Existence, uniqueness and continuous dependence on control of solutions to generalized neural field equations Текст научной статьи по специальности «Математика»

УДК 517.988.6, 517.968.48, 51-76

EXISTENCE, UNIQUENESS AND CONTINUOUS DEPENDENCE ON CONTROL OF SOLUTIONS TO GENERALIZED NEURAL FIELD

EQUATIONS

© E.O. Burlakov, E.S. Zhukovskiy

Key words: nonlinear integral equations; neural field equations; control; well-posedness. We consider nonlinear integral equation involving essentially bounded control. We obtain conditions for existence and uniqueness of solution to this equation and continuous dependence of this solution on control.

1. Introduction

The problem of continuous dependence on parameters of various classes of operator equations has been considered in numerous papers (see, e.g. the review [1] as well as the monograph [2] (pp. 203-210) and the references therein). Apart from theoretical interest, the problem also has an application importance connected to real processes and phenomena modeling. As constants and functions involved in models can be found only approximately, the crucial property of a "good" model is it's well-posedness, i.e. unique solvability and continuous dependence of the solution on the model parameters. The present work is utilizing the general results on existence, uniqueness and continuous dependence of solutions to operator Volterra equations on parameters obtained in [3] and applying these results to investigation of well-posedness of a controllable integral equation describing a wide class of models arising in neural field theory. Typical representative of this class is the Amari neural field equation [4]

w,it.x) = -W(t.x) + j,(x-yWMt.mv, t > cu £(i.i)

R

Here the function w(t,x) denotes the activity of a neural element at time t and position x. The connectivity function (spatial convolution kernel) w(x) determines the coupling between the elements and the non-negative function f (w) gives the firing rate of a neuron with activity w. Neurons at position x and time t are said to be active if f (w(t, x)) > 0 .

The literature on the model (1.1) and its extensions is vast (see, e.g. [5-8]). The key issues in most of the published papers on these models are existence and stability of localized stationary solutions (so-called bumps).

The problems of therapy of Epilepsy, Parkinson's disease, and other disorders of the central nervous system were recently investigated in [9-12] using neural activity models which incorporate brain electrical stimulation. In these papers the stimulation is considered as control and the corresponding optimization problems are posed. The results we offer below allow to investigate well-posedness of such models and, particulary, can serve as justification of the numerical optimization procedure used in [13](as the authors base their research on the Amari framework (1.1), where f is taken to be sigmoidal) as well as the starting point for implementation of numerical optimization to more complex objects (as were considered e.g. in [5,6]).

We generalize the models from [4-6,8] by introducing the following neural field equation t

w(t,x) = J J f (t, s,x,y,w(s,y),u(s,y))dyds, t £ [a, m), x £ Rm, (1.2)

a rm

with respect to the unknown continuous function w : [a, œ) xRm ^ Rra , which is spatially localized, i.e.

lim w(t,x) = 0, t € [a, œ). (1.3)

We obtain results on unique solvability of (1.2), (1.3) and continuous dependence of the solution on the Lebesgue measurable control u : [a, œ) x Rm ^ Rk involved in (1.2).

2. Existence, uniqueness and continuous dependence of solutions to Volterra operator equations on parameters

Let us introduce the following notation:

Rm is the m -dimensional real vector space with the norm | • | ; A is some metric space;

for any & c A , r > 0 , we denote BA(&, r) = (J [X € A | pA(X,s) < r} ;

see

B is some Banach space with the norm || • \\B ; 1 is the Lebesgue measure;

M([a, to) x Rm, ¡, Rk, u0) is a metric space of measurable functions u : [a, to) x Rm ^ Rk such that vrai sup lu(t,x) — u0(t, x)| < to , where u0 : [a, to) x Rm ^ Rk is some given

(t,i)e[a,»)xrm

measurable function, with the distance

PM ([a,»)xr-,mlr' ,«o)(u1,u2) = vrai suP |u1(t,x) — u2(t,x)|;

(t,x)e[a,<x)xrm

Y([a, b], B) is a Banach space of functions u : [a, b] — B with the norm || • ||r ;

L(Q,ß, Rra) is the space of all measurable and integrable functions % : Q — Rra with the norm

Ml^R") = / \x(s)\ds;

n

C0(Q, Rra) is the space of all continuous functions § : Q — Rra satisfying the additional condition lim §(x) = 0 in the case if Q is unbounded, with the norm ||$||c0(n r™) = max\§(x)\ ;

C([a, b], Co(Q, Rra)) is the space of all continuous functions v : [a, b] — Co(Q, Rra), with the norm

llv^c([a,6],c0(n,r")) = max ||v(i)|c0(n,r") •

t€[a,b\

In the notation for functional spaces we will not indicate the definition domains and the image sets of functions, provided that this leads to no ambiguity.

Definition 2.1. An operator ^ : Y ^ Y is said to be a Volterra operator (in the sense of A.N. Tikhonov [14]) if for any £ € (0,b—a) and any v1,v2 € Y the fact that u1(t) = u2(t) on [a, a+£] implies that (^ui)(t) = (^t>2)(t) on [a,a+£] .

In what follows we assume that in the space Y the following condition is fulfilled:

V -condition [15]: For arbitrary v € Y, {vi} C Y such that \vi — v\\t ^ 0 and for any £ € (0, b—a) if vi(t) =0 on [a, a+£] , then u(t) = 0 on [a, a+£] .

For any £ e (0,b—a) let Yg = Y([a,a+£],B) denote the linear space of restrictions ug of functions u e Y to [a, a+£] which implies that for each ug e Yg there exists at least one extension u e Y of the function ug . Then we can define the norm of Yg by ||ug||y5 = inf ||u||y , where the infimum is taken over all extensions u e Y of the function ug . Hence, the space Yg becomes a Banach space.

For an arbitrary £ e (0, b—a) let an operator Pg : Y ^ Y takes each ug e Yg to some extension u e Y of ug . Moreover, we define the operators Eg : Y ^ Yg by (Egu)(t) = u(t), t e [a, a+£] and ^g : Yg ^ Yg by ^gug = Eg^Pgug , respectively. Note that for any Volterra operator ^ : Y ^ Y the operator ^g : Yg ^ Yg is also a Volterra operator and it is independent of the way u = Pg ug extends ug .

Definition 2.2. A Volterra operator ^ : Y ^ Y is called locally contracting if there exists q < 1 such that for any r > 0 one can find 5 > 0 such that the following two conditions are satisfied for all u1,u2 e Y , such that ||ui||Y < r , ||u2||r < r :

qi) \\Estfui - Es< q\\Esv\ — EsV2\\rs ,

q2) for any 7 € (0, b—a—ö] , the condition EYu1 = EYu2 implies that

\\E1+s^L>1 — ey+s^u2\t7+5 < q\\EY+sVi — ey+&ui\\r7+ä•

Let us now consider the equation

u(t) = (tfu)(t), t e [a,b], (2.1)

where with ^ : Y ^ Y is a Volterra operator.

Definition 2.3. We define a local .solution to Eq. (2.1) on [a, a+7] , 7 € (0, b—a) to be a function vY € Yy that satisfies the equation vY = vY on [a, a+7] . We define a maximally extended solution to Eq. (2.1) on [a, a+Z) , Z € (0, b—a] to be a function vz : [a, a+Z) ^ B, whose restriction vY to [a, a+7] is a local solution of Eq. (2.1) for any y<Z and lim \\vY \\t7 = œ .

We define a global solution to Eq. (2.1) to be a function v € Y that satisfies this equation on the entire interval [a, b] .

Let us now consider the equation

u(t) = (F(u,A))(t),t e [a, b] (2.2)

with a parameter A e A, where for each A e A a Volterra operator F(-,A) : Y ^ Y satisfies the property: F(-,A0) = ^ for some Ao e A. Our aim is to formulate conditions for existence and uniqueness of solutions to Eq. (2.2) on a certain fixed set [a, a + £] C [a, b] (We, naturally, also apply Definition 2.3 to Eq. (2.2) at each fixed A e A); and convergence of these solutions to solution to Eq. (2.1) in the norm of Yg as A approaches Ao . This means, that the problem (2.2) is wellposed.

Definition 2.4. For any A € Ao C A , let the Volterra operator F(■, A) : Y — Y be given. This family of operators is called uniformly locally contracting if there exist q > 0 and ö > 0, such that for each A € A0 C A the operator F(-,A) : Y — Y is locally contracting with the constants q

and 5(r) independent of the choice of A € A0 . This means existence of a such q < 1 that for any r > 0 one can find 5 > 0 such that the operator F(■, A) : Y — Y (with any A € Ao ) satisfies the conditions qi), q2) for all v1,v2 € Y , Hv^t < r , ||v2||r < r .

The following theorem (see [3]) represents our main tool to study of the wellposedness of the problem (1.2), (1.3).

Theorem 2.1. Assume that the following two conditions are satisfied:

1) There is a neighborhood A0 of A0 where the operators F(■, A) : Y — Y, A € A0 are uniformly locally contracting;

2) For arbitrary v € Y , the mapping F : Y x A — Y is continuous at (v, A0) .

Then for each A € A0 , Eq. (2.2) has a unique global or maximally extended solution, and each local solution is a restriction of this solution.

If Eq. (2.2) has a global solution v0 at A = A0 , then for each A (sufficiently close to A0 ) it also has a global solution v = v(A), and ||u(A) — v0Hy — 0 as A — A0 .

If Eq. (2.2) has a maximally extended solution v0z defined on [a,a+() at A = A0 , then for any y € (0, () one can find a neighborhood of A0 such that for any A in this neighborhood Eq. (2.2) has a local solution vY = vY(A) defined on [a,a+j] and HvY(A) — v0l||y7 — 0 as A — A0 .

Remark 2.1. If the constant 5 in the condition 1) of Theorem 2.1 is independent of r, then Eq. (2.2) has a global solution. This holds truee.g. in the case of t -Volterra operators, when for all operators the corresponding delays t > 50 for some 50 > 0 .

3. Existence, uniqueness and continuous dependence on control of solutions to generalized neural field equations

We assume that for some r0 > 0, the following conditions on the system (1.2), (1.3) are fulfilled:

(i) For any t € [a, x>) , w € Mra , x € Mm and any ball u € Rk , the function f (t, ■,x, ■,w,u) : [a, x>) x Rm — Rra is measurable.

(ii) For almost all (s,y) € [a, x>) x Rm and any u € Rk , the function f (■, s, ■, y, ■,u) is continuous.

(iii) For any b € (a, x>) and any r > 0 , it holds true that

lim sup

i€[a,6],£€Rm\Biim (0,r)

t

J j f (t,s,x,y,w,u0(s,y) + A)dyds

for any w € BRn (0, r) , uniformly for all A € BRk (0, r0) .

(iv) For any b € (a, x>) and any r > 0 , there exists such g(b,r) € L([a, b] x Rm, ¡i, R) that

\f(t,s,x,y,w,u)\ < g(btr)(s,y)

for all x € Rm, w € BRn (0,r) , t € [a,b] , u € BRk ({u0(t,x)},r0) and almost all (s,y) € [a, b] x Rm .

Definition 3.1. Choose an arbitrary u € BM(u0, r0). We define a local solution to the system (1.2), (1.3), defined on [a,a+Y]xRm , y € (0, x>), to be a function w7 € C([a, a+Y],C0(Rm, Rra)), that satisfies the equation (1.2) on [a, a+Y] x Rm. We define a maximally extended solution to

0

the system (1.2), (1.3), defined on [a, a+n) x Rm , n G (0, œ), to be a function wn : [a, a+n) x Rm ^ Rra, whose restriction wY on [a, a+7] x Rm with any 7 < n is its local solution and lim \\wY(\aa+Y] c0(wm Rn)) = œ • We define a global solution to the system (1.2), (1.3) to be a

y^n—0 ' '

function w : [a, œ) x Rm ^ Rra , whose restriction wY on [a, a+7] x Rm for any 7 G (0, œ) is its local solution.

Theorem 3.1. Let the assumptions (i) - (iv) hold true. Assume that the following conditions are satisfied:

1) For the given r0 > 0 and any r > 0 there exists fr(s,y) € L([a, to) x R) such that for which \f (t,s,x,y,w\,u) — f (t, s,x,y,w2,u)\ < fr (s,y)\w\ — w2\ for all w\,w2 € BRn (0, r), u € BRk (uo(s,y),ro), t € [a, to) , x € Br™ (0,r) .

2) For any w € Rra , t € [a, to) , x € Rm, A ^ 0 it holds true that:

\f (t, •,x, •,w,uo(•, •) + A) — f (t, •,x, •,w,uo)\ ^ 0 in measure on [a, to) x Rm .

Then for each u € Bm(u0,r0), the system (1.2), (1.3) has a unique global or maximally extended solution, and each local solution is a restriction of this solution. Moreover, if at u = u0 the system (1.2), (1.3) has a local solution w0l defined on [a, a+7] x Rm, then for any {ui} C M([a, to) x Rfc, u0), pM(ui,u0) ^ 0 one can find number I such that for all i > I the

system (1.2), (1.3) has a local solution wY = wY(ui) defined on [a, a+7] x Q and

\\wy (ui) — w0y \\o ([a , a+y], Co (r™, r")) ^ 0.

Proof. We are going to apply Theorem 2.1, so we represent (1.2) in terms of operator equation in the following way

w = F(w, u),

t

(F(w,u))(t,x) = J J f (t,s,x,y,w(s,y),u(s,y))dyds.

Here, for each u € BM(uo, ro), F : C([a, b], Co(Rm, Rra)) ^ C([a, b], C0(Rm, Rra)) provided that the conditions (i)-(iv) are fulfilled.

Choose an arbitrary b € (a, to) , q0 < 1, r > 0. Let 7 € (0, b — a) and w\(t, •) = w2(t, •), t € [a, a+7] , where w\,w2 € Bo(\ayb],o0(r™, r"))(0, r) . Using assumptions (i)-(iv) and condition 1) of Theorem 3.1, we get the following estimates

sup

t&\a, a+Y+S], x€R"

t

J J f (t,s,x,y,wi(s,y),u)dyds-

t

J J f (t,s,x,y,w2(s,y),u)dyds

<

a+y+S

e/2 + sup

t&\a,a+Y+S], x£BRm (0 ,r£)

f (t, s, x, y, wi(s, y),u)dyds-

a+y

-f (t,s,x,y,W2(s,y),u) dyds < a+Y+à

e/2 + sup / / fr(s,y)\\wi - W2\\o([a,b],BC(wm,wn))dyds < e.

t£\a,a+Y+S\,x£BRm (0,rs) J J

a+Y rm

Here, r£ > 0, ô > 0 can be chosen in a such way that e < q0 . Thus, we checked that condition q2) is satisfied. The verification of condition qi) is analogous.

Next, we take arbitrary e > 0 , W € C([a,b],Co(Rm, Rra)), Wi C C([a,b],Co(Rm, Rra)), Ui C M ([a, œ) x Rm, ¡i, Rfc ,uo), \\W - Wi\c([a,b],c0(rm,r")) , Pm (ui,uo) ^ 0 ( i ), and estimate

\\F(w,u0) - F(wi,ui)\c([a,b],0o(wm,rn)) =

= sup

te[a,b],xer"

J J f (t,s,x,y,w(s,y),uo(s,y))dyds-

a i"

-// f(t,s,x,y,Wi(s,y),Ui(s,y))dyds <

a i"

t

< e/3+ sup / f (t,s,x,y,w(s,y),uo(s,y))dyds-

te[a,b],xeBRm(o,re) J J

a i"

t

- 11 f (t,s,x,y,Wi(s,y),Ui(s,y))dyds =

a i" t

= e/3+ sup / (\f(t,s,x,y,W(s,y),uo(s,y)) - f(t,s,x,y,w(s,y),Ui(s,y))\ +

t&[a,b],x&BR" (0,rE)J J V a i"

+ \f (t, s, x, y, w(s, y),Ui(s, y)) - f (t, s, x, y, Wi(s, y),Ui(s, y))ty dyds. Estimating the first summand of the integrand, we get

\f(t,s,x,y,w(s,y),Uo(s,y)) - f(t,s,x,y,w(s,y),Ui(s,y))\ < < \f(t,s,x,y,w(s,y),Uo(s,y)) - f (t£,s,x£,y,w£,Uo(s,y))\ +

+ \f (t£, s, t£,y, w£, Uo(s, y)) - f (t£, s, x£,y, w£, Ui(s, y))\ +

+ \f (t£, s, x£,y, w£, Ui(s, y)) - f (t, s, x, y, w(s, y), Ui(s, y))\.

Here t£, x£ , w£ are approximations of t, x, w(s,y), taking finite number of values (on their compact ranges of definition). Thus, using the condition 2) of Theorem 3.1 and the assumptions (i)-(iv), the first and third summands on the right-hand side of the inequality go to 0 uniformly with respect to (s, y) € [a, b] x Im and the second summand go to 0 in measure on [a, b] x Im .

Next, estimation of \f (t, s, x, y, w(s, y), Ui(s, y))-f (t, s, x, y, wi(s, y),Ui(s, y)) \ using the condition 1) of Theorem 3.1, gives uniform convergence of this expression to 0 on [a, b] x Im . Thus, we can find such I that for any i > I, we get t

sup / / (\f(t,s,x,y,w(s,y),Uo(s,y)) - f(t,s,x,y,w(s,y),Ui(s,y))\ +

t&[a,b],x&BRm (0,rE) J J V a i"

+ \f (t,s,x,y,w(s,y),Ui(s,y)) - f (t,s,x,y,Wi(s,y),Ui(s,y))tydyds < 2e/3, which concludes the verification of Theorem 2.1 conditions and completes the proof.

t

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ACKNOWLEDGEMENTS: The present research is supported by Russian Fund for Fundamental Research (Projects № 14-31-50184, № 14-01-97504).

Поступила в редакцию 21 ноября 2014 г.

Бурлаков Е.О., Жуковский Е.С. СУЩЕСТВОВАНИЕ, ЕДИНСТВЕННОСТЬ И НЕПРЕРЫВНАЯ ЗАВИСИМОСТЬ ОТ УПРАВЛЕНИЯ РЕШЕНИЙ ОБОБЩЕННЫХ УРАВНЕНИЙ НЕЙРОННЫХ ПОЛЕЙ.

Изучено нелинейное интегральное уравнение с ограниченным в существенном управлением. Получены условия существования и единственности решения такого уравнения, а также условия непрерывной зависимости решения от управления.

Ключевые слова: нелинейные интегральные уравнения; уравнения нейронных полей; управление; корректность.

Бурлаков Евгений Олегович, Норвежский университет естественных наук, г. Аас, Норвегия, аспирант, e-mail: eb_@bk.ru

Burlakov Evgenii Olegovich, Norwegian University of Life Sciences, As, Norway, PhD-candidate, e-mail: eb_@bk.ru

Жуковский Евгений Семенович, Тамбовский государственный университет имени Г.Р. Державина, г. Тамбов, Российская Федерация, доктор физико-математических наук, профессор, директор института математики, физики и информатики, е-mail: zukovskys@mail.ru

Zhukovskiy Evgeny Semenovich, Tambov State University named after G.R. Derzhavin, Tambov, Russian Federation, Doctor of Physics and Mathematics, Professor, Director of the Institute of Mathematics, Physics and Informatics, е-mail: zukovskys@mail.ru