Let the function f :J×Os×Gν →Rn satisfy the following conditions: for almost all t∈ J function f(t, x1

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Mem. Differential Equations Math. Phys. 31(2004), 145–148

I. Ramishvili

THE LINEARIZED MAXIMUM PRINCIPLE FOR QUASI-LINEAR NEUTRAL OPTIMAL PROBLEMS WITH DISCONTINUOUS INITIAL

CONDITION AND VARIABLE DELAYS IN CONTROLS (Reported on November 17, 2003)

Let J = [a, b] ⊂ Rbe a finite interval, O ⊂ Rⁿ, G ⊂ R^r be open sets; Let the function f :J×O^s×G^ν →Rⁿ satisfy the following conditions: for almost all t∈ J function f(t, x1, . . . , xs, u1, . . . , uν) is continuously differentiable with respect to xi ∈ O, i = 1, . . . , s, um ∈ G,m = 1, . . . , ν; for any fixed (x1, . . . , xs, u1, . . . , uν) ∈ O^s× G^ν the functions f(t, x1, . . . , xs, u1, . . . , uν), fx_i(·)i = 1, . . . , s,fu_m(·), m = 1, . . . , ν are measurable on J; for arbitrary compacts K ⊂O, N ⊂ Gthere exists a function mK,N(·)∈L(J, R+),R+= [0,∞),such that for any (x1, . . . , xs, u1, . . . , um)∈K^s×N^m and for almost allt∈J,the following inequality is fulfilled

|f(t, x1, . . . , xs, u1, . . . , uν)|+ Xs i=1

|fx_i(·)|+ Xν m=1

|fu_m(·)|≤mK,N(t).

Let the scalar functions τi(t), i = 1, . . . , s, θm(t), m = 1, . . . , ν, t ∈ R and ηj(t), j= 1, . . . , k,t∈R, be absolutely continuous and continuously differentiable , respectively, and satisfy the conditions: τi(t) ≤ t, ˙τi(t) > 0, i = 1, . . . , s; θm(t) ≤ t, ˙θm(t) > 0, m = 1, . . . , ν ηj(t) < t, ˙ηj(t) > 0, j = 1, . . . , k. Let Φ be the set of continuously differentiable functionsϕ:J1 = [τ, b]→M, τ= min{η1(a), . . . , ηk(a), τ1(a), . . . , τs(a)}, whereM ⊂Ois a convex set,kϕk= sup{|ϕ(t)|+|ϕ(t)˙ |:t ∈J}; Ω be the set of measurable functionsu:J2= [θ, b]→U,such thatcl{u(t) :t∈J}is a compact lying in G, θ= min{θ1(a), . . . , θν(a)}, whereU⊂Gis a convex set,kuk= sup{|u(t)|:t∈J2};

Ai(t), t∈J,i= 1, . . . , k, be continuousn×nmatrix functions. The scalar functions qⁱ(t0, t1, x0, x1),i= 1, . . . , l, are continuously differentiable on the setJ²×O².

To every elementλ= (t0, t1, x0, ϕ, u)∈E =J²×O×Φ×Ω let us correspond the differential equation

˙ x(t) =

Xk j=1

Aj(t) ˙x(ηj(t)) +f(t, x(τ1(t)), . . . , x(τs(t)), u(θ1(t)), . . . , u(θm(t))) (1) with discontinuous initial condition

x(t) =ϕ(t), t∈[τ, t0), x(t0) =x0. (2)

Definition 1. Letλ= (t0, t1, x0, ϕ, u)∈E,t0< b.The functionx(t) =x(t;λ)∈O, t∈[τ, t1],t1∈(t0, b] is said to be a solution corresponding to the elementλ,defined on the interval [τ, t1],if on the interval [τ, t0] the functionx(t) satisfies the condition (2), while on the interval [t0, t1] it is absolutely continuous and almost everywhere satisfies the equation (1).

2000Mathematics Subject Classification.49K25.

Key words and phrases. Neutral differential equation, Necessary conditions of optimality.

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146

Definition 2. The elementλ∈Eis said to be admissible if the corresponding solution x(t) =x(t;λ) satisfies the conditions

qⁱ(t0, t1, x0, x(t1)) = 0, i= 1, . . . , l. (3) The set of admissible elements will be denoted byE0.

Definition 3. The elementeλ= (et0,et1,ex0,ϕ,eu)e ∈E0 is said to be locally optimal, if there exists a numberδ >0 such that for an arbitrary elementλ∈E0 satisfying

|et0−t0|+|et1−t1|+|ex0−x0|+kϕe−ϕk+keu−uk≤δ the inequality

q⁰(et0,et1,xe0,x(eet1))≤q⁰(t0, t1, x0, x(t1)) (4) holds, wherex(t) =e x(t;λ).e

The problem (1)-(4) is said to be optimal problem with discontinuous initial condition and it consists in finding a locally optimal element.

In order to formulate the main results, we will introduce the following notation σi= (te0,fx0, . . . ,xf0

| {z }

i

,ϕ(eet0), . . . ,ϕ(eet0)

| {z }

(p−i)

,ϕ(τe p+1(te0)), . . . ,ϕ(τe s(te0))), i= 0, . . . , p;

σi= (γi,x(τe 1(γi)), . . . ,ex(τi−1(γi)),xe0,ϕ(τe i+1(γi)), . . . ,ϕ(τe s(γi))), σ⁰_i = (γi,ex(τ1(γi)), . . . ,ex(τi−1(γi)),ϕ(eet0),ϕ(τe i+1(γi)), . . . ,ϕ(τe s(γi))),

i=p+ 1, . . . , s;σs+1= (et1,ex(τ1(et1)), . . . ,x(τe s(et1))), γi=γi(et0), ρj =ρj(et0), γi(t) =τ_i⁻¹(t), ρj(t) =η⁻¹_j (t); ω= (t, x1, . . . , xs),

fe(ω) =f(ω,u(θe 1(t)), . . . ,u(θe ν(t))),

ffx_i[t] =fx_i(t,x(τe 1(t)), . . . ,ex(τs(t)), . . . ,u(θe 1(t)), . . . ,u(θe ν(t)));

Q= (q⁰, . . . , q^l)⁰, Qet0=Qt0(te0,te1,xe0,ex(et1)).

Theorem 1. Let the elementeλ= (et0,et1,xe0,ϕ,e eu)∈E0,et0 > abe locally optimal and the following conditions be fulfilled:

1)γi=et0,i= 1, . . . , p γp+1<· · ·< γs<et1,ρj<et1 j= 1, . . . , k;

2)there exists a numberδ >0such that

γ1(t)≤ · · · ≤γp(t), t∈(te0−δ,te0];

3)there exist the finite limits: γ˙⁻_i = ˙γi(et0−),i= 1, . . . , s,x(ηe˙ j(et1−)),j= 1, . . . , k;

ω→σlim_ife(ω) =f_i⁻, ω∈(et0−δ,et0]×O^s, i= 0, . . . , p, lim

(ω1,ω2)→(σ_i,σ⁰_i)

hf(ωe 1)−f(ωe 2)i

=f_i⁻, ω1, ω2∈(γi−δ, γi]×O^s, i=p+ 1, . . . , s,

ω→σlim_s+1f(ω) =e f_s+1⁻ , ω∈(et1−δ,et1]×O^s.

Then there exist a non-zero vector π = (π0, . . . , πl), π0 ≤ 0, and a solution χ(t) = (χ1(t), . . . , χn(t)),ψ(t) = (ψ1(t), . . . , ψn(t))of the system







˙

χ(t) =−Ps

i=1ψ(γi(t))fex_i[γi(t)] ˙γi(t), ψ(t) =χ(t) +Pk

j=1ψ(ρj(t)Aj(ρj(t)) ˙ρj(t), t∈[et0,et1], ψ(t) = 0, t >et1,

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such that the following conditions are fulfilled:

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147

a)the linearized maximum principles:

Zet1 et0

ψ(t) Xν m=1

feu_m[t]u(θe m(t))dt≥ Z et1

et0

ψ(t) Xν m=1

feu_m[t]u(θm(t))dt, ∀u∈Ω, Xs

i=p+1

Z et0 τ_i(et0)

ψ(γi(t))fex_i[γi(t)] ˙γi(t)ϕ(t)dte + Xk j=1

Z et0 η_j(et0)

ψ(ρj(t))Aj[ρj(t)] ˙ρj(t) ˙ϕ(t)dte ≥

≥ Xs i=p+1

Zte0 τ_i(et0)

ψ(γi(t))fex_i[γi(t)] ˙γi(t)ϕ(t)dt+

+ Xk j=1

Z et0 η_j(et0)

ψ(ρj(t))Aj[ρj(t)] ˙ρj(t) ˙ϕ(t)dt, ∀ϕ∈Φ;

b)the conditions for the moments et0,et1: πQet0≥ −ψ(et0−)[ ˙ϕ(eet0)−

Xk j=1

Aj(et0) ˙ϕ(ηe j(et0)) + Xp i=0

(bγ_i+1⁻ −bγ_i⁻)f_i⁻]+

+ Xs p+1

ψ(γi−)f_i⁻γ˙⁻_i +χ(et0) ˙ϕ(eet0),

πQet1 ≥ −ψ(et1) X^k

j=1

Aj(et1) ˙ex(ηj(et1−)) +f_s+1⁻

;

c)the condition for the solutionχ(t),ψ(t)

πQex0=−χ(et0), πQex1=ψ(et1) =χ(et1).

Here

b

γ₀⁻= 1, γb_i⁻= ˙γ⁻_i , i= 1, . . . , p, bγ_p+1⁻ = 0;

Theorem 2. Let the elementeλ= (et0,et1,xe0,ϕ,eu)e ∈E0,et1< bbe locally optimal and the condition1)of Theorem 1and the following conditions be fulfilled:

4)there exists a numberδ >0such that

γ1(t)≤ · ≤γp(t), t∈[te0,te0+δ);

5)there exist the finite limits :γ˙_i⁺= ˙γi(et0+),i= 1, . . . , s,x(ηe˙ j(et1+)),j= 1, . . . , k;

ω→σlim_if(ω) =e f_i⁺, ω∈[et0,et0+δ)×O^s, i= 0, . . . , p, lim

(ω1,ω2)→(σi,σ⁰_i)

hf(ωe 1)−f(ωe 2)i

=f_i⁺, ω1, ω2∈[γi, γi+δ)×O^s, i=p+ 1, . . . , s

ω→σlim_s+1fe

ω) =f_s+1⁺ , ω∈[et1,et1+δ

×O^s.

Then there exists a non-zero vectorπ= (π0, . . . , πl), π0≤0and a solutionχ(t),ψ(t)of the system(5)such that the conditionsa)andc)are fulfilled. Moreover,

πQet0≤ −ψ(et0+)[ ˙ϕ(eet0)− Xk j=1

Aj(et0) ˙ϕ(ηe j(et0)) + Xp i=0

(γb_i+1⁺ −bγ⁺_i)f_i⁺]+

+ Xs i=p+1

ψ(γi+)f_i⁺γ˙⁺_i +χ(et0) ˙ϕ(eet0),

πQet0≤ −ψ(et1)



 Xk j=1

Aj(et1) ˙ex(ηj(et1+)) +f_s+1⁺



.

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148

Here

b

γ₀⁺= 1, bγ⁺_i = ˙γ_i⁺, i= 1, . . . , p, bγ_p+1⁺ = 0.

Theorem 3. Let the element eλ = (et0,et1,xe0,ϕ,eu)e ∈ E0, et0,et1 ∈ (a, b)be locally optimal and the conditions of Theorems 1, 2 and the following conditions be fulfilled:

the functionsex(η˙ j(et1)),j= 1, . . . , k, are continuous;

γi,et0∈/

ηk1(ηk2(. . .(ηke(et1)), . . . ,))∈(a,et1) : e= 1,2, . . . , m= 1, . . . , e, km= 1, . . . , k , i=p+ 1, . . . , s;

Xp i=0

(bγ_i+1⁻ −bγ_i⁻)f_i⁻= Xp i=0

(bγ_i+1⁺ −bγ_i⁺)f_i⁺=f0, f_i⁻γ˙⁻_i =f_i⁺γ˙⁺_i =fi, i=p+ 1, . . . , s, f_s+1⁻ =f_s+1⁺ =fs+1.

Then there exists a non-zero vectorπ= (π0, . . . , πl),π0≤0and a solutionχ(t),ψ(t) of the system(5)such that the conditiona)andc)are fulfilled. Moreover,

πQet0=−ψ(et0)[ ˙ϕ(eet0)−

Xk j=1

Aj(et0) ˙ϕ(ηe j(et0))+f0]+

Xs i=p+1

ψ(γi)fi+χ(et0) ˙ϕ(eet0),

πQet0=−ψ(et1) Xk

j=1

Aj(et1) ˙ex(ηj(et1)) +fs+1

.

Finally we note that the optimal control problems for various classes of delay and neutral differential equations with discontinuous initial condition are considered in [1]–

[4].

References

1. G. Kharatishvili and T. Tadumadze, Nonlinear optimal control problem with variable delays,non-fixed initial moment,and a piecewise-continuous prehistory. (Russian) Tr. Mat. Inst. Steklova 220(1998), 236–255; translation inProc. Steklov Inst. Math.

220(1998), 233–252.

2. T. Tadumadze, On new necessary condition of optimality of the initial moment in control problems with delay. Mem. Differential Equations Math. Phys. 17(1999), 157–159.

3.T. Tadumadze and L. Alkhazishvili, Necessary conditions of optimality for optimal problems with delays and with a discontinuous initial condition. Mem. Differential Equations Math. Phys. 22(2001),154–158.

4.N. Gorgodze, Necessary conditions of optimality in neutral type optimal problems with non-fixed initial moment. Mem. Differential Equations Math. Phys. 19(2000), 150–153.

Author’s address:

Department of Mathematics No. 99 Georgian Technical University 77, M. Kostava St., Tbilisi 0175 Georgia