ~ Þ of U into Wð0; T; V; V 0 Þ is well-defined. Wð0; T; V; V 0 Þ. Furthermore, the solution map y ! yðu Consider terminal observation of activation functions at time T, and quadratic cost function associated with control system (3) is given by
~ Þ ytarget k2L2 ðXÞn þ ðu ~; u ~ ÞU ; JðuÞ ¼ kyf ðu
8u~ 2 U;
~; u ~ ÞU ¼ where ytarget ¼ ðytarget ; ytarget ; . . . ; ytarget Þt 2 L2 ðXÞn is desired values, and ðu n 1 2
ð5Þ Xn i¼1
~i ; u ~ i Þ. ðu
Let U ad be admissible subsets of U. Two fundamental theoretic optimal control problems for (3) with quadratic cost (5) are in here, also see [10] and [14]. ~ ¼ ðu ~ 1 ; u ~ 2 ; . . . ; u ~ n Þ 2 U ad such that (a)Find an element u
~Þ ¼ Jðu ~ Þ; inf Jðu
u2U ad
8u~ ¼ ðu~ 1 ; u~ 2 ; . . . ; u~ n Þ 2 U ad :
~. (b)Characterization of such u ~ is so-called optimal control of perturbed system (3) subject to cost function (5). Such u Theorem 2. Suppose U ad is a non-empty bounded closed convex set of U. Then there exists at least one optimal control ~ ¼ ðu ~1 ; u ~2 ; . . . ; u ~ n Þ for control problem (3) subject to cost function (5). u ~ composed of u and du, if one consider u ~ as a new control input in domain U, then the proof can be found in Although u [20]. ~ is given by the variational inequality The first order optimality condition for u
~ Þðu ~u ~ Þ P 0; J 0 ðu 0
8u~ 2 U ad ;
~
~ Þ in (5) at u ~ . where J ðu Þ denotes the Gâteaux derivative of Jðu ~ 2 U ad for (1) with cost function (5) is Theorem 3. Let all assumptions be satisfied, then the optimal distributed control u characterized by optimality systems
8 ~ Þ ~ Þ @yðu yðu > > ~ ÞÞ þ u ~ Þ ¼ ~ ðtÞ in Q ; > DDyðu þ F f ðyðu C > > @t R < ~ Þ @yðu ¼ 0 on R; > > > @g > > : ~ yðu ; 0Þ ¼ y0 onX; 8 ~ Þ ~ Þ @pðu pðu > > ~ Þ ¼ ~ ÞÞpðu ~ Þ in Q; DDpðu þ f 0 F ðyðu C > > > @t R < ~ Þ @pðu ¼ 0 on R; > > @g > > > : ~ Þ ¼ ðyf ðu ~ Þ ytarget Þ on X; pðt f ; u Z T Z ~ 2 U ad : ~ Þ þ u ~ ðtÞ; u ~ ðtÞ u ~ ðtÞÞ dx dt P 0; 8u ðpðu 0
ð6Þ
ð7Þ
X
~ . Clearly, it is well known that inequality (7) is the necessary optimality condition of u
4. Computational algorithm For meeting the two dimension demonstration, consider x ¼ ðx1 ; x2 Þ 2 X, where X ¼ ð0; LÞ ð0; LÞ in this section. Let n n L ~ h ¼ ðu ~ h1 ; u ~ h2 ; . . . ; u ~ hn Þ and yh ¼ ðyh1 ; yh2 ; . . . ; yhn Þ. be the partition number of domain ½0; L ½0; L, and the interval length h ¼ . Set u n h h Let tf be the final observed time, and 0 6 t f 6 T. Thus for yi;f ¼ yi ðtf Þ, citing (4) to formulate the minimization problem for approximate cost function
Jðuh Þ ¼
n Z X i¼1
X
yhi;f ytarget i
~hi y ~i ðuh Þ 2 V h satisfying with y
2
dx þ
Z 0
T
h 2 ~ i dt u
ð8Þ
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Ci
Z
T
Z
0
for all
h
X
dyi dt
v h dx dt þ di
Z 0
T
Z X
ryhi rv h dx dt ¼
1 Ri
Z 0
T
Z X
yhi v h dx dt þ
n Z X j¼1
0
T
Z X
F ij fj ðyhj Þv h dx dt þ
Z 0
T
~ hi v h dt; u
ð9Þ
v h 2 V h.
Theorem 4. From Theorem 2, there exists at least a minimizer to finite element problems (8)–(9). The Gâteaux derivative for the discrete solution yhi at any direction v h , denotes as yh0 i v h , satisfying (9) for 8v h 2 V h . Let phi pi ðyh Þ, hence, for quadratic cost to have that
J 0 ðyh Þv h ¼ 2
n Z X i¼1
0
T
Z
n Z X pi ðyh Þv h dx dt þ 2
X
i¼1
0
T
~ hi ðtÞv h ðtÞ dt: u
2 ~ h The necessary optimality condition (7) for optimal control pairing ðyh i ; ui Þ 2 V h L ð0; TÞ can be obtained as
n Z X i¼1
T 0
Z X
pi ðyh Þðuhi uh i Þ dx dt þ
n Z X i¼1
0
T h h uh i ðtÞðui ðtÞ ui ðtÞÞ dt P 0;
~ h ¼ ðu ~ h1 ; u ~ h2 ; . . . ; u ~ hn Þ 2 U ad , where U ad is the admissible set of U. for all u In order to solve such a optimization problem, let us to give the following minimization theorem, and omit its proof. ~ hk g be a minimizing sequence to finite element problems (8)–(9). Then each sequence of fu ~ hk g has a Theorem 5. Set fu n 2 hk ~ g, in L ð0; TÞ to minimize approximate cost function (8). subsequence, rewritten as fu Using modified conjugate gradient method to solve above minimization problems by Lasdon in [9]. The convergency in the order of OðhÞ can be proved as in our previous papers [18–20]. 5. Numerical demonstration Set L ¼ 100 and X ¼ ½0; 100 ½0; 100 is spatial domain for two dimensions case. Three neurons P1 ; P 2 ; P 3 are located at ð30; 30Þ; ð70; 30Þ; ð50; 50Þ, corresponding to activities potential functions y1 ; y2 ; y3 respectively. Let iteration number
~ 10 ðx1 ; x2 ; tÞ (red curve), t 2 ½0; 1. (For interpretation of the references Fig. 1. No perturbed initial control u10 ðx1 ; x2 ; tÞ (black curve), perturbed initial control u to color in this figure legend, the reader is referred to the web version of this article.)
~ 20 ðx1 ; x2 ; tÞ (green curve), t 2 ½0; 1. (For interpretation of the Fig. 2. No perturbed initial control u20 ðx1 ; x2 ; tÞ (black curve), perturbed initial control u references to color in this figure legend, the reader is referred to the web version of this article.)
~ 3 ðx1 ; x2 ; tÞ (blue curve), t 2 ½0; 1. (For interpretation of the references Fig. 3. No perturbed initial control u3 ðx1 ; x2 ; tÞ (black curve), perturbed initial control u to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 4. y1 ðx1 ; x2 ; 0Þ; y1 ðx1 ; x2 ; TÞ and their contour plots for ðx1 ; x2 Þ 2 ½0; 100 ½0; 100.
Fig. 5. y2 ðx1 ; x2 ; 0Þ; y2 ðx1 ; x2 ; TÞ and their contour plots for ðx1 ; x2 Þ 2 ½0; 100 ½0; 100.
N ¼ 10, partition number n ¼ 100; h ¼ L=n; t0 ¼ 0:0; T ¼ 10:0; dt ¼ 1:0 and take t f ¼ T. Consider diffusion HNN simultaneously systems for three neurons.
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Fig. 6. y3 ðx1 ; x2 ; 0Þ; y3 ðx1 ; x2 ; TÞ and their contour plots for ðx1 ; x2 Þ 2 ½0; 100 ½0; 100.
Fig. 7. Mexican hat functions at t ¼ 0 : y1 ð30; x2 ; 0Þ in x2 direction (red curve), y1 ðx1 ; 30; 0Þ in x1 direction (blue curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 8. Mexican hat functions at t ¼ 0 : y2 ð70; x2 ; 0Þ in x2 direction (red curve), y2 ðx1 ; 30; 0Þ in x1 direction (blue curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 9. Mexican hat functions at t ¼ 0 : y3 ð50; x2 ; 0Þ in x2 direction (blue curve), y3 ðx1 ; 30; 0Þ in x1 direction (red curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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8 3 X @y1 > > > c1j tanhðxyj Þ þ u1 ðtÞ in Q ; d1 Dy1 ¼ a1 y1 þ > > @t > > j¼1 > > > > 3 X > @y2 > > > c2j tanhðxyj Þ þ u2 ðtÞ in Q ; d2 Dy2 ¼ a2 y2 þ > > @t > j¼1 < 3 X @y3 > > c3j tanhðxyj Þ þ u3 ðtÞ in Q; d3 Dy3 ¼ a3 u3 þ > > > @t > j¼1 > > > > > > y1 ðx1 ; x2 ; 0Þ ¼ y10 ðx1 ; x2 Þ on X; > > > > > y2 ðx1 ; x2 ; 0Þ ¼ y20 ðx1 ; x2 Þ on X; > : y3 ðx1 ; x2 ; 0Þ ¼ y30 ðx1 ; x2 Þ on X:
Here, d1 ¼ d2 ¼ d3 ¼ 0:0001, a1 ¼ a2 ¼ a3 ¼ 1:0; c1j ¼ c2j ¼ c3j ¼
2
pk
and x ¼ p2 ; k ¼ 1:4. Then give the starting distributed
control by using coefficients b ¼ 0:1 102; f ¼ 1:5 102 in term of
u10 ðtÞ ¼ bð2 þ 3 sinðftÞÞ; u20 ðtÞ ¼ 2bð2 þ 3 sinðftÞÞ; u30 ðtÞ ¼ bð2 þ 3 cosðftÞÞ:
~ 1 ðtÞ in t 2 ½0; t f . Fig. 10. Optima control u
~ 2 ðtÞ in t 2 ½0; tf . Fig. 11. Optimal control u
~ 3 ðtÞ in t 2 ½0; tf . Fig. 12. Optimal control u
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~1 ðx1 ; x2 ; tÞ at each iteration n ¼ 1; 2; . . . ; 10 for ðx1 ; x2 Þ 2 ½0; 100 ½0; 100; t 2 ½0; tf . Fig. 13. Activity function y
The perturbations in control variables are given by amplitude b ¼ 0:2b and frequency f ¼ 0:2f . Then perturbed system with 0 the amplitude b ¼ ð1 þ 0:2Þb and frequency f 0 ¼ ð1 þ 0:2Þf in u1 ; u2 ; u3 appeared at each equation of y1 ; y2 ; y3 . See Figs. 1–3 for two starting controls of no perturbed and perturbed cases.
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~2 ðx1 ; x2 ; tÞ at each iteration n ¼ 1; 2; . . . ; 10 for ðx1 ; x2 Þ 2 ½0; 100 ½0; 100; t 2 ½0; tf . Fig. 14. Activity function y
The initial activities potential functions are given by
p y10 ¼ 240sech 0:01ðx1 30Þ2 þ 0:01ðx2 30Þ2 þ ; 5 p y20 ¼ 240sech 0:01ðx1 70Þ2 þ 0:01ðx2 30Þ2 þ ; 5 p y30 ¼ 240sech 0:01ðx1 50Þ2 þ 0:01ðx2 70Þ2 þ : 5
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~3 ðx1 ; x2 ; tÞ at each iteration n ¼ 1; 2; . . . ; 10 for ðx1 ; x2 Þ 2 ½0; 100 ½0; 100; t 2 ½0; tf . Fig. 15. Activity function y
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~1 ð30; x2 ; t f Þ in x2 direction (red curve), y ~1 ðx1 ; 30; t f Þ in x1 direction (blue curve). (For interpretation of the references to Fig. 16. Mexican hat functions at t f : y color in this figure legend, the reader is referred to the web version of this article.)
~2 ð70; x2 ; t f Þ in x2 direction (blue curve), y ~2 ðx1 ; 30; tf Þ in x1 direction (red curve). (For interpretation of the references to Fig. 17. Mexican hat functions at t f : y color in this figure legend, the reader is referred to the web version of this article.)
~3 ð50; x2 ; t f Þ in x2 direction (red curve), y ~3 ðx1 ; 70; t f Þ in x1 direction (blue curve). (For interpretation of the references to Fig. 18. Mexican hat functions at t f : y color in this figure legend, the reader is referred to the web version of this article.)
~1 ð30; 30; tÞ (red curve), y ~2 ð70; 30; tÞ (green curve), y ~3 ð50; 70; tÞ (blue curve) for t 2 ½0; tf . (For interpretation of the references to color in this figure Fig. 19. y legend, the reader is referred to the web version of this article.)
The target activities functions are supposed by
20 ðy þ 0:7y20 Þ; 240 10 90 y ; y2t ¼ 90 þ 240 20 20 ðy þ 0:7y20 Þ: y3t ¼ 240 30 y1t ¼
The start and target activities functions of y1 ; y2 ; y3 and their contour plots are shown in Figs. 4–6. As is well known, the activation potential is like Mexican hat function, before the control processing, their graphics at points P 1 ; P 2 ; P 3 in direction x1 ; x2 are given in Figs. 7–9.
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eJ 2.0 108
J 1.8 108
1.5 108
1.6 108 1.0 108
1.4 108 1.2 108
5.0 107
2
4
6
8
10
n
8.0 107
2
4
6
8
10
n
~ Þ; eeJ ¼ eJðu ~ Þ with JðuÞ; eJðuÞ (green, red points represent no perturbation, black point is perturbed one). (For interpretation Fig. 20. The comparison of eJ ¼ Jðu of the references to color in this figure legend, the reader is referred to the web version of this article.)
~ 1 ðtÞ (purple curve is perturbed one) for t 2 ½0; 1. (For interpretation of the references to color in Fig. 21. Optimal controls u1 ðtÞ (blue curve is original one), u this figure legend, the reader is referred to the web version of this article.)
~ 2 ðtÞ (purple curve is perturbed one) for t 2 ½0; 1. (For interpretation of the references to color in Fig. 22. Optimal controls u2 ðtÞ(blue curve is original one), u this figure legend, the reader is referred to the web version of this article.)
~ 3 ðtÞ (purple curve is perturbed one) for t 2 ½0; 1. (For interpretation of the references to color in Fig. 23. Optimal controls u3 ðtÞ (blue curve is original one), u this figure legend, the reader is referred to the web version of this article.)
Through the limited iterations steps, the optimal controls will be obtained as the form of
~1 ðtÞ ¼ 19:8486 þ 30:0ð2 þ 3 sinð150tÞÞ; u ~2 ðtÞ ¼ 0:275766 þ 59:9981ð2 þ 3 sinð150tÞÞ; u ~3 ðtÞ ¼ 19:8532 þ 30:0ð2 þ 3 sinð150tÞÞ: u The perturbed optimal control inputs are displayed in Figs. 10–12.
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~ 1 ðtÞ for t 2 ½0; 10. Fig. 24. Error function eu
~ 2 ðtÞ for t 2 ½0; 10. Fig. 25. Error function eu
~ 3 ðtÞ for t 2 ½0; 10. Fig. 26. Error function eu
~ 1 ðtÞ (blue line) for t 2 ½0; 1. (For interpretation of the references to color in this figure legend, the reader is Fig. 27. Error functions eu1 ðtÞ (red line), eu referred to the web version of this article.)
~1 ; y ~2 ; y ~3 (corresponding to u ~1 ; u ~2 ; u ~ 3 ) are changed with time at each iterAfter controlling process, the activities function y ation in following. In last iteration, the activities potential in presence of perturbation are computed in grouped graphics, see Figs. 13–15. These Mexican hat functions are changed and simulated at final iteration time tf in Figs. 16–18. The calculated Mexican hat functions verified the Theorem 1 and 2 numerically. At point ð30; 30Þ; ð70; 30Þ and ð50; 70Þ, ~i ðtÞ; i ¼ 1; 2; 3 are changed with time t, their graphics are plotted in Fig. 19. activation potential functions y
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~ 2 ðtÞ (blue line) for t 2 ½0; 1. (For interpretation of the references to color in this figure legend, the reader is Fig. 28. Error functions eu2 ðtÞ (red line), eu referred to the web version of this article.)
~ 3 ðtÞ (blue line) for t 2 ½0; 1. (For interpretation of the references to color in this figure legend, the reader is Fig. 29. Error functions eu3 ðtÞ (red line), eu referred to the web version of this article.)
It is accordant with the results in [5]. Optimal cost values attain J½u ¼ 8:33629 107 and eJ½u ¼ 8:70052 107 for no ~ Þ are calculated at each iteration. perturbation case and perturbed case. Cost functions values JðuÞ and eJ ¼ Jðu
J½1 ¼ 1:68294 108 ; J½2 ¼ 8:35024 107 ; J½3 ¼ 8:34863 107 ; 7
J½4 ¼ 8:34938 10 ;
eJ½1 ¼ 1:93494 108 ; eJ½2 ¼ 8:7219 107 ; eJ½3 ¼ 8:72139 107 ; eJ½4 ¼ 8:7156 107 ;
eJ½5 ¼ 8:71734 107 ; J½6 ¼ 8; 34379 107 ; eJ½6 ¼ 8:71167 107 ; J½7 ¼ 8:33859 107 ; eJ½7 ¼ 8:71043 107 ;
J½5 ¼ 8:34332 107 ;
J½8 ¼ 8:33741 107 ; J½9 ¼ 8:33673 107 ; J½10 ¼ 8:33629 107 ;
eJ½8 ¼ 8:70567 107 ; eJ½9 ¼ 8:70669 107 ; eJ½10 ¼ 8:70052 107 :
Two cases error values eJ ¼ J½n þ 1 J½n; eeJ ¼ eJ½n þ 1 eJ½n of cost functions are obtained as
eJ½1 ¼ 1:68294 108 ;
eeJ½1 ¼ 1:93494 108 ; eeJ½2 ¼ 1:06275 108 ;
eJ½2 ¼ 8:47918 107 ; eJ½3 ¼ 16086:6; eeJ½3 ¼ 5054:44; eJ½4 ¼ 7479:55; eJ½5 ¼ 60597:8; eJ½6 ¼ 4670:06; eJ½7 ¼ 51987:6; eJ½8 ¼ 11761:3; eJ½9 ¼ 6818:04; eJ½10 ¼ 4396:42;
eeJ½4 ¼ 57920:0; eeJ½5 ¼ 17437:0; eeJ½6 ¼ 56783:0; eeJ½7 ¼ 12310:0; eeJ½8 ¼ 47619:0; eeJ½9 ¼ 10186:8; eeJ½10 ¼ 61735:7:
The comparison graphics of two cost values and two error values at each iteration see Fig. 20.
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~ i ; i ¼ 1; 2; 3 are given in Figs. 21–23. The comparison of optimal control functions ui ; u ~i ¼ u ~ i ½n þ 1 u ~ ½n; i ¼ 1; 2; 3 at final iteration calculated as: Errors of control functions eu
~ 1 ¼ 0:0000517268 8:17772 109 sinð150tÞ; eu ~ 2 ¼ 0:00883129 4:25091 107 sinð150tÞ; eu ~ 3 ¼ 0:0000505376 8:07544 109 cosð150tÞ: eu Their graphs are simulated in Figs. 24–26. The comparison with error of no perturbed case eui ¼ ui ½n þ 1 ui ½n; i ¼ 1; 2; 3 are displayed in Figs. 27–29 for time interval t 2 ½0; 1, The numerous results directly illustrate Theorem 3. Obviously, the stability and convergence of nonlinear scheme are evidenced. For no perturbation case and perturbed case, the total running CPU times are 1314.6 s, and 1720.77 s, the used maximum memory are 165330.200 bytes, and 167774.368 bytes, respectively. Remark 1. Our results can be extended to three dimensions case clearly. In this work, the control input ui ðtÞ is depended time t, it would be efficient to solve the spatial depended case ui ðt; xÞ, see [20] to render details. It would be working for timedelayed HNNs certainly.
6. Conclusions This work presented the conclusion for control of Hopfield neural network (HNN) with diffusion term in the presence of perturbations occurred in control field. It would be a new consideration in the realization of control HNN. Future perspective research would be connected with the neuroscience and relative fields. Acknowledgments The authors thank 2010 Annual Conference of Japan Society for Industrial and Applied Mathematics (JSIAM) for presentation ‘‘Sensitivity of optimal control for diffusion Hopfield Neural Network in the presence of perturbation’’. References [1] R. Dautray, J.L. Lions, Mathematical analysis and numerical methods for science and technology, Evolution Problems I, vol. 5, Springer-Verlag, Berlin– Heidelberg–New York, 1992. [2] R. Fitz-Hugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophys. 17 (1955) 257–278. [3] A.L. Hodgkin, A.F. Huxley, A qualitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117 (1952) 500–544. [4] J.J. Hopfield, Neural network and physics system with collective computational abilities, Proc. Natl. Acad. Sci. USA 79 (1982) 2554–2558. [5] J.J. Hopfield, Neurons with graded response have collective properties like those of two state neurons, Proc. Natl. Acad. Sci. USA 81 (1984) 3088–3092. [6] J.J. Hopfield, D.W. Tank, Computing with neural circuits: a model, Science 233 (1986) 625–632. [7] D. Ji, J. Koo, S. Won, S. Lee, J. Park, Passivity-based control for Hopfield neural networks using convex representation, Appl. Math. Comput. 217 (2011) 6168–6175. [8] S. Lakshmanan, J. Park, H. Jung, P. Balasubramaniam, Design of state estimator for neural networks with leakage, discrete and distributed delays, Appl. Math. Comput. 218 (2012) 11297–11310. [9] L.S. Lasdon, S.K. Mitter, A.D. Warren, The conjugate gradient method for optimal control problems, IEEE Trans. Autom. Control 12 (1967) 132–138. [10] J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin–Heidelberg–New York, 1971. [11] J.L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I, II, Springer-Verlag, Berlin–Heidelberg–New York, 1972. [12] S. Nakagiri, Q.F. Wang, Optimal control problems for distributed Hopfield-type neural networks, Nonlinear Funct. Anal. Appl. 7 (2) (2002) 167–186. [13] J. Park, O. Kwon, On improved delay-dependent criterion for global stability of bidirectional associative memory neural networks with time-varying delays, Appl. Math. Comput. 2008 (1999) 435–446. [14] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematica Sciences, second ed., 68, Springer-Verlag, Berlin– Heidelberg–New York, 1997. [15] M. Velazco Fontova, A. Oliveira, C. Lyra, Hopfield neural networks in large-scale linear optimization problems, Appl. Math. Comput. 218 (2012) 6851– 6859. [16] Q.F. Wang, Boundary bang–bang control of distributed diffusion Hopfield neural network equations, Int. J. Comput. Numer. Anal. Appl. 6 (1) (2004) 33– 63. [17] Q.F. Wang, S. Nakagiri, Numerical solution of distributed diffusion Hopfield neural network equations, Appl. Math. Comput. 174 (2) (2006) 1343–1362. [18] Q.F. Wang, Theoretical and computational issue of optimal control for distributed Hopfield neural network equations with diffusion term, SIAM J. Sci. Comput. 29 (2) (2007) 890–911. [19] Q.F. Wang, Numerical approximate of optimal control for distributed diffusion Hopfield neural networks, Int. J. Numer. Methods Eng. 69 (3) (2007) 443–468. [20] Q.F. Wang, Computation issues of pointwise control for diffusion Hopfield neural network system, Appl. Math. Comput. 207 (2009) 148–163. [21] Q.F. Wang, Boundary pointwise control for diffusion Hopfield neural network, Int. J. Nanotechnol. Mol. Comput. 2 (1) (2010) 13–29.