THE PROBLEM OF OPTIMAL CONTROL

Memoirs on Dierential Equations and Mathematical Physics

Volume 11, 1997, 67{88

Guram Kharatishvili and Tamaz Tadumadze

THE PROBLEM OF OPTIMAL CONTROL

FOR NONLINEAR SYSTEMS WITH VARIABLE STRUCTURE, DELAYS AND PIECEWISE

CONTINUOUS PREHISTORY

initial functions are proved in the form of an integral maximum principle and conditions of transversality for nonlinear systems with a variable structure, delays and a piecewise continuous initial function in the case where values of the initial function (prehistory) and of the trajectory at a non-xed initial moment and at a moment of variation of the structure do not, generally speaking, coincide.

1991 Mathematics Subject Classication. 49K25.

Key words and Phrases. Maximum principle, transversality conditions, delays, variable structure systems.

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1. Introduction

Necessary conditions of optimality for the problem given below (see 2) are derived from a necessary condition of criticality [1,2,3] the basis of which forms the notion of quasi-convex set introduced by R.V. Gamkrelidze [4].

Variation of the structure of a system means that the system at some beforehand unknown moment may go over from one law of movement to another. After variation of the structure, the values of the initial function of the system depend on its previous state. This joins them into a single system with variable structure.

In conclusion, it should be noted that some particular cases of the prob- lem under consideration have been studied in [5].

2. Statement of the Problem. Necessary Conditions of Optimality

Let Oi Rⁿⁱ, Gi R^ri, i = 1;:::;m, be open sets, Rⁿ be an n- dimensional Euclidean space, J = [a;b] be a nite interval; let the func- tionsfi :J O²_i G²_i ^!Rⁿⁱ,i= 1;:::;m, be continuously dierentiable with respect to (x_i;z_i) ² O²_i, i = 1;:::;m, respectively, let _i : R^{1 !} R¹, _i : R^{1 !} R¹, i = 1;:::;m, be continuously dierentiable functions satisfying the conditions _i(t) t, __i(t) > 0, _i(t) < t, __i(t) > 0; let q^k(t¹;:::;tm⁺¹;x¹⁰;x¹;:::;xm⁰;xm),k= 0;:::;l, be scalar functions con- tinuously dierentiable in all argumentsti ²J,i= 1;:::;m+ 1, (xi⁰;xi)² O_i², i = 1;:::;m; let the functions gi : J Oi ^{1 !} Oi, i = 2;:::;m be continuous and continuously dierentiable with respect to xi ¹²Oi ¹, i= 1;:::;m; respectively, let i= (Ji¹;Ni) be the set of piecewise contin- uous functions'i:Ji^1!Niwith a nite number of points of discontinuity, Ji¹[i(a);b], Ni Oi be a convex bounded set,^k'i^k= sup^f'i(t)^{jt 2}Ji^1g; let i be the set of measurable functions ui :Ji^2!Ui satisfying the con- dition: clui(Ji²)-is a compactum lying inGi,Ji²= [i(a);b], UiGi is an arbitrary set.

Consider the sets Ai=J^1+iYi

p⁼¹Op^Yi

p⁼¹i^Yi

p⁼¹p; i= 1;:::;m;

with the elements i = (t¹;:::;ti⁺¹;x¹⁰;:::;xi⁰;'¹;:::;'i;u¹;:::;ui),i= 1;:::;m;^Qi_p⁼¹Op=O¹Oi, respectively.

To every element m ² Am, ti < ti⁺¹, i = 1;:::;m, we assign the dierential equation of variable structure

x_i(t) =fi(t;xi(t);xi(i(t));ui(t);ui(i(t))); t²[ti;ti⁺¹]; (1i) xi(t) ='i(t) +gi(t;xi ¹(t));t²[i(ti);ti);xi(ti) =xi⁰; (2i)

i= 1;:::;m:

Here and in the sequel we assume that g¹ = 0, that is, x¹(t) = '¹(t);t ² [¹(t¹);t¹).

Denition 1. The set of functions^fxi(t) =xi(t;i);t ²[i(ti);ti⁺¹];i= 1;:::;m^g, is said to be a solution of the equation with variable structure which correspond to an elementm²Am, if the functionxi(t)²Oi on the interval [i(ti);ti] satises the condition (2i), while on the interval [ti;ti⁺¹] it is absolutely continuous and satises the equation (1i) almost everywhere.

Denition 2. The elementm²Amis said to be admissible if the corre- sponding solution ^fx_i(t);t ²[_i(t_i);t_i⁺¹];i= 1;:::;m^gsatises the condi- tions

q^k(t¹;:::;tm⁺¹;x¹⁰;x¹(t²);:::;sm⁰;xm(tm⁺¹)) = 0; k= 1;:::;l:

The set of admissible elements will be denoted byA⁰_m.

Denition 3. The element ^em = (^et¹;:::;^etm⁺¹;x^e10;:::;x^em⁰;'^e1;:::;'^em; ue¹:::;u^em)²A⁰_m is said to be locally optimal if there exist a number >0 and a compact setKi 0i;i= 1;:::;m, such that for arbitrary elements m²A⁰_m satisfying

m^X+1 i⁼¹

j

et_i t_i^j+^Xm

i⁼¹

fjex_i⁰ x_i^0j+^ke' '_i^k+^kf^e_i f_i^k_Ki^g the inequality

q⁰(^et¹;:::;^etm⁺¹;x^e10;x^e1(^et²);:::;^exm⁰;^exm(^etm⁺¹))

q⁰(t¹;:::;tm⁺¹;x¹⁰;x¹(t²);:::;xm⁰;xm(tm⁺¹)) is fullled.

Here

fei fiKi =

Z

J H t;fi;Kidt;H t;fi;Ki=

= supf^ei t;xi;yi fi t;xi;yi+@f^ei()

@xi

@fi()

@xi

+@f^ei()

@yi @fi()

@yi

xi;yi²K_i² ; fei t;xi;yi=fi t;xi;yi;^eui(t);^eui(i(t)));fi(t;xi;yi) =

=fi(t;xi;yi;ui(t);ui(i(t)); x^ei=xi(t;^ei); xi(t) =xi(t;i): The problem of optimal control consists in nding a locally optimal ele- ment.

Theorem 1. (Necessary conditions of optimality). Let ^em ² A⁰_m be a locally optimal element; a < ^et¹ < < ^etm⁺¹ < b, ^eti < i = i(^eti) <

i⁺¹(^eti⁺¹) i = 1;:::;m ¹, ^etm < m = m(^etm) ⁶=^etm⁺¹; let the functions (^eui(t);u^ei(i(t)))i= 1;:::;m, be continuous at the points^eti;i;^eti⁺¹, respec- tively,i= 1;:::;m, and the functions('^ei(t);'^ei(i(t))),i= 1;:::;mbe con- tinuous at the points^eti, i= 1;:::;m, respectively. Then there exist a non- zero vectorn= (n⁰;:::;nl), n⁰ 0, and a solution i(t), t²[^eti;i(^eti⁺¹)]

of the conjugate equation

__i(t) = i(t)@f^e_i(t)

@xi i((t))@f^e_i(_i(t))

@yi _i(t) _i(t) i⁺¹(_i⁺¹(t))@f^e_i⁺¹(_i⁺¹(t))

@yi⁺¹ @^eg_i⁺¹(t)

@xi __i⁺¹(t) (3i) t²[^eti;^eti⁺¹]; i(t) = 0; t²(^eti⁺¹;i(^eti⁺¹)];

i=m;:::;1; such that the following conditions are fullled:

1) the integral maximum principle

eti

Z

i^(eti⁾

i(i(t))@f^e_i(_i(t))

@yi _i(t)'^ei(t)dt

eti

Z

i^(eti⁾

i(i(t))@f^e_i(_i(t))

@yi _i(t)'i(t)dt; ⁸'i²i; i= 1;:::;m; (4)

et^Zi⁺¹

eti

i(t)f^ei(t)dt

et^Zi⁺¹

eti

i(t)fi(t;^exi(t);x^ei(i(t)ui(t);ui(i(t)))dt;

8'i²i; i= 1;:::;m: (5) 2) transversality conditions

@Q@t_i ⁼ ai i ¹(^eti)f^ei ¹(t^ei) +bi^f i(^eti)f^ei(^eti) + i(i) [f^e_i(_i;x^e_i(_i);^ex_i⁰) f^e_i(_i;^ex_i(_i);'^e_i(^et_i) +^eg_i(^et_i))]__i(^et_i)^g;

i= 1;:::;m+ 1; (6)

@Q@xi⁰ = i(^et_i); @Q@xi = i(^et_i⁺¹); i= 1;:::;m: (7)

Here

fei(t) =f^ei(t;x^ei(t);^exi(i(t))); @f^ei(t)

@x_i ⁼ @

@x_ifi(t;^exi(t);^exi(t));

egi(t) =gi(t;^exi ¹(t)); @^egi⁺¹(t)

@xi = @

@xigi⁺¹(t;^exi(t));

i(t) is the characteristic function of the segment [i(^eti⁺¹);^eti⁺¹)]; gm⁺¹= 0, i.e., the last summand in the right-hand side of the equation (3m) equals zero; _i(t) is the function inverse to _i(t); a¹ = 0, a² == a_m⁺¹ = 1, b¹ = = bm = 1, bm⁺¹ = 0; tilde over Q denotes that the gradient is calculated at the point (^et¹;:::;^etm⁺¹;x^e10;x^e1(^et²);:::, ^exm⁰;^exm(^etm⁺¹),

Remark 1. If rank

@Q

@t¹;::: @Q@tm⁺¹; @Q@x¹⁰;::: @Q@xm⁰; @Q@x¹;:::; @Q@xm

= 1 +l;

then _m

X

i⁼¹max^fj i(t)^jjt²[^eti;^eti⁺¹]^jg6= 0:

Remark 2. From the integral maximum principle 1) one can obtain in a standard way the pointwise maximum principle with respect to the functions 'ei(t),i= 1;:::;m:

i(i(t))@f^ei(i(t))

@yi _i(t)'^ei(t) i(i(t))@f^ei(i(t))

@yi _i(t)'i;

8'i²Ni; a.e. on [i(^eti);^eti]; i= 1;:::;m; with respect to the controlsu^ei(t),i= 1;:::;m,

pi

X

p⁼¹_^p_i ¹(t) i(^p_i ¹(t))f^ei(_i^{p 1}(t)) +^{Xi pi}

p⁼¹ :q__pi(t) i(q_pi(t))f^ei(q^p(t))

pi

X

p⁼¹_^p_i ¹(t) i(^p_i ¹(t))fi(^{p 1}(t);^exi(_i^{p 1}(t));x^ei(i(^p_i ¹(t))); ui;pi p⁺¹;ui;pi p) +^{Xi pi}

p⁼¹ _^p_i(t) i(^p_i(t))fi(^p_i(t);^exi(^p_i(t));x^ei(i(^p_i(t))); ui;pi⁺p;ui;pi⁺p ¹); ⁸(u¹⁰;:::;u¹;i)²u¹⁺_i ⁱ; a.e. on

[i;pi;i;pi⁺¹]; pi= 1;:::;i; i= 1;:::;m:

Here

i;pi =i(i;pi ¹); pi= 1;:::;i; i;⁰=i(^eti); i;pi⁺¹=^eti⁺¹;

⁰_i(t) =t, _pi(t) =i(^p_i ¹(t));i(t) is the function inverse toi(t),¹_i(t) = i(t); we assume that^P0_p⁼¹p= 0.

Theorem 2. Let ^e_m ²A⁰_m be a locally optimal element; a <^et¹ <<

etm⁺¹ < b, ^eti < i < i^+1eti⁺¹, i= 1;:::;m 1, ^etm < m⁶= ^etm⁺¹; let the functions (u^ei(t);u^ei(i(t))), i = 1;:::;m, be left- (right-) continuous at the points ^eti;i;^eti⁺¹, i = 1;:::;m, respectively. Then there exist a non-zero vector(+) = (⁰(+);:::;q(+)) and a solution i(+)(t), t²[^eti;i(^eti⁺¹)]

of the conjugate equation (3i), i = 1;:::;m, such that the conditions in which we have to substitute and i(t) instead of (+) and ⁽⁺⁾_i , respec- tively, i = 1;:::;m, are fullled. Moreover, the equality (6) is replaced by the inequality

@^e

@ti ai i ¹(^eti)f^ei ¹(^eti ) +bi^f i (^eti)f^ei(^eti ) + _i (i)[f^ei(i ;^exi(i);

ex¹⁰) f^ei(i ;x^ei(i);'i(^eti ) +g^ei(^eti))]_i(^eti)^g; i= 1;:::;m+ 1

⁺@^e

@ti a_i ⁺_i ¹(t)f^e_i ¹(^et_i+) +b_i^f _i⁺(^et_i)f^e_i(^et_i+) _i⁺(_i)[f^e_i(_i+;x^e_i(_i); xe¹⁰) f^ei(i ;^exi(i);'^ei(^eti+) +gi(^eti))]_i(^eti)^g;

where

fe_i(^et⁽⁺⁾_i ) =f^e(^et⁽⁺⁾_i ;x^e_i(^et_i);x^e_i(^e_i(^et⁽⁺⁾_i )):

Consider now for (1i) and (2i), i= 1;:::;m, where g² ==g_m = 0, the problem with the boundary conditions

q¹(t¹;x¹⁰) = 0;qi(ti;xi ¹(ti)) xi⁰= 0; i=r;:::;m;

q_m⁺¹(t_m⁺¹;x_m(t_m⁺¹)) = 0; (8) and with the functional

q⁰(tm⁺¹;xm(tm⁺¹))^!min: (9) The functions

q¹:J0^1!R^l1;qi:J0i^!Rⁿⁱ; i= 2;:::;m;

qm⁺¹:J0m^!R^l2; q⁰:J0m^!R¹ are assumed to be continuously dierentiable in all arguments.

The functionQfor the problem under consideration is of the form Q= (q⁰;q¹;q² x²⁰;:::;q_m x_m⁰;q_m⁺¹): (10) Taking into account (10), from Theorem 1 there follows

Theorem 3. Letmbe a locally optimal element of the problem(1i), (2i), i= 1;:::;m, (8), (9) and let the conditions of Theorem 1 be fullled. Then there exist a non-zero vector = (⁰;¹;:::;m⁺¹), ⁰ 0, ^{1 2} R^l1, i ²Rⁿⁱ, i= 1;:::;m, m^{+1 2}R^l2 and a solution i(t), t²[^eti;i(^eti⁺¹)], of the conjugate equation(3i)i= 1;:::;m, where the last term in the right- hand side equals zero, such that the conditions (4), (5) are fullled, while the conditions of transversality take the form

i@q^ei

@ti = ai i ¹(^eti)f^ei ¹(^eti) +bi^f i(^eti)f^ei(^eti) +

+ i(i)[f^ei(i;^exi(i);x^ei⁰) f^ei(i;x^ei(i);'^ei(^eti))]i(^eti)^g ci⁰ @q^e0_i

@tm⁺¹; i= 1;:::;m+ 1;¹ @q^e1

@x^{10 = 1(e}t¹); i(^eti) =i; i= 2;:::;m;

i @qi

@x_i ^{1 = i 1(e}ti); i= 2;:::;m;

⁰@q^e0

@xm +m⁺¹@q^em⁺¹

@xm = m(^etm⁺¹): . Here

c¹==cm= 0; cm⁺¹⁼¹; x^ei⁰=qi(^eti;^exi ¹(^eti); i= 2;:::;m:

3. Proof of Theorem 1.

The necessary conditions of optimality are proved by the scheme given in [1, 2, 3]. When applying this scheme, the principal moment is the construc- tion of a continuous and dierentiable mapping which plays an important role in deriving the necessary conditions of optimality. To this end we present below and prove (see 3.1 and 3.2) the appropriate theorems.

3.1. Continuous Dependence and Dierentiability of the Solution. LetO Rⁿ be an open set: E(JO²;Rⁿ) be a space of n-dimensional functions f;J0^2!Rⁿ satisfying the following conditions:

3) for a xed t ² J the function f(t;x;y) is continuously dierentiable with respect to (x;y)²O²;

4) for a xed (x;y) ² O² the functions f;fx;fy are measurable with respect tot; for an arbitrary compactumKO and an arbitrary function f there exists a functionmf;k(t)²L¹(J;R¹⁺),R¹⁺= [0;¹) such that

jf(t;x;y)^j+^jf_x(_)^j+^jf_y(_)^jm_f;K(t) ⁸(t;x;y)²JK²:

In the space E(J O²;Rⁿ), let us introduce by means of the bases of neighborhoods of zero two locally convex separable topologies [1, 2, 3],

fVs(K;) E(J 0²;Rⁿ) a compactum K and a number > 0 are arbitrary^g,s= 1;2, where

V¹(K;) =^ff²E(JO²;Rⁿ)^jh(f;K)^g; (11) V²(K;) =^ff²E(JO²;Rⁿ)^j

Z

J H⁰(t;f;K)dt^g; (12) h(f;K) = sup

^Zt⁰⁰

t⁰ f(t;x;y)dt^j(t⁰;t⁰⁰;x;y)²J²K²

; H⁰(t;f;K) = sup^njf(t;x;y)^j+^jfx()^j+^jfy()^j(x;y)²K^2o: Consider the sets

B_i=J^1+iYi

p⁼¹O_p^Yi

p⁼¹(J_i¹;O_i)^Yi

p⁼¹E(JO_i²;Rⁿⁱ); i= 1;:::;m;

with the elementsi = (t¹;:::;ti⁺¹;x¹⁰;:::;xi⁰;'¹;:::;'i;f¹;:::;fi),i= 1;:::;m respectively. In what follows we will assume that the topologies T¹, T² are prescribed in the spaces E(J O_i²;Rⁿⁱ) and i = 1;:::;m (see (11) and (12)).

To every elementm²Bmthere corresponds the dierential equation of variable structure:

x_i(t) =fi(t;xi(t);xi((t))); t²[ti;ti⁺¹]; (13i) xi(t) ='i(t) +gi(t;xi ¹(t)); t²[i;(ti);ti⁺¹);

xi(ti) =xi⁰; i= 1;:::;m: (14i) Denition 4. A set of functions ^fxi(t) = xi(t;i);t ² [i(ti);ti⁺¹];i = 1;:::;m^gis said to be a solution of the equation of variable structure cor- responding to the element m ² Bm, if the function xi(t) ² 0i on the segment [i(ti);ti] satises the condition (14i) and on the interval [ti;ti⁺¹] is absolutely continuous and satises the equation (13i) a.e.

Theorem 4. Let ^fexi(t);t ² [Ji(^eti);^eti⁺¹];i = 1;:::;m^g, a < ^et¹ < <

etm⁺¹< b, be a solution corresponding to the element^em=(^et¹;:::;^etm⁺¹;^ex¹⁰; :::;^exm⁰;'^e1;:::;'^em;f^e1;:::;f^em) ² Bm and let Ki¹ 0i be a compactum containing some neighborhood of the set

K_i⁰=cl[^fex_i(t)^jt²[^et_i;^et_i⁺¹]^g[f'^e_i(t)^j2J_i^1g]:

Then for every" >0 there exists a number =(")>0 such that to every element

m²V(^em;K¹¹;:::;Km¹;;c⁰) =^mY+1

i⁼¹ V(^eti;)^Ym

i⁼¹V('^ei;)

m

Y

i⁼¹V('^ei;)^Ym

i⁼¹V¹(f^ei;Ki¹)^\V²(f^ei;Ki¹;c⁰));

there corresponds a solution^fxi(t;i);t²[Ji(ti);ti⁺¹];i= 1;:::;m^g. More- over, the function xi(t;i) is dened on [Ji(ti);^eti⁺¹+] (Ji(a);b), and satises almost everywhere on [ti;^eti+] the equation (13i).

If _sm²V(^em;K¹¹;:::;Km¹;;c⁰); s= 1;2; then

jxi(t;¹_i) xi(t;²_i)^j"; t²[max(t¹_i;t²_i);^eti⁺¹+];

i= 1;:::;m: (15)

Here V(^eti;), V(x^ei⁰;),V('^ei;), are the -neighborhoods of the points

eti,x^ei⁰, '^ei, in the spaces R¹, Rⁿ, respectively; (Ji¹;Rⁿⁱ); V¹(f^ei;Ki;) = fei+V¹(Ki¹;),V²(f^ei;Ki¹c⁰) =f^ei+V²(Ki¹;c⁰),V¹(Ki⁰;)E(J0²_i;Rⁿⁱ, V²(Ki¹;c⁰)E(J0²_i;Rⁿⁱ;c⁰>0 is a xed number.

Theorem 4 can be proved by the method given in [6] (see also [7]) and is used in proving the continuity of the mapping (see 3.3).

Remark 3. There exists a number^e2[0;] (see (12)) such that V²(f^ei;Ki¹;^e)V¹(f^ei;Ki;^e)^\V²(f^ei;Ki¹;c⁰); i= 1;:::;m:

Consequently, the inequality (15) is the more so valid for _sm^2mY+1

i⁼¹ V(^eti;^e)^Ym

i⁼¹V('^ei;^e)^Ym

i⁼¹V²(f^ei;Ki¹;^e): This fact is used in proving the openness of the setD⁰ (see 3.2).

Introduce the set

Vi=^fi= (t¹;:::;ti⁺¹;x¹⁰;:::;xi⁰;'¹;:::;'i;f¹;:::;fi)²

2Bi ^ei^jti^jci; xi⁰c¹;^k'i^kci;fp=^Xs

j⁼¹j"f_jp;p= 1;:::;i;

jj^jci^g; i= 1;:::;m;

f_jp ² E(J 0²_p;R^np), p = 1;:::;i, j = 1;:::;s, are xed points and s, ci>0 are xed numbers.

From Theorem 4 we have

Theorem 5. There exist numbers"⁰>0,⁰>0 such that for an arbitrary (";m)²[0;"⁰]Vm, to the element^em+"mthere corresponds solution

fxi(t;^ei⁺"i);t²[i(ti);ti⁺¹]; i= 1;:::;m^g; ti=^eti+"ti: Moreover,xi(t;^ei+"i) is dened on [i(ti);^eti⁺¹+⁰].

Remark 4. Due to the uniqueness,xi(t;^ei) on the interval [i(^eti);^eti⁺¹+ ⁰] is a continuation of x^ei(t). Therefore the functionx^ei(t) in the sequel is assumed to be dened on the whole interval [i(^eti);^eti⁺¹+⁰].

Using the numbers⁰and"⁰(see Theorem 5), we introduce the notation xi(t;"i) =xi(t;i+"i) x^ei(t);t²[max(ti;^eti);^eti⁺¹+⁰];"²[0;"⁰]: Theorem 6. Let a <^et¹ <<^et_m⁺¹ < b, ^et_i < _i =_i(^et_i)< _i⁺¹(^et_i⁺¹), i+ 1;:::;m 1,^etm< m=m(^etm)⁶=^etm⁺¹; let the functionsf^ei(t;xi;yi), i = 1;:::;m, be continuous respectively at the points (^eti;^exi⁰;'^ei(i(^eti))), (i;x^ei(i);^exi⁰), (i;x^ei(i);'^ei(^eti)), i = 1;:::;m, and the functions '^ei(t), i= 1;:::;m, be continuous respectively at the points^eti, i= 1;:::;m. Then there exist numbers"¹²[0;"⁰],¹²[0;⁰] such that the following formula is valid:

xi(t;"i) ="xi(t;i) +oi(t;"i); ⁸(t;";i)²[i⁺¹(^eti⁺¹);

eti⁺¹+¹][0;"¹]Vi; i= 1;:::;m; (16) where

xi(t;i) =Yi(^eti;t)^hxi⁰ f^ei(^eti)ti

i+

eti

Z

i^(eti⁾

Yi(i(s);t)@f^ei(i(s))

@y_i

h'_i(s) +@^eg_i(s)

@xi ¹x_i ¹(s)ⁱ__i(s)ds+

et^Zi⁺¹

eti

Y_i(s;t)f^e_ids Y_i(_i;t)

fei(i;x^ei(i);^exi⁰) f^ei(i;^exi(i);'^ei;(^eti) +^egi(^eti))_i(^eti)ti =

=x¹_i(t;i) x²_i(t;i); i= 1;:::;m; (17) Yi(s;t) is a matrix function satisfying the equation

@Yi(s;t)

@s ⁼ Yi(s;t)@f^ei(s)

@xi Yi(i(s);t)@f^ei(i(s))

@yi _i(s); s²[^eti;^eti⁺¹] and also the condition

Yi(t;t) =E; Yi(s;t) = 0; s > t; next,

"lim^!0oi(t;"i)="= 0; uniformly with respect to (t;i)²[i⁺¹(^eti⁺¹);

eti⁺¹+¹]Vi;f^ei(t) =fi(t;x^ei(t);^exi(i(t))); m⁺¹(^etm⁺¹) =^etm^{+1 1}: The proof of Theorem 6 is conducted in a way described in [2, 7]. In the same way one can prove more general

Theorem 7. Let a < ^et¹ < < ^etm⁺¹ < b, ^eti < i < i⁺¹(^eti⁺¹), i = 1;:::;m 1,^etm< m⁶=^etm⁺¹, and let the conditions

(t;xi)^!(ti(+);x^ei⁰)

limf^e_i(t;x_i;'^e_i(_i(t))) =f^e_i(^et_i(+));^ex_i⁰;'^e(_i(^et_i(+)))<¹; (t;xi;yi)^!(i(+);x^ei();x^ei⁰)

limf^ei(t;xi;yi) =f^ei(^ei(+);x^ei(i);^exi⁰)<¹;

be fullled. Then there exist numbers"¹²[0;"⁰],¹²[0;⁰], such that the formula (16) is valid for an arbitrary point (t;";i) ²[i⁺¹(^eti⁺¹);^eti⁺¹+ ¹][0;"¹]V_i⁽⁺⁾, while in the formula (17) before ti there take place respectively the expressions

Y(^eti;t)f^ei(^eti(+)) +Y(i;t)f^ei(i;(+);x^ei(i);^exi⁰) fei(i(+);x^ei(i);'^ei(^eti(+)) +g(^eti))_i(^eti): Here

V_i ^fi²Vi^j tK0; K= 1;:::;i+ 1^g; V_i^+fi²Vi^j tK 0; K= 1;:::;i+ 1^g:

Theorem 8 ([The Cauchy formula [8]]). Let A(t), B(t), t ²J¹ = [s¹;s²] be summable matrix functions of dimensionnn; letF(t), t²J¹be an n- dimensional summable vector function and(t) satisfy the same conditions asi(t) do; let'(t), t²[(s¹);s²] be a piecewise continuous function. Then the solution of the equation

x_(t) =A(t)x(t) +B(t)x((t)) +F(t); t²J¹; x(t) ='(t); t²[(s¹);s¹);x(s¹) =x⁰²Rⁿ; can be represented as

x(t) =Y(s¹;t)x⁰+ ^s

1

Z

⁽s¹⁾ Y((s);t)B((s))_(s)'(s)ds+ ^t

Z

s¹ Y(s;t)F(s)ds;

t²J¹;

whereY(s;t) is a matrix function satisfying the equation

@Y(s;t)

@s ⁼ Y(s;t)A(s) Y((s);t)B((s))_(s); s²[s¹;t]

and also the condition

Y(t;t) =E; Y(s;t) = 0; s > t; (s) is the function inverse to(s).

On the basis of Theorem 8 we can conclude that the functionx¹_i(t;i) (see (17)) satises the equation

x_¹_i(t) =@f^e_i(t)

@xi x¹_i(t) +@f^e_i(t)

@yi x¹_i(_i(t)) +f^e_i(t);

t²[^eti;^eti⁺¹] (18) with the initial condition

x¹_i(t) ='i(t) +@^egi(t)

@xi ¹xi ¹(t); t²[i(^eti);^eti);

x¹_i(^eti) =xi⁰ f^ei(^et¹)ti; (19) while the functionx²_i(t;i) (see (17)) satises the equation

x_²_i(t) =@f^ei(t)

@xi x²_i(t) +@f^ei(t)

@yi x²_i(i(t)); t²[i;^eti⁺¹]; (20) with the initial condition

x²_i(t) = 0; t < i x²_i(i) = [f^ei(i;^exi(i);x^ei⁰)

fei(i;x^ei(i);'^ei(^eti))]_i(^eti)ti: (21) For the sake of brevity we denote the function,xi(t;i),x¹_i(t;i) and x²_i(t;i), respectively by xi(t), x¹_i(t) andx²_i(t).

Theorem 9. The following formula is valid:

m

X

i⁼¹

@Q^e

@xixi(^eti⁺¹) =^Xm

i⁼¹(Yi(^eti)x¹_i(^eti) Yi(i)x²_i(i) +i); (22) where

i=

eti

Z

i^(eti⁾

Yi(i(t))@f^ei(i(t))

@yi _i(t)'i(t)dt+ +

et^Zi⁺¹

eti

Yi(t)f^ei(t)dt; (23)